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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

Harvey and van der Hoeven established an $O(n\log n)$ algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

A positive $\kappa$ beats $n\log n$ asymptotically; larger $\kappa$ is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.

NoneFormalized record≥ 0.00003666565558019Open frontier

2 provers on it0 of 2 missions formalized

3SUM Exponent

Classical algorithms solve 3SUM in $O(n^2)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic $O(n^{1.9992})$ algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.999074Formalized record

3 provers on it4 of 4 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Classical algorithms solve all-pairs shortest paths in $O(n^3)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic $O(n^{2.99942})$ algorithm. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.103205334138Formalized record≤ 2Open frontier

9 provers on it7 of 8 missions formalized

Sharp diagonal Hlawka constant

The sharp Hlawka inequality for Schatten $p$ -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real $p\ge256$ . We conjecture that the same formula holds for all $p\ge2$ .

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 80Formalized record≤ 70Open frontier

3 provers on it7 of 8 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

Schnirelmann (1930) showed some finite $k$ works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved $k = 5$ unconditionally. Helfgott (2013) proved that every odd number greater than $5$ is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: $27$ is neither prime nor $2$ + prime.

≤ 27Formalized record≤ 5Open frontier

35 provers on it13 of 15 missions formalized

Matrix multiplication exponent

Schoolbook matrix multiplication takes $n^3$ operations. The exponent $\omega$ is the infimum of all $\tau$ such that two $n \times n$ matrices can be multiplied in $O(n^{\tau})$ arithmetic operations; trivially $\omega \geq 2$ , and $\omega = 2$ is conjectured but open.

Strassen gave the first nontrivial bound, $\omega < 2.81$ , in 1969, and introduced the laser method in 1986 to reach $\omega < 2.48$ . Coppersmith and Winograd's 1990 bound of $2.376$ stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached $\omega < 2.371339$ in 2025, and the current record is $\omega < 2.371177$ , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

NoneFormalized record≥ 0.00003666565558019Open frontier

2 provers on it0 of 2 missions formalized

3SUM Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.999074Formalized record

3 provers on it4 of 4 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.

≤ 7.103205334138Formalized record≤ 2Open frontier

9 provers on it7 of 8 missions formalized

Sharp diagonal Hlawka constant

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 80Formalized record≤ 70Open frontier

3 provers on it7 of 8 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

≤ 27Formalized record≤ 5Open frontier

35 provers on it13 of 15 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Convex Optimization Operations Research Optimal Transport+1·Captain: mikedeng1

Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes 2: The Dual Objective f_δ(β, λ) = E_P0[ℓ_rob(β, λ; X)] Is Proper and ConvexResearch Paper

Motivation

Distributionally robust optimization (DRO) replaces the expected loss $E_{P_0}[\ell(\beta^{\mathsf T}X)]$ of a decision $\beta$ under a baseline distribution $P_0$ by its worst case over all distributions $P$ that are close to $P_0$ . When closeness is measured by an optimal transport cost, the worst case over this infinite-dimensional ball has a one-dimensional dual: by Theorem 1 of Blanchet, Murthy and Zhang (arXiv:1810.02403v3, Math. Oper. Res. 47(2), 2022), building on the strong duality of Blanchet and Murthy (arXiv:1604.01446),

\sup_{P : D_c(P_0, P) \le \delta} E_P\big[\ell(\beta^{\mathsf T}X)\big] = \inf_{\lambda \ge 0} f_\delta(\beta, \lambda).

The robust problem $\inf_{\beta \in B} \sup_P E_P[\ell(\beta^{\mathsf T}X)]$ thus becomes a joint minimization of the dual objective $f_\delta$ over $(\beta, \lambda)$ . Whether that minimization is a convex program, and on which set the objective is finite, decides whether first-order methods such as the stochastic gradient schemes of the same paper apply at all. This mission formalizes the paper's answer, Theorem 2.

Setting

The data $X$ take values in $\mathbb{R}^d$ with law $P_0$ , a Borel probability measure. A decision is a vector $\beta$ in a convex set $B \subseteq \mathbb{R}^d$ , the loss is $\ell(\beta^{\mathsf T}x)$ for $\ell : \mathbb{R} \to \mathbb{R}$ , and $\delta > 0$ is the transport budget.

The transport cost is the state-dependent Mahalanobis cost $c(x, x') = (x - x')^{\mathsf T}A(x)(x - x')$ for a map $A$ from $\mathbb{R}^d$ to positive definite $d \times d$ matrices. Assumption 1: $c$ is lower semicontinuous (1a), and $\rho_{\min}\|v\|^2 \le v^{\mathsf T}A(x)v \le \rho_{\max}\|v\|^2$ for $P_0$ -almost every $x$ , with $\rho_{\min} > 0$ (1b). Write $q_\beta(x) = \beta^{\mathsf T}A(x)^{-1}\beta$ .

Assumption 2: $\ell$ is convex, its growth exponent

\kappa = \inf\Big\{ s \ge 0 : \sup_{u \in \mathbb{R}} \big(\ell(u) - s u^2\big) < \infty \Big\}

is finite, and $E_{P_0}\|X\|^4 < \infty$ .

For $\gamma \in \mathbb{R}$ , $\lambda \ge 0$ and $x \in \mathbb{R}^d$ define, as in display (7),

F(\gamma, \beta, \lambda; x) = \ell\big(\beta^{\mathsf T}x + \gamma\sqrt{\delta}\, q_\beta(x)\big) - \lambda\sqrt{\delta}\big(\gamma^2 q_\beta(x) - 1\big),

the robust loss $\ell_{rob}(\beta, \lambda; x) = \sup_{\gamma \in \mathbb{R}} F(\gamma, \beta, \lambda; x) \in \mathbb{R} \cup \{+\infty\}$ , its set of maximizers $\Gamma^*(\beta, \lambda; x)$ , and the dual objective

f_\delta(\beta, \lambda) = E_{P_0}\big[\ell_{rob}(\beta, \lambda; X)\big] \in \mathbb{R} \cup \{\pm\infty\}.

The effective domain is $\mathbb{U} = \{(\beta, \lambda) \in B \times \mathbb{R}_+ : f_\delta(\beta, \lambda) < \infty\}$ , and with the threshold $\lambda_{thr}(\beta) = \kappa\sqrt{\delta}\,\operatorname{ess\,sup}_{P_0} q_\beta$ the paper sets $\mathbb{U}_1 = \{\lambda > \lambda_{thr}(\beta)\}$ and $\mathbb{U}_2 = \{\lambda \ge \lambda_{thr}(\beta)\}$ inside $B \times \mathbb{R}_+$ .

Formalization targets

Goal: Theorem 2 (p. 10)

Under Assumptions 1 and 2, $f_\delta : B \times \mathbb{R}_+ \to \mathbb{R} \cup \{\infty\}$ is proper and convex: $f_\delta > -\infty$ on $B \times \mathbb{R}_+$ ; for each $\beta \in B$ some $\lambda \ge 0$ has $f_\delta(\beta, \lambda) < \infty$ ; and for $\theta_1, \theta_2 \in B \times \mathbb{R}_+$ , $\alpha \in [0, 1]$ ,

f_\delta\big(\alpha\theta_1 + (1 - \alpha)\theta_2\big) \le \alpha f_\delta(\theta_1) + (1 - \alpha) f_\delta(\theta_2).

Milestones

Lemma 3 (p. 32). For each $\varepsilon > 0$ there are $C_1, C_2 > 0$ , not depending on $x$ , $\beta$ , $\lambda$ , such that whenever $\lambda \ge (\kappa + \varepsilon)\sqrt{\delta}\,q_\beta(x)$ we have $\sqrt{\delta}|g|q_\beta(x) \le 1 + C_1\varepsilon^{-1}(1 + |\beta^{\mathsf T}x|)$ for every $g \in \Gamma^*(\beta, \lambda; x)$ , and $\ell_{rob}(\beta, \lambda; x) \le \lambda\sqrt{\delta} + C_2(1 + \varepsilon + \varepsilon^{-1})(1 + |\beta^{\mathsf T}x|)^2$ .
Display (25) (p. 32). $\sup_{\Delta \in \mathbb{R}^d}\{\ell(\beta^{\mathsf T}(x + \Delta)) - \tfrac{\lambda}{\sqrt\delta}(\Delta^{\mathsf T}A(x)\Delta - \delta)\} = \ell_{rob}(\beta, \lambda; x)$ .
Lemma 2 (p. 16). $(\beta, \lambda) \mapsto \ell_{rob}(\beta, \lambda; x)$ is convex on $B \times \mathbb{R}_+$ for every $x$ .
Lemma 1 (p. 16). $\Gamma^* \neq \emptyset$ and $\ell_{rob}$ finite if $\lambda > \kappa\sqrt{\delta}\,q_\beta(x)$ ; $\Gamma^* = \emptyset$ and $\ell_{rob} = \infty$ if $\lambda < \kappa\sqrt{\delta}\,q_\beta(x)$ ; consequently $\mathbb{U}_1 \subseteq \mathbb{U} \subseteq \mathbb{U}_2$ .

Lemma 3 is stated with constants depending only on $\varepsilon$ (and the fixed data), as its proof produces them and as the paper's later arguments integrate bound (b) over $x$ .

Significance

Theorem 2 is the structural fact behind the paper's algorithms: with Theorem 1 it turns a minimax problem over probability measures into a convex minimization in $d + 1$ real variables, on which the paper's strong convexity results (Theorems 3–5) and its stochastic gradient schemes are built. Lemma 1 identifies the effective domain up to its boundary, which tells an algorithm where the dual multiplier must lie, and Lemma 3 gives the growth control of the robust loss used throughout the appendix.

The results are proved in the paper; none of them has a machine-checked proof. The mission produces a Lean statement of the dual objective as an extended-real integral and of its convexity and properness, reusable by the other missions of this series (strong convexity near the optimizers, worst-case distributions, comparative statics in $\delta$ ), and contributions of formal proofs of the milestones and the goal.

Difficulty

The objective takes the value $+\infty$ on part of its domain: Lemma 1(b) shows that $\ell_{rob} = +\infty$ below the threshold, so the naive reading " $f_\delta$ is a real convex function" is false, and convexity must be handled for functions with values in $\mathbb{R} \cup \{\infty\}$ , including the conventions $0 \cdot \infty = 0$ at the endpoints of the segment. Properness requires a bound on $\ell_{rob}$ that is integrable in $x$ , so the constants of Lemma 3 must not depend on $x$ ; a bound that is only pointwise finite does not suffice. The identity (25) relates a supremum over $\mathbb{R}^d$ to a supremum over $\mathbb{R}$ through a change of the dual variable, and the two multipliers must be kept apart. Finally, the expectation in $f_\delta$ is an integral of an extended-real function whose measurability is not assumed but has to be derived from Assumption 1.

Formalization scope

$\mathbb{R}^d$ is EuclideanSpace ℝ (Fin d) and parameters $(\beta, \lambda)$ live in EuclideanSpace ℝ (Fin d) × ℝ. $\ell_{rob}$ is an EReal supremum; $f_\delta$ is the published ModelRiskOT.Duality.extIntegral, $\int \varphi^+\,dP_0 - \int \varphi^-\,dP_0$ with lower Lebesgue integrals. $\kappa$ is a real infimum and $\lambda_{thr}$ uses the real essential supremum; both are used only under Assumptions 1–2, which make them meaningful. Convexity of extended-real maps is written as the explicit inequality in EReal, since ConvexOn does not apply.

Standing assumptions carried by every statement: $P_0$ a probability measure, $\delta > 0$ , $B$ convex (the paper's standing assumption). Conventions committed to: "for any $x$ " in Lemmas 1–2 is every $x$ , not almost every; Lemma 2 assumes only Assumption 1a and the convexity of $\ell$ ; the identity (25) is stated with the old multiplier written as $\lambda/\sqrt{\delta}$ ; the typo " $\beta A(x)^{-1}\beta$ " in Lemma 1 is read as $\beta^{\mathsf T}A(x)^{-1}\beta$ ; the goal's "finite somewhere" is stated for each $\beta \in B$ , so the goal is vacuous only when $B = \emptyset$ . The fourth-moment condition of Assumption 2 is kept as printed.

A trivializing formalization is ruled out: "proper" is not encoded as "finite everywhere" (false by Lemma 1(b)), and pointwise convexity of $\ell_{rob}$ (Lemma 2) is a milestone, not the goal, which is about the expectation $f_\delta$ .

A complete development needs: positive definite matrices and the quadratic minimization behind (25); continuity of convex functions and growth bounds; lower Lebesgue integrals of extended-real functions and their monotonicity and additivity; measurability of a supremum over $\gamma$ of continuous functions. The definitions layer is shared with the other four missions of this series. Proofs of any milestone and of the goal are welcome.

Selected references

J. Blanchet, K. Murthy, F. Zhang, Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes, Mathematics of Operations Research 47(2), 2022. arXiv:1810.02403v3, doi:10.1287/moor.2021.1178
J. Blanchet, K. Murthy, Quantifying Distributional Model Risk via Optimal Transport, Mathematics of Operations Research 44(2), 2019. arXiv:1604.01446, doi:10.1287/moor.2018.0936

8 thms1 active userReviewed

Operations Research Optimization Probability+1·Captain: mikedeng1

From Data to Decisions: Distributionally Robust Optimization Is Optimal 1: The Relative-Entropy Robust Predictor Is the Least Conservative Predictor Whose Disappointment Decays at Rate rResearch Paper

Motivation

A decision maker who must choose $x$ before observing an uncertain outcome $\xi$ usually does not know the distribution of $\xi$ ; only past observations $\xi_1,\dots,\xi_T$ are available. Data-driven optimization replaces the unknown expected cost by an estimate computed from the data. The simplest estimate, the sample average, is optimistically biased: the cost realized out of sample is, with probability close to one half, higher than predicted. Many corrections have been proposed: robust optimization over moment, $\phi$ -divergence or Wasserstein ambiguity sets (Delage & Ye 2010; Ben-Tal et al. 2013; Bertsimas, Gupta & Kallus 2018), and empirical-likelihood methods (Lam 2019; Duchi, Glynn & Namkoong 2021). Each is justified by a guarantee it satisfies. Van Parys, Mohajerin Esfahani and Kuhn (arXiv:1704.04118; Management Science 67(6), 2021) ask a different question: among all estimates that satisfy a given guarantee, which one is the least conservative? Their answer singles out one distributionally robust predictor, built on the relative entropy, as the best possible one in a precise sense. This mission formalizes that answer for the prediction problem with finitely many outcomes.

Setting

The outcome $\xi$ takes values in $\Xi=\{1,\dots,d\}$ . Decisions range over a compact set $X\subseteq\mathbb R^n$ , and the cost $\gamma(x,i)$ is continuous in $x$ for each $i$ . These are the standing assumptions of §2 (p. 5).

A model is a point $\mathbb P$ of the probability simplex $\mathcal P=\{\mathbb P\in\mathbb R^d_+:\sum_i\mathbb P(i)=1\}$ , with the topology inherited from $\mathbb R^d$ . Under a model the expected cost is $c(x,\mathbb P)=\sum_i\mathbb P(i)\gamma(x,i)$ .

The data are $T$ independent draws from an unknown model. They enter only through the empirical distribution $\hat{\mathbb P}_T(i)=\frac1T\#\{t\le T:\xi_t=i\}$ . Write $\mathbb P^\infty(\cdot)$ for probabilities when the samples are drawn from $\mathbb P$ .

A data-driven predictor is a continuous function $\hat c:X\times\mathcal P\to\mathbb R$ ; the number $\hat c(x,\hat{\mathbb P}_T)$ estimates $c(x,\mathbb P)$ . The set of all of them is $\mathcal C$ . Its out-of-sample disappointment is the probability $\mathbb P^\infty\big(c(x,\mathbb P)>\hat c(x,\hat{\mathbb P}_T)\big)$ that the true cost exceeds the prediction.

The relative entropy of $\mathbb P'$ with respect to $\mathbb P$ is

I(\mathbb P',\mathbb P)=\sum_{i}\mathbb P'(i)\log\frac{\mathbb P'(i)}{\mathbb P(i)}\in[0,\infty],

with $0\log(0/p)=0$ and $p'\log(p'/0)=+\infty$ for $p'>0$ .

For a threshold $r\ge0$ , the distributionally robust predictor is

\hat c_r(x,\mathbb P')=\sup_{\mathbb P\in\mathcal P}\{c(x,\mathbb P):I(\mathbb P',\mathbb P)\le r\}.\tag{10}

It is the worst expected cost over all models under which the observed frequencies $\mathbb P'$ are not exponentially unlikely at rate more than $r$ . The observed frequencies are the first argument of $I$ .

Formalization targets

Goal: Theorem 4 (strong optimality)

The paper's meta-optimization problem (5) is

\min_{\hat c\in\mathcal C}{}^{\preceq_{\mathcal C}}\ \hat c\quad\text{s.t.}\quad\limsup_{T\to\infty}\frac1T\log\mathbb P^\infty\big(c(x,\mathbb P)>\hat c(x,\hat{\mathbb P}_T)\big)\le-r\quad\forall x\in X,\ \mathbb P\in\mathcal P,

where $\hat c_1\preceq_{\mathcal C}\hat c_2$ means $\hat c_1\le\hat c_2$ pointwise on $X\times\mathcal P$ . A feasible $\hat c^\star$ is strongly optimal if $\hat c^\star\preceq_{\mathcal C}\hat c$ for every feasible $\hat c$ . The goal is:

\text{If } r>0,\ \text{then } \hat c_r \text{ is strongly optimal in (5).}

This statement contains feasibility (continuity and the decay rate) and pointwise domination of every feasible competitor.

Milestones

Proposition 1(ii) and (iii): $I$ is jointly convex and lower semicontinuous on $\mathcal P\times\mathcal P$ .
The supremum in (10) is attained.
Theorem 2: the strong LDP $\mathbb P^\infty(\hat{\mathbb P}_T\in\mathcal D)\le(T+1)^de^{-T\inf_{\mathcal D}I(\cdot,\mathbb P)}$ .
Theorem 1: the weak LDP, upper bound (7a) and, for $\mathbb P>0$ , lower bound (7b) with the interior of $\mathcal D$ .
Proposition 2: the dual representation $\hat c_r(x,\mathbb P')=\min_{\alpha\ge\bar\gamma(x)}\alpha-e^{-r}\prod_i(\alpha-\gamma(x,i))^{\mathbb P'(i)}$ , with a minimizer in an explicit interval.
Proposition 3 and Theorem 3: $\hat c_r$ is continuous, and is feasible in (5) for every $r\ge0$ .
Display (15): a maximizer in the closed relative entropy ball can be approximated in cost by a strictly positive model in the open ball.

Two further statements are included without milestones: Proposition 1(i), the information inequality, and Theorem 5, the finite-sample bound $(T+1)^de^{-rT}$ on the disappointment of $\hat c_r$ .

Significance

The theorem turns a choice among robust estimators into a uniqueness statement. If a decision maker asks only for a disappointment probability that decays exponentially at rate $r$ under every possible model, then $\hat c_r$ is the unique least conservative continuous predictor. Any predictor that is smaller anywhere fails the guarantee for some model. The reverse predictor (12) and the restricted predictor (13), which fix the other argument of $I$ or hedge only against models absolutely continuous with respect to the observed frequencies and are common in the literature, differ from $\hat c_r$ (Remark 3, pp. 15–16), so the theorem does not single them out. The companion missions of this series extend the result to predictor–prescriptor pairs (mission 2) and to compact continuous outcome spaces (mission 3).

The result is proved in the paper; nothing here is open. To our knowledge none of it has been machine-checked. A formal development would also give Mathlib-level statements of the method of types: Sanov's theorem for finite alphabets in both its finite-sample and asymptotic forms, together with the convexity and semicontinuity of the relative entropy with the $+\infty$ convention. These are reusable far beyond this paper.

Difficulty

Feasibility (Theorem 3) is a direct consequence of the LDP upper bound, once the disappointment set is seen to lie outside the relative entropy ball. The difficulty is the optimality half. Showing that a competitor $\hat c$ with $\hat c(x,\mathbb P_0')<\hat c_r(x,\mathbb P_0')$ at a single point must be infeasible requires a lower bound on a disappointment probability. The obvious approach evaluates the LDP lower bound at a maximizer $\mathbb P_0$ of (10). That fails: the lower bound (7b) needs a strictly positive model, the maximizer usually lies on the boundary of the simplex and on the boundary of the ball $I(\mathbb P_0',\cdot)\le r$ , and the bound is only in terms of the interior of the disappointment set. The argument has to move to a nearby model, and this uses the continuity of $\hat c$ , the convexity of $I$ , and the $+\infty$ convention of $I$ in its second argument. The asymptotic statements also require care with the logarithm of a vanishing probability.

Formalization scope

Outcomes and models. $\Xi$ is Fin d. $\mathcal P$ is the subtype of stdSimplex ℝ (Fin d), so interiors and continuity are relative to $\mathcal P$ , as footnote 1 (p. 12) requires.
Decisions and cost. $X$ is a compact subset of EuclideanSpace ℝ (Fin n), and $\gamma : X\to\mathbb R^d$ is continuous in $x$ for each outcome.
Relative entropy. $I$ is EReal-valued and is $+\infty$ whenever $\mathbb P(i)=0<\mathbb P'(i)$ . A real-valued sum would give a finite value there under Lean's log 0 = 0, and so change $\hat c_r$ .
Probabilities. Sampling probabilities are finite sums over sample paths $\Xi^T$ . The empirical distribution is defined for $T\ge1$ ; finite-sample statements assume $T\ge1$ .
Decay rates. Rates are stated without logarithms: $\limsup\frac1T\log p_T\le-r$ is written "for every $r'<r$ , eventually $p_T\le e^{-r'T}$ ". The form through Real.log would give a never-disappointed predictor rate $0$ , which makes Theorem 3 false.
Infima. Infima of $I$ are taken in the extended reals.
Competitors. They range over all jointly continuous functions on $X\times\mathcal P$ . Dropping continuity makes Theorem 4 false.
Sets $\mathcal D$ . The LDP statements require Borel $\mathcal D\subseteq\mathcal P$ , represented by MeasurableSet D in the simplex subtype.
Display (15). The clause $0<r_2$ is dropped: it is unused, and it fails for $d=1$ .

Welcome contributions: the method of types on Fin d (type classes, their cardinality bounds and probabilities); joint convexity and lower semicontinuity of the finite relative entropy with the $+\infty$ convention; and Berge's maximum theorem in the form used for Proposition 3. These are reusable beyond this mission. The definitions duplicate those of mission 2 of this series and are meant to be merged once published.

The source is the arXiv preprint arXiv:1704.04118v3 (22 Dec 2019); all theorem, display and page numbers refer to it.

Selected references

B. P. G. Van Parys, P. Mohajerin Esfahani, D. Kuhn, From Data to Decisions: Distributionally Robust Optimization is Optimal, Management Science 67(6), 2021. Preprint arXiv:1704.04118v3. https://arxiv.org/abs/1704.04118v3 — https://doi.org/10.1287/mnsc.2020.3678
T. M. Cover, J. A. Thomas, Elements of Information Theory, 2nd ed., Wiley, 2006 (Theorems 2.6.3, 2.7.2, 11.4.1). https://doi.org/10.1002/047174882X
A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Springer, 1998. https://doi.org/10.1007/978-1-4612-5320-4
A. Ben-Tal, D. den Hertog, A. De Waegenaere, B. Melenberg, G. Rennen, Robust Solutions of Optimization Problems Affected by Uncertain Probabilities, Management Science 59(2), 2013. https://doi.org/10.1287/mnsc.1120.1641
E. Delage, Y. Ye, Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems, Operations Research 58(3), 2010. https://doi.org/10.1287/opre.1090.0741
C. Berge, Topological Spaces, Oliver & Boyd, 1963 (pp. 115–116, maximum theorem).

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Machine Learning Optimal Transport Probability+1·Captain: mikedeng1

Minimax Statistical Learning with Wasserstein Distances II: Excess Local Minimax Risk of Wasserstein ERM for a Uniformly Lipschitz ClassResearch Paper

Motivation

In statistical learning a hypothesis $f$ is chosen from a class $\mathcal F$ using $n$ samples drawn from an unknown distribution $P$ , and its quality is measured by the risk $R(P,f) = \mathbf E_P[f(Z)]$ . When the distribution at deployment time may differ from the one that generated the data (domain drift, adversarial perturbation, model misspecification), a natural replacement is the worst-case risk over a neighbourhood of $P$ . Distributionally robust optimization with Wasserstein neighbourhoods takes these neighbourhoods to be balls in an optimal-transport distance, which compare distributions through the geometry of the instance space instead of through likelihood ratios. Wasserstein balls also contain distributions with support different from that of $P$ , which divergence balls do not.

J. Lee and M. Raginsky, Minimax statistical learning with Wasserstein distances (NeurIPS 2018, arXiv:1705.07815), give generalization guarantees for the empirical version of this procedure, the local minimax ERM. Their analysis rests on the strong duality theorem of Gao and Kleywegt (arXiv:1604.02199), which turns the supremum over a Wasserstein ball into a minimization over one real multiplier $\lambda \ge 0$ . This mission formalizes their Theorem 2, the excess-risk bound for a class of uniformly Lipschitz hypotheses, together with the steps of its proof. The source is the arXiv v2 preprint, which contains the NeurIPS camera-ready text and the supplementary appendices.

Setting

The instance space $\mathcal Z$ is a Polish space (complete separable metric space) with metric $d_{\mathcal Z}$ , and it is bounded: $\operatorname{diam}(\mathcal Z) = \sup_{z,z'} d_{\mathcal Z}(z,z') < \infty$ (Assumption 1). Fix $p \ge 1$ . The $p$ -Wasserstein distance between Borel probability measures is

W_p(P,Q) = \Big(\inf_{M} \mathbf E_M\big[d^p_{\mathcal Z}(Z,Z')\big]\Big)^{1/p},

the infimum over couplings $M$ of $P$ and $Q$ . For a radius $\varrho > 0$ the ball is $B^W_{\varrho,p}(P) = \{Q : W_p(P,Q) \le \varrho\}$ .

The hypothesis class $\mathcal F$ consists of upper semicontinuous functions $f : \mathcal Z \to \mathbb R$ with $0 \le f \le M$ (Assumption 2). The local worst-case risk and the local minimax risk are

R_{\varrho,p}(P,f) = \sup_{Q \in B^W_{\varrho,p}(P)} R(Q,f), \qquad R^*_{\varrho,p}(P,\mathcal F) = \inf_{f \in \mathcal F} R_{\varrho,p}(P,f).

Given i.i.d. samples $Z_1,\dots,Z_n$ from $P$ with empirical distribution $P_n = \frac1n\sum_i \delta_{Z_i}$ , the local minimax ERM is $\hat f \in \arg\min_{f \in \mathcal F} R_{\varrho,p}(P_n,f)$ .

Assumption 3 asks that every $f \in \mathcal F$ be $L$ -Lipschitz with one constant $L$ : $f(z') - f(z) \le L\,d_{\mathcal Z}(z',z)$ . The complexity of $\mathcal F$ is measured by the entropy integral

\mathfrak C(\mathcal F) = \int_0^\infty \sqrt{\log \mathcal N(\mathcal F,\|\cdot\|_\infty,u)}\,du,

where $\mathcal N(\mathcal F,\|\cdot\|_\infty,u)$ is the covering number of $\mathcal F$ in the uniform metric. The dual integrand of Gao–Kleywegt duality is $\varphi_{\lambda,f}(z) = \sup_{z'}\{f(z') - \lambda d^p_{\mathcal Z}(z,z')\}$ .

Formalization targets

Goal: Theorem 2 (10)

Under Assumptions 1–3, for every $\delta \in (0,1)$ , with probability at least $1-\delta$ ,

R_{\varrho,p}(P,\hat f) - R^*_{\varrho,p}(P,\mathcal F) \le \frac{48\,\mathfrak C(\mathcal F)}{\sqrt n} + \frac{48 L\cdot\operatorname{diam}(\mathcal Z)^p}{\sqrt n\cdot\varrho^{p-1}} + 3M\sqrt{\frac{\log(2/\delta)}{2n}}.

Milestones, in the order the proof uses them

Proposition 4 (8): $R_{\varrho,p}(Q,f) = \min_{\lambda\ge0}\{\lambda\varrho^p + \mathbf E_Q[\varphi_{\lambda,f}(Z)]\}$ .
Lemma 1: the optimal dual multiplier satisfies $\tilde\lambda \le L\varrho^{-(p-1)}$ .
(C.2): $R_{\varrho,p}(P_n,f^*) - R_{\varrho,p}(P,f^*) \le \int \varphi_{\lambda^*,f^*}\,d(P_n - P)$ for an achiever $f^*$ .
(C.3) with $\Lambda = [0, L\varrho^{-(p-1)}]$ and $\Phi = \{\varphi_{\lambda,f} : \lambda \in \Lambda, f \in \mathcal F\}$ : $R_{\varrho,p}(P,\hat f) - R_{\varrho,p}(P_n,\hat f) \le \sup_{\varphi\in\Phi}\int\varphi\,d(P - P_n)$ .
(C.4): with probability at least $1-\delta/2$ , $R_{\varrho,p}(P,\hat f) - R_{\varrho,p}(P_n,\hat f) \le 2\mathfrak R_n(\Phi) + M\sqrt{2\log(2/\delta)/n}$ .
(C.5): with probability at least $1-\delta/2$ , $R_{\varrho,p}(P_n,f^*) - R_{\varrho,p}(P,f^*) \le M\sqrt{\log(2/\delta)/(2n)}$ .
The Rademacher bound of Appendix C.3: $\mathfrak R_n(\Phi) \le \frac{24}{\sqrt n}\mathfrak C(\mathcal F) + \frac{24L\operatorname{diam}(\mathcal Z)^p}{\sqrt n\,\varrho^{p-1}}$ .

Significance

Theorem 2 says that robust ERM over a uniformly Lipschitz class is nearly optimal for the robust risk at the unknown distribution, with excess of order $1/\sqrt n$ and explicit constants. The dependence on $\varrho$ enters only through $\varrho^{-(p-1)}$ ; at $p = 1$ the bound does not depend on $\varrho$ and recovers the usual rate of ordinary ERM. The paper derives Corollaries 1 and 2 (regression with a Lipschitz link, and Gaussian-kernel classes) and the domain-adaptation bound of Theorem 4 from it. Lemma 1, which confines the dual multiplier to a fixed interval, is also used outside the theorem: Remark 3 proposes it as the search range for the fixed- $\lambda$ adversarial training of Sinha et al.

The result is proved in the paper; to our knowledge none of it has been machine-checked. What this mission adds is a verified version of a complete distributionally robust generalization argument: Gao–Kleywegt strong duality on a bounded Polish space, the multiplier bound, symmetrization and bounded differences for a class indexed by a function and a real parameter, and a Dudley entropy bound for that joint class. The paper's appendix states the Rademacher bound with a stray constant $C_0$ that belongs to a different assumption (Assumption 4); the milestone states the bound without it, which is what the proof gives and what the constant $48L$ of (10) needs.

Difficulty

The obvious route bounds $\sup_{f}|R_{\varrho,p}(P,f) - R_{\varrho,p}(P_n,f)|$ uniformly over $\mathcal F$ . That needs uniform control of the dual multiplier for every $f$ and every empirical distribution, and without it the class $\{\varphi_{\lambda,f}\}$ is indexed by an unbounded $\lambda$ , so its complexity is infinite. Lemma 1 resolves this only for the minimizers, which is why the argument is organized around $\hat f$ and an achiever $f^*$ rather than a uniform bound.

The formal difficulties are separate. Strong duality with attainment of the minimum is a measure-theoretic theorem about couplings on a Polish space (existence of optimal couplings, measurable selection of near-maximizers), none of which is in Mathlib for Wasserstein balls. The supremum over $\Phi$ must be shown measurable before its expectation means anything. Dudley's chaining bound over a separable, infinite index set is also missing from Mathlib in the form needed here. Finally, the theorem does not assume that $R^*_{\varrho,p}(P,\mathcal F)$ is attained, while the paper's proof uses an achiever $f^*$ , so a complete proof needs an approximation argument that the paper does not write out.

Formalization scope

$\mathcal Z$ is a MetricSpace with BorelSpace and PolishSpace instances, and Assumption 1 is Bornology.IsBounded (Set.univ : Set 𝒵). $\operatorname{diam}(\mathcal Z)$ is Metric.diam univ.
$W_p$ and the ball are the published definitions WassersteinLinOpt.Ball.wassersteinDist and wassersteinBall. Membership in $\mathcal P_p(\mathcal Z)$ is automatic on a bounded space.
The standing assumptions $p \ge 1$ and $\varrho > 0$ are binders of every item.
Risks, the dual integrand $\varphi_{\lambda,f}$ and the Rademacher average are real suprema, infima and Bochner integrals. Under the binders each is over a nonempty set that is bounded on the relevant side, and each integrand is bounded and measurable.
The covering number is the internal one (centres in $\mathcal F$ ), valued in $\mathbb N\cup\{\infty\}$ , and $\mathfrak C(\mathcal F)$ is valued in $[0,\infty]$ . Every item whose bound contains $\mathfrak C(\mathcal F)$ assumes $\mathfrak C(\mathcal F) < \infty$ and uses its real value. This is a disclosed pin: otherwise the bound is $+\infty$ .
Assumption 3 is ∀ f ∈ ℱ, ∀ z z', f z' - f z ≤ L * dist z' z, with one $L$ quantified before $\mathcal F$ , and $0 \le L$ is a disclosed pin.
The sample is ω : Fin n → 𝒵 with law Measure.pi (fun _ => P), and $n \ge 1$ . "With probability at least $1-\delta$ " means the outer measure of the failure set is at most $\delta$ , for $0 < \delta < 1$ .
The ERM is a sample-indexed selection $\hat f$ with membership and minimality hypotheses, never Classical.choose. Such a selection exists exactly when empirical minimizers exist, which display (7) presupposes.
"min" over $\lambda$ in Proposition 4 includes attainment, and the argmins of Lemma 1 and (C.2) are minimality hypotheses.

Trivializing formalizations are ruled out:

Assumption 3 is not written with $L$ chosen after $f$ , and $L \ge 0$ is not omitted.
The stray $C_0$ is not in the Lean.
The goal is about $R^*_{\varrho,p}(P,\mathcal F)$ , not about a fixed competitor $f$ .
No probability is converted with toReal, and an infinite $\mathfrak C(\mathcal F)$ is never converted to $0$ .

A complete development needs:

Gao–Kleywegt duality for bounded upper semicontinuous losses;
McDiarmid's and Hoeffding's inequalities on product measures;
symmetrization;
Dudley's entropy integral bound for sub-Gaussian processes indexed by a separable pseudometric space.

The last three are reusable well beyond this mission. Contributions of any of these as stand-alone results are welcome, as are proofs of the milestones.

Selected references

J. Lee and M. Raginsky, Minimax statistical learning with Wasserstein distances, Advances in Neural Information Processing Systems 31 (NeurIPS 2018); arXiv:1705.07815v2. https://arxiv.org/abs/1705.07815
R. Gao and A. J. Kleywegt, Distributionally robust stochastic optimization with Wasserstein distance, Mathematics of Operations Research 48(2), 2023; arXiv:1604.02199. https://arxiv.org/abs/1604.02199
A. Sinha, H. Namkoong and J. Duchi, Certifying some distributional robustness with principled adversarial training, ICLR 2018; arXiv:1710.10571. https://arxiv.org/abs/1710.10571
M. Talagrand, Upper and Lower Bounds for Stochastic Processes, Springer, 2014. https://doi.org/10.1007/978-3-642-54075-2

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Linear Optimization Operations Research Optimization·Captain: mikedeng1

Constrained Assortment Optimization for the Nested Logit Model 7: Under Space Constraints, for Every α > 1, O(⌈α/(α−1)⌉ n^(⌈α/(α−1)⌉+2)) Candidate Assortments Include an α-Approximate SolutionResearch Paper

Motivation

A retailer that sells products in categories must decide which products to offer in each category. When customers choose among the offered products according to the nested logit model, the expected revenue depends on the offered assortment in a nonlinear way, and the retailer's shelf space adds a knapsack constraint in every category. Gallego and Topaloglu (Management Science, 2014) show that the resulting space-constrained assortment problem is NP-hard even with a single nest (via Lemma 2.1 of Rusmevichientong, Shen and Shmoys, cited in the paper as 2009), give a factor-2 approximation through a linear program, and then, in Online Supplement C of the paper, refine the approach into an approximation scheme: for every desired guarantee $\alpha>1$ , a polynomially sized family of candidate assortments per nest suffices.

The scheme follows the partial-enumeration idea of Frieze and Clarke (1984) for multi-dimensional knapsack problems, but has to work for a whole one-parameter family of knapsack problems, one for every value of a scalar $u\ge0$ , with a single collection of candidates fixed in advance.

Setting

There are nests $i\in M$ and products $j\in N=\{1,\dots,n\}$ . Product $j$ of nest $i$ has a preference weight $v_{ij}>0$ , a revenue $r_{ij}\in\mathbb R$ (no sign or order is assumed), and a space requirement $w_{ij}>0$ ; nest $i$ has capacity $c_i$ with $w_{ij}\le c_i$ . An assortment of nest $i$ is a set $S\subseteq N$ ; it is feasible, $S\in\mathcal C_i$ , when $\sum_{j\in S}w_{ij}\le c_i$ . Write

V_i(S)=\sum_{j\in S}v_{ij},\qquad R_i(S)=\frac{\sum_{j\in S}r_{ij}v_{ij}}{V_i(S)}\quad(R_i(\emptyset)=0).

For $u\ge0$ , problem (7) is $\max_{S\in\mathcal C_i}V_i(S)(R_i(S)-u)$ , which equals the knapsack problem (10) $\max_{S\in\mathcal C_i}\sum_{j\in S}v_{ij}(r_{ij}-u)$ . The quantity $v_{ij}(r_{ij}-u)$ is the utility of product $j$ ; $z^\ast(u)$ denotes the optimal value of (10). An assortment $\hat S\in\mathcal C_i$ is $\alpha$ -approximate at $u$ when $V_i(S)(R_i(S)-u)\le\alpha V_i(\hat S)(R_i(\hat S)-u)$ for all $S\in\mathcal C_i$ .

For a set $J\subseteq N$ , problem (17) is the linear program

\max\Big\{\sum_{j}v_{ij}(r_{ij}-u)x_{ij} : \sum_j w_{ij}x_{ij}\le c_i,\ x_{ij}=1\ (j\in J),\ 0\le x_{ik}\le\mathbf 1\big(v_{ik}(r_{ik}-u)\le\min_{j\in J}v_{ij}(r_{ij}-u)\big)\ (k\notin J)\Big\},

the LP relaxation of (10) with the products of $J$ fixed in the knapsack and every product whose utility exceeds the smallest utility in $J$ removed. Its optimal value is $\zeta^\ast(u,J)$ . Rounding a solution $x$ down gives the assortment $\lfloor x\rfloor=\{j:x_{ij}=1\}$ . $\wp_q$ is the family of subsets of $N$ with at most $q$ elements.

Formalization targets

Goal: Theorem 9 (p. 40)

For every $\alpha>1$ , with $q=\lceil\alpha/(\alpha-1)\rceil$ , there is a collection $\{A_i^t:t\in\mathcal T_i\}\subseteq\mathcal C_i$ such that

|\mathcal T_i|\le 5(q+1)\,n^{q+2}+1\quad\text{and}\quad\forall u\ge0\ \exists t:\ A_i^t\text{ is }\alpha\text{-approximate for (7) at }u .

The paper writes $|\mathcal T_i|=O(\lceil\alpha/(\alpha-1)\rceil n^{\lceil\alpha/(\alpha-1)\rceil+2})$ ; the explicit bound above is of that order.

Milestones

Whenever (17) is feasible, it has an optimal solution with at most one fractional component (p. 39).
If an optimal $S^\ast_i$ of (10) at $\hat u$ has $|S^\ast_i|\le q$ , then rounding down an optimal solution of (17) with $J=S^\ast_i$ solves (10) at $\hat u$ (p. 41).
If $|S^\ast_i|>q$ and $J^\ast_i$ holds $q$ products of $S^\ast_i$ of largest utility, a fractional component $j'$ of an optimal solution of (17) at $(\hat u,J^\ast_i)$ satisfies $v_{ij'}(r_{ij'}-\hat u)\le z^\ast(\hat u)/q$ (pp. 41–42).
In the same situation $S^\ast_i$ is feasible for (17), so $z^\ast(\hat u)\le\zeta^\ast(\hat u,J^\ast_i)$ (p. 42).
Lemma 8 (p. 40), per $u$ : for $q\ge2$ and every $u\ge0$ some $J\in\wp_q$ makes every optimal solution of (17) with at most one fractional component round down to a $q/(q-1)$ -approximate solution of (10).

Significance

Together with Theorem 4 of the paper (an $\alpha$ -approximate candidate per nest for every $u$ yields an $\alpha$ -approximate assortment for the whole problem) and Theorem 2 (the best combination of candidates solves a linear program with $1+m$ variables), Theorem 9 gives, for every $\alpha>1$ , an $\alpha$ -approximation of the space-constrained nested logit assortment problem by a single linear program with $O(m\lceil\alpha/(\alpha-1)\rceil n^{\lceil\alpha/(\alpha-1)\rceil+2})$ constraints; for $\alpha=3/2$ this is $O(mn^5)$ . It shows that the factor 2 of Section 5 is not a barrier of the method.

The result is proved in the paper. As far as the platform's records show, none of it is formalized; this mission produces a machine-checked proof of Lemma 8 and Theorem 9, including the explicit counting of parameter intervals that the paper only sketches.

A related but different construction, with large and small products and a continuous knapsack whose weights are preference weights, appears in the unconstrained variants of Davis, Gallego and Topaloglu (items NestedLogitVariants.PowersDelta.*); it is not reused here.

Difficulty

Two points resist the obvious argument. First, rounding down the plain LP relaxation of (10) drops one fractional product whose utility can be as large as the whole optimum, which is why Section 5 only reaches a factor 2; no choice of rounding of that relaxation alone does better. The guarantee has to come from the partially fixed problems (17), and their indicator constraint is delicate: it excludes only products of strictly larger utility than the least utility in $J$ , and a version that also excludes ties can cut off the optimal assortment of (10), and the comparison with $z^\ast(u)$ is then lost.

Second, the collection must be fixed before $u$ . The optimal solution of (17) depends on $u$ through the ordering of the utilities, the ordering of the utility-to-space ratios and the signs of the utilities, which change only at the intersection points of $2(n+1)$ lines. Because the feasible set of (17) itself depends on $u$ through the indicator constraint, the solution on an open interval between two such points need not remain optimal at its endpoints; the count must treat the breakpoints as pieces of their own.

Formalization scope

Nests are a finite type ι; products are Fin n (0-based); assortments are Finset (Fin n), with $\emptyset$ for $\bar 0$ .
The published model NestedLogitVariants.LP.Model is referenced; its within-nest no-purchase weight is set to zero (I.vnp i = 0), which makes V and R the paper's $V_i$ and $R_i$ . The paper's extension $V_i(S_i)=v_{i0}\mathbf 1(S_i\ne\bar 0)+\dots$ is not formalized.
Added hypotheses, all flagged in the items: $v_{ij}>0$ , $w_{ij}>0$ , $c_i\ge0$ (the empty assortment is feasible even when $n=0$ ), $q\ge2$ in Lemma 8 and $q\ge1$ in milestone 3. The paper's $w_{ij}\le c_i$ is kept in the goal.
The paper's $O(\cdot)$ is pinned to $5(q+1)n^{q+2}+1$ , with a constant that does not depend on $\alpha$ , as the paper's explicit factor $\lceil\alpha/(\alpha-1)\rceil$ indicates: for $n\ge1$ , at most $(q+1)n^q$ sets in $\wp_q$ times at most $2n(n+1)+1\le5n^2$ pieces of $[0,\infty)$ ; the $+1$ covers $n=0$ .
$q=\lceil\alpha/(\alpha-1)\rceil$ is Nat.ceil; $\alpha>1$ is strict.
The collection is quantified before $u$ ; problem (17) is a real LP over $[0,1]^n$ ; its indicator constraint excludes only products of strictly larger utility.
Lemma 8 is stated per $u$ : the interval solutions $x_i^g(J)$ are not defined, and their counting belongs to Theorem 9.
Trivializing formalizations are ruled out: the size bound is part of the goal (without it $A=\mathcal C_i$ works, and a bound like $2^n$ would do the same); $\alpha$ is arbitrary in $(1,\infty)$ , not fixed; and (17) carries its indicator constraint, without which the $q/(q-1)$ argument fails.
Welcome contributions: a reusable greedy theorem for the fractional knapsack LP (one fractional component), and a general lemma counting the pieces on which a finite family of affine functions has constant order and signs.

Selected references

G. Gallego and H. Topaloglu, Constrained Assortment Optimization for the Nested Logit Model, Management Science 60(10), 2014. https://doi.org/10.1287/mnsc.2014.1931 (cited from the authors' manuscript of Sept. 11, 2013).
A. M. Frieze and M. R. B. Clarke, Approximation algorithms for the m-dimensional 0–1 knapsack problem: worst-case and probabilistic analyses, European Journal of Operational Research 15(1), 1984. https://doi.org/10.1016/0377-2217(84)90053-5
P. Rusmevichientong, Z.-J. M. Shen and D. B. Shmoys, Dynamic assortment optimization with a multinomial logit choice model and capacity constraint, Operations Research 58(6), 2010. https://doi.org/10.1287/opre.1100.0866
J. M. Davis, G. Gallego and H. Topaloglu, Assortment Optimization Under Variants of the Nested Logit Model, Operations Research 62(2), 2014. https://doi.org/10.1287/opre.2014.1256

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Complexity Theory Operations Research Optimization+2·Captain: mikedeng1

A Comment on "Computational Complexity of Stochastic Programming Problems" 1: δ-Accurate Expected Recourse Values of Problem (2) Determine the #Parity Count of a Knapsack PolytopeResearch Paper

Motivation

A two-stage stochastic program chooses a decision before uncertainty is observed and a recourse decision afterward. Its objective can involve the expected optimal value of the recourse problem. Even with linear constraints, evaluating that expectation can be difficult when the random input has many coordinates. Dyer and Stougie's 2006 complexity paper studied this question for stochastic programming. Hanasusanto, Kuhn and Wiesemann identified a false volume identity in one fixed-recourse argument, then supplied a quantitative replacement in their 2015 preprint. The replacement shows that sufficiently accurate expected-recourse values determine an exact counting answer.

The distinction matters to researchers designing numerical methods for stochastic programs. A hardness statement at a specified absolute accuracy identifies the scale at which a general exact-information guarantee would have strong complexity consequences. It does not say that useful approximations at coarser accuracy are unavailable. The present mission records the mathematical reduction behind the paper's Theorem 1 and its quantitative tolerance.

Setting

Let $k\ge1$ . A realization $\xi=(\xi_1,\ldots,\xi_k)$ lies in the unit cube $C=[0,1]^k$ , equipped with Lebesgue measure. This cube has volume one, so integration over it is expectation under the uniform law. Let $\alpha\in\mathbb R^k_+$ be nonnegative weights and $\beta\ge0$ a budget. The second-stage value $Q(\xi;\alpha,\beta)$ is the maximum of

\sum_{j=1}^k\xi_jy_j-\beta z

over $0\le z\le1$ and $0\le y_j\le\alpha_jz$ for each $j$ . Its expected recourse value is $\mathcal Q(\alpha,\beta)=\int_C Q(\xi;\alpha,\beta)\,d\xi$ . There is no first-stage decision in this instance of the model.

The knapsack polytope is $P(\alpha,\beta)=\{\xi\in C:\sum_j\alpha_j\xi_j\le\beta\}$ , and $V(\alpha,\beta)$ denotes its volume. For integer weights and budget, its binary points correspond to subsets $S\subseteq\{1,\ldots,k\}$ with $\sum_{j\in S}\alpha_j\le\beta$ . The #Parity count $D$ is the number of such subsets of even size minus the number of odd size. The paper invokes the counting complexity of this problem from Dyer and Frieze's 1988 volume paper.

For $i=0,\ldots,k$ , define $\gamma_i=\beta+i/(k+1)$ and a square matrix $F$ by $F_{ic}=\gamma_i^{k-c}$ for $c=0,\ldots,k$ . The first column has power $k$ and the last has power zero. This order matters: the first coordinate of the coefficient vector is the parity count. The matrix inverse is controlled by a column-sum bound of the kind studied by Gautschi in 1962.

Formalization targets

Goal: recourse values determine the count

Write $\varepsilon_3(\alpha)=1/[2k!(\|\alpha\|_1+2)^k(k+1)^{k+1}\prod_j\alpha_j]$ and $\delta_7(\alpha)=[\alpha_k\varepsilon_3(\alpha)/(1+\alpha_k)]^2$ , exactly the right sides of (3) and (7). For positive integer weights, $\beta\le\sum_j\alpha_j$ , and $0<\delta<\delta_7(\alpha)$ , take any oracle $Q_\delta(t)$ with $|Q_\delta(t)-\mathcal Q(\alpha,t)|\le\delta$ at every nonnegative $t$ . Set $h=2\sqrt\delta$ and

\widetilde g_i=\frac{Q_\delta(\gamma_i+h)-Q_\delta(\gamma_i)}{h}+1.

The target is that every solution of

F\widetilde x=\left(k!\prod_{j=1}^k\alpha_j\right)\widetilde g

satisfies $|\widetilde x_0-D|<1/2$ . Nearest-integer rounding therefore recovers $D$ .

Supporting targets

The milestones establish the value formula for the second-stage program, the inclusion–exclusion volume formula (8), the integer coefficient identity for the exact volume system, the inverse bound (4), the perturbation bound (6), the derivative identity in Lemma 2, a global volume-growth bound, and the finite-difference estimate in Theorem 1. Together they retain the paper's constants and its sequence from expected values to volume approximations to an exact count. Proposition 1 separately records why the earlier proposed identity $1-\mathcal Q=V$ fails.

Significance

The result certifies a concrete information transfer. A function returning expected-recourse values to the stated absolute tolerance also contains enough information to distinguish the exact even-minus-odd count of feasible binary points. The bound states how accurate those values must be as dimension and weights change; a qualitative assertion of hardness alone would hide that dependence. The paper concludes #P-hardness after accounting for the polynomial-time operations of the reduction. That complexity-class conclusion is part of the paper, while the Lean goal isolates the reduction's numerical correctness.

A complete formalization would give separately reusable results about cube integrals, knapsack volumes, perturbations of Vandermonde systems, and finite differences of expected hinge functions. The source paper proves the mathematical assertions; this proposal stages their Lean statements as open proof obligations. No machine-checked proof is claimed here. The companion counterexample also keeps the corrected argument anchored to the precise error it repairs.

Difficulty

The tempting identity $1-\mathcal Q(\alpha,\beta)=V(\alpha,\beta)$ fails: the expectation of the second-stage optimum is a positive-part expectation, while volume is a distribution function. The paper's Proposition 1 makes the discrepancy explicit for all-ones weights. Recovering volume requires differentiation with respect to the budget, and an approximate function value does not directly provide an accurate derivative. The difficulty is controlling both oracle error and the change of volume over a finite budget interval tightly enough that the final coefficient error is strictly below one half.

There are further exactness points. The column order of $F$ fixes which coefficient equals $D$ . The $1$ -norm in (4) is the maximum column sum of the inverse, not a generic matrix norm. The perturbation threshold contains $k!$ , every weight, and the power $(k+1)^{k+1}$ ; losing one factor can invalidate the rounding guarantee.

Formalization scope

Lean uses vectors $\mathrm{Fin}(k)\to\mathbb R$ , with zero-based indices; the paper's $\alpha_k$ is the final coordinate. Lebesgue measure is restricted to $C$ , whose volume is one. The LP value is a real supremum of its feasible objective values. For the nonnegative weights used in the theorems, the feasible set is nonempty and bounded, so this represents the actual maximum. The expected value is a real integral over the cube. The matrix $F$ has descending powers, and its $1$ -norm estimate is written as a bound on each inverse column sum. Binary vectors are represented by subsets; $D$ keeps the paper's even-minus-odd convention.

Positive weights are required where (3) and (7) divide by their product, and $\delta>0$ where $h$ divides a finite difference. The case $\beta>\sum_j\alpha_j$ is excluded from the main target because the paper first answers it directly with $D=0$ . Lemma 2 excludes $\alpha=0,\beta=0$ , where the claimed derivative fails to exist. The volume-growth milestone gives the content of the paper's $\mathcal Q''\le1/\alpha_k$ argument without requiring a second derivative at a kink; it applies across the actual interval starting at $\beta$ . The printed determinant sign is also adjusted to the descending columns by asserting nonsingularity, which is what the proof uses. The second-stage value milestone retains only the optimal value, because the optimal decisions the page calls unique need not be unique at ties. The mission must use the LP and counting definitions above; replacing them with an arbitrary function assumed to have the desired derivative would empty the reduction.

The formal work requires measure and integration theory, finite polytope volume, finite sums over subsets, matrix inverses, and real inequalities. Contributions proving the eight milestones and any honest supporting lemmas are welcome. The setup does not encode polynomial-time computability, bit lengths, or a #P complexity class, so closure of the Lean goal alone is not a formal proof of the full complexity statement.

Selected references

G. A. Hanasusanto, D. Kuhn and W. Wiesemann, A comment on “computational complexity of stochastic programming problems”, Optimization Online preprint 2015/03/4825, version of October 6, 2015; published in Mathematical Programming, 2016. Preprint.
M. Dyer and L. Stougie, Computational complexity of stochastic programming problems, Mathematical Programming 106 (2006), 423–432. DOI.
M. E. Dyer and A. M. Frieze, On the complexity of computing the volume of a polyhedron, SIAM Journal on Computing 17 (1988), 967–974. DOI.
W. Gautschi, On inverses of Vandermonde and confluent Vandermonde matrices, Numerische Mathematik 4 (1962), 117–123. Scan.

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Operations Research Optimization Probability+1·Captain: mikedeng1

Bootstrap Robust Prescriptive Analytics 1: The Entropic Robust Budget Is Broken on Bootstrap Data with Probability at Most Σⱼ exp(−n·max{r, rⱼⁿ})Research Paper

Motivation

Prescriptive analytics chooses a decision $z$ from data after observing a context $x_0$ : a retailer sets an order quantity after seeing the weather forecast, a hospital schedules staff after seeing the day of the week. A common recipe estimates the expected cost of each decision from the training observations closest to $x_0$ (nearest neighbours, or a kernel-weighted Nadaraya–Watson average) and then minimizes that estimate (Bertsimas and Kallus, 2020). The decision that minimizes an estimate tends to look better on the training data than it is: its estimated cost is optimistically biased. Bertsimas and Van Parys (arXiv:1711.09974v2) measure that optimism on resampled data. Bootstrap data are $n$ independent draws from the training distribution, in the sense of Efron (1979). A cost budget computed on the training data disappoints when the estimate on a bootstrap sample exceeds it. The paper replaces the nominal estimate by a robust budget, a worst case over all distributions within relative-entropy distance $r$ of the training distribution. Its Theorem 6 bounds the probability of disappointment by an explicit sum of exponentials in $n$ .

This mission formalizes that theorem, together with the structural results its proof rests on and the Nadaraya–Watson special case, Corollary 2. The same paper's dual reformulation of the budget (Lemma 2) and its optimality statement for the relative entropy (Proposition 1) are the subject of two sibling missions.

Setting

The $n$ training points have a finite set $\Omega_n$ of distinct values. A distribution on $\Omega_n$ is a vector $D=(D_i)_{i\in\Omega_n}$ in the standard simplex; these form $\mathcal D_n$ . The training distribution $D_{\mathrm{tr}}\in\mathcal D_n$ gives every point of $\Omega_n$ positive mass. The bootstrap distributions are

\mathcal D_{n,n}=\{D\in\mathcal D_n:\ nD_i\in\{0,1,\dots,n\}\ \forall i\}.

These are the possible empirical distributions $D_{\mathrm{bs}[n]}$ of $n$ independent draws from $D_{\mathrm{tr}}$ .

Around the context $x_0$ the points of $\Omega_n$ are grouped into nested neighbourhoods $\emptyset=N^0\subseteq N^1\subseteq\dots\subseteq N^n=\Omega_n$ . Each point $i$ carries a positive weight $w_i$ and, for a decision $z$ , a loss $\ell_i=L(z,\bar y_i)$ . Fix $k\in\{1,\dots,n\}$ . The estimator (18) of a distribution $D$ averages the loss with weights $w_iD_i$ over the smallest neighbourhood $N^{j}$ with $D$ -mass at least $k/n$ . For $j\in[n]$ , the partial estimator $E^{n,j}_D$ is the value of the linear program (22). Its domain is contained in

\mathcal D^j_n=\Big\{D\in\mathcal D_n:\ \sum_{i\in N^{j-1}}D_i\le\tfrac{k-1}{n},\ \ \tfrac kn\le\sum_{i\in N^j}D_i\Big\},

and off that domain $E^{n,j}_D=-\infty$ .

The bootstrap distance is the relative entropy

B(D,D')=\sum_{i}D_i\log\frac{D_i}{D'_i}.

The partial robust budgets are $c^j_n=\sup\{E^{n,j}_D:\ D\in\mathcal D_n,\ B(D,D_{\mathrm{tr}})\le r\}$ , the robust budget is $\bar c_n=\max_{j\in[n]}c^j_n$ (26), and the minimum radii are $r^j_n=\inf\{B(D,D_{\mathrm{tr}}):D\in\mathcal D^j_n\}$ (29).

Formalization targets

Goal: Theorem 6 (p. 15)

For every decision $z$ ,

\mathbb P\Big[E^n_{D_{\mathrm{bs}[n]}}[L(z,y)\,|\,x=x_0]>\bar c_n(z)\Big]\ \le\ \sum_{j\in[n]}\exp\big(-n\cdot\max\{r,r^j_n\}\big).

The bound holds for every $n$ , $k$ , neighbourhood chain, weight vector and radius, with no asymptotics.

Milestones

The domain inclusion (23), with $E^{n,j}_D=-\infty$ off $\mathcal D^j_n$ (p. 11).
The closed form of $E^{n,j}_D$ on $\mathcal D^j_n$ as a weighted average over $N^j$ (B.1, p. 26).
The sets $\mathcal D^j_n\cap\mathcal D_{n,n}$ , $j\in[n]$ , partition $\mathcal D_{n,n}$ (B.1, p. 26).
Theorem 1: $E^n_D=\max_{j\in[n]}E^{n,j}_D$ on $\mathcal D_{n,n}$ (p. 11).
Theorem 5, Csiszár's inequality: for every convex $\mathcal C\subseteq\mathcal D_n$ and $n\ge1$ ,

\mathbb P[D_{\mathrm{bs}[n]}\in\mathcal C]\le\exp\Big(-n\inf_{D\in\mathcal C}B(D,D_{\mathrm{tr}})\Big)\quad\text{(p. 14)}.

Each $\mathcal C_j=\{D\in\mathcal D^j_n:E^{n,j}_D>\bar c\}$ is convex (p. 15).
On $\mathcal D_{n,n}$ , $E^n_D>\bar c$ if and only if $D\in\bigcup_{j}\mathcal C_j$ (p. 15).
$\inf_{D\in\mathcal C_j}B(D,D_{\mathrm{tr}})\ge\max\{r,r^j_n\}$ when $\bar c=\bar c_n$ (p. 15).
For Nadaraya–Watson, the disappointment set is convex and lies at bootstrap distance at least $r$ (B.6, p. 30).
Corollary 2: with $k=n$ the disappointment is at most $\exp(-n\cdot r)$ (p. 16).

Significance

Theorem 6 makes the radius $r$ an explicit statistical dial. The sum $\sum_j\exp(-n\max\{r,r^j_n\})$ can be computed from the training data, because each $r^j_n$ is a convex program. So a practitioner can choose $r$ for a target disappointment $b$ , and the bound decays exponentially in $n$ at rate at least $r$ . Corollary 2 gives a single exponential bound for Nadaraya–Watson: $r\ge\log(1/b)/n$ gives disappointment at most $b$ . Together with the paper's Proposition 1, which shows that no smaller distance keeps the rate $-r$ , the result explains why the relative entropy is the natural ambiguity measure for bootstrap guarantees.

On the formal side, the results are proved in the paper. Theorem 5 is cited there from Csiszár (1984), not reproved. The local platform search found no matching machine-checked statement of Theorem 6, Theorem 1 or Csiszár's convex-set inequality. A formal proof of Theorem 5 on a finite alphabet would be a reusable large-deviation tool: a finite-sample Sanov upper bound without polynomial prefactor, for empirical distributions of i.i.d. samples in any convex set. Theorem 1 and the partition lemma are the combinatorial facts behind the nearest-neighbours robust formulation of this paper.

Difficulty

Two steps carry the weight. The first is Theorem 5. The method of types alone gives $\mathbb P[D_{\mathrm{bs}[n]}\in\mathcal C]\le(n+1)^{|\Omega_n|}\exp(-n\inf_{\mathcal C}B)$ , and its polynomial factor is fatal here: the goal has none. Removing it needs convexity of $\mathcal C$ in an essential way; the bound is false for general sets. The set $\mathcal C$ need not be closed and its infimum need not be attained. The second is Theorem 1. The estimator selects its neighbourhood depending on $D$ , so it is neither linear nor concave in $D$ . The disappointment event is therefore not convex, and Theorem 5 cannot be applied to it directly. Splitting it over the sets $\mathcal D^j_n$ needs the partition of $\mathcal D_{n,n}$ . That partition fails off the grid $\mathcal D_{n,n}$ : a mass strictly below $k/n$ is at most $(k-1)/n$ only for multiples of $1/n$ .

Formalization scope

The support $\Omega_n$ is a finite type $\iota$ , and a distribution is a vector ι → ℝ in stdSimplex ℝ ι. Covariates, responses, the context and the distance function of Definition 2 do not appear. The neighbourhoods are a chain N : ℕ → Finset ι with Monotone N, N 0 = ∅ and N n = univ, which abstracts Definition 2 and its tie-breaking. Weights are positive, losses are real-valued (Assumption 1 allows $+\infty$ ; no planned statement needs it), and nonnegativity and convexity of the loss are not assumed. The relative entropy is EReal-valued, with $0\log0=0$ and $B(D,D')=+\infty$ when $D'_i=0<D_i$ . Every supremum and infimum that may be empty or unbounded is in EReal: an infeasible program (22) is $-\infty$ , an empty $\mathcal D^j_n$ has $r^j_n=+\infty$ , and $\exp(-n\cdot(+\infty))=0$ . The bootstrap law is the product measure Measure.pi of $n$ copies of $\sum_iD_{\mathrm{tr},i}\delta_i$ . The robust budget is the formulation (26), $\max_jc^j_n$ , which the paper computes and which never exceeds (24). The theorem is stated for every decision $z$ , not only for the robust prescriptor. The printed strict inequalities for the infima over $\mathcal C_j$ are stated as $\ge$ .

Trivializing encodings are ruled out explicitly. Real sSup/sInf would turn infeasible programs and empty sets into a finite $0$ . A real logarithm with a vanishing reference would give $B$ a finite junk value. A single draw or a non-product law would make Theorem 5 false. A chain without $N^0=\emptyset$ and $N^n=\Omega_n$ would break the partition. expNeg treats $+\infty$ explicitly so that an empty $\mathcal D^j_n$ contributes $0$ , not $1$ .

A complete development needs the method of types on a finite alphabet, I-projections onto convex sets of the simplex, and the Pythagorean inequality. The latter two are reusable well beyond this mission. Contributions to any milestone, or to a standalone finite-alphabet Csiszár inequality, are welcome.

Selected references

D. Bertsimas and B. Van Parys, Bootstrap robust prescriptive analytics, arXiv:1711.09974v2, 2021. https://arxiv.org/abs/1711.09974
I. Csiszár, Sanov property, generalized I-projection and a conditional limit theorem, Annals of Probability 12(3), 1984. https://doi.org/10.1214/aop/1176993227
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Springer, 2nd ed., 2010. https://doi.org/10.1007/978-3-642-03311-7
B. Efron, Bootstrap methods: another look at the jackknife, Annals of Statistics 7(1), 1979. https://doi.org/10.1214/aos/1176344552
D. Bertsimas and N. Kallus, From predictive to prescriptive analytics, Management Science 66(3), 2020. https://doi.org/10.1287/mnsc.2018.3253

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Convex Optimization Dynamical Systems Optimization·Captain: mikedeng1

First-Order Optimization Algorithms via Inertial Systems with Hessian Driven Damping I: Under (G₂)–(G₃), (DIN-AVD)α,β,b Trajectories Satisfy f(x(t)) − min f = O(1/(t²w(t)))Research Paper

Motivation

Accelerated first-order methods for convex minimization, starting with Nesterov's 1983 scheme, reach the rate $f(x_k)-\min f=\mathcal O(1/k^2)$ , which is optimal for the class of convex functions with Lipschitz gradient. Su, Boyd and Candès (NIPS 2014; JMLR 2016) showed that Nesterov's method is a discretization of the second-order ODE $\ddot x+\frac3t\dot x+\nabla f(x)=0$ , and since then continuous-time inertial dynamics have been the main tool for designing and analysing accelerated algorithms: a Lyapunov argument for the ODE is found first, and then transcribed to the discrete scheme.

Nesterov-type methods oscillate. Adding to the dynamic a damping term driven by the Hessian, $\beta(t)\nabla^2f(x(t))\dot x(t)$ , attenuates these oscillations, and because $\nabla^2f(x(t))\dot x(t)$ is the time derivative of $\nabla f(x(t))$ , time discretizations of such dynamics remain first-order methods. Attouch, Chbani, Fadili and Riahi (arXiv:1907.10536, Math. Program. 2020) develop this programme. This mission formalizes its first result: a convergence-rate theorem for the continuous dynamic with general damping and time-scale functions.

Timeline.

2014: Su, Boyd and Candès identify $\ddot x+\frac3t\dot x+\nabla f(x)=0$ as the continuous limit of Nesterov's method and prove $\mathcal O(1/t^2)$ .
2016: Attouch, Peypouquet and Redont (J. Differential Equations 261; reference [12] of arXiv:1907.10536) add the constant Hessian damping $\beta\nabla^2 f(x)\dot x$ , prove $\mathcal O(1/t^2)$ for $\alpha>3$ and $\int t^2\|\nabla f(x(t))\|^2dt<\infty$ .
2019–2020: Attouch, Chbani, Fadili and Riahi allow time-dependent $\beta(t)$ and a time scale $b(t)$ , and isolate the growth conditions $(\mathcal G_2)$ , $(\mathcal G_3)$ under which the Lyapunov argument works (Theorem 1 here).

Setting

Let $\mathcal H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$ . Let $f:\mathcal H\to\mathbb R$ be a convex function of class $\mathcal C^2$ that attains its minimum at some point $x^\star$ . Fix a starting time $t_0>0$ , a constant $\alpha\ge1$ , and two non-negative continuous functions $\beta,b:[t_0,+\infty[\to\mathbb R_+$ . These are the paper's standing assumptions (H).

A solution trajectory of

(\mathrm{DIN\text{-}AVD})_{\alpha,\beta,b}\qquad \ddot x(t)+\frac{\alpha}{t}\dot x(t)+\beta(t)\nabla^2 f(x(t))\dot x(t)+b(t)\nabla f(x(t))=0

is a twice differentiable curve $x:[t_0,+\infty[\to\mathcal H$ satisfying this equation at every $t\ge t_0$ . The coefficient $\alpha/t$ is an asymptotically vanishing viscous damping, $\beta(t)$ weights the Hessian-driven damping, and $b(t)$ is a time-scale factor in front of the gradient.

When $\beta$ is differentiable, set

w(t):=b(t)-\dot\beta(t)-\frac{\beta(t)}{t},\qquad \delta(t):=t^2w(t).

The rate in Theorem 1 is expressed through $w$ .

Formalization targets

Goal: Theorem 1

Assume $w$ is differentiable and the growth conditions

(\mathcal G_2)\quad b(t)>\dot\beta(t)+\frac{\beta(t)}{t},\qquad (\mathcal G_3)\quad t\dot w(t)\le(\alpha-3)w(t)

hold for all $t\ge t_0$ . Then $w(t)>0$ and

f(x(t))-\min_{\mathcal H}f=\mathcal O\!\Big(\frac1{t^2w(t)}\Big),\qquad \int_{t_0}^{\infty}t^2\beta(t)w(t)\|\nabla f(x(t))\|^2dt<\infty,

\int_{t_0}^{\infty}t\big((\alpha-3)w(t)-t\dot w(t)\big)\big(f(x(t))-\min f\big)dt<\infty .

The goal leaves $\beta$ and $b$ general. Specific choices are the companions below.

Milestones

The paper's own proof steps, in order:

the derivative (3) of the energy $E(t)=\delta(t)(f(x(t))-f(x^\star))+\frac12\|v(t)\|^2$ , where $v(t)=(\alpha-1)(x(t)-x^\star)+t(\dot x(t)+\beta(t)\nabla f(x(t)))$ ;
the identity $\dot v(t)=t[\dot\beta(t)+\beta(t)/t-b(t)]\nabla f(x(t))$ ;
the differential inequality (4);
the monotonicity of $E$ via (5)–(6);
the two integral bounds by $E(t_0)$ .

Companions

Theorem 3: $\beta(t)\equiv\beta$ , $b(t)=1+\beta/t$ , $\alpha\ge3$ give $\mathcal O(1/t^2)$ and, for $\beta>0$ , $\int t^2\|\nabla f(x(t))\|^2<\infty$ .
Theorem 2: the Attouch–Peypouquet–Redont case $b\equiv1$ , $\alpha>3$ , with the same conclusions.

Significance

The result. Theorem 1 contains the $\mathcal O(1/t^2)$ rate of the continuous Nesterov dynamic as the case $\beta\equiv0$ , $b\equiv1$ , $\alpha\ge3$ . It recovers the Attouch–Peypouquet–Redont theorem and the vanishing-coefficient system of Case 2 as special cases. The time scale $b$ gives faster rates $\mathcal O(1/(t^2w(t)))$ when $w$ grows. For $\beta>0$ it also gives the integral estimate on $\|\nabla f(x(t))\|^2$ , which is the continuous counterpart of the fast decay of gradients proved for the algorithms (IPAHD) and (IGAHD) later in the paper.

Formalizing it. The result is proved in the paper, and nothing about it has been machine-checked. The mission produces:

a reusable predicate for solutions of second-order ODEs with Hessian damping on a half-line;
the Lyapunov identity and inequality for that predicate;
a checked version of the integration step that turns a pointwise differential inequality into finite integrals.

Difficulty

The energy estimate is an exact computation that has to match term by term. The cross terms $\langle\nabla f(x(t)),\dot x(t)\rangle$ cancel only because $\delta=t^2w$ is chosen to match $\dot v$ . That computation needs the chain rule $\frac{d}{dt}\nabla f(x(t))=\nabla^2 f(x(t))\dot x(t)$ in a Hilbert space.

The naive conclusion " $\dot E\le0$ , so integrate (4)" hides two analytic steps:

monotonicity from a one-sided derivative on a closed half-line;
integrability on an unbounded interval of non-negative functions that need not be continuous, because $\dot w$ is only assumed to exist.

The (i) rate needs the positivity of $w$ from $(\mathcal G_2)$ to divide by $\delta(t)$ .

Formalization scope

Lean conventions:

$\mathcal H$ is a real inner product space that is complete. $\nabla f$ is Mathlib's gradient f, and $\nabla^2 f(x)u$ is fderiv ℝ (gradient f) x u.
Trajectories are maps $\mathbb R\to\mathcal H$ with explicit velocity and acceleration maps, tied to them by HasDerivWithinAt on $[t_0,+\infty[$ . Values before $t_0$ are irrelevant, and derivatives at $t_0$ are one-sided.
$\dot\beta$ and $\dot w$ are explicit maps with derivative hypotheses. The paper writes them, so it presupposes they exist.
$\min f$ is $f(x^\star)$ for a given global minimizer. $\mathcal O$ is =O[atTop], with constants depending on all data.
" $\int<\infty$ " is IntegrableOn on $[t_0,+\infty[$ , never a bound on a Bochner integral that could be the junk value $0$ .

These choices rule out trivial readings: Lean's zero value for the derivative of a non-differentiable function, an ODE with an unlinked velocity, and an integral that is zero because the integrand is not integrable. The goal assumes nothing about $E$ , $v$ or (4)–(6), which appear only in milestones.

Companion items:

Theorems 2 and 3 add $\beta>0$ to the gradient-integral claim. The paper derives it from Theorem 1 (ii), which carries a factor $\beta$ , so it says nothing at $\beta=0$ .
Theorem 2 must cope with growth conditions that hold only for large $t$ .

Contributions are welcome: the chain rule for gradient along a curve, a monotonicity lemma for one-sided derivatives on Ici, and the integrability lemma for non-negative functions bounded by a non-increasing energy.

Selected references

H. Attouch, Z. Chbani, J. Fadili, H. Riahi, First-order optimization algorithms via inertial systems with Hessian driven damping, Math. Program. (2020); arXiv:1907.10536v2. https://arxiv.org/abs/1907.10536
H. Attouch, J. Peypouquet, P. Redont, Fast convex minimization via inertial dynamics with Hessian driven damping, J. Differential Equations 261, No. 10 (2016) 5734–5783 (as cited in arXiv:1907.10536v2, ref. [12]). https://arxiv.org/abs/1907.10536
W. Su, S. Boyd, E. J. Candès, A differential equation for modeling Nesterov's accelerated gradient method: theory and insights, NIPS 27 (2014) 2510–2518; journal version J. Mach. Learn. Res. 17 (2016). https://jmlr.org/papers/v17/15-084.html
Y. Nesterov, A method of solving a convex programming problem with convergence rate O(1/k²), Soviet Math. Dokl. 27 (1983) 372–376.

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Algebra Complexity Theory Theoretical Computer Science·Captain: mikedeng1

The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs II: BLP+Affine Solves PCSP(A, B) Exactly When Pol(A, B) Has Block-Symmetric Polymorphisms of Arbitrarily High WidthResearch Paper

Motivation

A promise constraint satisfaction problem $\mathrm{PCSP}(\mathbf A,\mathbf B)$ asks to distinguish instances that are satisfiable in a structure $\mathbf A$ from instances that are not even satisfiable in a weaker structure $\mathbf B$ . Approximate graph colouring (tell 3-colourable graphs from graphs that are not 5-colourable) and $(1\text{-in-}3\text{-SAT},\ \mathrm{NAE\text{-}SAT})$ are standard examples. The algebraic approach to CSPs explains the complexity of such problems through polymorphisms, the multi-argument maps $A^L\to B$ that preserve every relation.

Two polynomial-time relaxations dominate the algorithmic side: the Basic LP relaxation, solved over $\mathbb Q_{\ge0}$ , and the affine relaxation, the same linear system solved over $\mathbb Z$ . Barto, Bulín, Krokhin and Opršal (BBKO19) characterized each one separately: BLP solves $\mathrm{PCSP}(\mathbf A,\mathbf B)$ exactly when $\mathrm{Pol}(\mathbf A,\mathbf B)$ has symmetric polymorphisms of all arities, and the affine relaxation exactly when it has alternating polymorphisms of all odd arities. Brakensiek and Guruswami (BG19) used ad hoc combinations of the two. Brakensiek, Guruswami, Wrochna and Živný (arXiv:1907.04383) combine them into a single algorithm, BLP+Affine, and characterize its power exactly. This mission formalizes that characterization, Theorem 4 of the paper.

Setting

A signature assigns to each symbol $R$ an arity $\mathrm{ar}(R)$ . A relational structure $\mathbf A$ on a finite domain $A$ fixes relations $R^{\mathbf A}\subseteq A^{\mathrm{ar}(R)}$ . A homomorphism $\mathbf A\to\mathbf B$ is a map $\sigma:A\to B$ sending every tuple of $R^{\mathbf A}$ into $R^{\mathbf B}$ , and $(\mathbf A,\mathbf B)$ is a promise template if one exists. An instance $X$ has variables $x_1,\dots,x_n$ and constraints $(R_j,\bar x_j)$ ; it is satisfiable in $\mathbf C$ (written $X\to\mathbf C$ ) if some assignment sends each $\bar x_j$ into $R_j^{\mathbf C}$ .

A map $f:A^L\to B$ is a polymorphism if applying $f$ column-wise to any $L\times\mathrm{ar}(R)$ matrix whose rows are in $R^{\mathbf A}$ gives a tuple of $R^{\mathbf B}$ . It is block-symmetric for a partition $[L]=B_1\sqcup\dots\sqcup B_\kappa$ if it is invariant under permutations of its arguments that preserve every block, and its width is the largest minimum block size over such partitions.

The Basic LP $\mathrm{LP}_{\mathbb Q}(X,\mathbf A)$ has a probability distribution $w_i$ on $A$ for each variable and $p_j$ on $R_j^{\mathbf A}$ for each constraint, with the marginal of $p_j$ at each position of $\bar x_j$ equal to the distribution of the variable in that position. The affine relaxation $\mathrm{Aff}_{\mathbb Z}(X,\mathbf A)$ is the same system over $\mathbb Z$ with unknowns $r_i$ , $q_j$ and no sign constraints. BLP+Affine (Figure 1) takes a relative interior point $(w,p)$ of the LP, keeps only the integer solutions $(r,q)$ that vanish wherever $(w,p)$ does, and accepts iff one remains. It correctly solves $\mathrm{PCSP}(\mathbf A,\mathbf B)$ if it accepts every $X\to\mathbf A$ and rejects every $X\not\to\mathbf B$ .

The minor of $f:A^L\to B$ along $\pi:[L]\to[L']$ is $f_{/\pi}(x_1,\dots,x_{L'})=f(x_{\pi(1)},\dots,x_{\pi(L)})$ . The minion $\mathcal M_{\mathrm{BLP+Aff}}$ has as $L$ -ary objects the pairs $(w,r)$ with $w:[L]\to\mathbb Q_{\ge0}$ , $r:[L]\to\mathbb Z$ , both summing to $1$ , and $w(i)=0\Rightarrow r(i)=0$ ; the minor along $\pi$ sums both over fibres of $\pi$ . A minion homomorphism $\xi:\mathcal M_{\mathrm{BLP+Aff}}\to\mathrm{Pol}(\mathbf A,\mathbf B)$ sends $L$ -ary objects to $L$ -ary polymorphisms and commutes with minors. The free structure $F_{\mathcal M_{\mathrm{BLP+Aff}}}(\mathbf A)$ has the $|A|$ -ary objects as its domain, and a tuple is in $R^F$ iff it is the family of coordinate minors of one $|R^{\mathbf A}|$ -ary object.

Formalization targets

Goal: Theorem 4

For a promise template $(\mathbf A,\mathbf B)$ on finite domains, the following are equivalent:

\text{BLP+Affine correctly solves } \mathrm{PCSP}(\mathbf A,\mathbf B) \iff \forall N\ \exists f\in\mathrm{Pol}(\mathbf A,\mathbf B):\ \mathrm{width}(f)\ge N \iff \forall L\ \exists f\in\mathrm{Pol}^{(2L+1)}\ \text{symmetric on blocks of sizes } L,\ L+1.

Milestones

The paper proves Theorem 4 through Lemmas 7, 8 and 9. The milestones follow that proof:

$\mathcal M_{\mathrm{BLP+Aff}}$ is a minion (p. 10);
Lemma 8: a minion homomorphism $\mathcal M_{\mathrm{BLP+Aff}}\to\mathrm{Pol}(\mathbf A,\mathbf B)$ yields the $(2L+1)$ -ary two-block polymorphisms;
the finite-subset construction in the proof of Lemma 9: a symmetric polymorphism of arity $L^*\ge M\ell^2$ , $L^*=u\ell+v$ , gives a minion homomorphism on the finite set $\mathcal M_{\ell,M}$ by repeating coordinate $i$ exactly $W_i=u\ell w(i)+v r(i)$ times;
Lemma 9: block-symmetric polymorphisms of arbitrarily high width yield a minion homomorphism;
Observation 15, Lemma 16 (compactness for structures) and Lemma 17 (free structures versus minion homomorphisms, for $\mathcal M_{\mathrm{BLP+Aff}}$ ), which together give
Lemma 7: BLP+Affine correctly solves $\mathrm{PCSP}(\mathbf A,\mathbf B)$ iff $\mathcal M_{\mathrm{BLP+Aff}}\to\mathrm{Pol}(\mathbf A,\mathbf B)$ .

A companion item, Example 10, records that the disjoint union of a directed 2-cycle and a directed 3-cycle has no block-symmetric polymorphism of width at least $2$ .

Significance

Theorem 4 is an exact criterion: the power of an algorithm is read off from the polymorphisms of the template. Its sufficiency direction (Theorem 3, the subject of the companion mission) says that one fixed algorithm decides every PCSP with block-symmetric polymorphisms of unbounded width, which covers nearly all tractable Boolean PCSPs known at the time, including those with threshold and alternating-threshold polymorphisms. The necessity direction, new in Theorem 4, gives a method for proving that BLP+Affine fails on a template: exhibit, for some $L$ , the absence of a $(2L+1)$ -ary polymorphism with blocks of sizes $L$ and $L+1$ . Example 10 is an instance: a tractable template that the algorithm does not solve. The minion $\mathcal M_{\mathrm{BLP+Aff}}$ and its free structure are the template for later characterizations of relaxation hierarchies.

The result is proved in the paper. As far as is known it has not been machine-checked. A formalization adds a checked development of minions, minion homomorphisms and free structures for PCSPs, a compactness theorem for relational structures, and a checked version of the one-block argument whose many-block extension the paper only asserts.

Difficulty

The equivalence passes through three different kinds of object: an algorithm, an infinite algebraic object (a homomorphism defined on all of $\mathcal M_{\mathrm{BLP+Aff}}$ ), and finite families of polymorphisms. From polymorphisms one obtains homomorphisms only on finite pieces $\mathcal M_{\ell,M}$ , each from a different polymorphism, and these do not agree with one another; a single homomorphism on the whole minion needs a compactness argument over the finitely many restrictions, which uses the finiteness of $A$ and $B$ . The passage from "every instance satisfiable in the free structure is satisfiable in $\mathbf B$ " to a homomorphism from the infinite free structure is a second compactness step. The relative interior point of Figure 1 must be kept: accepting whenever both relaxations are merely feasible is a different, weaker algorithm.

Formalization scope

All objects are in the definitions file PCSPBLPAff.Characterization.Setting, built on Mathlib only.

Signatures are (τ : Type) (ar : τ → ℕ), with no finiteness of $\tau$ ; the paper's requirement that arities be positive is dropped, and no statement needs it. Domains are finite types; instance variables are Fin n and arities Fin L.
The LP is over $\mathbb Q$ and the affine system over $\mathbb Z$ . Constraint distributions are functions on all tuples that vanish outside $R^{\mathbf A}$ , and marginal conditions are imposed per position of a constraint's scope.
The relative interior point is encoded by the only property the algorithm uses: a solution whose zero coordinates are exactly those that vanish on the whole polytope. The algorithm is formalized by its acceptance condition; the polynomial-time claims are out of scope.
Width requires at least one block, so a nullary function does not have every width.
There is no abstract minion structure. A minion homomorphism $\mathcal M_{\mathrm{BLP+Aff}}\to\mathrm{Pol}(\mathbf A,\mathbf B)$ is spelled out as a family of maps from objects of each arity to polymorphisms of that arity, commuting with minors. Lemma 17 is stated for $\mathcal M=\mathcal M_{\mathrm{BLP+Aff}}$ . Free-structure objects are indexed by $A$ and by tuples, identifying $A$ with $[|A|]$ .
Lemma 16 drops the hypothesis that $\mathbf F$ is infinite, which is unnecessary.

Two trivializing formalizations are ruled out: the acceptance condition keeps the maximal-support requirement, and the free structure's relation requires the witnessing pair to be an object of $\mathcal M_{\mathrm{BLP+Aff}}$ supported in $R^{\mathbf A}$ .

A complete development needs finite sums over fibres, transport of objects along Fintype.equivFin, Tychonoff or König's lemma (Mathlib.Order.KonigLemma, Mathlib.Combinatorics.Compactness), and the existence of a maximal-support point of a rational polytope. The compactness theorem for relational structures and the free-structure lemma are reusable beyond this mission. Proofs of any milestone, and of the general Lemma 17 for an abstract minion, are welcome.

Selected references

J. Brakensiek, V. Guruswami, M. Wrochna, S. Živný, The Power of the Combined Basic LP and Affine Relaxation for Promise CSPs, arXiv:1907.04383v3, 2020 (also SIAM J. Comput., 2020, https://doi.org/10.1137/20M1312745). https://arxiv.org/abs/1907.04383
L. Barto, J. Bulín, A. Krokhin, J. Opršal, Algebraic approach to promise constraint satisfaction, arXiv:1811.00970, 2019; J. ACM 68(4), 2021. https://arxiv.org/abs/1811.00970
J. Brakensiek, V. Guruswami, An Algorithmic Blend of LPs and Ring Equations for Promise CSPs, SODA 2019, pp. 436–455. https://arxiv.org/abs/1807.05194
R. Diestel, Graph Theory, 5th ed., Springer, 2016, Theorem 8.1.3 (de Bruijn–Erdős compactness). https://doi.org/10.1007/978-3-662-53622-3

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Optimization Statistics·Captain: mikedeng1

Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms II: CD-PSI(k) Terminates Finitely at a Partial Swap Inescapable Minimum of Order kResearch Paper

Motivation

Best subset selection asks for the linear model that fits a response with as few predictors as possible. It is a basic problem of statistics and a standard test case for mixed-integer optimization. The penalized form

\min_{\beta\in\mathbb R^p}\ \tfrac12\|y-X\beta\|^2+\lambda_0\|\beta\|_0

is NP-hard in general. Exact mixed-integer methods (Bertsimas, King, Mazumder 2016) certify global optimality but scale to moderate $p$ only. Hazimeh and Mazumder (arXiv:1803.01454, Operations Research 2020) proposed a hierarchy of local minima for the $L_0$ -penalized problem, together with algorithms that reach each class. These are coordinate descent for coordinate-wise minima, and coordinate descent combined with small combinatorial swap searches for the finer swap inescapable minima. The software built on these algorithms, L0Learn, runs on problems with $p$ in the millions.

The companion mission (part I) proves that the coordinate descent stage converges. This mission proves the guarantee of the second stage: the local combinatorial search stops after finitely many rounds, and it stops at a partial swap inescapable minimum.

Setting

Data and objective. Let $X\in\mathbb R^{n\times p}$ have columns $X_1,\dots,X_p$ of unit Euclidean norm, let $y\in\mathbb R^n$ , and let $\lambda_0>0$ , $\lambda_1,\lambda_2\ge0$ . Problem (2) is

\min_{\beta\in\mathbb R^p}F(\beta)=f(\beta)+\lambda_0\|\beta\|_0,\qquad f(\beta)=\tfrac12\|y-X\beta\|^2+\lambda_1\|\beta\|_1+\lambda_2\|\beta\|_2^2,

where $\|\beta\|_0$ is the number of nonzero entries. The paper names three cases: (L0), with $\lambda_1=\lambda_2=0$ ; (L0L1), with $\lambda_1>0=\lambda_2$ ; and (L0L2), with $\lambda_1=0<\lambda_2$ . $\operatorname{Supp}(\beta)=\{i:\beta_i\neq0\}$ , and $U_S\beta$ keeps the entries of $\beta$ in $S$ and zeroes the others.

Classes of minima. A stationary solution has nonnegative lower directional derivative $F'(\beta;d)=\liminf_{\alpha\downarrow0}(F(\beta+\alpha d)-F(\beta))/\alpha$ in every direction $d$ . A CW minimum is a point where each coordinate $\beta_i$ minimizes $F$ with the other coordinates held fixed. For a positive integer $k$ , a PSI( $k$ ) minimum is a stationary $\beta^*$ with support $S$ such that, for all $S_1\subseteq S$ and $S_2\subseteq S^c$ with $|S_1|,|S_2|\le k$ ,

F(\beta^*)\le\min_{\beta_{S_2}}F\big(\beta^*-U_{S_1}\beta^*+U_{S_2}\beta\big).

In words, dropping at most $k$ active coordinates, adding at most $k$ new ones and optimizing only over the added ones never lowers the objective.

Algorithm 1 (CDSS) is cyclic coordinate descent. Each step sets one coordinate to the minimizer given by the thresholding operator $T$ of (12). Every time a support has been visited $Cp$ times, the algorithm also performs a spacer step: one pass of coordinate descent on $f$ restricted to that support.

Algorithm 2 (CD-PSI( $k$ )) alternates two steps. It runs Algorithm 1 from $\hat\beta^\ell$ to obtain $\beta^{\ell+1}$ . It then searches the swap problem (14),

\min_{\beta,S_1,S_2}F\big(\beta^{\ell+1}-U_{S_1}\beta^{\ell+1}+U_{S_2}\beta\big)\ \ \text{s.t.}\ \ S_1\subseteq S,\ S_2\subseteq S^c,\ |S_1|,|S_2|\le k,

with $S=\operatorname{Supp}(\beta^{\ell+1})$ , for a feasible $\hat\beta$ with $F(\hat\beta)<F(\beta^{\ell+1})$ . If it finds one, it sets $\hat\beta^{\ell+1}=\hat\beta$ and continues; otherwise it terminates.

Formalization targets

Goal: Theorem 4

Assume the (L0), (L0L1) or (L0L2) setting. For (L0) and (L0L1), assume also that every $\min\{n,p\}$ columns of $X$ are linearly independent (Assumption 1) and, when $p>n$ , that the start $\beta^0$ satisfies the objective bound of Assumption 2. Then every run of CD-PSI( $k$ ) satisfies

\exists L:\quad \text{(14) is improving at }\beta^1,\dots,\beta^L,\quad \text{(14) is not improving at }\beta^{L+1},\quad \beta^{L+1}\in\mathrm{PSI}(k).

No bound on $L$ is part of the goal, and the paper states none.

Milestones

Theorem 2 (p. 12): the iterates of Algorithm 1 have eventually constant support and converge to a CW minimum with that support. This is the goal of the companion mission, restated here on this mission's definitions.
Lemma 3 (p. 7): CW minima are characterized by the thresholding conditions (8).
Lemma 1 (p. 6): $\beta$ is stationary iff $\nabla_{\operatorname{Supp}(\beta)}f(\beta)=0$ .
§A.8, p. 44, same objective: a CW minimum minimizes $f$ over the vectors supported in its support, so two CW minima with the same support have the same $F$ .
§A.8, p. 44, distinct supports: along a run that has not yet terminated, $F(\beta^{\ell})$ strictly decreases and no support repeats.

Significance

The result. Theorem 4 makes CD-PSI( $k$ ) an algorithm with a guarantee. It always stops, and its output satisfies an optimality condition that is strictly stronger than coordinate-wise optimality: the hierarchy (4) of the paper places PSI( $k$ ) minima inside CW minima, and for large $k$ they coincide with global minimizers. The same termination argument underlies the full-optimization variant (FSI minima) and the swap heuristics of L0Learn, which the paper treats "by the same argument".

Formalizing it. The theorem is proved on paper, in a short argument in Appendix A.8 that rests on the convergence of Algorithm 1 (Theorem 2, the companion mission). No part of it has been machine-checked. This mission produces checked definitions of PSI minima, of the swap problem and of the two-level algorithm; a formal proof of the finite termination; and the bridge lemmas between CW minima, stationarity and the restricted least-squares problem, which any formal treatment of $L_0$ -penalized regression needs.

Difficulty

The obvious argument is that each round lowers the objective, there are finitely many supports, so the algorithm stops. A strict decrease of $F$ alone gives no finiteness: a real-valued sequence can decrease forever. What makes the argument work is that the value of $F$ at an output of Algorithm 1 depends only on its support. That needs every output to be a CW minimum, which is Theorem 2, a nontrivial convergence theorem for a discontinuous objective. It also needs every CW minimum to minimize the convex function $f$ over its support, which uses the characterization (8) and convexity.

A second difficulty is bookkeeping. Algorithm 1 never stops ("while not converged"), so its output is a limit. Each call of Algorithm 1 inside Algorithm 2 must satisfy the hypotheses of Theorem 2, in particular Assumption 2 for the new starting point, and that holds only because the objective has decreased since $\beta^0$ .

Formalization scope

Vectors are Fin p → ℝ, so indices are $0$ -based. All norms are explicit sums, never Lean's sup norm. $\operatorname{sign}$ is Real.sign, with $\operatorname{sign}(0)=0$ . The lower directional derivative is EReal-valued. The theorems are stated for the three named problems, via the hypothesis $\lambda_1=0\lor\lambda_2=0$ . Assumptions 1–2 are required only when $\lambda_2=0$ .

Algorithm 1 is a deterministic state machine with a cyclic pointer, a table $\mathrm{Count}$ and a pending-spacer flag, following the box on p. 11. A run of Algorithm 2 is a pair of sequences: $\beta^{\ell+1}$ is the limit of the CDSS iterates from $\hat\beta^\ell$ , and $\hat\beta^{\ell+1}$ is any improving feasible point of (14), required only while the run has not terminated. The predicate is not vacuous: on the instance $n=p=1$ , $X=y=1$ , $\lambda_0=1/4$ , $\lambda_1=\lambda_2=0$ , $C=k=1$ , $\beta^0=0$ , a run exists and terminates at $\ell=0$ . " $\le\min_{\beta_{S_2}}$ " in Definition 3 is read as " $\le$ every value".

A statement that assumes the outputs of Algorithm 1 are CW minima, or that the run terminates, or that only finitely many supports are visited, would trivialize the goal. All three are conclusions of the mission, not hypotheses.

The development needs the CDSS model shared with the companion mission, convexity of the restricted least-squares objective and a finiteness argument over Finset (Fin p). The definitions of Problem (2), Algorithm 1 and the stationarity notions are reusable for any later mission on $L_0$ -penalized regression. Proofs of the milestones independently of the goal are welcome; Theorem 2 can also be imported once the companion mission proves it. Theorem and page numbers are those of arXiv:1803.01454v3.

Selected references

H. Hazimeh, R. Mazumder, Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms, Operations Research 68(5), 2020. arXiv:1803.01454v3. https://arxiv.org/abs/1803.01454 , https://doi.org/10.1287/opre.2019.1919
D. Bertsimas, A. King, R. Mazumder, Best subset selection via a modern optimization lens, Annals of Statistics 44(2), 2016. https://doi.org/10.1214/15-AOS1388
A. Beck, Y. C. Eldar, Sparsity Constrained Nonlinear Optimization: Optimality Conditions and Algorithms, SIAM J. Optim. 23(3), 2013. https://doi.org/10.1137/120869778

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Convex Optimization Numerical Analysis Optimization·Captain: mikedeng1

First-Order Optimization Algorithms via Inertial Systems with Hessian Driven Damping II: The Inertial Proximal Algorithm (IPAHD) Satisfies f(x_k) − min f = O(1/δ_k)Research Paper

Motivation

Accelerated first-order methods, such as Nesterov's method and FISTA, reach a convergence rate of $\mathcal O(1/k^2)$ on the objective values of a smooth convex function, against $\mathcal O(1/k)$ for plain gradient descent. Since the work of Su, Boyd and Candès (arXiv:1503.01243), these methods are studied through their continuous-time limits, second-order differential equations with a vanishing viscous damping $\frac{\alpha}{t}\dot x$ . The trajectories of these equations, and the iterates of the corresponding algorithms, oscillate strongly around the minimizers.

Adding a Hessian-driven damping term $\beta(t)\nabla^2 f(x(t))\dot x(t)$ to the dynamic damps these oscillations while keeping the fast rate. The idea goes back to Alvarez, Attouch, Bolte and Redont (2002) and was developed for fast convex minimization by Attouch, Peypouquet and Redont (arXiv:1601.07113, J. Differential Equations 2016). Shi, Du, Jordan and Su (arXiv:1810.08907) showed that the gradient-correction term of Nesterov's method is exactly such a Hessian damping, observed in a high-resolution limit. Because $\nabla^2 f(x(t))\dot x(t)$ is the time derivative of $\nabla f(x(t))$ , the Hessian never has to be evaluated after discretization, and the resulting algorithms are genuinely first-order.

Attouch, Chbani, Fadili and Riahi (arXiv:1907.10536, Math. Program. 2020) study the system (DIN-AVD) $_{\alpha,\beta,b}$ , $\ddot x+\frac{\alpha}{t}\dot x+\beta(t)\nabla^2f(x)\dot x+b(t)\nabla f(x)=0$ , with general time-dependent coefficients, and derive algorithms from it. This mission formalizes their first discrete result: the convergence rate of the implicit (proximal) discretization.

Setting

Let $\mathcal H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$ , and let $f:\mathcal H\to\mathbb R$ be a convex function of class $\mathcal C^1$ with gradient $\nabla f$ , attaining its minimum at some point $x^\star$ . Fix $\alpha\ge1$ , a step size $h>0$ with $s=h^2$ , and two non-negative sequences $(\beta_k)_{k\in\mathbb N}$ (the Hessian damping) and $(b_k)_{k\in\mathbb N}$ (the time scaling of the gradient).

For $\mu\ge0$ and $y\in\mathcal H$ , the proximal point $\mathrm{prox}_{\mu f}(y)$ is the minimizer of $z\mapsto\mu f(z)+\tfrac12\|z-y\|^2$ .

The Inertial Proximal Algorithm with Hessian Damping (IPAHD) generates a sequence $(x_k)_{k\in\mathbb N}$ by, for $k\ge1$ ,

\mu_k=\frac{k}{k+\alpha}\big(\beta_k\sqrt s+sb_k\big),\qquad y_k=x_k+\Big(1-\frac{\alpha}{k+\alpha}\Big)(x_k-x_{k-1})+\beta_k\sqrt s\Big(1-\frac{\alpha}{k+\alpha}\Big)\nabla f(x_k),

x_{k+1}=\mathrm{prox}_{\mu_kf}(y_k).

It is the implicit time discretization, with step $h$ , of (DIN-AVD) $_{\alpha,\beta,b}$ :

k(x_{k+1}-2x_k+x_{k-1})+\alpha(x_{k+1}-x_k)+\beta_khk\big(\nabla f(x_{k+1})-\nabla f(x_k)\big)+b_kh^2k\nabla f(x_{k+1})=0. \tag{8}

The rate is measured by the sequence

\delta_k:=h\big(b_khk-\beta_{k+1}-k(\beta_{k+1}-\beta_k)\big)(k+1), \tag{9}

under two growth conditions: $(\mathcal G_2^{\rm dis})$ $b_khk-\beta_{k+1}-k(\beta_{k+1}-\beta_k)>0$ and $(\mathcal G_3^{\rm dis})$ $\delta_{k+1}-\delta_k\le(\alpha-1)\frac{\delta_k}{k+1}$ . They are the discrete counterparts of the conditions $(\mathcal G_2)$ , $(\mathcal G_3)$ of the continuous analysis.

Formalization targets

Goal: Theorem 4 (p. 11)

If $(\mathcal G_2^{\rm dis})$ and $(\mathcal G_3^{\rm dis})$ hold for every $k\ge k_0$ , for some $k_0\ge1$ , then every sequence generated by (IPAHD) satisfies: $\delta_k>0$ for $k\ge k_0$ , and

f(x_k)-\min_{\mathcal H}f=\mathcal O\Big(\frac1{\delta_k}\Big),\qquad \sum_k\delta_k\beta_{k+1}\|\nabla f(x_{k+1})\|^2<+\infty.

The rate is stated in terms of $\delta_k$ , with no constant fixed in advance, so it covers every admissible choice of $(\beta_k)$ and $(b_k)$ at once.

Milestones (the steps of the proof, pp. 10–13)

Every run of (IPAHD) satisfies (8).
Under (8), $v_{k+1}-v_k=h C_k\nabla f(x_{k+1})$ , where $v_k:=(\alpha-1)(x_k-x^\star)+k(x_k-x_{k-1}+\beta_kh\nabla f(x_k))$ and $C_k=\beta_{k+1}+k(\beta_{k+1}-\beta_k)-b_khk$ .
$\tfrac12\|v_{k+1}\|^2-\tfrac12\|v_k\|^2\le-(\alpha-1)hC_k(f(x^\star)-f(x_{k+1}))-hC_k(k+1)(f(x_k)-f(x_{k+1}))$ .
For the energy $E_k:=\delta_k(f(x_k)-f(x^\star))+\tfrac12\|v_k\|^2$ : $E_{k+1}-E_k\le\big(\delta_{k+1}-\delta_k-(\alpha-1)\frac{\delta_k}{k+1}\big)(f(x_{k+1})-f(x^\star))$ .
$f(x_k)-\min f\le E_{k_0}/\delta_k$ for $k\ge k_0$ .
$E_{k+1}-E_k+h\big(\tfrac h2C_k^2+\delta_k\beta_{k+1}\big)\|\nabla f(x_{k+1})\|^2\le0$ .

Significance

With $\beta_k\equiv\beta>0$ and $b_k=1+\beta/(hk)$ one gets $\delta_k=h^2k(k+1)$ , so Theorem 4 gives the accelerated rate $\mathcal O(1/k^2)$ for a proximal algorithm with Hessian damping, together with the summability of $k^2\|\nabla f(x_{k+1})\|^2$ , a fast decay of the gradients that the plain inertial proximal algorithm does not provide. The time-dependent coefficients let the user trade damping against speed, and Remark 2 of the paper gives $\mathcal O(1/(k(k+1)))$ under a uniform version of $(\mathcal G_2^{\rm dis})$ . The proximal scheme only needs $f$ of class $\mathcal C^1$ ; the non-smooth case (Theorem 5) is obtained from it through the Moreau envelope.

The result is proved in the paper. As far as the platform's catalog shows, neither the algorithm nor any Lyapunov analysis of an inertial algorithm with Hessian-driven damping has been machine-checked. A formal proof would certify a discrete energy argument whose printed version contains an index slip (a dropped factor $k+1$ in an intermediate display on p. 12), and the definitions are reused by the other missions of this series.

Difficulty

The proof is a discrete Lyapunov argument, and the difficulty is bookkeeping rather than a single deep step. The coefficient $\delta_k$ is not chosen in advance: it is forced by the requirement that the terms $f(x_k)-f(x_{k+1})$ cancel in $E_{k+1}-E_k$ , and the inequality for $\tfrac12\|v_{k+1}\|^2-\tfrac12\|v_k\|^2$ only closes after the signs of $C_k$ , $\beta_{k+1}$ and $\alpha-1$ are used in the right places. The convexity of $f$ enters twice, at different points ( $x^\star$ and $x_k$ ). Reading the proximal step as the first-order optimality condition $x_{k+1}+\mu_k\nabla f(x_{k+1})=y_k$ and converting it into (8) requires matching $\sqrt s$ with $h$ and the coefficient $1-\frac{\alpha}{k+\alpha}$ with $\frac{k}{k+\alpha}$ . A naive direct estimate of $f(x_k)-\min f$ from the proximal inequality alone gives only the unaccelerated rate.

Formalization scope

The space is {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H], $\nabla f$ is Mathlib's gradient f, convexity is ConvexOn ℝ Set.univ f and $\mathcal C^1$ is ContDiff ℝ 1 f. The minimizer is a point $x^\star$ with $f(x^\star)\le f(y)$ for all $y$ , the concrete form of the paper's standing hypothesis $\mathrm{argmin}f\neq\emptyset$ . The proximal step is the published predicate GoldenRatioVI.Shared.IsProxPoint: $x_{k+1}$ minimizes $\mu_kf(z)+\tfrac12\|z-y_k\|^2$ . No proximal function with a junk value is defined. Sequences are indexed by $\mathbb N$ , the recursion is required for $k\ge1$ , and $x_0$ , $x_1$ are free. $\mathcal O$ is =O[Filter.atTop] and the series is Summable in $\mathbb R$ .

Hypotheses added relative to the printed theorem, all disclosed in the statements:

$h>0$ , $s=h^2$ ;
$\beta_k,b_k\ge0$ for every $k$ , the discrete analogue of the standing hypothesis (H), needed for $\mu_k\ge0$ and used in the proof;
the growth conditions are assumed for $k\ge k_0$ , with $k_0\ge1$ .

At $k=0$ , $(\mathcal G_2^{\rm dis})$ reads $-\beta_1>0$ , so reading the conditions "for all $k$ " would make every hypothesis set unsatisfiable and the theorem empty. A sorry-free check shows the hypotheses hold for $\beta\equiv0$ , $b\equiv1$ , $h=1$ , $\alpha=4$ , $k_0=2$ . The goal statement mentions only $f$ , the run and $\delta_k$ ; it does not refer to $v_k$ , $E_k$ or $C_k$ , and a formalization that assumes an energy inequality would not prove Theorem 4.

The definitions $\mu_k$ , $y_k$ , the run predicate, $\delta_k$ , $C_k$ , $v_k$ and $E_k$ live in one definition file. Proofs of the milestones are welcome individually; the identities (milestones 1–2) are algebra plus the first-order condition of the proximal step, and the inequalities (3–6) need the gradient inequality of a differentiable convex function. Theorem 5 (the non-smooth case) is not part of this mission.

Selected references

H. Attouch, Z. Chbani, J. Fadili, H. Riahi, First-order optimization algorithms via inertial systems with Hessian driven damping, Math. Program., 2020; arXiv:1907.10536v2. https://arxiv.org/abs/1907.10536
H. Attouch, J. Peypouquet, P. Redont, Fast convex minimization via inertial dynamics with Hessian driven damping, J. Differential Equations 261 (2016), 5734–5783. https://arxiv.org/abs/1601.07113
B. Shi, S. S. Du, M. I. Jordan, W. J. Su, Understanding the acceleration phenomenon via high-resolution differential equations, Math. Program., 2021. https://arxiv.org/abs/1810.08907
W. Su, S. Boyd, E. J. Candès, A differential equation for modeling Nesterov's accelerated gradient method: theory and insights, J. Mach. Learn. Res. 17 (2016). https://arxiv.org/abs/1503.01243

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Convex Optimization Optimization·Captain: mikedeng1

From Error Bounds to the Complexity of First-Order Descent Methods for Convex Functions 1: Subgradient Descent Sequences of Convex KL Functions Obey f(x_k) ≤ ψ(α_k) for a 1-D Worst-Case Prox SequenceResearch Paper

Motivation

Many first-order methods for convex minimization — the forward–backward (proximal gradient) method, alternating minimization, trust-region and majorization–minimization schemes — produce sequences that decrease the objective by an amount proportional to the squared step and whose step controls a subgradient at the new point. Luo and Tseng (Ann. Oper. Res. 1993) used such conditions together with error bounds to derive convergence rates, and Attouch, Bolte and Svaiter (Math. Program. 2013) isolated them as an abstract definition of descent sequences analysed through the Kurdyka–Łojasiewicz (KL) inequality. Those analyses give convergence and asymptotic rates, but not explicit complexity bounds with the constants in view.

Bolte, Nguyen, Peypouquet and Suter (arXiv:1510.08234; Math. Program. 165, 2017) show that, for convex functions, the KL inequality yields such bounds: the values and iterates of every subgradient descent sequence are dominated by an explicit one-dimensional proximal recursion built from the desingularizing function. This is the paper's main theorem (Theorem 16), and the target of this mission.

Setting

Let $H$ be a real Hilbert space and $f : H \to (-\infty, +\infty]$ proper, lower semicontinuous and convex, with $\operatorname{argmin} f \neq \emptyset$ and, after a shift, $\min f = 0$ . The subdifferential is

\partial f(x) = \{u \in H : f(y) \ge f(x) + \langle u, y - x\rangle \ \text{for all } y\},

and $\partial^0 f(x)$ denotes its least-norm element, with $\|\partial^0 f(x)\| = +\infty$ when $\partial f(x) = \emptyset$ .

For $\bar r > 0$ , the class $\mathcal K(0, \bar r)$ consists of the functions $\varphi$ continuous on $[0, \bar r)$ and $C^1$ on $(0, \bar r)$ with $\varphi(0) = 0$ , $\varphi$ concave and $\varphi' > 0$ . The function $f$ has the KL property on $[0 < f < \bar r]$ with desingularizing function $\varphi \in \mathcal K(0, \bar r)$ if

\varphi'(f(x))\,\|\partial^0 f(x)\| \ge 1 \qquad \text{whenever } 0 < f(x) < \bar r.

A sequence $(x_k)_{k \in \mathbb N}$ is a subgradient descent sequence with constants $a, b > 0$ if $x_0 \in \operatorname{dom} f$ and, for every $k \ge 1$ ,

(H1) $f(x_k) + a\|x_k - x_{k-1}\|^2 \le f(x_{k-1})$ ;
(H2) some $\omega_k \in \partial f(x_k)$ satisfies $\|\omega_k\| \le b\|x_k - x_{k-1}\|$ .

Fix $0 < r_0 < \bar r$ , put $\alpha_0 = \varphi(r_0)$ , and let $\psi = (\varphi|_{[0, r_0]})^{-1} : [0, \alpha_0] \to [0, r_0]$ , an increasing convex profile. Assumption (A): $\psi'$ is Lipschitz continuous on $[0, \alpha_0]$ with constant $\ell > 0$ , and $\psi'(0) = 0$ . With

\zeta = \frac{\sqrt{1 + 2\ell a b^{-2}} - 1}{\ell} > 0, \qquad \alpha_{k+1} = \operatorname{argmin}\Big\{\psi(u) + \tfrac{1}{2\zeta}(u - \alpha_k)^2 : u \ge 0\Big\},

$(\alpha_k)$ is the one-dimensional worst-case proximal sequence.

Formalization targets

Goal: Theorem 16

If $(x_k)$ is a subgradient descent sequence with $f(x_0) = r_0$ and (A) holds, then $x_k$ converges strongly to a minimizer $x^*$ and

f(x_k) \le \psi(\alpha_k) \quad (k \ge 0), \qquad \|x_k - x^*\| \le \frac{b}{a}\,\alpha_k + \sqrt{\frac{\psi(\alpha_{k-1})}{a}} \quad (k \ge 1).

Milestones

(20), in the proof of Theorem 14: while $f(x_k) > 0$ and $x_k \neq x_{k-1}$ , $\varphi(f(x_k)) - \varphi(f(x_{k+1})) \ge \frac{a}{b}\big(2\|x_k - x_{k+1}\| - \|x_{k-1} - x_k\|\big)$ .
Theorem 14: with only $f(x_0) \le r_0 < \bar r$ , $x_k \to x^* \in \operatorname{argmin} f$ strongly and $\|x_k - x^*\| \le \frac{b}{a}\varphi(f(x_k)) + \sqrt{f(x_{k-1})/a}$ for $k \ge 1$ .
(23): $(\alpha_k)$ exists, is positive, satisfies $\alpha_{k+1} = (I + \zeta\psi')^{-1}(\alpha_k)$ , decreases strictly to $0$ , and $\psi(\alpha_k) \to 0$ .
(27): with $\beta_j = \varphi(f(x_j))$ , every $k \ge 1$ with $f(x_k) > 0$ has $(\beta_{k-1} - \beta_k)/\psi'(\beta_k) \ge \zeta$ .
Claim 1: for $\lambda^0 > \lambda^1 > 0$ and $\gamma > 0$ , $(I + \lambda^0\psi')^{-1}(\gamma) < (I + \lambda^1\psi')^{-1}(\gamma)$ .
Claim 2: proximal sequences with step sizes $\lambda^0_k \ge \lambda^1_k > 0$ from a common start in $(0, \alpha_0]$ satisfy $\beta^0_k \le \beta^1_k$ .

Significance

Theorem 16 turns the qualitative KL convergence theory into explicit complexity: once a desingularizing function is known, the number of iterations needed to reach a given accuracy, for both the values and the iterates, can be read off a scalar recursion with constants $a$ , $b$ , $\ell$ taken from the method and the function. The authors apply it, through its corollaries, to obtain the linear rate of ISTA for $\ell^1$ -regularized least squares and complexity bounds for projection methods for convex feasibility problems. The bound on $\|x_k - x^*\|$ holds even when $f$ has a continuum of minimizers.

The theorem is proved in the paper and, to our knowledge, has no machine-checked proof. This mission asks for a Lean proof of Theorem 16 and of the six intermediate results its proof uses, each stated with the paper's exact constants.

Difficulty

The convergence part (Theorem 14) is set in an arbitrary Hilbert space: summability of the steps gives only a Cauchy sequence, identifying its limit as a minimizer needs a closedness property of $\partial f$ that is not available in Mathlib for EReal-valued functions, and the degenerate cases ( $x_k = x_{k-1}$ or $f(x_k) = 0$ at some step) need separate treatment.

The complexity part requires one-dimensional analysis that Mathlib does not package: derivatives of the inverse $\psi = \varphi^{-1}$ on a closed interval, a descent inequality for $\psi$ when $\psi'$ is Lipschitz only on $[0, \alpha_0]$ , and monotonicity of the scalar resolvents $(I + \lambda\psi')^{-1}$ . The obvious approach, comparing $f(x_k)$ with $\psi(\alpha_k)$ one step at a time with the same step size, fails: the sequence $\varphi(f(x_k))$ is itself a proximal sequence for $\psi$ , but with step sizes that vary with $k$ and are not known in advance.

Formalization scope

$f$ is a function $H \to$ EReal satisfying the published GammaZero (never $-\infty$ , somewhere finite, convex epigraph, lower semicontinuous), with CompleteSpace H. $\partial f$ is the published subgrad; $\min f = 0$ with $\operatorname{argmin} f \neq \emptyset$ is the hypothesis " $f \ge 0$ and $f$ vanishes somewhere".
The KL inequality is required for every $v \in \partial f(x)$ , which is the least-norm form with the convention $\|\partial^0 f(x)\| = +\infty$ off $\operatorname{dom}\partial f$ . The class $\mathcal K(0, \bar r)$ is the published IsDesingularizer; $\varphi'$ is deriv φ, used only on $(0, \bar r)$ .
$\psi$ is a real function given as data with $\psi(\varphi(s)) = s$ on $[0, r_0]$ ; only its values on $[0, \alpha_0]$ matter. $\psi'$ is the derivative within $[0, \alpha_0]$ , and the minimization in (22) runs over $[0, \alpha_0]$ , the domain of $\psi$ . The argmin is unique, so (22) is a predicate on sequences; (23) proves that the predicate is satisfiable, so Theorem 16 is not vacuous. Resolvent points $(I + \lambda\psi')^{-1}(\gamma)$ are likewise predicates.
$\alpha_0 = \varphi(r_0)$ , $\zeta$ and the coefficient $b/a$ are fixed by the data; none is existentially quantified. The inequality $f(x_k) \le \psi(\alpha_k)$ is stated in EReal.
Claim 2 is printed with $\beta^0_0 = \beta^1_0 \in (0, r_0]$ ; the resolvents act on $[0, \alpha_0]$ and the claim is applied with $\beta_0 = \alpha_0$ , so it is formalized on $(0, \alpha_0]$ .
Theorem numbers, equation numbers and pages refer to the arXiv version arXiv:1510.08234v3, not to the journal version.

Useful infrastructure, reusable beyond this mission: the inverse-function derivative on an interval, the one-dimensional descent lemma with a derivative Lipschitz on a closed interval, and monotonicity of scalar resolvents. Contributions of these as separate lemmas are welcome.

Selected references

J. Bolte, T. P. Nguyen, J. Peypouquet, B. W. Suter, From error bounds to the complexity of first-order descent methods for convex functions, Math. Program. 165 (2017) 471–507. arXiv:1510.08234v3, doi:10.1007/s10107-016-1091-6
H. Attouch, J. Bolte, B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems, Math. Program. 137 (2013) 91–129. doi:10.1007/s10107-011-0484-9
Z.-Q. Luo, P. Tseng, Error bounds and convergence analysis of feasible descent methods: a general approach, Ann. Oper. Res. 46 (1993) 157–178. doi:10.1007/BF02096261
J. Bolte, A. Daniilidis, A. Lewis, The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim. 17 (2007) 1205–1223. doi:10.1137/050644641

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Convex Optimization Numerical Analysis Optimization·Captain: mikedeng1

First-Order Optimization Algorithms via Inertial Systems with Hessian Driven Damping III: The Inertial Gradient Algorithm (IGAHD) Satisfies f(x_k) − min f = O(1/k²) and Σ k²‖∇f(x_k)‖² < ∞Research Paper

Motivation

Accelerated gradient methods use information from earlier iterates to reduce the objective gap of a smooth convex optimization problem faster than ordinary gradient descent. Their behavior depends on the precise momentum rule. Attouch, Chbani, Fadili, and Riahi study a discrete rule with an additional difference of consecutive gradients, motivated by an inertial differential equation with Hessian driven damping. This mission concerns their inertial gradient algorithm with Hessian damping, called IGAHD, and the rates proved for it in Theorem 6 of the source paper. The result matters when one wants a rate for objective values together with control of the gradients seen during the run.

The paper appeared as a 2020 preprint and subsequently in Mathematical Programming. The mission follows the pinned arXiv version, whose theorem and equation numbers supply all citations here. The proximal and strongly convex algorithms elsewhere in the paper have separate missions; their dynamics and rate statements differ from IGAHD.

Setting

Let $H$ be a real Hilbert space and let $f:H\to\mathbb R$ be convex and differentiable. Assume that its gradient $\nabla f$ is $L$ -Lipschitz: $\|\nabla f(u)-\nabla f(v)\|\le L\|u-v\|$ for every $u,v\in H$ , where $L>0$ . Choose a minimizer $x^*$ , so $f(x^*)=\min_H f$ . The positive step size $s$ satisfies $s\le1/L$ . The momentum parameter $\alpha$ is at least $3$ , and the damping parameter $\beta$ lies in $[0,2\sqrt{s})$ .

Given $x_0,x_1\in H$ , the algorithm takes the following two steps for every integer $k\ge1$ :

\begin{aligned} y_k={}&x_k+\left(1-\frac{\alpha}{k}\right)(x_k-x_{k-1}) -\beta\sqrt{s}\bigl(\nabla f(x_k)-\nabla f(x_{k-1})\bigr) -\frac{\beta\sqrt{s}}{k}\nabla f(x_{k-1}),\\ x_{k+1}={}&y_k-s\nabla f(y_k). \end{aligned}

The paper sets $t_{k+1}=k/(\alpha-1)$ and $v_k=x_{k-1}-x^*+t_k(x_k-x_{k-1}+\beta\sqrt{s}\nabla f(x_{k-1}))$ . Its Lyapunov quantity is $E_k=t_k^2(f(x_k)-f(x^*))+(2s)^{-1}\|v_k\|^2$ for $k\ge1$ . These expressions are fixed by the source formulas (14)–(15), p. 15, not chosen to simplify a later proof.

Formalization targets

The central target is the full rate statement of Theorem 6. In the range where the paper's monotonicity argument applies, $E_k$ is eventually nonincreasing. For every IGAHD run,

f(x_k)-\min_H f=O(k^{-2})\quad(k\to\infty).

When $\beta>0$ , the theorem also gives both weighted gradient summability claims:

\sum_{k\ge0} k^2\|\nabla f(y_k)\|^2<\infty, \qquad \sum_{k\ge0} k^2\|\nabla f(x_k)\|^2<\infty.

The milestone list follows claims the authors state while proving that target. It begins with the extended descent inequality (Lemma 1, (28)), records the identities for $t_k$ and $v_k$ , and then captures the displayed energy and $\varepsilon$ inequalities. Remark 4 asks for the $O(k^{-3})$ bound on the smallest squared gradient norm among the first $k$ iterates; Remark 5 asks for the $O(k^{-2})$ objective gap at $y_k$ . Those are companion statements and are not substituted for the main theorem.

Significance

The rate bounds give three kinds of information about one trajectory. The objective gap controls the values attained at the iterates. The two summable series imply that the gradients at both the extrapolated points and the iterates become small in a weighted aggregate sense when damping is positive. The paper then uses this series estimate to derive the best-iterate estimate of Remark 4 and the extrapolated-value estimate of Remark 5, all on p. 18.

The result is proved in the cited paper; the open work here is its Lean proof and the supporting smooth convex analysis. The development makes the precise discrete algorithm, the Lyapunov quantity, and the descent inequality reusable as mathematical interfaces. A complete formal proof would also make explicit where the author's finite-index monotonicity wording needs correction.

Difficulty

The usual smooth descent estimate alone gives a decrease after one gradient step, but the extrapolated point $y_k$ also carries a momentum term and a gradient difference. Its objective value is therefore not the same quantity as $f(x_k)$ , and a direct comparison of successive objective values does not give the claimed summability. The source uses a stronger descent inequality with a squared difference of gradients. The energy change then contains a quadratic expression in $\nabla f(y_k)$ and $\nabla f(x_k)$ whose sign depends on $\beta<2\sqrt{s}$ and, at first, on how large $k$ is. Establishing that sign uniformly over the relevant tail of the run is the central obstruction.

Formalization scope

The Lean space is a complete real inner product space. Convexity is ConvexOn ℝ Set.univ f; differentiability is explicit because the phrase “whose gradient is $L$ -Lipschitz” presupposes a gradient. The existence of a minimizer is represented by a selected $x^*$ with $f(x^*)\le f(z)$ for every $z$ . Positive $L$ and positive $s$ make $s\le1/L$ meaningful. The algorithm is a predicate on sequences $x,y:\mathbb N\to H$ , imposed from $k=1$ so no coefficient is divided by zero. The first two $x$ terms are free, and $y_0$ is unused. The $O$ statements use the filter at infinity, so finite initial terms do not affect them; summability uses the ordinary series over natural indices.

The printed claim that $(E_k)$ is nonincreasing from its first term is false for the paper's own definitions. On $H=\mathbb R$ with $f(u)=u^2/2$ , $L=s=1$ , $\alpha=3$ , $\beta=0$ , $x_0=0$ , and $x_1=1$ , one gets $E_1=0$ and $E_2=1/8$ . The formal goal keeps eventual monotonicity, as supported by the proof on pp. 16–18. This correction is disclosed rather than treated as a new hypothesis. No energy decrease, gradient bound, or conclusion is assumed in the run predicate. The milestone definitions also specify $B_k$ exactly as on p. 17. Contributions to the goal, to the extended descent lemma, or to the explicit energy estimates are all useful; the definition layer and smooth convex lemmas can serve later optimization formalizations.

Selected references

H. Attouch, Z. Chbani, J. Fadili, and H. Riahi, First-order optimization algorithms via inertial systems with Hessian driven damping, Mathematical Programming, 2020; arXiv:1907.10536v2, especially §3.2, pp. 15–18, and Appendix A.1, p. 34.

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Convex Optimization Dynamical Systems Optimization·Captain: mikedeng1

First-Order Optimization Algorithms via Inertial Systems with Hessian Driven Damping IV: For µ-Strongly Convex f, (DIN)2√µ,β Trajectories Satisfy f(x(t)) − min f ≤ Ce^{−√µ(t−t₀)/2}Research Paper

Motivation

Accelerated first-order methods for convex minimization, such as Nesterov's method and Polyak's heavy ball method, are now routinely studied through their continuous-time limits: second-order differential equations whose trajectories descend toward a minimizer. In the time-continuous setting the convergence analysis is a Lyapunov argument, and a good Lyapunov function for the ODE usually suggests one for the algorithm obtained by discretizing it.

Attouch, Chbani, Fadili and Riahi (arXiv:1907.10536v2, Math. Program. 2020) study inertial dynamics with Hessian-driven damping: besides the viscous friction term $\gamma\dot x$ , the equation contains a term $\beta\nabla^2 f(x(t))\dot x(t)$ , which equals $\beta\frac{d}{dt}\nabla f(x(t))$ . This damping is known to reduce the oscillations of the heavy ball trajectory, and because it is a time derivative of the gradient, explicit and implicit time discretizations of it produce first-order algorithms with no Hessian evaluation. This mission covers the strongly convex continuous-time result of the paper, Theorem 7 (§4.1, pp. 19–21). For $\beta=0$ it reduces to the heavy ball equation with friction $2\sqrt\mu$ , whose exponential rate is due to Siegel (arXiv:1903.05671, Theorem 2.2), as the paper's Remark 7 records. For $\beta>0$ , Remark 7 points to a result on a related but different dynamical system by Wilson, Recht and Jordan (arXiv:1611.02635, Theorem 1), whose rate is slightly worse. The algorithms (IPAHD-SC) and (IGAHD-SC) of §5 of the paper are the discrete counterparts of this theorem.

Setting

Let $\mathcal H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$ . Let $f:\mathcal H\to\mathbb R$ be a convex function of class $\mathcal C^2$ , with gradient $\nabla f$ and Hessian $\nabla^2 f$ . For $\mu>0$ , $f$ is $\mu$ -strongly convex (Definition 1) if $f-\frac\mu2\|\cdot\|^2$ is convex. Such an $f$ is assumed to have a minimizer $x^\star$ , and $\min_{\mathcal H} f=f(x^\star)$ .

Fix $t_0>0$ and a real constant $\beta$ . A solution trajectory of

\ddot x(t)+2\sqrt{\mu}\,\dot x(t)+\beta\nabla^2 f(x(t))\dot x(t)+\nabla f(x(t))=0 \tag{19}

is a twice differentiable map $x:[t_0,+\infty[\to\mathcal H$ satisfying the equation at every $t\ge t_0$ . This is the dynamic $(\mathrm{DIN})_{\gamma,\beta}$ of the paper with the constant viscous damping $\gamma=2\sqrt\mu$ .

The proof uses two auxiliary functions. The first is the Lyapunov function

\mathcal E(t)=f(x(t))-\min_{\mathcal H}f+\tfrac12\big\|\sqrt\mu(x(t)-x^\star)+\dot x(t)+\beta\nabla f(x(t))\big\|^2 .

The second is $Z(t)=2\beta(f(x(t))-f(x^\star))+\sqrt\mu\|x(t)-x^\star\|^2$ . Both appear in the milestones under the names lyap and Zfun.

Formalization targets

Goal: Theorem 7 (p. 19)

Assume $0\le\beta\le\frac1{2\sqrt\mu}$ . Then three statements hold. First, for all $t\ge t_0$ ,

\frac\mu2\|x(t)-x^\star\|^2\le f(x(t))-\min_{\mathcal H}f\le C e^{-\frac{\sqrt\mu}{2}(t-t_0)},\qquad C:=f(x(t_0))-\min_{\mathcal H}f+\mu\|x(t_0)-x^\star\|^2+\|\dot x(t_0)+\beta\nabla f(x(t_0))\|^2 .

Second, there is $C_1>0$ such that for all $t\ge t_0$

e^{-\sqrt\mu t}\int_{t_0}^t e^{\sqrt\mu s}\|\nabla f(x(s))\|^2\,ds\le C_1e^{-\frac{\sqrt\mu}{2}t}.

Third,

\int_{t_0}^\infty e^{\frac{\sqrt\mu}{2}t}\|\dot x(t)\|^2\,dt<+\infty .

Milestones

The milestones are the displayed steps of the paper's proof, in order:

the strong convexity inequality $\langle\nabla f(y),y-x^\star\rangle\ge f(y)-f(x^\star)+\frac\mu2\|y-x^\star\|^2$ ;
the first differential inequality for $\mathcal E$ ;
the nonnegativity of $\frac\mu4X^2+\frac{\beta}{2\sqrt\mu}Y^2-\beta\sqrt\mu XY$ ;
$\frac{d}{dt}\mathcal E+\frac{\sqrt\mu}2\mathcal E+\frac{\sqrt\mu}2\|\dot x\|^2\le0$ ;
$\mathcal E(t)\le\mathcal E(t_0)e^{-\frac{\sqrt\mu}2(t-t_0)}$ ;
the differential inequality for $Z$ .

Companion: the case $\beta=0$

The last sentence of Theorem 7 states that for $\beta=0$ , $f(x(t))-\min f=\mathcal O(e^{-\sqrt\mu t})$ . The paper gives no proof of it and attributes it to Siegel. It is a separate item, listed as the last milestone, and is not part of the goal.

Significance

Theorem 7 gives exponential decay, at the rate $e^{-\frac{\sqrt\mu}2 t}$ , of the values and of the distance to the minimizer for the whole range $0\le\beta\le1/(2\sqrt\mu)$ of Hessian damping. It also gives an averaged exponential decay of $\|\nabla f(x(t))\|^2$ , which the authors describe as new in the literature, and weighted integrability of the squared velocity. These estimates guide the design of the strongly convex algorithms of §5, whose linear rates $q=1/(1+\frac12\sqrt{\mu s})$ mirror the continuous rate.

Formalizing it builds machine-checked continuous-time Lyapunov arguments in a Hilbert space: differentiation of energy functions along a trajectory that is only one-sidedly differentiable at $t_0$ , the chain rule through $\nabla f$ (the Hessian term), and Grönwall-type integration of differential inequalities. To our knowledge none of Theorem 7 has a machine-checked proof; the paper's proof is complete apart from the gap at $\beta=0$ in part (ii) noted under Difficulty.

Difficulty

The algebra of the first differential inequality is a direct but long expansion, and the quadratic form is where the upper bound on $\beta$ enters. The main analytic steps are elsewhere. First, the energy must be shown differentiable along the trajectory, which requires the derivative of $t\mapsto\nabla f(x(t))$ , a chain rule through the Fréchet derivative of the gradient. Second, the differential inequalities must be integrated on $[t_0,+\infty[$ with one-sided derivatives at $t_0$ . Third, part (ii) must be derived from the $Z$ -inequality. The paper's derivation divides by $\beta^2$ , so it covers only $\beta>0$ . At $\beta=0$ part (ii) is still true, but it needs a separate argument, for example local Lipschitz continuity of $\nabla f$ near $x^\star$ together with the convergence $x(t)\to x^\star$ from part (i). A proof that only follows the printed text does not cover the case $\beta = 0$ .

Formalization scope

Space and function. $\mathcal H$ is a real inner product space that is complete. $f$ is ConvexOn ℝ univ and ContDiff ℝ 2, $\nabla f$ is Mathlib's gradient f, and the Hessian applied to $v$ at $x$ is fderiv ℝ (gradient f) x v.
Strong convexity is Definition 1 verbatim, with $\mu>0$ as a separate hypothesis. Convexity of $f$ is kept from the standing hypothesis (H) of p. 2 even though strong convexity implies it.
The minimizer. $\operatorname{argmin}f\neq\emptyset$ is a point xstar with $f(x^\star)\le f(y)$ for all $y$ . $\min f$ is written f xstar.
The trajectory is a triple $x,\dot x,\ddot x:\mathbb R\to\mathcal H$ . At every $t\ge t_0$ , $x$ has derivative $\dot x(t)$ and $\dot x$ has derivative $\ddot x(t)$ within $[t_0,+\infty[$ , and (19) holds. Values before $t_0$ play no role, and $\dot x(t_0)$ in the constant $C$ is the right derivative. $\beta$ is a real constant and $t_0>0$ , as in (H).
Derivatives in milestones are stated as the existence of a number that is the one-sided derivative and satisfies the inequality. Such a number is unique.
Integrals. Part (ii) is an interval integral of a function continuous on $[t_0,t]$ . The "Moreover" clause is stated as integrability on $[t_0,+\infty[$ of the nonnegative integrand. The constant $C_1$ is existentially quantified after the trajectory, so it may depend on it, as on the page.
Ruling out trivial readings. The goal never mentions $\mathcal E$ or $Z$ : its conclusions are about $f(x(t))$ , $\|x(t)-x^\star\|$ , $\nabla f(x(t))$ and $\dot x(t)$ only. The solution predicate is satisfiable: the constant trajectory at $x^\star$ satisfies it.
Infrastructure needed: the chain rule for $\nabla f\circ x$ with one-sided derivatives, differentiation of $\|\cdot\|^2$ and $\langle\cdot,\cdot\rangle$ along curves, and a Grönwall-type lemma $u'\le-cu\Rightarrow u(t)\le u(t_0)e^{-c(t-t_0)}$ on $[t_0,\infty)$ with one-sided derivatives. These are reusable for the other continuous-time missions of the series. Contributions of these general lemmas are welcome as separate theorems.

Selected references

H. Attouch, Z. Chbani, J. Fadili, H. Riahi, First-order optimization algorithms via inertial systems with Hessian driven damping, Math. Program. (2020); arXiv:1907.10536v2. https://arxiv.org/abs/1907.10536
J. W. Siegel, Accelerated first-order methods: differential equations and Lyapunov functions, 2019. https://arxiv.org/abs/1903.05671
A. C. Wilson, B. Recht, M. I. Jordan, A Lyapunov analysis of momentum methods in optimization, J. Mach. Learn. Res. 22 (2021); arXiv:1611.02635. https://arxiv.org/abs/1611.02635
B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys. 4 (1964). https://doi.org/10.1016/0041-5553(64)90137-5

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Convex Optimization Machine Learning Optimization·Captain: mikedeng1

Stochastic Dual Coordinate Ascent Methods for Regularized Loss Minimization 3: Under Refined Dual Strong Convexity, SDCA Reaches Expected Dual Sub-optimality ε after 2(n/s)·log(2/ε) IterationsResearch Paper

Motivation

Regularized loss minimization covers support vector machines, logistic and ridge regression, and the absolute-deviation regression of robust statistics. Stochastic dual coordinate ascent (SDCA) solves these problems by maximizing the Fenchel dual one randomly chosen coordinate at a time. Shalev-Shwartz and Zhang (arXiv:1209.1873, JMLR 2013) gave the first analysis of SDCA in terms of the duality gap. For smooth losses their rate is linear; for Lipschitz but non-smooth losses such as the hinge loss it is of order $1/\epsilon$ , which is no better than stochastic gradient descent, although SDCA is observed to converge much faster than SGD on SVM problems asymptotically.

Section 5 of the paper explains that observation. Many non-smooth losses are smooth at most points: the hinge loss $\max(0,1-uy_i)$ fails to be smooth only at $uy_i=1$ . If, at the optimum, most margins stay away from the kink, the dual is strongly concave in most coordinates, and SDCA converges linearly in the dual objective. Earlier linear-rate results for dual coordinate ascent on SVMs (Luo and Tseng, 1992) depend on the smallest nonzero eigenvalue of the data Gram matrix, which can be arbitrarily small when two data points are close; the condition of Section 5 avoids the Gram matrix.

Setting

Data $x_1,\dots,x_n\in\mathbb R^d$ (Euclidean norm), scalar convex losses $\varphi_1,\dots,\varphi_n:\mathbb R\to\mathbb R$ and $\lambda>0$ define the primal problem

\min_{w\in\mathbb R^d} P(w),\qquad P(w)=\frac1n\sum_{i=1}^n\varphi_i(w^\top x_i)+\frac\lambda2\|w\|^2 .

The convex conjugate of $\varphi_i$ is $\varphi_i^*(u)=\sup_z(zu-\varphi_i(z))\in(-\infty,+\infty]$ . The dual problem is $\max_{\alpha\in\mathbb R^n}D(\alpha)$ with

D(\alpha)=\frac1n\sum_{i=1}^n-\varphi_i^*(-\alpha_i)-\frac\lambda2\|w(\alpha)\|^2,\qquad w(\alpha)=\frac{1}{\lambda n}\sum_{i=1}^n\alpha_ix_i .

A dual variable is feasible when every $\varphi_i^*(-\alpha_i)$ is finite. A function is $L$ -Lipschitz if $|\varphi_i(a)-\varphi_i(b)|\le L|a-b|$ . The standing assumptions are $\|x_i\|\le1$ , $\varphi_i\ge0$ and $\varphi_i(0)\le1$ .

Procedure SDCA starts from $\alpha^{(0)}=0$ . At iteration $t$ it picks $i$ uniformly at random, independently of the past, chooses $\Delta\alpha_i$ maximizing $-\varphi_i^*(-(\alpha_i^{(t-1)}+\Delta\alpha_i))-\frac{\lambda n}{2}\|w(\alpha^{(t-1)})+(\lambda n)^{-1}\Delta\alpha_ix_i\|^2$ , and sets $\alpha^{(t)}=\alpha^{(t-1)}+\Delta\alpha_ie_i$ . Let $\alpha^*$ be a dual optimum and $w^*=w(\alpha^*)$ .

The refined dual strong convexity condition (Definition 2, (4)) asks for functions $\gamma_i(\cdot)\ge0$ with

\varphi_i^*(-a)-\varphi_i^*(-b)+u(a-b)\ge\gamma_i(u)|a-b|^2\qquad\text{for all }a,b\text{ and }u\in\partial\varphi_i^*(-b).

For the hinge loss one may take $\gamma_i(u)=|uy_i-1|$ . With constants $\gamma_i\ge0$ , the dual strong convexity inequality (5) is

D(\alpha^*)-D(\alpha)\ge\frac1n\sum_{i=1}^n\gamma_i|\alpha_i-\alpha_i^*|^2+\frac\lambda2\|w(\alpha)-w^*\|^2\quad\text{for every feasible }\alpha,

and $N(u)=\#\{i:\gamma_i<u\}$ counts the coordinates with small modulus.

Formalization targets

Goal: Theorem 5 (p. 9)

Assume convex $L$ -Lipschitz losses, the standing assumptions, and (5) with constants $\gamma_i\ge0$ . For every $\epsilon_D>0$ and $s\in(0,1]$ with $\epsilon_D\ge 8L^2\frac{s}{\lambda n}N(\frac{s}{\lambda n})/n$ ,

t\ge2\frac ns\log\frac2{\epsilon_D}\quad\Longrightarrow\quad\mathbb E\bigl[D(\alpha^*)-D(\alpha^{(t)})\bigr]\le\epsilon_D .

The constants $2$ and $8$ are the paper's.

Milestones

Proposition 1 (pp. 8–9): condition (4), with $\gamma_i=\gamma_i(w^{*\top}x_i)$ , implies (5), and $|(w^*-w)^\top x_i|\ge\gamma_i|a_i-\alpha_i^*|$ whenever $-a_i\in\partial\varphi_i(w^\top x_i)$ (6).
Lemma 5 (p. 19): under (5), one step raises the expected dual by at least $\frac{s}{2n}$ times the dual sub-optimality, plus $\frac{3s\lambda}{4n}\|w^*-w\|^2$ , minus $(\frac sn)^2G_*(s)/(2\lambda)$ with $G_*(s)=\frac1n\sum_i(\|x_i\|^2-\gamma_i\lambda n/s)(\alpha_i^*-\alpha_i)^2$ .
Lemma 3 (p. 15): the conjugate of an $L$ -Lipschitz function is $+\infty$ outside $[-L,L]$ .
Lemma 6 (p. 20): $G_*(s)\le4L^2N(s/(\lambda n))/n$ .
Dual recursion (§7.6, p. 20): $\epsilon_D^{(t)}\le(1-\frac s{2n})\epsilon_D^{(t-1)}+(\frac sn)^2\frac{G_*(s)}{2\lambda}$ and $\epsilon_D^{(t)}\le(1-\frac s{2n})^t\epsilon_D^{(0)}+\frac sn\frac{G_*(s)}\lambda$ , with $G_*(s)=4L^2N(s/(\lambda n))/n$ .
Lemma 2 (p. 13): $D(\alpha)\le P(w^*)\le P(0)\le1$ and $D(0)\ge0$ .

Significance

Theorem 5 gives a linear rate for the dual sub-optimality of SDCA on non-smooth losses whenever few constants $\gamma_i$ are small. If $N(s_0/(\lambda n))=0$ for some $s_0>0$ (for the SVM, $\lambda n|w^{*\top}x_iy_i-1|\ge s_0$ for all $i$ ), the iteration count is $(2n/s_0)\log(2/\epsilon_D)$ , compared with order $n+L^2/(\lambda\epsilon)$ from the Lipschitz analysis of Theorem 1 of the same paper. The paper uses Theorem 5 to explain the observed asymptotic advantage of SDCA over SGD, and its Theorem 6 converts it into a duality-gap bound.

The result is proved in the paper. It has not been formalized: no SDCA convergence result is on Prove2Me, and Mathlib has no conjugate-duality theory for this problem class. The mission produces a formal account of the dual objective in extended reals, of the exact coordinate maximization step, and of the averaging over uniformly random coordinate sequences, together with a checked proof of the rate. The companion missions of this series cover Theorem 1 (Lipschitz losses) and Theorem 2 (smooth losses).

Difficulty

Condition (5) gives a modulus $\gamma_i$ per coordinate, and some $\gamma_i$ may be zero, so the dual is not strongly concave and the standard linear-rate argument for strongly concave coordinate ascent does not apply. The per-step progress has a variance term with the sign of $\|x_i\|^2-\gamma_i\lambda n/s$ , which is positive exactly on the coordinates with small modulus; that term must be controlled by bounding the dual variables themselves, which requires the Lipschitz structure through the conjugate. The extended-real-valued conjugate is a second source of work: every quantity must be shown finite along the run before it can be manipulated as a real number.

Formalization scope

Lean works in EuclideanSpace ℝ (Fin d), with $n$ data points indexed by Fin n, $n>0$ . The conjugate is an EReal supremum and $D$ is EReal-valued, $-\infty$ exactly at infeasible points. Real-valued expectations of dual values appear only together with a conjunct stating their finiteness. The SDCA step is a map $\Delta$ with the hypothesis that $\Delta(\alpha,i)$ is an exact maximizer of the coordinate objective. The run is a fold over the sequence of chosen coordinates, and the expectation is the uniform average over all sequences of length $t$ (SAGA.Convex.expectIdx, a published definition). The optimum $\alpha^*$ is a hypothesis $D(\beta)\le D(\alpha^*)$ for all $\beta$ ; in (5), $w^*$ is $w(\alpha^*)$ . Inequality (5) quantifies over feasible $\alpha$ only. Lemmas 5 and 6 are stated for an arbitrary feasible current state, which is stronger than the history-averaged form printed. The paper's range $s\in[0,1]$ becomes $s\in(0,1]$ , because at $s=0$ the iteration bound $2(n/s)\log(2/\epsilon_D)$ is undefined. A formalization that admits $s=0$ would evaluate it to $0$ and claim the bound at $t=0$ , which is false.

The development needs elementary facts about conjugates of scalar functions in EReal (convexity, the Fenchel–Young inequality, the Lipschitz domain bound), the weak duality inequality $D\le P$ , and finite averages over coordinate sequences. The conjugate facts and the expectation calculus are reusable by the other two missions of the series. Proofs of individual milestones are welcome independently.

Selected references

S. Shalev-Shwartz and T. Zhang, Stochastic Dual Coordinate Ascent Methods for Regularized Loss Minimization, Journal of Machine Learning Research 14 (2013), 567–599; preprint arXiv:1209.1873v2. https://arxiv.org/abs/1209.1873
Z.-Q. Luo and P. Tseng, On the convergence of coordinate descent method for convex differentiable minimization, Journal of Optimization Theory and Applications 72 (1992), 7–35. https://doi.org/10.1007/BF00939948
C.-J. Hsieh, K.-W. Chang, C.-J. Lin, S. S. Keerthi and S. Sundararajan, A dual coordinate descent method for large-scale linear SVM, ICML 2008. https://doi.org/10.1145/1390156.1390208

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Algorithmic Game Theory Probability Stochastic Systems·Captain: mikedeng1

Probabilistic Analysis of Mean-Field Games 2: The Mean-Field Strategies Form an ε_N-Nash Equilibrium of the N-Player Game, with ε_N ≤ cN^{−1/(d+4)}Research Paper

Motivation

Mean-field games were introduced by Lasry and Lions (2007) and by Huang, Malhamé and Caines (2006) as limits of stochastic differential games with many symmetric players, each interacting with the others only through the empirical distribution of their states. The limit replaces a game of $N$ coupled players by one representative player facing a deterministic flow of distributions that it must reproduce. That flow is the equilibrium condition. The limit object is easier to solve. Whether it says anything about the finite game it came from is a separate question: a mean-field equilibrium is useful only if the strategies it produces are nearly optimal for each of $N$ actual players.

Carmona and Delarue (SIAM J. Control Optim. 51, 2013) solve the mean-field game through the stochastic maximum principle. For drifts that are affine and costs that are convex, the equilibrium is the solution of a forward–backward SDE of McKean–Vlasov type. Its existence is the subject of the first mission of this series. This mission formalizes §4 of the paper. The strategies read off from that solution are an approximate Nash equilibrium of the $N$ -player game, with an explicit rate in $N$ .

Setting

The state space is $\mathbb R^d$ and the control space is $\mathbb R^k$ , both with the Euclidean norm. Each player $i$ has a private state $U^i_t$ , driven by its own $m$ -dimensional Brownian motion $W^i$ , the motions being independent. The dynamics are

dU^i_t = b(t, U^i_t, \bar\nu^N_t, \beta^i_t)\,dt + \sigma\,dW^i_t,\qquad U^i_0 = x_0,\qquad \bar\nu^N_t = \frac1N\sum_{j=1}^N\delta_{U^j_t},

where $b(t,x,\mu,\alpha) = b_0(t,\mu) + b_1(t)x + b_2(t)\alpha$ is affine and $\sigma$ is a constant matrix. Player $i$ chooses a strategy $\beta^i$ , a process that is progressively measurable for the filtration of $(W^1,\dots,W^N)$ and satisfies $\mathbb E\int_0^T|\beta^i_t|^2dt<\infty$ . It pays

\bar J^{N,i}(\beta^1,\dots,\beta^N) = \mathbb E\Big[g(U^i_T,\bar\nu^N_T) + \int_0^T f(t,U^i_t,\bar\nu^N_t,\beta^i_t)\,dt\Big].

The costs $f$ , $g$ satisfy the paper's assumptions (A.1)–(A.7): joint convexity in $(x,\alpha)$ with strong convexity in $\alpha$ , $C^{1,1}$ regularity, local Lipschitz continuity in the measure argument for the 2-Wasserstein distance $W_2$ , and growth conditions.

On the mean-field side, let $(X_t,Y_t,Z_t)$ solve the McKean–Vlasov FBSDE (3.1), let $\mu_t = \mathbb P_{X_t}$ , and let $u$ be the FBSDE value function. This is a function that is Lipschitz in $x$ with linear growth and satisfies $Y_t = u(t,X_t)$ . Let $\hat\alpha(t,x,\mu,y)$ minimize the Hamiltonian $\langle b,y\rangle + f$ over $\alpha$ , and let $J$ be the mean-field cost (4.1). The candidate strategies are

\bar\alpha^{N,i}_t = \hat\alpha\big(t, X^i_t, \mu_t, u(t,X^i_t)\big),

where $(X^1,\dots,X^N)$ solves the particle system (4.2): the dynamics above, with each player using this feedback against the frozen flow $\mu$ . Each player needs only its own state.

Formalization targets

Goal: Theorem 4.2

There exist $c>0$ and positive numbers $(\epsilon_N)_{N\ge1}$ with $\epsilon_N\le cN^{-1/(d+4)}$ such that, for every $N\ge1$ , every player $i$ and every admissible deviation $\beta^i$ ,

\bar J^{N,i}(\bar\alpha^{N,1},\dots,\beta^i,\dots,\bar\alpha^{N,N})\ \ge\ \bar J^{N,i}(\bar\alpha^{N,1},\dots,\bar\alpha^{N,N}) - \epsilon_N.

The constant and the sequence come before $N$ , the player and the deviation.

Milestones

The milestones follow the proof, in attack order:

Lemma 4.1 (Horowitz–Karandikar): $\mathbb E[W_2^2(\bar\mu^N,\mu)]\le CN^{-2/(d+4)}$ for i.i.d. samples from $\mu\in\mathcal P_{d+5}$ .
Well-posedness of (4.2).
The moment bound (4.4).
The transport inequality (4.13).
Propagation of chaos (4.11) and its consequence (4.12) for the empirical measure.
The equilibrium cost estimate (4.14): $\bar J^{N,i}(\bar\alpha) = J + O(N^{-1/(d+4)})$ .
For a deviation of bounded energy $\mathbb E\int|\beta^1|^2\le A$ : (4.17), (4.20) and the lower bound (4.23), $\bar J^{N,1}\ge J - c_AN^{-1/(d+4)}$ .
The coercivity (4.25) for deviations of large energy.

Companion: Corollary 4.3

Large deviations are unprofitable for every player (4.27). A bounded deviation of one player keeps every player's cost above $J-\epsilon_N$ (4.28).

Significance

The theorem justifies the mean-field game as a model of large finite games. Solving one McKean–Vlasov FBSDE produces distributed closed-loop strategies. No player can improve its cost by more than $O(N^{-1/(d+4)})$ by deviating, even with a strategy that uses all players' noises. The rate gives a quantitative meaning to "large $N$ ". The milestones (4.11)–(4.12) are a quantitative propagation-of-chaos result for the particle system driven by the decoupling field. Such results are used on their own in the numerics and statistics of interacting particle systems.

The result is proved in the paper. No part of it has been machine-checked, as far as the platform's records show. The work that remains is to formalize the known proof. That needs Wasserstein estimates for empirical measures (Lemma 4.1, cited by the paper without proof), Gronwall arguments for SDE systems, and the optimality of the mean-field control (Theorem 2.2, the first mission's milestone, used here through $J\le J(\beta^1)$ ). Sharper rates are known in low dimension (Fournier and Guillin, 2015). They would be welcome as variants, but they are not the target.

Difficulty

The obvious argument compares the deviating player's cost with the mean-field cost of the same strategy. It fails in two places. The deviation changes the empirical measure that every other player sees, so the other players' states are no longer independent copies of $X$ . It must be shown that one player of bounded energy moves the population's empirical measure by only $O(N^{-1/(d+4)})$ in $W_2$ ((4.17)–(4.20)). The second problem is that the deviation $\beta^1$ is adapted to the filtration of all $N$ Brownian motions, larger than the one in which the mean-field control problem is posed. The comparison $J\le J(\beta^1)$ must survive this enlargement. Finally, nothing bounds the energy of a deviation a priori. Deviations of large energy have to be excluded separately by a coercivity estimate ((4.25)), and only then can the rate be obtained uniformly.

Formalization scope

The data live on EuclideanSpace ℝ (Fin n), so every norm, $M_2$ and $W_2$ is Euclidean. Matrix norms are operator norms. $W_2$ is the published WassersteinDRO.Duality.wassersteinDistance 2, valued in $[0,\infty]$ . Measures carry Mathlib's σ-algebra on Measure, which on probability measures on $\mathbb R^d$ coincides with the paper's Borel σ-field of weak convergence. The SDEs and BSDEs are the published Itô-process predicates of Peng (Peng1990.SMP.SolvesSDE, SolvesBSDE). Solutions also have almost surely continuous paths on $[0,T]$ . The game filtration is generated by $(W^1,\dots,W^N)$ and augmented by the null sets. Players are indexed by Fin N, so the paper's player $1$ is index $0$ .

The mean-field solution and the game live on two probability spaces. Only $\mu_t=\mathbb P_{X_t}$ , $J$ and $u$ pass between them. The value function $u$ enters through its properties (3.3), the identity $Y_t=u(t,X_t)$ and joint measurability, as in Lemma 3.5. $\hat\alpha$ is the chosen minimizer of the Hamiltonian, with default $0$ ; by Lemma 2.1 the default is never used on $[0,T]\times\mathcal P_2$ . The assumptions also contain a standing convention the page does not write: $f$ and $g$ are Borel measurable. Without it the expected costs would not be expectations of measurable integrands. Moments and expected energies are lower Lebesgue integrals in $[0,\infty]$ . Costs are real expectations, finite under (A.5) for admissible strategies.

The statements exclude the trivializing readings:

the flow is the law of the mean-field state, not an arbitrary flow;
distances are Euclidean, not sup-norm;
the empirical measure is used only for $N\ge1$ ;
$W_2^2$ is never converted to a real number where it could be infinite;
the constants $c$ , $(\epsilon_N)$ , $c_A$ , $N_0$ sit exactly where the page's quantifiers put them;
in a deviation the other players keep their processes $\bar\alpha^{N,j}$ , while the empirical measure follows the new states.

The goal does not mention the decoupled copies or Lemma 4.1.

A complete development needs:

Wasserstein estimates for empirical measures, which are reusable well beyond this mission;
existence, uniqueness and moment bounds for Lipschitz SDE systems with measure-dependent drift;
the stochastic-maximum-principle comparison of the first mission.

Contributions on any of these are welcome. Well-posedness of the deviated system (4.5) is used but is not stated on the page; a proof of it is also welcome.

Selected references

R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim. 51(4):2705–2734, 2013. https://doi.org/10.1137/120883499
J.-M. Lasry and P.-L. Lions, Mean field games, Japanese Journal of Mathematics 2:229–260, 2007. https://doi.org/10.1007/s11537-007-0657-8
M. Huang, R. P. Malhamé and P. E. Caines, Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle, Communications in Information and Systems 6(3):221–252, 2006. https://doi.org/10.4310/CIS.2006.v6.n3.a5
J. Horowitz and R. L. Karandikar, Mean rates of convergence of empirical measures in the Wasserstein metric, J. Comput. Appl. Math. 55(3):261–273, 1994. https://doi.org/10.1016/0377-0427(94)90033-7
A.-S. Sznitman, Topics in propagation of chaos, École d'Été de Probabilités de Saint-Flour XIX, Lecture Notes in Math. 1464, Springer, 1991. https://doi.org/10.1007/BFb0085169
N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probab. Theory Related Fields 162:707–738, 2015. https://doi.org/10.1007/s00440-014-0583-7

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Operations Research Optimization Probability+1·Captain: mikedeng1

From Data to Decisions: Distributionally Robust Optimization Is Optimal 3: On a Compact Continuous State Space the Relative-Entropy Robust Predictor Is Feasible and Strongly OptimalResearch Paper

Motivation

A decision maker who chooses $x$ to minimize an expected cost $c(x,\mathbb P)=\mathbb E_{\mathbb P}[\gamma(x,\xi)]$ rarely knows the distribution $\mathbb P$ of the random parameter $\xi$ . What is available is a sample $\xi_1,\dots,\xi_T$ . Any rule that turns the sample into a cost estimate can disappoint: the realized expected cost may exceed the estimate. In a cost-minimization context such underestimates are more harmful than overestimates, which is why distributionally robust optimization replaces the unknown $\mathbb P$ by a worst case over a set of distributions consistent with the data.

Van Parys, Mohajerin Esfahani and Kuhn (arXiv:1704.04118, Management Science 2021) asked which such rule is best. They define a data-driven predictor to be least conservative among all predictors whose probability of disappointment decays exponentially at a prescribed rate $r$ , and they prove that a single predictor, the worst-case expected cost over a relative-entropy ball around the empirical distribution, is the unique strong solution. Their main results are for a finite set of outcomes $\Xi=\{1,\dots,d\}$ . Section 5 extends the predictor result to an arbitrary compact $\Xi\subseteq\mathbb R^d$ ; this mission formalizes that extension. The version used throughout is the arXiv preprint arXiv:1704.04118v3 (22 December 2019), and every index below is the preprint's.

Setting

Fix compact sets $X\subseteq\mathbb R^n$ (decisions) and $\Xi\subseteq\mathbb R^d$ (outcomes), and a cost $\gamma:X\times\Xi\to\mathbb R$ that is jointly continuous.

The model class $\mathcal P$ is the set of Borel probability distributions on $\Xi$ , with the topology of weak convergence. $X\times\mathcal P$ carries the product topology.
The model-based predictor is $c(x,\mathbb P)=\int_\Xi\gamma(x,\xi)\,\mathrm d\mathbb P(\xi)$ .
The relative entropy of $\mathbb P'$ with respect to $\mathbb P$ (Definition 8) is $I(\mathbb P',\mathbb P)=\int_\Xi\log\frac{\mathrm d\mathbb P'}{\mathrm d\mathbb P}\,\mathrm d\mathbb P'$ if $\mathbb P'\ll\mathbb P$ , and $+\infty$ otherwise.
Given independent samples $\xi_1,\dots,\xi_T$ from $\mathbb P$ , the empirical distribution is $\hat{\mathbb P}_T=\frac1T\sum_{t=1}^T\delta_{\xi_t}$ , and $\mathbb P^\infty$ denotes the law of the sample path.
A data-driven predictor is a continuous function $\hat c:X\times\mathcal P\to\mathbb R$ ; the estimate after $T$ samples is $\hat c(x,\hat{\mathbb P}_T)$ . Its out-of-sample disappointment is $\mathbb P^\infty\big(c(x,\mathbb P)>\hat c(x,\hat{\mathbb P}_T)\big)$ .
Problem (5): $\hat c$ is feasible if, for all $x\in X$ and $\mathbb P\in\mathcal P$ ,

\limsup_{T\to\infty}\frac1T\log\mathbb P^\infty\big(c(x,\mathbb P)>\hat c(x,\hat{\mathbb P}_T)\big)\le -r,

and strongly optimal if it is feasible and $\hat c(x,\mathbb P')\le\hat c'(x,\mathbb P')$ for all $(x,\mathbb P')$ and every feasible $\hat c'$ .

The distributionally robust predictor is $\hat c_r(x,\mathbb P')=\sup\{c(x,\mathbb P):\mathbb P\in\mathcal P,\ I(\mathbb P',\mathbb P)\le r\}$ .
$\bar\gamma(x)=\max_{\xi\in\Xi}\gamma(x,\xi)$ is the worst-case cost and $\Xi^\star(x)$ the set of its maximizers.

Formalization targets

Goal: Theorem 10 (p. 26)

r\ge0\ \Longrightarrow\ \hat c_r\text{ is feasible in (5)},\qquad r>0\ \Longrightarrow\ \hat c_r\text{ is strongly optimal in (5)}.

Both halves are in the goal. The case $r=0$ of feasibility is the statement that $\hat c_0=c$ is continuous.

Milestones

Lemma 1 (p. 23): $c$ is continuous on $X\times\mathcal P$ .
Lemma 2 (33), p. 29: $\hat c_r$ equals a supremum over pairs $(\mathbb P_c,p)$ with $\mathbb P'\ll p\,\mathbb P_c\ll\mathbb P'$ , where the mass $1-p$ is placed at cost $\bar\gamma(x)$ .
Lemma 3 (p. 31): the perturbed problem (34), $\hat c_{r,\epsilon}$ , satisfies $\hat c_r\le\hat c_{r,\epsilon}\le\hat c_r+\epsilon$ .
Lemma 4 (35), p. 31:

\hat c_{r,\epsilon}(x,\mathbb P')\le\min_{\alpha\ge\bar\gamma(x)+\epsilon}\ \alpha-e^{-r}\exp\Big(\int_\Xi\log(\alpha-\gamma(x,\xi))\,\mathrm d\mathbb P'(\xi)\Big),

with a minimizer $\alpha^\star\le(\bar\gamma(x)+\epsilon-e^{-r}c(x,\mathbb P'))/(1-e^{-r})$ , and with equality for $\epsilon>0$ .

Proposition 6 (p. 25): $\hat c_r$ is continuous on $X\times\mathcal P$ for $r\ge0$ .
Theorem 9 (24a)/(24b), p. 25: the weak large deviation principle for $\hat{\mathbb P}_T$ in the weak topology, with the closure $\operatorname{cl}\mathcal D$ in the upper bound and the interior in the lower bound.
(41), p. 35: if $\mathbb P_0(\Xi^\star(x))<1$ and $c(x,\mathbb P_0)\ge\hat c_r(x,\mathbb P')$ , then $I(\mathbb P',\mathbb P_0)\ge r$ .
Proposition 1(ii), (iii) in the general setting (p. 24): $I$ is jointly convex and jointly lower semicontinuous on $\mathcal P\times\mathcal P$ .

Proposition 5, the dual representation (23) of $\hat c_r$ itself, is included as a further item; the proof of Theorem 10 goes through Lemmas 3–4 and does not use it.

Significance

Theorem 10 says that, on any compact outcome space, no continuous predictor is less conservative than $\hat c_r$ while keeping its disappointment probability below $e^{-rT}$ to first order in the exponent. Relative-entropy distributionally robust optimization is therefore not one choice among many ambiguity sets but the optimal one for this criterion, and the dual representation turns its evaluation into a one-dimensional convex problem. The paper also shows that the continuous-state analogue of its prescriptor result (Theorem 11) only holds up to an arbitrarily small shift, so the predictor statement formalized here is the clean part of the extension.

The results are proved in the paper; none is formalized. A complete development yields a formal Sanov-type weak large deviation principle for empirical measures in the weak topology, joint convexity and lower semicontinuity of the Kullback–Leibler divergence for probability measures on a compact metric space, and a measure-theoretic dual representation of a relative-entropy worst-case expectation. Mathlib has the Kullback–Leibler divergence klDiv and the weak topology on probability measures, but none of these three results.

Difficulty

The finite-state proof of feasibility bounds the disappointment through the large deviations upper bound over the disappointment set $\mathcal D$ itself. On a continuous space that bound only holds over the closure $\operatorname{cl}\mathcal D$ in the weak topology (24a), and the paper notes that this invalidates the finite-state argument (p. 25). The rate must then be controlled on a larger, closed set of empirical distributions, where the strict inequality defining a disappointment is lost; this is where the mass that $\mathbb P_0$ puts on the worst-case scenarios $\Xi^\star(x)$ starts to matter, as the hypothesis of (41) shows.

Continuity of $\hat c_r$ in the weak topology is the second obstacle. The supremum runs over an infinite-dimensional set of distributions, a worst-case distribution may put mass outside the support of $\mathbb P'$ , and the integrals involved can diverge or take the value $-\infty$ . Weak convergence controls integrals of bounded continuous functions only, and $\log(\alpha-\gamma)$ is unbounded below at $\alpha=\bar\gamma(x)$ .

Formalization scope

$\mathcal P$ is ProbabilityMeasure ↥Ξ for the Borel σ-algebra of the subtype, with Mathlib's weak-convergence topology. $X$ and $\Xi$ are compact subsets of Euclidean spaces, $\gamma:↥X\to↥\Xi\to\mathbb R$ is jointly continuous; these standing assumptions of §2 (p. 5) and §5 (p. 23) appear as hypotheses of every theorem. No other hypothesis is added. Nonemptiness of $\Xi$ is automatic wherever a distribution on $\Xi$ is in scope.
$I$ is Mathlib's InformationTheory.klDiv, valued in $[0,\infty]$ . It equals Definition 8, including the value $+\infty$ when $\mathbb P'\not\ll\mathbb P$ or when the log-likelihood ratio is not integrable.
The empirical distribution is defined for $T\ge1$ . The probability of an event about $\hat{\mathbb P}_T$ is the outer measure under the product measure $\mathbb P^{\otimes T}$ , and is $0$ at $T=0$ ; all statements are asymptotic in $T$ .
Decay rates are stated without logarithms: $\limsup\frac1T\log p_T\le-s$ becomes "for every $r'<s$ , eventually $p_T\le e^{-r'T}$ ", which is equivalent and handles $p_T=0$ . A naive Real.log encoding would be wrong, since Lean sets $\log0=0$ . Predictors must be continuous for the weak topology; dropping continuity, or replacing the weak topology by a finer one, changes the problem.
The constraint $\int\log(\frac1p\frac{\mathrm d\mathbb P'}{\mathrm d\mathbb P_c})\,\mathrm d\mathbb P'\le r$ in (33)–(34) carries an explicit integrability requirement, so that a divergent integral fails the constraint instead of evaluating to $0$ . The geometric mean $\exp\int\log(\alpha-\gamma)\,\mathrm d\mathbb P'$ follows the paper's convention $\log0=-\infty$ ; it is defined as $\inf_{\delta>0}\exp\int\log(\alpha+\delta-\gamma)\,\mathrm d\mathbb P'$ , which is the same value for $\alpha\ge\bar\gamma(x)$ .
At the end of the proof of Theorem 10 the page writes "when $\epsilon>0$ " for strong optimality; the theorem statement says $r>0$ , which is what is formalized.

Contributions are welcome at every level: a Sanov upper and lower bound for empirical measures on compact metric spaces, convexity and lower semicontinuity of klDiv, and the measure-theoretic duality of Lemma 4 are each reusable beyond this mission.

Selected references

B. P. G. Van Parys, P. Mohajerin Esfahani, D. Kuhn, From Data to Decisions: Distributionally Robust Optimization is Optimal, Management Science 67(6), 2021. Preprint arXiv:1704.04118v3. https://arxiv.org/abs/1704.04118
I. Csiszár, A simple proof of Sanov's theorem, Bulletin of the Brazilian Mathematical Society 37(4), 2006. https://doi.org/10.1007/s00574-006-0023-1
T. van Erven, P. Harremoës, Rényi divergence and Kullback–Leibler divergence, IEEE Transactions on Information Theory 60(7), 2014. https://doi.org/10.1109/TIT.2014.2320500
A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, 2nd ed., Springer, 1998. https://doi.org/10.1007/978-1-4612-5320-4

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Control Theory Probability Stochastic Systems·Captain: mikedeng1

Probabilistic Analysis of Mean-Field Games 1: The McKean–Vlasov FBSDE of a Convex Mean-Field Game Has a Solution, with a Lipschitz FBSDE Value FunctionResearch Paper

Motivation

A mean-field game describes a population of agents whose individual dynamics and costs depend on the distribution of the population's states. In a large population, an individual agent has negligible influence on that distribution but still responds to it. The mathematical problem is to find a control whose induced state law agrees with the distribution used to choose the control. Carmona and Delarue's 2013 analysis formulates this matching condition through a coupled stochastic system. Mean-field games were introduced independently by Lasry and Lions (2006–2007), through a coupled Hamilton–Jacobi–Bellman/Fokker–Planck system of partial differential equations, and by Huang, Malhamé and Caines (2006). Carmona and Delarue replace the PDE system by a probabilistic one, which handles costs of quadratic growth and degenerate noise without a uniform ellipticity assumption. Its existence theorem supplies the state-dependent control used later in the paper to construct approximate Nash equilibria for finite-player games.

The article first treats a control problem with a prescribed flow of probability measures, then imposes the requirement that the flow equal the law of the resulting state process. This mission concerns that self-consistent system and the regularity of its backward component. Proposition 3.7, a companion target, adds a uniqueness conclusion under the paper's monotonicity condition.

Setting

Fix a horizon $T$ , an initial state $x_0\in\mathbb R^d$ , an $m$ -dimensional Brownian motion $W$ , and unrestricted controls $a\in\mathbb R^k$ . The state $X_t$ has values in $\mathbb R^d$ . Its drift is affine in state and control,

b(t,x,\mu,a)=b_0(t,\mu)+b_1(t)x+b_2(t)a,

where $\mu$ is a probability measure on $\mathbb R^d$ with finite second moment. The volatility $\sigma\in\mathbb R^{d\times m}$ is constant and uncontrolled. A running cost $f(t,x,\mu,a)$ and a terminal cost $g(x,\mu)$ determine the control problem.

The Hamiltonian is $H(t,x,\mu,y,a)=\langle b(t,x,\mu,a),y\rangle+f(t,x,\mu,a)$ . Under convexity, it has a unique minimizer $\hat a(t,x,\mu,y)$ . The forward-backward stochastic differential equation (FBSDE) couples a forward state $X$ to an adjoint process $Y$ and a noise coefficient $Z$ . The forward equation uses the feedback $\hat a(t,X_t,\mu_t,Y_t)$ ; the backward equation has driver $\partial_xH$ and terminal condition $\partial_xg(X_T,\mu_T)$ . The McKean–Vlasov requirement sets $\mu_t=\mathcal L(X_t)$ at every time, giving the system (3.1):

\begin{cases} dX_t=b\big(t,X_t,\mathcal L(X_t),\hat a(t,X_t,\mathcal L(X_t),Y_t)\big)\,dt+\sigma\,dW_t, & X_0=x_0,\\ dY_t=-\partial_xH\big(t,X_t,\mathcal L(X_t),Y_t,\hat a(t,X_t,\mathcal L(X_t),Y_t)\big)\,dt+Z_t\,dW_t, & Y_T=\partial_xg\big(X_T,\mathcal L(X_T)\big).\end{cases}

This law matching is the extra condition beyond solvability for a prescribed flow.

All seven standing assumptions of Theorem 3.2 are retained. (A.1)–(A.3) specify measurable, bounded affine drift and $C^{1,1}$ running costs with a positive control-convexity constant $\lambda$ . (A.4) gives local boundedness, differentiability, and convexity of the terminal cost. (A.5) controls growth and dependence on the measure argument using the 2-Wasserstein distance $W_2$ ; (A.6) bounds the control gradient at zero; and (A.7) is a weak mean-reverting condition on the spatial gradients at Dirac laws. A measure flow is bounded when its second moments are uniformly bounded over $[0,T]$ .

Formalization targets

The goal is Theorem 3.2. It asserts existence of $(X,Y,Z)$ solving the self-consistent system. For every solution, there is an FBSDE value function $u:[0,T]\times\mathbb R^d\to\mathbb R^d$ and a constant $c\ge0$ such that

\|u(t,x)\|\le c(1+\|x\|),\qquad \|u(t,x)-u(t,x')\|\le c\|x-x'\|,\qquad Y_t=u(t,X_t)\quad\text{a.s. for all }t\in[0,T].

It also gives $\mathbb E[\sup_{0\le t\le T}\|X_t\|^\ell]<\infty$ for every real $\ell\ge1$ . The constant $c$ belongs to each solution; the theorem does not assert a single universal value for all models.

The milestone sequence follows the article's stated results: Lemma 2.1 characterizes the Hamiltonian minimizer; Theorem 2.2 and Proposition 2.5 compare controlled costs; Lemma 3.5 establishes a frozen-flow solution and a decoupling function; Proposition 3.8 handles bounded spatial gradients; Lemmas 3.9 and 3.10 state the approximation and stability results; and the first sentence of §3.7 records solvability under stronger state convexity. The companion Proposition 3.7 asks for uniqueness when the running cost separates into a measure-dependent and a control-dependent part and the measure-dependent costs satisfy the Lasry–Lions monotonicity inequalities.

Significance

A solution of the self-consistent FBSDE identifies a law flow consistent with the feedback chosen from that same law. The value function represents the adjoint state as a deterministic function of time and the forward state. Its linear growth and Lipschitz regularity control how feedback changes when states move. Those properties are used in the article's subsequent approximate-Nash analysis for the finite-player game; without them, its estimates cannot be stated in the same form. The moment conclusion gives the integrability needed for those comparisons.

The paper proves these results. The formalization task is to express its stochastic equations, probability laws, measure dependence, and analytic assumptions precisely enough that the known argument can be checked in Lean. The reusable outputs include a Euclidean mean-field control model, a finite-moment measure-flow interface, and a componentwise stochastic solution predicate compatible with published Brownian-motion and Itô-process definitions.

Difficulty

For a prescribed measure flow, the control problem produces a forward-backward system, but this does not by itself make the resulting state law equal to the prescribed flow. Standard Lipschitz contraction arguments may give short-time solvability; the theorem allows an arbitrary fixed horizon. The argument must also preserve control convexity and quantitative estimates while passing from models with bounded spatial gradients to the stated growth conditions. The adjoint value function must work simultaneously across the full time interval, and the theorem requires moment bounds of every order at least one.

Formalization scope

Lean uses EuclideanSpace ℝ (Fin d) for states and EuclideanSpace ℝ (Fin k) for controls, so all norms in the costs, moment conditions, and $W_2$ are Euclidean. A joint state-control norm is computed from the sum of the two squared Euclidean norms. Time uses $\mathbb R_{\ge0}$ ; measures with finite second moment use extended nonnegative moments, and $W_2$ is the published coupling-based definition. The measure-space sigma algebra is Mathlib's Giry sigma algebra, agreeing with the weak-convergence Borel sigma algebra on the probability measures here. Matrix bounds use operator norms.

The stochastic layer reuses published predicates for a standard Brownian motion, progressive $L^2$ processes, Itô integrals, SDEs, and BSDEs. The filtration is the Brownian filtration augmented by null sets. Peng's componentwise stochastic predicates use coordinate functions, so the Lean development converts between those and Euclidean states. Solutions include continuous trajectories and the square-integrability condition (2.14). The real cost expectations are used only in settings where the hypotheses ensure integrability. The Hamiltonian minimizer has a total-function fallback outside its existence domain, while Lemma 2.1 ensures that branch is never used in the theorems. Laws are formed only from measurable state variables. An arbitrary prescribed flow cannot replace the state law in the goal.

The following readings are fixed. (A.5)'s joint display for $(f,g)$ is two inequalities, one for $f$ and one for $g$ without the time and control arguments. Every assumption is imposed for $t\in[0,T]$ and for measures in $\mathcal P_2(\mathbb R^d)$ only; measurability of $f$ , $g$ and $b$ is required on that domain. Gradients of $f$ and $g$ are computed from the functions, not supplied as separate data. A solution of (2.13) or (3.1) is a progressively measurable triple with a.s. continuous forward and adjoint paths satisfying (2.14). Costs are real expectations; under the theorem's hypotheses the integrands are integrable, so no default value of the integral is used. In Proposition 2.5 the term $\langle x_0'-x_0,Y_0\rangle$ is written as an expectation, equal to the page's term because $Y_0$ is a.s. constant for the augmented Brownian filtration.

The statements exclude the trivializing encodings: the flow in (3.1) is the law of the forward process in every coefficient, not a free flow; distances and moments use Euclidean norms, not the coordinate supremum norm; the minimizer is the minimizer of $H$ , not a free feedback; moment and admissibility conditions are extended-valued integrals compared with $\infty$ ; and constants declared uniform by the article (Lemma 2.1's Lipschitz constant, Lemma 3.5's $c$ , Lemma 3.10's $\lambda',c'_L$ ) are quantified before the data they are uniform over.

Definitions of the model and solution class, proofs of the eight source-indexed milestones, the goal theorem, and the companion uniqueness result are in scope. The result concerns the source's full class of models satisfying (A.1)–(A.7), rather than a convenient special case.

Selected references

René Carmona and François Delarue, Probabilistic analysis of mean-field games, SIAM Journal on Control and Optimization 51(4), 2705–2734, 2013. DOI: 10.1137/120883499.
Jean-Michel Lasry and Pierre-Louis Lions, Mean field games, Japanese Journal of Mathematics 2(1), 229–260, 2007. DOI: 10.1007/s11537-007-0657-8.
Minyi Huang, Roland P. Malhamé and Peter E. Caines, Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle, Communications in Information and Systems 6(3), 221–252, 2006. DOI: 10.4310/CIS.2006.v6.n3.a5.
Shige Peng, A general stochastic maximum principle for optimal control problems, SIAM Journal on Control and Optimization 28(4), 966–979, 1990. DOI: 10.1137/0328054.

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Machine Learning Operations Research Optimal Transport+1·Captain: mikedeng1

Regularization via Mass Transportation I: Wasserstein-Robust Linear Classification with Label Flipping Is a Finite Convex ProgramResearch Paper

Motivation

A classifier trained from finitely many examples can perform well on those examples while responding poorly to changes in the feature distribution or to mislabeled observations. Distributionally robust learning addresses this by choosing a classifier against every probability distribution within a specified distance of the empirical distribution. The resulting optimization appears to range over an infinite-dimensional family of distributions. Shafieezadeh-Abadeh, Kuhn, and Mohajerin Esfahani show that, for linear classification with a convex Lipschitz loss, this problem has an exact finite convex reformulation. That result is Theorem 3.11(ii) of the 2019 arXiv preprint, which is the source used here.

The authors' transport metric allows a data point's features to move and its binary label to flip, with separate prices for those two changes. This matters when robustness to inaccurate labels is part of the model. A label-preserving metric cannot express that choice. The paper builds on its Section 2 learning model and the transport duality developed in Appendix A.1; the present mission isolates the classification result and the intermediate statements used for it. The preprint's Corollary 3.14 specializes the result to logloss, while its Corollary 3.12 treats hinge loss. A logloss-specific reformulation and a label-preserving classification result already have proved statements on Prove2Me; the general convex-Lipschitz, finite label-cost theorem remains the target of this mission.

Setting

Let $V$ be a finite-dimensional real vector space with a chosen norm $\|\cdot\|$ . Features are $x\in V$ and labels are $y\in\{-1,+1\}$ . A linear classifier is a continuous linear functional $w:V\to\mathbb R$ , with score $\langle w,x\rangle=w(x)$ . Its dual norm $\|w\|_*$ is the operator norm induced by $\|\cdot\|$ . Given $N\ge1$ training pairs $(\hat x_i,\hat y_i)$ , their empirical distribution is $\hat P_N=N^{-1}\sum_{i=1}^N\delta_{(\hat x_i,\hat y_i)}$ .

For $\kappa>0$ , the transport cost between pairs is

d((x,y),(x',y'))=\|x-x'\|+\kappa\mathbf1_{y\ne y'}.

The Wasserstein distance $W(Q,P)$ is the least expected transport cost among joint distributions whose two marginals are $Q$ and $P$ . For a radius $\rho\ge0$ , the Wasserstein ball $\mathbb B_\rho(\hat P_N)$ contains probability distributions $Q$ satisfying $W(Q,\hat P_N)\le\rho$ . The loss is $L(y\langle w,x\rangle)$ , where $L:\mathbb R\to[0,\infty)$ is convex and Lipschitz continuous. The Lipschitz modulus $\operatorname{lip}(L)$ is the smallest Lipschitz bound for $L$ , defined as the supremum of its difference quotients. These are the objects and standing conventions of Sections 1.1, 2, and 3.2 of the preprint.

Formalization targets

The goal is Theorem 3.11(ii): for every fixed $w$ , the worst-case expected loss equals the value of program (18),

\sup_{Q\in\mathbb B_\rho(\hat P_N)}\mathbb E^Q[L(y\langle w,x\rangle)] = \inf_{\lambda,s}\left\{\lambda\rho+\frac1N\sum_{i=1}^{N}s_i: \begin{array}{l} L(\hat y_i\langle w,\hat x_i\rangle)\le s_i,\\ L(-\hat y_i\langle w,\hat x_i\rangle)-\kappa\lambda\le s_i\quad(i=1,\ldots,N),\\ \operatorname{lip}(L)\|w\|_*\le\lambda \end{array}\right\}.

Taking the infimum of both sides over $w$ gives equality between problem (4) and the full program (18). The fixed- $w$ identity is stated explicitly because it is the stronger assertion established in the proof. The milestone list follows the paper's appendix: Lemma A.1 converts the robust expectation into a scalar transport-price infimum; the label split isolates the two possible labels; Lemma A.3 evaluates the feature supremum; the ensuing display gives the program in terms of conjugate slopes; and the final identity identifies those slopes with $\operatorname{lip}(L)$ . Each milestone is tied to its printed page in the preprint.

Significance

The theorem replaces a search over probability distributions by a finite convex program with one scalar $\lambda$ and one slack variable $s_i$ per sample, in addition to the classifier $w$ . It makes the price of label changes explicit through $\kappa\lambda$ and the price of feature changes through $\operatorname{lip}(L)\|w\|_*$ . Consequently, different convex Lipschitz classification losses can be treated within the same transport model rather than by deriving a new distributional optimization problem for each loss. The preprint applies the result to hinge, smoothed hinge, and logloss examples in the paragraphs following Theorem 3.11.

The result is proved in the cited paper. Its general statement and the paper's appendix milestones are not yet machine-checked in this mission. A verified development would connect the published definitions of the Wasserstein ball and Lipschitz modulus to an exact program-value theorem that later work can import. The loss-specific proved statements already available on Prove2Me provide useful comparisons, but their narrower losses or label-preserving transport costs do not replace this theorem.

Difficulty

The visible finite constraints contain no probability distributions, while the left-hand side optimizes over all probability measures in a Wasserstein ball. Establishing that the two values coincide requires control of the transport budget and the inner worst-case loss at the same time. The straightforward comparison of feasible values gives only one inequality. It does not show that every improvement allowed by a distribution can be represented by the scalar transport price and the samplewise constraints. The case $\rho=0$ also needs care: Lemma A.1's strong-duality proof cites a result for strictly positive radius, whereas the theorem itself allows zero radius and states an infimum rather than an attained minimum.

Formalization scope

Lean represents $V$ by an abstract finite-dimensional real normed space with its Borel measurable structure. This retains the arbitrary norm of $\mathbb R^n$ in the paper. Labels are Bool, interpreted as $+1$ and $-1$ ; the published DRLogReg_Reformulation_Core definition supplies their sign map, the metric, Wasserstein distance, ball, and empirical distribution. The published WassersteinDRO_Duality_lipschitzModulus definition supplies the exact Lipschitz modulus. The mission's own definitions add the general loss, the feasible set of (18), the fixed-classifier program value, and the effective domain $\Theta=\{\theta:L^*(\theta)<\infty\}$ of the convex conjugate. The same objects and conventions are used across all milestones.

All potentially infinite robust values are represented in $[0,\infty]$ ; the unbounded feature suprema in Lemma A.3 and the label split use extended reals. The standing loss condition $L\ge0$ comes from the paper's Section 2.1 loss codomain. It also makes the lower Lebesgue integral and the finite program's extended-nonnegative objective faithful. The sample size is positive, $\kappa>0$ , and the goal allows $\rho=0$ . Lemma A.1 separately assumes $\rho>0$ , matching the strong-duality condition used on page 29. The printed extra index $j\in[J]$ in the page-35 classification display is carried in its verbatim milestone provenance but not in the Lean statement, because no $j$ exists in part (ii). The formalization does not obtain the theorem by restricting $L$ to a constant loss, by taking a singleton feature space, or by substituting an arbitrary Lipschitz bound for its modulus. Contributions that establish transport duality, the conjugate-domain slope identity, or the zero-radius case are directly reusable.

Selected references

S. Shafieezadeh-Abadeh, D. Kuhn, and P. Mohajerin Esfahani, Regularization via Mass Transportation, arXiv preprint arXiv:1710.10016v3, 2019. Preprint.
S. Shafieezadeh-Abadeh, P. Mohajerin Esfahani, and D. Kuhn, Distributionally Robust Logistic Regression, Advances in Neural Information Processing Systems 28, 2015. Published Prove2Me formalization.

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Linear Optimization Operations Research Optimization·Captain: mikedeng1

Constrained Assortment Optimization for the Nested Logit Model 4: Under Space Constraints, O(n²) Rounded Knapsack-Relaxation Assortments Include a min{2, 1/(1−ε)}-Approximate Solution for Every u ≥ 0Research Paper

Why space-constrained assortments matter

A retailer choosing which products to display faces a physical limit: different products consume different amounts of shelf space. Under a nested logit choice model, the value of an assortment also depends on how customers substitute among products within a nest and on whether they leave without buying. Gallego and Topaloglu study this combination of choice and feasibility constraints in Constrained Assortment Optimization for the Nested Logit Model. Their authors' manuscript of September 11, 2013 is the numbered source for this mission. Its §5 isolates a finite family of candidate assortments for each nest, despite the per-nest space-constrained problem being a knapsack problem. A small family matters because the paper's earlier Theorem 4 can then combine the families across nests while controlling expected revenue.

The paper contrasts this case with cardinality constraints. When every product consumes one unit, the per-nest optimization can be solved exactly by a short list of candidates. General space requirements turn it into a knapsack problem, so the paper gives quantitative approximation factors instead. The factor improves when no single product occupies much of a nest's capacity. These statements and their dependencies occur in §§5.1–5.2, pp. 19–22 of the manuscript.

Products, nests, and the parameterized problem

There is a finite set of nests $M$ and a nonempty finite product set $N=\{1,\ldots,n\}$ in each nest. In nest $i$ , product $j$ has preference weight $v_{ij}>0$ , real revenue $r_{ij}$ , and space requirement $w_{ij}>0$ . The nest's capacity is $c_i$ , and each product fits by itself: $w_{ij}\le c_i$ . An assortment $S_i\subseteq N$ is feasible when its total space requirement is at most $c_i$ . The empty assortment is feasible and gives zero revenue. The full nested logit model has a no-purchase preference weight $v_0$ and a dissimilarity parameter $\gamma_i\in(0,1]$ for each nest; these parameters are relevant to the cross-nest expected revenue, but the local problem of this mission uses only one nest's product data.

For an assortment $S_i$ , write $V_i(S_i)=\sum_{j\in S_i}v_{ij}$ and $R_i(S_i)=\sum_{j\in S_i}v_{ij}r_{ij}/V_i(S_i)$ , with $R_i(\varnothing)=0$ . The paper's parameterized local problem, problem (7), maximizes $V_i(S_i)(R_i(S_i)-u)$ over feasible assortments for each $u\ge0$ . Since the within-nest no-purchase weight is zero, equation (8) gives its equivalent sum of product utilities:

V_i(S_i)(R_i(S_i)-u)=\sum_{j\in S_i}v_{ij}(r_{ij}-u).

Problem (10) is this objective written as a zero-one knapsack: each product decision is zero or one and $\sum_jw_{ij}x_{ij}\le c_i$ . Its LP relaxation permits $0\le x_{ij}\le1$ . At parameter $u$ , the product's utility-to-space ratio is $f_{ij}(u)=v_{ij}(r_{ij}-u)/w_{ij}$ . Rounding a relaxation solution down retains precisely the products with $x_{ij}=1$ . The candidate family also contains each singleton assortment $\{j\}$ .

Formalization targets

Theorem 6 asserts that a collection $\mathcal A_i$ of at most quadratic size includes a two-approximate solution to problem (7) for every $u\ge0$ . Here an assortment $S$ is $\alpha$ -approximate at $u$ if it is feasible and $\alpha V_i(S)(R_i(S)-u)$ is at least the objective of every feasible assortment. The $O(n^2)$ statement is pinned to the explicit bound $|\mathcal A_i|\le(n+1)^2$ .

\forall u\ge0,\quad \exists S\in\mathcal A_i:\quad \forall T\in C_i,\quad V_i(T)(R_i(T)-u)\le 2V_i(S)(R_i(S)-u).

The §5.2 target keeps the same collection and adds the finer guarantee whenever each product is small relative to capacity. For every $\epsilon\in[0,1)$ such that $w_{ij}\le\epsilon c_i$ for all $j$ , and for every $u\ge0$ , some member of $\mathcal A_i$ satisfies

\forall T\in C_i,\quad V_i(T)(R_i(T)-u)\le\frac{1}{1-\epsilon}V_i(S)(R_i(S)-u).

The two members need not be the same, but the finite family is fixed before $u$ and $\epsilon$ are chosen. Together the claims yield the paper's factor $\min\{2,1/(1-\epsilon)\}$ . The main goal formalizes both parts; Theorem 6 and the paper's intermediate statements are milestones.

What the guarantees supply

The local result converts infinitely many parameter values into a finite search domain. Theorem 4 of the same paper says that if each nest has candidates containing an $\alpha$ -approximate local solution for every $u\ge0$ , then some cross-nest combination obtains at least the optimal expected revenue divided by $\alpha$ . Theorem 2 relates the best combination of candidates to a linear program. Thus the finite collection in this mission supplies the local ingredient needed for the paper's global factor two, or the smaller factor under the product-size condition. These consequences are stated in the paragraphs immediately following Theorem 6 and the §5.2 conclusion.

The paper already proves these mathematical results. This mission asks for machine-checked proofs of its specific knapsack relaxation, rounding inequality, and approximation statements. It also creates reusable finite-knapsack definitions and explicit interfaces between the paper's local objective and the published nested logit model. The goal and milestones here remain open Lean statements until solvers provide proofs; compiling a statement with sorry checks its syntax and types, not its mathematical validity.

Where the argument is delicate

The parameter $u$ ranges over a continuum, yet one family of bounded size must work for all of it. Different values of $u$ change the objective coefficients and may change which products belong in an LP optimum. At ties or sign changes, the optimum need not be unique, so a statement about a particular chosen solution requires care. A second difficulty is the fractional LP coordinate: its value can exceed the rounded set's objective, and the refined factor uses a comparison between its utility and the space already occupied. The displayed chain in §5.2 contains quotients that can have zero denominators; the formal milestone states its endpoint inequality directly. A fractional product with zero utility need not force capacity to be filled, which is why the capacity-consumption milestone states positive utility explicitly.

Formalization scope

The existing NestedLogitVariants.LP.Model supplies the nested logit instance and its $V_i$ and $R_i$ functions. A general finite continuous-knapsack definition is separate from the paper-specific assortment data. Products use Fin n, nests use a finite type, and assortments use finite sets. The empty finite set represents $\bar 0$ . The model's within-nest no-purchase weight is fixed to zero, as in the manuscript's main local formulation. Revenues remain arbitrary real numbers; there is no assumption that they are nonnegative or sorted. The capacity and product space requirements are real numbers, with $w_{ij}>0$ and $w_{ij}\le c_i$ . Positive preference weights and nonempty $N$ are explicit. No condition on $\gamma_i$ or $v_0$ is needed for this local problem (7).

The candidate family is a finite set of rounded LP optima and all singletons. Its size is bounded before $u$ and $\epsilon$ are quantified. Each approximation compares against feasible integer assortments using problem (7), including the empty assortment. The LP optimum itself is never substituted for the integer comparator. General unequal space requirements remain in scope. Contributions that formalize the parametric LP behavior, its one-fractional-coordinate optimum, and the paper's two rounding guarantees fit this mission.

Selected references

Guillermo Gallego and Huseyin Topaloglu, Constrained Assortment Optimization for the Nested Logit Model, Management Science 60(10), 2014, DOI 10.1287/mnsc.2014.1931. Authors' manuscript, September 11, 2013, especially pp. 7–8 and 19–22.

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Combinatorics Graph Theory Theoretical Computer Science·Captain: mikedeng1

Generation and Properties of Snarks 3: A Snark on 26 Vertices All of Whose 2-Factors Consist of Odd Cycles Refutes the Abreu–Labbate–Sheehan ConjectureResearch Paper

Motivation

A cubic graph has chromatic index 3 or 4 (Vizing). Cubic graphs of chromatic index 4 — the uncolourable ones — are where most of the open problems on cubic graphs live: a minimal counterexample to the cycle double cover conjecture or to Tutte's 5-flow conjecture would be among them. Uncolourable graphs with small edge cuts or short cycles reduce to smaller ones, so attention concentrates on snarks, the uncolourable cubic graphs that are cyclically 4-edge connected and have girth at least 5.

Brinkmann, Goedgebeur, Hägglund and Markström (arXiv:1206.6690v3, J. Combin. Theory Ser. B 103, 2013) generated all snarks on up to 36 vertices and tested a series of published conjectures against them. One of these concerns the 2-factors of a snark, its spanning subgraphs in which every vertex has degree 2. A 3-edge-colouring of a cubic graph gives 2-factors made of even cycles, so a snark is a graph in which some 2-factors are forced to contain odd cycles. At the far end are the snarks in which every 2-factor consists of odd cycles only. Abreu, Labbate and Sheehan (European J. Combin. 33, 2012), studying pseudo and strongly pseudo 2-factor isomorphic graphs, conjectured that this happens only for the Petersen graph, the Blanuša-2 snark and the Flower snarks (paper, p. 11, Conjecture 4.11).

Timeline:

2012: Abreu, Labbate and Sheehan state the conjecture.
2012–2013: Brinkmann, Goedgebeur, Hägglund and Markström refute it with a snark on 26 vertices, printed in Appendix 8.1 of their paper (p. 29); by computer search they report it is the smallest counterexample and that one more exists on 34 vertices (Observation 4.12).

Setting

All graphs are finite and simple. A graph $G$ is cubic if every vertex has degree 3. It is colourable if its edges can be coloured with three colours so that edges sharing an endpoint get different colours, and uncolourable otherwise. The girth $g(G)$ is the number of vertices of a shortest cycle. $G$ is cyclically $k$ -edge connected if deleting fewer than $k$ edges never leaves two components that both contain a cycle. A snark is an uncolourable, cyclically 4-edge connected cubic graph with $g(G) \ge 5$ (paper, p. 4).

A 2-factor of $G$ is a spanning 2-regular subgraph $F$ : it uses every vertex, and every vertex has degree exactly 2 in $F$ . Each component of $F$ is a cycle. Say that all 2-factors of $G$ are odd if every component of every 2-factor of $G$ has an odd number of vertices.

The witness is the graph $G_{8.1}$ of Appendix 8.1, given in the paper as a list of higher-numbered neighbours of the vertices $1, \dots, 26$ :

\{2, 3, 4, 5, 6, 7, 8, 9, 10, 7, 9, 8, 10, 11, 12, 13, 14, 15, 16, 15, 17, 18, 19, 18, 20, 18, 21, 22, 21, 23, 23, 24, 22, 24, 25, 26, 26, 25, 26\}.

It has 26 vertices and 39 edges. In Lean it is G81 : SimpleGraph (Fin 26) with paper vertex $i$ as $i-1$ .

Formalization targets

Goal: Observation 4.12, existence half

G_{8.1} \text{ is a snark} \quad\text{and}\quad \text{every 2-factor of } G_{8.1} \text{ consists of odd cycles.}

Because $G_{8.1}$ has 26 vertices, it is not the Petersen graph (10 vertices), not the Blanuša-2 snark (18) and not a Flower snark $J_k$ ( $4k$ vertices). The goal therefore refutes Conjecture 4.11 without formalizing the conjecture's list of exceptions.

Milestones

The goal splits into five properties of the one graph $G_{8.1}$ :

$G_{8.1}$ is cubic;
$g(G_{8.1}) \ge 5$ ;
$G_{8.1}$ is cyclically 4-edge connected;
every 2-factor of $G_{8.1}$ consists of odd cycles;
$G_{8.1}$ is uncolourable.

A supporting statement, not from the paper, asserts that $G_{8.1}$ has a 2-factor, so that milestone 4 is not vacuous.

Significance

The result. Conjecture 4.11 would have characterised a natural extremal class of snarks by a short explicit list. The counterexample shows that the class is not that list: there are snarks outside the Petersen, Blanuša-2 and Flower families in which no 2-factor contains an even cycle. Such graphs are relevant to questions about 2-factor structure, oddness and cycle double covers, where the parity of 2-factor cycles is exactly what the arguments track.

Formalizing it. The result is proved in the paper by exhibiting the graph and checking it by computer; it has not been machine-checked in a proof assistant. A formal proof certifies every property of the printed graph — including that the printed adjacency list decodes to a graph with these properties — and leaves a reusable library of definitions (snarks, cyclic edge connectivity, 2-factors) that other snark statements can import.

Difficulty

Each property is a finite check, but some are large. Cubicity and girth are local. Cyclic 4-edge connectivity quantifies over all sets of at most three of the 39 edges (about ten thousand) and needs, for each, a component analysis of the remaining graph. The odd-2-factor property quantifies over all 2-factors; in a cubic graph these are the complements of the perfect matchings, and $G_{8.1}$ has 56 perfect matchings. The obvious route — decide over all subgraphs — is out of reach: there are $2^{39}$ edge subsets, and the statement is phrased through Mathlib's ConnectedComponent and IsRegularOfDegree, which do not evaluate by kernel computation at this size. Uncolourability has the same problem: the naive search space has $3^{39}$ colourings.

Formalization scope

Conventions committed to:

Vertices are Fin 26, paper vertex $i$ = Lean vertex $i-1$ ; adjacency is the symmetric closure of an explicit edge list G81Edges, so it is decidable.
Colourable means G.lineGraph.Colorable 3 (a proper 3-edge-colouring), never G.Colorable 3.
Girth is Mathlib's egirth : ℕ∞, so "girth at least 5" is 5 ≤ G.egirth.
A 2-factor is a graph F on the same vertex type with F ≤ G and F.IsRegularOfDegree 2 — spanning by construction. Oddness is per component: Odd (Nat.card c.supp) for every component c. It is not the parity of the whole vertex set (26 is even).
Cyclic $k$ -edge connectivity: for every edge set S with S.card < k, any two components of G.deleteEdges S that both contain a cycle are equal.

The odd-2-factor property is vacuous for a graph without 2-factors. The goal is not trivialised this way: $G_{8.1}$ has 2-factors, and a supporting theorem states it.

Out of scope, because they rest on exhaustive computer enumeration: that $G_{8.1}$ is the smallest counterexample, that there is exactly one further counterexample on 34 vertices, Observation 4.10 (all snarks on at most 36 vertices have oddness 2), and Tables 4 and 5. The Petersen graph, the Blanuša-2 snark and the Flower snarks are not defined here; the refutation uses only their orders.

Contributions welcome: proofs of any milestone; general lemmas linking 2-factors of cubic graphs to perfect matchings, and 3-edge-colourings to even 2-factors; decision procedures for cyclic edge connectivity of explicit graphs. The definitions are shared in shape with the other missions of this series and are meant to be reused.

Selected references

G. Brinkmann, J. Goedgebeur, J. Hägglund, K. Markström, Generation and properties of snarks, arXiv:1206.6690v3, 2013; J. Combin. Theory Ser. B 103 (2013). https://arxiv.org/abs/1206.6690v3 , https://doi.org/10.1016/j.jctb.2013.05.001
M. Abreu, D. Labbate, J. Sheehan, Pseudo and strongly pseudo 2-factor isomorphic regular graphs and digraphs, European J. Combin. 33 (2012), no. 8, 1847–1856. https://mathscinet.ams.org/mathscinet-getitem?mr=2950486
G. Brinkmann, K. Coolsaet, J. Goedgebeur, H. Mélot, House of Graphs: a database of interesting graphs, Discrete Appl. Math. 161 (2013), 311–314. https://mathscinet.ams.org/mathscinet-getitem?mr=2973372

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Numerical Analysis Optimization·Captain: mikedeng1

Proximal Gradient Method for Nonsmooth Optimization over the Stiefel Manifold: Every Limit Point of ManPG Is Stationary, and an ε-Stationary Point Is Reached Within ⌈2L(F(X₀)−F*)/(γᾱε²)⌉ IterationsResearch Paper

Motivation

Many problems in statistics and signal processing ask for an orthonormal frame $X\in\mathbb R^{n\times r}$ that is both a good fit to the data and sparse or otherwise structured. Sparse principal component analysis, compressed modes in physics, unsupervised feature selection and sparse blind deconvolution all take the form

\min_{X}\ F(X)=f(X)+h(X)\quad\text{s.t.}\quad X^\top X=I_r,

with $f$ smooth and $h$ convex but nonsmooth, typically an $\ell_1$ penalty. Before the work of Chen, Ma, So and Zhang, algorithms for this class fell into three groups: subgradient-oriented methods, which solve a quadratic program over sampled subgradients in every step; proximal point algorithms, whose subproblems are as hard as the original problem; and operator-splitting methods such as SOC and MADMM. The manifold proximal gradient method (ManPG) carries the Euclidean proximal gradient method over to the Stiefel manifold. The paper shows that ManPG keeps the global convergence and the $O(\varepsilon^{-2})$ iteration complexity that Riemannian gradient descent has for smooth objectives (Boumal, Absil and Cartis, 2019).

Setting

The feasible set is the Stiefel manifold

\mathcal M=\mathrm{St}(n,r)=\{X\in\mathbb R^{n\times r}: X^\top X=I_r\},\qquad r\le n.

The ambient space $\mathbb R^{n\times r}$ carries the Frobenius inner product $\langle A,B\rangle=\operatorname{Tr}(A^\top B)$ and norm $\|A\|_F$ . The tangent space at $X\in\mathcal M$ is $T_X\mathcal M=\{V: V^\top X+X^\top V=0\}$ . The orthogonal projection onto it is $\operatorname{Proj}_{T_X\mathcal M}(Y)=(I_n-XX^\top)Y+\tfrac12X(X^\top Y-Y^\top X)$ .

The standing assumptions on $F=f+h$ are imposed on all of $\mathbb R^{n\times r}$ :

(i) $f$ is differentiable, and its gradient $\nabla f$ is $L$ -Lipschitz;
(ii) $h$ is convex and $L_h$ -Lipschitz.

A point $X\in\mathcal M$ is stationary (Definition 3.3) if

0\in\operatorname{grad}f(X)+\operatorname{Proj}_{T_X\mathcal M}\partial h(X),\qquad \operatorname{grad}f(X)=\operatorname{Proj}_{T_X\mathcal M}\nabla f(X),

where $\partial h$ is the convex subdifferential.

A retraction $\mathrm{Retr}_X(\xi)$ maps tangent vectors at $X$ back to $\mathcal M$ , with $\mathrm{Retr}_X(0)=X$ . On the compact manifold $\mathcal M$ every retraction satisfies, for some $M_1,M_2>0$ and all $X\in\mathcal M$ , $\xi\in T_X\mathcal M$ (Fact 3.6),

\|\mathrm{Retr}_X(\xi)-X\|_F\le M_1\|\xi\|_F,\qquad \|\mathrm{Retr}_X(\xi)-(X+\xi)\|_F\le M_2\|\xi\|_F^2.

ManPG (Algorithm 1) takes $X_0\in\mathcal M$ , $\gamma\in(0,1)$ and $t>0$ . In iteration $k$ it solves the convex subproblem

V_k=\operatorname*{argmin}_{V\in T_{X_k}\mathcal M}\ \langle\nabla f(X_k),V\rangle+\frac1{2t}\|V\|_F^2+h(X_k+V).\tag{4.3}

It then backtracks $\alpha=1,\gamma,\gamma^2,\dots$ until $F(\mathrm{Retr}_{X_k}(\alpha V_k))\le F(X_k)-\alpha\|V_k\|_F^2/(2t)$ , and sets $X_{k+1}=\mathrm{Retr}_{X_k}(\alpha_kV_k)$ .

A point $X\in\mathcal M$ is $\varepsilon$ -stationary (Definition 5.4) if the solution of (4.3) at $X$ with $t=1/L$ has $\|V\|_F\le\varepsilon/L$ . The proof of Lemma 5.2 uses a bound $G$ on $\|\nabla f\|_F$ over $\mathcal M$ and the constants

c_0=M_2G+\frac{LM_1^2}{2},\qquad \bar\alpha=\frac{1}{2(c_0+L_hM_2)t}.

Formalization targets

Goal: Theorem 5.5

For every run of ManPG:

every limit point $\bar X$ of $\{X_k\}$ is stationary;
with $t=1/L$ , for every $\varepsilon>0$ some

k\le\left\lceil\frac{2L\,(F(X_0)-F^*)}{\gamma\bar\alpha\varepsilon^2}\right\rceil

has $\|V_k\|_F\le\varepsilon/L$ and $X_k$ is $\varepsilon$ -stationary.

Here $F^*$ is the optimal value of (1.1). The constant is the paper's, kept exactly.

Milestones, in the order the proof uses them

Lemma 5.1: $g(\alpha V_k)-g(0)\le\frac{(\alpha-2)\alpha}{2t}\|V_k\|_F^2$ for $\alpha\in[0,1]$ , where $g$ is the objective of (4.3).
(5.5): $f(\mathrm{Retr}_{X}(\alpha V))-f(X)\le\alpha\langle\nabla f(X),V\rangle+c_0\alpha^2\|V\|_F^2$ .
Lemma 5.2, split into three parts:
- sufficient decrease for every $0<\alpha\le\min\{1,\bar\alpha\}$ ;
- well-definedness of the line search: a run exists from every $X_0$ ;
- the run form $\gamma\min\{1,\bar\alpha\}\le\alpha_k\le1$ , with $F(X_{k+1})-F(X_k)\le-\frac{\alpha_k}{2t}\|V_k\|_F^2$ .
Lemma 5.3: $V_k=0$ implies $X_k$ is stationary.
Two displays of the proof of Theorem 5.5:
- $\|V_k\|_F^2\to0$ ;
- the telescoped chain $F(X_0)-F^*>\frac{tK\varepsilon^2}{2}\gamma\bar\alpha$ when $\|V_k\|_F>\varepsilon/L$ for all $k<K$ .

Significance

Theorem 5.5 is the paper's main result. It certifies ManPG as a first-order method for nonsmooth problems on the Stiefel manifold with the same $O(\varepsilon^{-2})$ complexity as Riemannian gradient descent in the smooth case. When $h=0$ the bound reduces to the Boumal–Absil–Cartis rate (Remark 5.6). Later proximal methods on matrix manifolds build on this analysis. Lemmas 5.1–5.3 are the reusable steps: descent of the subproblem, sufficient decrease after retraction, and the stopping test as a stationarity certificate.

The result is proved in the paper; it is not formalized anywhere. No proof assistant library contains the convergence theory of a proximal method on a matrix manifold. The mission produces a machine-checked version of the full chain from the subproblem to the iteration bound. Along the way it builds infrastructure that Mathlib lacks: the Frobenius geometry of $\mathbb R^{n\times r}$ as a trace formula, the tangent space and projection of the Stiefel manifold, and optimality conditions for convex problems restricted to a subspace.

Difficulty

The bookkeeping inequalities (Lemma 5.1, (5.5), the decrease estimate, the telescoping) are short computations once the right facts are available. The difficulty lies elsewhere.

Lemma 5.3 and the limit-point claim need the optimality condition of the subproblem (4.3): a subdifferential sum rule for a convex function restricted to the linear subspace $T_X\mathcal M$ , followed by the projection formula. The paper cites this condition rather than proving it.
The limit-point claim also needs a closedness argument. Subgradients of the Lipschitz function $h$ are bounded by $L_h$ , so along a convergent subsequence they have a limit point that is again a subgradient. The projection $X\mapsto\operatorname{Proj}_{T_X\mathcal M}$ must be shown continuous. The paper's proof leaves both steps implicit.
The well-definedness of the line search requires existence of a minimizer of (4.3), by strong convexity and coercivity on a subspace.
The descent lemma for $f$ has to be derived from the derivative along lines and the Lipschitz gradient.

A natural first idea is to restate everything for the Riemannian gradient and reuse smooth Riemannian results. That route does not work: $h$ is nonsmooth, and the argument runs in the ambient space through the Euclidean subdifferential.

Formalization scope

Representation.
- Matrices are Matrix (Fin n) (Fin r) ℝ.
- The inner product is trace (Aᵀ * B) and the norm is its square root. No Mathlib matrix norm instance is used.
- $\mathrm{St}(n,r)$ is the published definition ProjLikeRetr.Stiefel.stiefel.
Assumptions.
- $f$ comes with a gradient map gradf, tied to it by derivatives along every line.
- $\nabla f$ is $L$ -Lipschitz in the Frobenius norm, with $L>0$ .
- $h$ is convex and $L_h$ -Lipschitz on the whole ambient space.
- $r\le n$ .
Retraction. The retraction is any map sending tangent vectors at points of $\mathcal M$ into $\mathcal M$ , with $\mathrm{Retr}_X(0)=X$ and the bounds (3.1)–(3.2). By Fact 3.6 this covers every retraction; smoothness is never used.
Constants.
- $G$ , $M_1$ , $M_2$ , $L$ , $L_h$ are parameters, each with its defining hypothesis.
- $c_0$ and $\bar\alpha$ are the explicit formulas above.
- $F^*$ is an attained lower bound of $F$ on $\mathcal M$ .
The run. A run of Algorithm 1 is a predicate on sequences $(X_k,V_k,\alpha_k)$ . Each $V_k$ minimizes (4.3), $\alpha_k=\gamma^j$ for the least passing $j$ , and $X_{k+1}=\mathrm{Retr}_{X_k}(\alpha_kV_k)$ .
The goal's conventions.
- A limit point is a cluster point of the sequence.
- "Returns an $\varepsilon$ -stationary point within $N$ iterations" means some $k\le N$ passes the stopping test $\|V_k\|_F\le\varepsilon/L$ , with $N=\lceil\cdot\rceil$ as Nat.ceil.

Two encodings would trivialize the theorem, and the mission excludes both.

An unsatisfiable run predicate would make the goal vacuous. The well-definedness milestone asserts that a run exists from every $X_0\in\mathcal M$ .
A projection or stationarity condition that holds trivially would make the first claim empty. Stationarity uses the explicit projection formula and the convex subdifferential on all of $\mathbb R^{n\times r}$ .

Lemma 5.2 as printed fixes $\alpha\le\min\{1,\bar\alpha\}$ and asserts the decrease for the accepted iterate. What its proof establishes is three separate facts: decrease at every trial stepsize $\alpha\le\min\{1,\bar\alpha\}$ , termination of the line search, and $\alpha_k\ge\gamma\min\{1,\bar\alpha\}$ along a run. The mission states these three facts.

Contributions are welcome at every level: proofs of the milestones, and general lemmas such as the descent lemma for $C^{1,1}$ functions on $\mathbb R^{n\times r}$ , the optimality condition of a strongly convex problem on a subspace, and properties of the Stiefel tangent projection. Theorem numbers and pages refer to arXiv:1811.00980v2, the version read for this mission.

Selected references

S. Chen, S. Ma, A. M.-C. So, T. Zhang, Proximal gradient method for nonsmooth optimization over the Stiefel manifold, SIAM Journal on Optimization 30(1), 2020. https://doi.org/10.1137/18m122457x (arXiv:1811.00980v2: https://arxiv.org/abs/1811.00980)
N. Boumal, P.-A. Absil, C. Cartis, Global rates of convergence for nonconvex optimization on manifolds, IMA Journal of Numerical Analysis 39(1), 2019. https://doi.org/10.1093/imanum/drx080
P.-A. Absil, R. Mahony, R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, 2008. https://doi.org/10.1515/9781400830244

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Machine Learning Optimal Transport Probability+1·Captain: mikedeng1

Minimax Statistical Learning with Wasserstein Distances I: Data-Dependent High-Probability Bound on the Local Worst-Case RiskResearch Paper

Motivation

Training data describe one probability distribution, while a deployed prediction rule may be evaluated under a nearby distribution. Distributionally robust learning measures a rule by its worst expected loss over a specified set of nearby laws. This changes the question from how well the rule fits the observed sample to how well it performs when probability mass can move. Wasserstein distance supplies a geometric notion of “nearby”: moving mass farther across the instance space costs more. The setting is relevant to the domain drift discussed by Lee and Raginsky and to Wasserstein robust optimization as developed by Mohajerin Esfahani and Kuhn.

Theorem 1 of Lee and Raginsky's 2018 paper gives a finite-sample answer. It controls the local worst-case risk at the true population law using quantities computed from the empirical law, and also controls the empirical local worst-case risk using population quantities. Both assertions are simultaneous over the entire loss class. This mission concerns exactly those two probability inequalities and the displayed claims in Appendix C.1 that support them. The source is the pinned arXiv:1705.07815v2 preprint, containing the NeurIPS 2018 paper and its supplementary appendix.

Setting

The instance space $\mathcal Z$ is a bounded Polish metric space with distance $d_{\mathcal Z}$ . A probability measure $P$ generates $n\ge1$ independent observations $Z_1,\ldots,Z_n$ . Their empirical distribution is $P_n=n^{-1}\sum_{i=1}^n\delta_{Z_i}$ . A loss $f:\mathcal Z\to\mathbb R$ has risk $R(Q,f)=\int f\,dQ$ under a probability law $Q$ . The class $\mathcal F$ consists of upper semicontinuous losses satisfying $0\le f(z)\le M$ for every $f\in\mathcal F$ and $z\in\mathcal Z$ , as in Assumption 2 of the paper.

For $p\ge1$ , the $p$ -Wasserstein distance $W_p(Q,P)$ is the least $p$ th-power transport cost over couplings of $Q$ and $P$ , raised to the power $1/p$ . The Wasserstein ball $B^W_{\varrho,p}(P)$ consists of probability laws $Q$ with $W_p(Q,P)\le\varrho$ , where $\varrho>0$ . The local worst-case risk is

R_{\varrho,p}(P,f)=\sup_{Q\in B^W_{\varrho,p}(P)}R(Q,f).

The dual expression uses the envelope $\varphi_{\lambda,f}(z)=\sup_{z'\in\mathcal Z}\{f(z')-\lambda d_{\mathcal Z}(z,z')^p\}$ for $\lambda\ge0$ . To measure the size of $\mathcal F$ , let $\mathcal N(\mathcal F,\|\cdot\|_\infty,u)$ be its uniform-metric covering number at radius $u$ . The entropy integral is $\mathfrak C(\mathcal F)=\int_0^\infty\sqrt{\log\mathcal N(\mathcal F,\|\cdot\|_\infty,u)}\,du$ . These symbols match the local Lean setting and the source's notation.

Formalization targets

Theorem 1: two data-dependent risk bounds

For every $t>0$ , the mission goal is the conjunction of the paper's two displays:

\mathbb P\!\left\{\exists f\in\mathcal F:\ R_{\varrho,p}(P,f)>\inf_{\lambda\ge0}\left[(\lambda+1)\varrho^p+\mathbb E_{P_n}\varphi_{\lambda,f}+\frac{M\sqrt{\log(\lambda+1)}}{\sqrt n}\right]+\frac{24\mathfrak C(\mathcal F)+Mt}{\sqrt n}\right\}\le 2e^{-2t^2},

\mathbb P\!\left\{\exists f\in\mathcal F:\ R_{\varrho,p}(P_n,f)>\inf_{\lambda\ge0}\left[(\lambda+1)\varrho^p+\mathbb E_P\varphi_{\lambda,f}+\frac{M\sqrt{\log(\lambda+1)}}{\sqrt n}\right]+\frac{24\mathfrak C(\mathcal F)+Mt}{\sqrt n}\right\}\le 2e^{-2t^2}.

The seven milestones preserve displayed claims from Appendix C.1: the deterministic comparison through the supremum deviation $X_\lambda$ , its concentration and symmetrization bounds, the entropy-integral estimate with constant $24$ , the fixed- $\lambda$ probability bound, and the integer-grid union and comparison with real multipliers. They are statements from the author's argument, ordered by their appearance. The goal retains both directions of the published theorem and every printed constant.

Significance

The first bound provides a sample-dependent certificate for the worst-case population loss of every member of $\mathcal F$ at once. The second compares the robust empirical quantity with a population envelope. Together they make it possible to assess what a local Wasserstein ambiguity set buys in finite samples, rather than relying only on asymptotic convergence. The complexity term depends on the geometry of the loss class through its covering numbers, following a line of empirical-process bounds also seen in Bartlett and Mendelson's study of Rademacher complexity. Lee and Raginsky emphasize that this class complexity can reflect hypotheses outside the support of the data-generating law.

The mathematical theorem is proved in the 2018 paper; the particular declarations in this mission remain open Lean goals. A complete formal development would connect probability inequalities for supremum processes to the published Wasserstein distance and ball, make the entropy integral and its measurability consequences explicit, and verify both directions of the final bound. The reusable output includes a precise internal uniform covering number and a finite-sample Rademacher process for metric-space loss classes.

Difficulty

A fixed multiplier $\lambda$ is easier to control than the infimum over all $\lambda\ge0$ . A pointwise concentration bound alone does not produce a simultaneous statement over an uncountable multiplier set. The source's final formula contains a logarithmic multiplier penalty, and retaining its coefficient $M$ and the probability factor $2$ requires accounting for that parameter range exactly. Independently, the supremum over an arbitrary loss class is a genuine random variable only with enough regularity. The boundedness of individual losses does not by itself settle the measurability of their unrestricted supremum.

The envelope $\varphi_{\lambda,f}$ also mixes the metric geometry with empirical-process complexity. Estimating a class of envelopes must remain tied to the covering properties of the original class $\mathcal F$ . Replacing the envelope class by a finite hand-picked subclass or assuming the desired concentration inequality as a new hypothesis would change the theorem being formalized.

Formalization scope

Lean uses a metric space with its Borel measurable structure and a Polish-space instance, together with boundedness of the whole carrier. The Wasserstein distance and ball are imported from the published WassersteinLinOpt.Ball definitions. Because the carrier is bounded, every probability measure has a finite $p$ th moment, matching the source's $\mathcal P_p(\mathcal Z)$ . The ball is a subtype of probability measures; its supremum is taken only over feasible laws. Sample outcomes are functions Fin n → 𝒵, and the product measure gives their independent common law. The empirical law is a normalized finite sum of Dirac masses, while $\mathbb E_{P_n}$ is written as its equal finite average.

The paper does not specify whether covering centres must lie in $\mathcal F$ . This formalization uses the internal covering number, whose centres do; its value is at least the external covering number. The entropy integral is extended nonnegative and is required to be finite in statements that convert it to a real number. This disclosed restriction avoids the default conversion of $+\infty$ to zero and supports the separability needed for expected suprema. Milestones using $X_\lambda$ also require $\mathcal F$ to be nonempty, so a real supremum never acquires the empty-family default value. The empty-class case of Theorem 1 remains valid because its failure events are empty.

The source writes minima over nonnegative real multipliers and positive integers. The Lean statements use bounded-below real infima; for the final bound, the appendix's comparison holds at every multiplier, so this reading is at least as strong as the printed minimum. All radii are positive, $p\ge1$ , $n\ge1$ , and $t>0$ where it appears. Failure probabilities use outer measure of the sample event directly, with no conversion through toReal. The goal never quantifies over arbitrary distributions outside the Wasserstein ball, and expectations of supremum processes are guarded by the finite-entropy hypothesis. Contributions are welcome on the concentration, symmetrization, covering-number, and transport-risk interfaces as well as the goal itself.

Selected references

Jaeho Lee and Maxim Raginsky, Minimax Statistical Learning with Wasserstein Distances, Advances in Neural Information Processing Systems 31, 2018. arXiv:1705.07815v2.
Peyman Mohajerin Esfahani and Daniel Kuhn, Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations, Mathematical Programming 171, 2018. DOI:10.1007/s10107-017-1172-1.
Peter L. Bartlett and Shahar Mendelson, Rademacher and Gaussian Complexities: Risk Bounds and Structural Results, Journal of Machine Learning Research 3, 2002. JMLR paper.

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Optimization Statistics·Captain: mikedeng1

Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms I: Cyclic Coordinate Descent with Spacer Steps Converges to a Coordinate-Wise MinimumResearch Paper

Motivation

Best subset selection asks for the linear model that fits a response best using only a few of many candidate predictors. In its penalized form it is the least-squares problem with an $L_0$ penalty, the number of nonzero coefficients. The problem is NP-hard. Mixed-integer optimization can solve it to certified optimality at moderate sizes (Bertsimas, King, Mazumder 2016), but statisticians who fit whole regularization paths with tens of thousands of predictors need something faster. Hazimeh and Mazumder (arXiv:1803.01454, Operations Research 2020) proposed cyclic coordinate descent for the $L_0$ -penalized problem, the method that glmnet made standard for the lasso. Their algorithm underlies the L0Learn software package.

Coordinate descent on a convex smooth-plus-separable objective converges by classical arguments (Tseng 2001). Those arguments do not cover the $L_0$ penalty, which is discontinuous. The paper adds occasional spacer steps to plain cyclic coordinate descent and proves that the modified algorithm converges to a coordinate-wise minimum. This mission formalizes that convergence theorem.

Setting

Let $X\in\mathbb R^{n\times p}$ be a model matrix whose columns $X_1,\dots,X_p$ have unit Euclidean norm, and let $y\in\mathbb R^n$ . For $\beta\in\mathbb R^p$ let

f(\beta)=\tfrac12\|y-X\beta\|^2+\lambda_1\|\beta\|_1+\lambda_2\|\beta\|_2^2,\qquad F(\beta)=f(\beta)+\lambda_0\|\beta\|_0,

with $\lambda_0>0$ and $\lambda_1,\lambda_2\ge0$ ; $\|\beta\|_0$ counts the nonzero coordinates. Problem (2) is $\min_\beta F(\beta)$ . Its three named cases are (L0) ( $\lambda_1=\lambda_2=0$ ), (L0L1) ( $\lambda_1>0=\lambda_2$ ) and (L0L2) ( $\lambda_1=0<\lambda_2$ ). $\mathrm{Supp}(\beta)=\{i:\beta_i\ne0\}$ .

Fix all coordinates but the $i$ -th. With unit-norm columns, minimizing $F$ over $\beta_i$ amounts to minimizing a scalar function of $\tilde\beta_i=\langle y-\sum_{j\ne i}X_j\beta_j,X_i\rangle$ . The minimizer set is $\tilde T(\tilde\beta_i,\lambda_0,\lambda_1,\lambda_2)$ (7). The thresholding operator $T$ (12) picks one element of it:

T(a,\lambda_0,\lambda_1,\lambda_2)=\mathrm{sign}(a)\frac{|a|-\lambda_1}{1+2\lambda_2}\ \text{ if }\ \frac{|a|-\lambda_1}{1+2\lambda_2}\ge\sqrt{\frac{2\lambda_0}{1+2\lambda_2}},\qquad 0\ \text{ otherwise.}

A coordinate-wise (CW) minimum is a $\beta$ such that every $\beta_i$ minimizes $F$ in coordinate $i$ with the others fixed (Definition 2). A stationary solution is a $\beta$ whose lower directional derivative $F'(\beta;d)=\liminf_{\alpha\downarrow0}(F(\beta+\alpha d)-F(\beta))/\alpha$ is nonnegative in every direction $d$ (Definition 1).

Algorithm 1 (CDSS) takes an initial point $\beta^0$ and a positive integer $C$ . It sweeps the coordinates cyclically. Each non-spacer step sets the current coordinate to $T(\tilde\beta_i,\lambda_0,\lambda_1,\lambda_2)$ and increments a counter for the resulting support. When the counter of a support $S$ reaches $Cp$ , a spacer step follows: one sequential pass over the coordinates of $S$ with the $L_0$ term switched off, $\beta_i\leftarrow T(\tilde\beta_i,0,\lambda_1,\lambda_2)$ . The counter of $S$ is then reset. The iterates $\beta^k$ count both kinds of step.

For (L0) and (L0L1) the paper adds Assumption 1: every $\min\{n,p\}$ columns of $X$ are linearly independent. It also adds Assumption 2, which constrains the initial point when $p>n$ : $F(\beta^0)\le\lambda_0 n$ for (L0), and $F(\beta^0)\le\min_\beta f(\beta)+\lambda_0 n$ for (L0L1).

Formalization targets

Goal: Theorem 2 (p. 12)

For (L0), (L0L1) and (L0L2), under Assumptions 1–2 in the first two cases, there are $m$ and a support $S$ with

\mathrm{Supp}(\beta^k)=S\ \ (k\ge m),\qquad \beta^k\to B,\qquad B\text{ a CW minimum},\qquad \mathrm{Supp}(B)=S .

The two parts share $S$ : the support stabilizes, and the limit is a CW minimum on that support.

Milestones

Lemma 5, monotone descent: $F(\beta^{k+1})\le F(\beta^k)$ and $F(\beta^k)\downarrow F^*\ge0$ .
Lemma 6: $\|\beta^k\|_0\le\min\{n,p\}$ for (L0), (L0L1).
Lemma 8: $\{\beta^k\}$ is bounded.
Lemma 1: stationarity is equivalent to $\nabla_S f(\beta)=0$ on the support $S$ .
Lemma 9: along a support $S$ recurring in non-spacer steps, the spacer iterates eventually keep support $S$ . Every subsequence with support $S$ converges to the unique minimizer $\beta^*$ of $f$ on $S$ , and $|\beta^*_j|\ge\sqrt{2\lambda_0/(1+2\lambda_2)}$ on $S$ .
Lemma 10: the support of every limit point occurs infinitely often.
Lemma 2: closed form of $\tilde T$ .
Lemma 11: for two limit points whose supports differ by one index $j$ , $|B^{(2)}_j|$ exceeds $\sqrt{2\lambda_0/(1+2\lambda_2)}$ , or equals it, according to whether $X_j$ is correlated with the smaller support.
Lemma 12: zeroing a nonzero coordinate decreases $F$ by at least $\frac{1+2\lambda_2}{2}\big(|\beta^k_j|-\sqrt{2\lambda_0/(1+2\lambda_2)}\big)^2$ .
Lemma 3: the coordinatewise characterization (8) of CW minima, with the condition $|\tilde\beta_i|>\lambda_1$ of (5) on the support (see Formalization scope).

Significance

Theorem 2 guarantees that exact cyclic coordinate descent on Problem (2), with spacer steps, has a limit, and that the limit is a CW minimum. That class is strictly smaller than the stationary solutions and the fixed points of iterative hard thresholding ((4), p. 5). It also serves as the inner solver of the paper's local combinatorial search (Algorithm 2, Theorem 4). So the guarantee transfers to the partial-swap-inescapable minima studied in the companion mission. Support stabilization (part 1) is also what reduces the asymptotic analysis to smooth strongly convex coordinate descent on a fixed support (Theorem 3).

The result is proved in the paper; no machine-checked version exists. A formalization would check the convergence argument for a discontinuous objective, where the bookkeeping of supports, counters and step indices is easy to get wrong. It would also provide a reusable Lean model of $L_0$ -penalized least squares and its thresholding operators.

Difficulty

The standard proof for cyclic coordinate descent needs a uniform decrease per step or a continuous objective. Neither holds here. A coordinate may switch between zero and a nonzero value infinitely often while $F$ decreases by arbitrarily small amounts, so the descent property alone gives neither support stabilization nor convergence of the iterates. Coordinates dropped infinitely often and coordinates added infinitely often must both be excluded. The tie-breaking rule in $T$ matters here: with the other choice on ties the algorithm is different. The exclusion of coordinates added infinitely often is only sketched in the paper (p. 43), and a complete proof has to supply it.

Formalization scope

Vectors are Fin p → ℝ with 0-based indices. Every norm is written as an explicit sum; Mathlib's sup norm on function types is never used. Unit-norm columns, $\lambda_0>0$ , $\lambda_1,\lambda_2\ge0$ and $p\ge1$ are hypotheses. The case $\lambda_1,\lambda_2>0$ , which the paper never names, is excluded from the statements about the algorithm's limits. $\mathrm{sign}$ is Real.sign. $T$ chooses the nonzero solution on ties, as in (12). The lower directional derivative takes values in EReal, since it is $+\infty$ in directions leaving the support. Lemma 3 is stated with the condition $|\tilde\beta_i|>\lambda_1$ on the support, taken from (5). The printed condition (8) omits it, and without it the equivalence fails for $\lambda_1>0$ . Indices and pages throughout are those of arXiv:1803.01454v3.

Algorithm 1 is a deterministic state machine: iterate, cyclic pointer, the counter array Count, and a flag for a pending spacer step. The iterates are therefore defined from $\beta^0$ and $C$ , not quantified over. The pointer advances only on non-spacer steps, following the pseudocode box; the prose formula $i=(k+1)\bmod(p+1)$ on p. 10 contradicts it. A statement about an arbitrary sequence satisfying a step predicate, or one that adds a step size or a sufficient-decrease hypothesis, would describe a different algorithm and is ruled out.

Useful infrastructure includes the scalar thresholding lemmas and coercivity of $f$ under Assumption 1 on supports of size at most $\min\{n,p\}$ . Contributions of proofs for any milestone are welcome, as are lemmas on the state machine (descent of each step type, the counter bookkeeping).

Selected references

H. Hazimeh, R. Mazumder, Fast Best Subset Selection: Coordinate Descent and Local Combinatorial Optimization Algorithms, Operations Research 68(5), 2020; arXiv:1803.01454v3. https://arxiv.org/abs/1803.01454
D. Bertsimas, A. King, R. Mazumder, Best subset selection via a modern optimization lens, Annals of Statistics 44(2), 2016. https://doi.org/10.1214/15-AOS1388
P. Tseng, Convergence of a block coordinate descent method for nondifferentiable minimization, J. Optim. Theory Appl. 109, 2001. https://doi.org/10.1023/A:1017501703105
A. Beck, Y. C. Eldar, Sparsity constrained nonlinear optimization: optimality conditions and algorithms, SIAM J. Optim. 23(3), 2013. https://doi.org/10.1137/120869778
J. Friedman, T. Hastie, R. Tibshirani, Regularization paths for generalized linear models via coordinate descent, J. Stat. Softw. 33(1), 2010. https://doi.org/10.18637/jss.v033.i01

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Dynamical Systems Machine Learning Optimization·Captain: mikedeng1

Convergence and Dynamical Behavior of the ADAM Algorithm for Nonconvex Stochastic Optimization 3: Under a Łojasiewicz Condition the Adam ODE Converges to One Critical Point at Rate t^(−θ/(1−2θ))Research Paper

Motivation

Adam (Kingma and Ba, 2015) is the default optimizer for training neural networks. At each step it keeps an exponential moving average $m$ of stochastic gradients and an exponential moving average $v$ of their squared coordinates, corrects both for their initialization at zero (the bias correction), and moves the iterate by $-\gamma\,\hat m/(\varepsilon+\sqrt{\hat v})$ coordinatewise.

Barakat and Bianchi (arXiv:1810.02263v4, SIAM J. Math. Data Sci. 2021) study Adam through the ODE method: as the step size $\gamma\to0$ with $1-\alpha\sim a\gamma$ and $1-\beta\sim b\gamma$ , the interpolated iterates follow a non-autonomous ordinary differential equation. Their Theorem 3.2 shows that the trajectory of this ODE approaches the critical set of the objective. That leaves open whether the trajectory settles at a single critical point, and how fast. For gradient-like systems the classical answer goes through the Łojasiewicz gradient inequality (Łojasiewicz, 1963, for real-analytic functions), and the dissipative-systems framework of Haraux and Jendoubi (2015) turns such an inequality into convergence with a rate. This mission formalizes the paper's Theorem 3.4, which brings that conclusion to the continuous-time Adam dynamics.

Timeline.

1963: Łojasiewicz proves the gradient inequality for real-analytic functions and derives convergence of bounded gradient-flow trajectories.
2009: Attouch and Bolte show convergence of the proximal algorithm under the Łojasiewicz property, with rates according to the exponent.
2014: Bolte, Sabach and Teboulle prove the same kind of result for proximal alternating linearized minimization.
2015: Haraux and Jendoubi collect the continuous-time theory for dissipative autonomous systems.
2018–2021: Barakat and Bianchi prove the result for the Adam ODE, which is non-autonomous and has a singular field at $t=0$ .

Setting

Let $F:\mathbb R^d\to\mathbb R$ be continuously differentiable with locally Lipschitz gradient $\nabla F$ , and let $S:\mathbb R^d\to\mathbb R^d$ be locally Lipschitz. In the paper $F(x)=\mathbb E f(x,\xi)$ and $S(x)=\mathbb E\,\nabla f(x,\xi)^{\odot2}$ . Its §7 uses only the two properties above, and so does this mission. Assume $F$ is coercive ( $F(x)\to+\infty$ as $\|x\|\to\infty$ ) and $S(x)>0$ coordinatewise for every $x$ . Fix constants $a,b,\varepsilon>0$ with $b\le 4a$ .

The state is $z=(x,m,v)\in\mathcal Z=\mathbb R^d\times\mathbb R^d\times\mathbb R^d$ , and all vector operations are coordinatewise. The Adam field (3.3) is, for $t>0$ ,

h(t,z)=\Big(-\frac{(1-e^{-at})^{-1}m}{\varepsilon+\sqrt{(1-e^{-bt})^{-1}v}},\ a(\nabla F(x)-m),\ b(S(x)-v)\Big),

and (ODE) is $\dot z(t)=h(t,z(t))$ . A global solution with initial condition $(x_0,0,0)$ is a continuous $z:[0,+\infty)\to\mathbb R^d\times\mathbb R^d\times[0,+\infty)^d$ that is continuously differentiable on $(0,+\infty)$ , satisfies (ODE) for every $t>0$ , and has $z(0)=(x_0,0,0)$ . The critical set is $\mathcal S=\{x:\nabla F(x)=0\}$ .

A number $\theta$ is a Łojasiewicz exponent of $F$ at $x^*$ if there are $c>0$ and $\sigma>0$ with

\|\nabla F(x)\|\ \ge\ c\,|F(x)-F(x^*)|^{1-\theta}\qquad\text{whenever }\|x-x^*\|\le\sigma. \tag{3.7}

Assumption 3.3 (the Łojasiewicz property) requires an exponent $\theta\in(0,\tfrac12]$ at every $x^*\in\mathcal S$ . It holds for real-analytic and semialgebraic $F$ .

The proof uses the Lyapunov function $V(t,z)=F(x)+\tfrac12\sum_i m_i^2/U_i(t,v)$ , where $U_i(t,v)=a(1-e^{-at})(\varepsilon+\sqrt{v_i/(1-e^{-bt})})$ (3.4)–(3.5). It also uses its perturbation $\tilde W_\delta(t,z)=V(t,z)-\delta\langle\nabla F(x),m\rangle+\delta\|S(x)-v\|^2$ (7.11) and $w_\delta(t)=\tilde W_\delta(t,z(t))$ .

Formalization targets

Goal: Theorem 3.4

Assume Assumption 3.3 and that $F(\mathcal S)$ has empty interior in $\mathbb R$ . Let $z=(x,m,v)$ be a global solution with initial condition $(x_0,0,0)$ . Then there is $x^*\in\mathcal S$ with $x(t)\to x^*$ as $t\to\infty$ . Moreover, if $\theta\in(0,\tfrac12]$ is a Łojasiewicz exponent of $F$ at $x^*$ , then

\|x(t)-x^*\|\le C\,t^{-\frac{\theta}{1-2\theta}}\quad(t>0)\ \text{ if }0<\theta<\tfrac12,\qquad \|x(t)-x^*\|\le C\,e^{-\delta t}\quad(t\ge0)\ \text{ if }\theta=\tfrac12,

for a constant $C>0$ (and, in the second case, some $\delta>0$ ) depending on the trajectory, $x^*$ and $\theta$ but not on $t$ .

Milestones

The milestones follow the proof in §7.4:

the decrease of $V$ along the non-autonomous field (Lemma 7.5, $V$ part);
convergence to the critical set (Theorem 3.2, restated here);
the upper bound (7.12) on $w_\delta$ by the four residual terms $|F(x)|$ , $\|m\|^2$ , $\|\nabla F(x)\|^2$ , $\|S(x)-v\|^2$ ;
the absolute continuity of $w_\delta$ and the derivative bound (7.13);
the differential inequality $\dot w_\delta\le-c_4\,w_\delta^{2(1-\theta)}$ of step iv), after the limit value of $F(x(t))$ is normalised to $0$ .

Significance

Theorem 3.2 only places the limit points of the trajectory in $\mathcal S$ . When $\mathcal S$ contains a continuum of critical points, the trajectory could in principle drift along it forever. Theorem 3.4 excludes this under the Łojasiewicz property. It shows that continuous-time Adam has finite length and converges to one critical point, and it makes the speed explicit. The speed is polynomial with exponent $\theta/(1-2\theta)$ in general and exponential when $\theta=\tfrac12$ , which holds for instance at a nondegenerate critical point. The theorem places continuous-time Adam in the same class as gradient flow, heavy-ball dynamics and proximal methods.

The result is proved in the paper (§7.4, with the final integration delegated to arguments of Haraux and Jendoubi). It has not, to current knowledge, been machine-checked. The mission produces a formal proof, together with formal statements of the Lyapunov estimates (7.12)–(7.13) for the Adam ODE. The Łojasiewicz-rate step is reusable for other perturbed gradient-like systems.

Difficulty

The standard Łojasiewicz argument for gradient flow uses $F$ itself as the Lyapunov function, whose derivative is $-\|\nabla F\|^2$ . For the Adam ODE neither $F$ nor $V$ works. $V$ decreases only at the rate $\|m\|^2$ , which says nothing about $\|\nabla F(x)\|$ when the momentum is small. The system is also non-autonomous, with a field that is singular at $t=0$ and only locally Lipschitz in $z$ . As a result $w_\delta$ is only absolutely continuous, so all derivative inequalities hold only almost everywhere. A further obstacle is that the Łojasiewicz inequality at a single critical point does not by itself control the trajectory: the limit set is a connected compact subset of $\mathcal S$ , and the inequality has to be made uniform on a neighbourhood of it. Finally, the rate has to be stated for an arbitrary exponent at the limit point, not only for the uniform exponent the argument produces.

Formalization scope

$\mathbb R^d$ is EuclideanSpace ℝ (Fin d) and $\mathcal Z$ is the product E × E × E, whose Mathlib norm is the maximum of the three Euclidean norms. This does not change any statement here, which concern $x$ , $m$ , $v$ separately. $F$ and $S$ are abstract, under the hypotheses of §7 (ContDiff ℝ 1 F, LocallyLipschitz (gradient F), LocallyLipschitz S, $S>0$ coordinatewise, coercivity as Tendsto F (cocompact _) atTop). Under Assumption 2.2 the paper's $F$ and $S$ satisfy these, so the statements imply the printed ones. A solution is a map ℝ → E × E × E constrained only on $[0,\infty)$ . The field uses $|v|$ in place of $v$ (the paper's extension, p. 12) and is never evaluated at $t=0$ . Statements are made for every global solution; there is exactly one (Theorem 3.1, a separate mission).

The polynomial bound is stated for $t>0$ , because Lean's real power gives $0^{-r}=0$ while the paper's right-hand side is $+\infty$ at $t=0$ . The constant $C$ is quantified before $t$ . A version with $C$ chosen after $t$ , or one that assumes $x(t)$ converges, would be trivial and is not the theorem. (7.13) states absolute continuity of $w_\delta$ on every $[1,T]$ together with the almost-everywhere derivative bound, since an almost-everywhere bound alone says nothing about $w_\delta$ .

A complete development needs the Lyapunov calculus of §7.1, compactness of the trajectory, the limit-set theory behind Theorem 3.2, the uniformisation of the Łojasiewicz inequality on a compact connected set of critical points, and the Haraux–Jendoubi integration lemmas. Contributions to the last two are reusable beyond this mission and are welcome, as are proofs of the individual milestones.

Selected references

A. Barakat and P. Bianchi, Convergence and Dynamical Behavior of the ADAM Algorithm for Nonconvex Stochastic Optimization, SIAM J. Math. Data Sci. 3(1), 2021; arXiv:1810.02263v4. https://arxiv.org/abs/1810.02263v4
D. P. Kingma and J. Ba, Adam: A Method for Stochastic Optimization, ICLR 2015. https://arxiv.org/abs/1412.6980
S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Les Équations aux Dérivées Partielles, CNRS, 1963, pp. 87–89.
A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems, SpringerBriefs in Mathematics, 2015. https://doi.org/10.1007/978-3-319-23407-6
H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program. 116, 2009, pp. 5–16. https://doi.org/10.1007/s10107-007-0133-5
J. Bolte, S. Sabach and M. Teboulle, Proximal alternating linearized minimization for nonconvex and nonsmooth problems, Math. Program. 146, 2014, pp. 459–494. https://doi.org/10.1007/s10107-013-0701-9

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Convex Optimization Machine Learning Optimization·Captain: mikedeng1

Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization 1: All Four Frank-Wolfe Variants Reach Duality Gap ≤ 2βC_f(1+δ)/(K+2), β = 27/8, Within K IterationsResearch Paper

Why a duality-gap guarantee matters

The Frank–Wolfe method, also called the conditional-gradient method, minimizes a smooth convex objective over a compact convex set while asking only for a minimizer of a linear function over that set. In domains where a linear optimization oracle is easier than a projection, this distinction changes which algorithms are practical. Examples in Jaggi's paper include optimization over convex hulls of atomic sets, norm-constrained matrix problems, and domains where iterates can be stored as combinations of a few atoms (Jaggi 2013, §§1–2).

A bound on the unknown difference between the current objective and the optimum does not, by itself, tell an implementation when to stop. The duality gap in this paper can be computed from the same linear optimization problem the method already solves. It provides a certificate tied to the current iterate, so an observed small gap is meaningful even when the optimal objective value is unavailable. The paper's Theorem 2 guarantees that one of the first $K$ iterates of four variants has such a certificate, including variants whose linear subproblems are solved only approximately (Jaggi 2013, PDF pp. 2–4).

Setting and notation

Let $D$ be a compact convex subset of a real vector space and let $f$ be a convex, continuously differentiable real-valued function. The optimization problem is $\min_{x\in D}f(x)$ , and $x^*\in D$ denotes a minimizer. A Frank–Wolfe step begins at $x^{(k)}$ , chooses an atom $s^{(k)}\in D$ by approximately minimizing the linear function $s\mapsto\langle s,\nabla f(x^{(k)})\rangle$ , and forms a new point from the current point and the chosen atom. In Algorithm 1 the minimization is exact; Algorithm 2 allows an additive oracle error. Algorithm 3 uses a line search along the segment to the atom. Algorithm 4 minimizes over the convex hull of all atoms selected so far, including $s^{(0)}=x^{(0)}$ (Jaggi 2013, Algorithms 1–4).

For $x\in D$ , the paper defines the duality gap by

g(x)=\max_{s\in D}\langle x-s,\nabla f(x)\rangle.

Convexity gives $0\le f(x)-f(x^*)\le g(x)$ . An exact minimizer of the linear subproblem attains the maximum defining $g(x)$ , so its chosen atom gives the certificate directly (Jaggi 2013, equation (2) and §2).

The curvature constant measures the scaled departure of $f$ from its linearization along chords of $D$ :

C_f=\sup_{x,s\in D,\;0<\gamma\le1}\frac{2}{\gamma^2}\left[f(x+\gamma(s-x))-f(x)-\langle\gamma(s-x),\nabla f(x)\rangle\right].

The paper uses $C_f$ in both the oracle tolerance $\delta\gamma_k C_f/2$ and the convergence bounds, where $\gamma_k=2/(k+2)$ and $\delta\ge0$ measures linear-subproblem accuracy. For a linear objective $C_f=0$ ; for $f(x)=\|x\|_2^2/2$ , it is the squared Euclidean diameter of $D$ (Jaggi 2013, equation (3) and §3).

Formalization targets

The milestone path records the linearization inequality and its primal-error certificate, the identity between the exact linear subproblem and the gap, and the paper's primal convergence Theorem 1:

f(x^{(k)})-f(x^*)\le\frac{2C_f}{k+2}(1+\delta),\qquad k\ge1.

The mission goal is primal-dual convergence, Theorem 2. If one of Algorithms 1–4 runs for $K\ge2$ iterations, there is an index $1\le\widehat k\le K$ for which

g(x^{(\widehat k)})\le\frac{2\beta C_f}{K+2}(1+\delta),\qquad\beta=\frac{27}{8}.

Both targets retain the paper's constants and its approximate-oracle tolerance. The mission also states the paper's curvature examples, Lipschitz-gradient comparison, and invariance of curvature under affine reparameterization. These companions clarify what $C_f$ measures; the goal itself is the duality-gap theorem (Jaggi 2013, PDF pp. 3–4).

What the result supplies

Theorem 2 converts a rate expressed using an unknown optimum into a guarantee about a quantity available through the algorithm's linear oracle. Combined with the certificate inequality, a small observed gap also bounds the primal error. The same theorem covers an exact oracle, an approximate oracle, line search, and full correction over past atoms, with one numerical constant. That common guarantee is the reason the four update rules belong in one formal target rather than four unrelated asymptotic statements (Jaggi 2013, Theorems 1–2).

The paper proves these statements; this mission asks for machine-checked proofs of their precise Lean versions. The source PDF held here is the nine-page published proceedings paper, which states the results but does not contain its supplementary appendices with proofs. A complete formal development would therefore also be useful as a self-contained account of the gap, curvature, and four algorithms. The definitions of the gap and curvature, and the exact-oracle certificate, can be reused in later convex-optimization developments.

Where the difficulty lies

Theorem 1 bounds the objective difference at each iterate, but the bound on $g(x^{(\widehat k)})$ asks for a good certificate at some iterate in a finite window. A small primal error alone does not assert that the current linear oracle sees a small gap. The analysis has to connect the gap to progress across steps while accounting for the approximate linear oracle and for updates that differ from the fixed convex-combination step. The fully corrective variant also remembers all prior atoms, whereas the line-search variant chooses its step length from the current segment. These differences are present in the Lean step predicate and cannot be erased by assuming a gap bound on the run (Jaggi 2013, Algorithm 2–4 and Theorem 2).

Formalization scope

Lean represents $D$ as a set in a real normed vector space and writes the pairing with $\nabla f(x)$ as the Fréchet derivative $f'(x)$ applied to a displacement. The paper's footnote names a Hilbert space; the normed-space statement is a disclosed generalization because the results use the inner product only through the derivative pairing. Convexity is required on $D$ , rather than on the whole ambient space, while the paper's continuously differentiable hypothesis is retained. Compactness, convexity, feasibility of $x^{(0)}$ , and existence of $x^*$ appear explicitly in the convergence statements.

The curvature supremum excludes $\gamma=0$ , where the source formula contains division by zero. Its defining set is required to be bounded above when a convergence theorem treats $C_f$ as a finite real number; compactness and an existing feasible point make the duality-gap supremum meaningful. The selected variant is one of four explicit step rules. Algorithm 3's oracle tolerance uses $\gamma_k=2/(k+2)$ before line search, and Algorithm 4's atom hull contains $x^{(0)}$ . The iterates are zero-based; the goal constrains the source algorithm's steps $0,\ldots,K$ and selects $\widehat k$ from $1,\ldots,K$ . For Algorithm 1, the exact oracle is encoded for all $\delta\ge0$ ; $\delta=0$ recovers the page's stated case.

The construction relies on Mathlib's topology, convexity, continuous derivatives, real suprema, metric diameter, and continuous affine maps. Contributions that prove the milestone certificates, establish the curvature identities, or connect the four concrete step rules to the main finite-iteration bound are in scope. A run hypothesis that directly bounds its gap or primal error would remove the assertion being formalized and is excluded.

Selected references

Martin Jaggi, Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization, Proceedings of the 30th International Conference on Machine Learning, JMLR Workshop and Conference Proceedings 28, 2013. Published paper.

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