Probability

Motivation

The capacitated vehicle routing problem (CVRP) asks for delivery routes of minimum cost. Each route starts and ends at a depot, every customer is visited exactly once, and the demand served on a route does not exceed the vehicle capacity. The problem is central in logistics and one of the most studied problems in combinatorial optimization. Its standard exact methods are branch-and-cut algorithms built on the two-index vehicle flow formulation, a 0/1 program over arcs whose capacity constraints are the rounded capacity inequalities (RCIs); see Laporte, Nobert and Desrochers (1985) and Semet, Toth and Vigo (2014).

In practice customer demands are uncertain. A chance-constrained CVRP requires each route to respect its capacity with probability at least $1-\epsilon$ under a known distribution. That distribution is rarely known. Most solution methods also need independent demands. Ghosal and Wiesemann (Oper. Res. 68(3), 2020) study the distributionally robust chance-constrained CVRP. There the chance constraint must hold for every distribution in an ambiguity set $\mathcal P$ of plausible distributions. The ambiguity set may contain dependent distributions and uncountably many of them, so it is not clear a priori that the problem can be solved by the usual branch-and-cut machinery. This mission formalizes the paper's answer to that question: its Theorem 1 and the counterexample that precedes it.

Setting

The graph is complete and directed. Its nodes are $V=\{0,\dots,n\}$ and its arcs are $A=\{(i,j)\in V\times V:i\neq j\}$. Node $0$ is the depot and $V_C=\{1,\dots,n\}$ are the customers. There are $m$ vehicles, indexed by $K=\{1,\dots,m\}$, each of capacity $Q>0$. Traversing the arc $(i,j)$ costs $c(i,j)\ge 0$; costs may be asymmetric.

A route $\mathbf R_k=(R_{k,1},\dots,R_{k,n_k})$ is an ordered list of customers, with $R_{k,0}=R_{k,n_k+1}=0$. A route set $\mathbf R=(\mathbf R_1,\dots,\mathbf R_m)\in\mathfrak P(V_C,m)$ partitions $V_C$ into $m$ nonempty ordered routes. Its cost is $c(\mathbf R)=\sum_{k}\sum_{l=0}^{n_k}c(R_{k,l},R_{k,l+1})$.

The demand vector $\tilde{\boldsymbol q}\in\mathbb R^n$ is random. The ambiguity set $\mathcal P$ is a set of probability distributions of $\tilde{\boldsymbol q}$ and $\epsilon\in(0,1)$ is the risk level. The problem RVRP($\mathcal P$) minimizes $c(\mathbf R)$ over route sets such that

$$\mathbb P\Big[\textstyle\sum_{i\in\mathbf R_k}\tilde q_i\le Q\Big]\ge 1-\epsilon\qquad\forall\,\mathbb P\in\mathcal P,\ \forall\,k\in K .$$

With $\mathbb Q\text{-VaR}_{1-\epsilon}[\tilde X]=\inf\{x:\mathbb Q[\tilde X\le x]\ge1-\epsilon\}$, the demand estimator of the paper's Eq. (2) is

$$d_{\mathcal P}(S)=\max\left\{\left\lceil\frac1Q\sup_{\mathbb P\in\mathcal P}\mathbb P\text{-VaR}_{1-\epsilon}\Big[\sum_{i\in S}\tilde q_i\Big]\right\rceil,1\right\}\quad(S\neq\emptyset),\qquad d_{\mathcal P}(\emptyset)=0 .$$

The problem 2VF($\mathcal P$) minimizes $\sum_{(i,j)\in A}c(i,j)x_{ij}$ over $x\in\{0,1\}^A$ with in- and out-degree $1$ at every customer and $m$ at the depot, and with the RCIs

$$\sum_{i\in V\setminus S}\sum_{j\in S}x_{ij}\ge d_{\mathcal P}(S)\qquad\forall\,S\subseteq V_C,\ S\neq\emptyset .$$

A route set induces the arc vector with $x_{ij}=1$ exactly when $(i,j)=(R_{k,l},R_{k,l+1})$ for some $k,l$ (the paper's Eq. (3)). The estimator satisfies the subadditivity condition (S) if $d_{\mathcal P}(S\cup T)\le d_{\mathcal P}(S)+d_{\mathcal P}(T)$ for all $S,T\subseteq V_C$.

Formalization targets

Goal: Theorem 1

Assume $\tilde{\boldsymbol q}\ge\mathbf 0$ $\mathbb P$-a.s. for all $\mathbb P\in\mathcal P$, and assume $d_{\mathcal P}$ is real valued and satisfies (S). Then:

$$\begin{aligned} &\text{(i)}\ \ \mathbf R \text{ feasible in RVRP}(\mathcal P)\ \Longrightarrow\ x(\mathbf R)\text{ feasible in 2VF}(\mathcal P),\ \ c(x(\mathbf R))=c(\mathbf R);\\ &\text{(ii)}\ \ x\text{ feasible in 2VF}(\mathcal P)\ \Longrightarrow\ x=x(\mathbf R)\text{ for an RVRP}(\mathcal P)\text{-feasible }\mathbf R,\text{ unique up to reordering routes},\ c(x)=c(\mathbf R). \end{aligned}$$

Milestones

1. The chance constraint $\mathbb Q[\tilde X\le\tau]\ge1-\epsilon$ is equivalent to $\mathbb Q\text{-VaR}_{1-\epsilon}[\tilde X]\le\tau$ (p. 720).
2. Eq. (1): a route satisfies its robust chance constraint if and only if the worst-case VaR of its cumulative demand is at most $Q$.
3. Example 1: an instance with two customers where a route set is RVRP($\mathcal P$)-feasible, yet its induced flow violates the RCI for $S=\{1,2\}$, since $d_{\mathcal P}(\{1,2\})\ge3$.
4. Example 1 (continued): on that instance $d_{\mathcal P}$ violates (S).
5. Theorem 1 (i) and 6. Theorem 1 (ii), stated separately.

Significance

Theorem 1 separates the modeling question from the algorithmic one. Whenever the ambiguity set yields a subadditive estimator, the distributionally robust CVRP is solved exactly by a two-index flow branch-and-cut. The only change from the deterministic case is the right-hand side $d_{\mathcal P}(S)$ of the RCIs, however many distributions $\mathcal P$ contains. The companion missions of this series show that (S) holds for every moment ambiguity set (Theorem 2 of the paper) and compute $d_{\mathcal P}$ for several classes of such sets. Example 1 shows that the hypothesis cannot be dropped: ambiguity sets that pin down each customer's marginal distribution break the equivalence.

The paper's proofs are in its online supplement; no machine-checked version of these statements exists. Formalizing them produces a checked reduction between a stochastic routing model and an integer program. It also produces reusable definitions of route sets, induced arc flows and RCIs over directed graphs with a depot.

Difficulty

Direction (ii) is a graph decomposition. A 0/1 vector with the prescribed degrees splits into $m$ depot cycles plus possibly depot-free subtours. The RCIs, through the $\max\{\cdot,1\}$ in $d_{\mathcal P}$, must exclude the subtours, and the RCI on the customers of a single route must enforce that route's chance constraint. Uniqueness up to reordering requires that directed routes are recovered from arcs.

Direction (i) is where (S) enters. The naive argument bounds the number of vehicles entering $S$ by $d_{\mathcal P}(S)$ directly from the chance constraints. It fails because the chance constraints control each route separately, while $d_{\mathcal P}(S)$ looks at the joint worst case of the demands in $S$; Example 1 is exactly this failure. A set $S$ is typically visited by several routes, each covering only part of it. Relating the per-route guarantees to the joint quantity $d_{\mathcal P}(S)$ needs both hypotheses of the theorem: nonnegative demands and (S).

Formalization scope

Customers are Fin n (0-based; the paper's customer $i$ is i - 1). Nodes are Fin (n+1) with the depot 0 and customer i at i.succ, and vehicles are Fin m. A route set is R : Fin m → List (Fin n): every route is nonempty and the concatenated routes are a permutation of all customers. Arc vectors are ℕ-valued functions on ordered node pairs, with values in $\{0,1\}$ and the non-arcs $(i,i)$ fixed to $0$.

Distributions are measures on Fin n → ℝ, and the ambiguity set is a set of probability measures. Chance constraints are written ENNReal.ofReal (1 - ε) ≤ P {q | …}. Value-at-risk is the published MultistageStochastic.valueAtRisk at level 1 - ε. The worst-case VaR is a real sSup and $d_{\mathcal P}$ is integer valued.

Two conventions implicit on the page are explicit hypotheses:

• $Q>0$, because (2) divides by $Q$;
• boundedness of the VaR values for every customer set, which encodes the paper's declaration $d_{\mathcal P}:2^{V_C}\to\mathbb R_+$.

A real sSup of an unbounded set is $0$ in Lean. Without the boundedness hypothesis every such estimator would silently equal $1$ and (ii) would fail. For an empty ambiguity set the Lean estimator equals $1$ on nonempty sets, as the paper's does.

The RCIs range over all nonempty customer sets with the depot on the outside. The estimator keeps the ceiling and the $\max\{\cdot,1\}$. 2VF feasibility mentions neither routes nor chance constraints. RVRP feasibility does not mention $d_{\mathcal P}$. A formalization in which either side refers to the other, or in which $d_{\mathcal P}$ drops the $\max\{\cdot,1\}$, is not this theorem.

Useful contributions include lemmas on the decomposition of degree-constrained 0/1 arc vectors into depot cycles, monotonicity of VaR under almost-sure ordering, and the CDF right-continuity behind milestone 1.

Related platform work: SupplyChainTheory_vrp formalizes a different, symmetric, unit-demand VRP and is not reused.

Selected references

• S. Ghosal, W. Wiesemann, The Distributionally Robust Chance-Constrained Vehicle Routing Problem, Operations Research 68(3):716–732, 2020. https://doi.org/10.1287/opre.2019.1924
• G. Laporte, Y. Nobert, M. Desrochers, Optimal routing under capacity and distance restrictions, Operations Research 33(5):1050–1073, 1985. https://doi.org/10.1287/opre.33.5.1050
• F. Semet, P. Toth, D. Vigo, Classical exact algorithms for the capacitated vehicle routing problem, in P. Toth, D. Vigo (eds.), Vehicle Routing: Problems, Methods, and Applications, 2nd ed., SIAM, 2014, 37–57. https://doi.org/10.1137/1.9781611973594.ch2
• J. Lysgaard, A. N. Letchford, R. W. Eglese, A new branch-and-cut algorithm for the capacitated vehicle routing problem, Mathematical Programming 100(2):423–445, 2004. https://doi.org/10.1007/s10107-003-0481-8

Motivation

In best arm identification with fixed confidence, a learner samples $K$ unknown distributions ("arms") one at a time, and must eventually stop and name the arm with the largest mean, with an error probability at most a prescribed risk $\delta$, while using as few samples as possible. The problem goes back to the sequential design of experiments (Chernoff, 1959; Even-Dar, Mannor and Mansour, 2006) and underlies adaptive A/B testing, clinical trial design and hyperparameter selection.

Any fixed-confidence strategy consists of three parts: a sampling rule, a stopping rule and a decision rule. Garivier and Kaufmann (arXiv:1602.04589, COLT 2016) proposed the Track-and-Stop strategy, the first shown to match the asymptotic lower bound on the expected sample complexity. Its stopping rule is a generalized likelihood ratio (GLR) test, Chernoff's stopping rule. Its correctness, the guarantee that the recommended arm is wrong with probability at most $\delta$, must hold whatever the sampling rule, which is what allows the sampling rule to be tuned freely for efficiency. This mission formalizes that guarantee for Bernoulli arms, Theorem 10 of the paper, with its explicit threshold $\beta(t,\delta) = \log(2t(K-1)/\delta)$.

Setting

The arms are $\mathcal A = \{1,\dots,K\}$. A Bernoulli bandit model is a mean vector $\boldsymbol\mu = (\mu_1,\dots,\mu_K) \in [0,1]^K$: pulling arm $a$ returns reward $1$ with probability $\mu_a$ and $0$ otherwise, independently of the past. The class $\mathcal S$ contains the models with a unique optimal arm $a^*(\boldsymbol\mu)$, i.e. $\mu_{a^*} > \mu_i$ for all $i \ne a^*$.

At round $t$ the learner chooses an arm $A_t$ as a (possibly randomized) function of the past observations and observes a reward $X_t$. Write $N_a(t)$ for the number of pulls of arm $a$ among the first $t$ rounds, $s_a(t)$ for the number of those pulls that returned $1$, and $\hat\mu_a(t) = N_a(t)^{-1}\sum_{s \le t} X_s \mathbb 1\{A_s = a\}$ for the empirical mean. The likelihood of arm $a$'s observations under mean $u$ is $p_u(\underline X^a_{N_a(t)}) = u^{s_a(t)}(1-u)^{N_a(t)-s_a(t)}$.

The GLR statistic for "arm $a$ is at least as good as arm $b$" is

$$Z_{a,b}(t) = \log \frac{\max_{\mu'_a \ge \mu'_b} p_{\mu'_a}(\underline X^a_{N_a(t)})\, p_{\mu'_b}(\underline X^b_{N_b(t)})}{\max_{\mu'_a \le \mu'_b} p_{\mu'_a}(\underline X^a_{N_a(t)})\, p_{\mu'_b}(\underline X^b_{N_b(t)})}.$$

Chernoff's stopping rule with exploration rate $\beta(t,\delta)$ is

$$\tau_\delta = \inf\{t \ge 1 : \exists a \in \mathcal A,\ \forall b \ne a,\ Z_{a,b}(t) > \beta(t,\delta)\},$$

and the decision rule recommends $\hat a_{\tau_\delta} \in \operatorname{argmax}_a \hat\mu_a(\tau_\delta)$.

The Krichevsky–Trofimov (KT) distribution on binary sequences $x \in \{0,1\}^n$ is $\mathrm{kt}(x) = \int_0^1 \big(\pi\sqrt{u(1-u)}\big)^{-1} p_u(x)\,\mathrm du$, the Bernoulli likelihood mixed over the Beta$(1/2,1/2)$ prior.

Formalization targets

Goal: Theorem 10

For every $\delta \in (0,1)$, every sampling strategy, and the threshold $\beta(t,\delta) = \log\big(2t(K-1)/\delta\big)$,

$$\forall \boldsymbol\mu \in \mathcal S,\qquad \mathbb P_{\boldsymbol\mu}\big(\tau_\delta < \infty,\ \hat a_{\tau_\delta} \ne a^*\big) \le \delta .$$

Milestone: Lemma 11 (Willems, Shtarkov and Tjalkens, 1995)

$\mathrm{kt}$ is a probability law on $\{0,1\}^n$, and for $n \ge 1$,

$$\sup_{x\in\{0,1\}^n}\ \sup_{u\in[0,1]} \frac{p_u(x)}{\mathrm{kt}(x)} \le 2\sqrt n .$$

Milestone: the pairwise crossing bound of Appendix C.1

With $T_{a,b} = \inf\{t : Z_{a,b}(t) > \beta(t,\delta)\}$, for all arms with $\mu_a < \mu_b$,

$$\mathbb P_{\boldsymbol\mu}(T_{a,b} < \infty) \le \frac{\delta}{K-1}.$$

Significance

Theorem 10 decouples correctness from efficiency. Because the guarantee holds for every sampling strategy, any sampling rule, including the C-Tracking and D-Tracking rules of Track-and-Stop, the uniform rule, or a heuristic, inherits $\delta$-correctness as soon as it is paired with Chernoff's stopping rule at this threshold. The asymptotic optimality result of the paper (Theorem 14) then only has to control the sample complexity. The threshold is explicit, with no unspecified constant, in contrast to the deviational threshold of Proposition 12.

The result is proved in the paper, in Appendix C.1, and rests on Lemma 11, which the paper quotes from the universal coding literature without proof. As far as the platform's catalog shows, none of these results is formalized. The platform holds a machine-checkable statement of the analogous result for Gaussian arms with the Lattimore–Szepesvári threshold (BanditAlgorithm.chernoff_stopping_rule_sound, Lemma 33.7 of Bandit Algorithms), which is a different model and a different threshold. Formalizing Theorem 10 adds a proof of Lemma 11 (the KT regret bound, reusable in information theory and universal prediction), the Bernoulli GLR statistic, and a change of measure from the true bandit law to a Bayesian mixture law on the trajectory space.

Difficulty

The obvious approach bounds, for each fixed $t$, the probability that $Z_{a,b}(t)$ exceeds $\beta(t,\delta)$ by a concentration inequality and sums over $t$. This fails: the sampling strategy is arbitrary and adaptive, so $N_a(t)$ and $N_b(t)$ are random and depend on the past rewards, and a fixed-sample-size deviation bound does not apply; a union bound over the possible values of the counts loses more than the threshold allows. The maximum likelihood in the numerator of $Z_{a,b}$ is also not a probability density, so the likelihood ratio cannot directly be read as a change of measure. The argument must control the whole trajectory law under an arbitrary randomized policy, and must handle empty samples (an arm never pulled contributes likelihood $1$) and the boundary means $0$ and $1$.

Formalization scope

The formalization is in Lean 4 with Mathlib and reuses the platform's canonical bandit model: StochasticBandit, BanditPolicy (a Markov kernel per round from the observed history to the next arm, so randomized strategies are included), banditTrajMeasure (the law of the infinite trajectory, where coordinate $s$ is round $s+1$) and IsSoundBAI from BanditTrajectory; bernoulliBandit from bernoulliRelativeEntropy; and only the pull counts trajPullCount and empirical means trajEmpiricalMean from TrackAndStop. The Gaussian GLR and threshold of TrackAndStop are not used.

Conventions committed to:

• Arms are Fin K. Bernoulli means range over $[0,1]$, degenerate laws included; the paper's exponential-family mean space is $(0,1)$, so the $[0,1]$ statement implies the paper's.
• $Z_{a,b}(t)$ is defined as the ratio of the two maxima over $[0,1]^2$, not by the closed form (7), which holds only when $\hat\mu_a(t) \ge \hat\mu_b(t)$. Both maxima are attained and positive.
• The stopping rule ranges over $t \ge 1$; at $t = 0$ there is no observation and the paper's $\beta(0,\delta) = \log 0$ is undefined. $\tau_\delta = \infty$ when the rule never fires.
• The decision rule is quantified: the goal holds for every recommendation that maximizes the empirical mean at $\tau_\delta$, whatever the tie-breaking.
• Probabilities of events are outer measures under the trajectory law; no measurability is assumed.
• $K \ge 1$ only. For $K = 1$ the statement is trivially true (no suboptimal arm).

Disclosed deviations from the page: Lemma 11's ratio bound is stated for $n \ge 1$, since at $n = 0$ the printed bound reads $1 \le 0$; the use of the lemma in Appendix C.1 is unaffected. Appendix C.1 calls the result "Proposition 10" (a slip for Theorem 10) and prints the KT density as $1/\sqrt{\pi u(1-u)}$ (a slip for $1/(\pi\sqrt{u(1-u)})$ of Lemma 11, which is the normalized one); the formalization follows Lemma 11.

A trivializing formalization is ruled out: stating the result for Gaussian arms or with the Lattimore–Szepesvári threshold is the platform's existing Lemma 33.7 and is not Theorem 10, and a free decision rule without the argmax hypothesis would make the claim false rather than faithful.

Contributions welcome: a proof of Lemma 11; a change-of-measure lemma for banditTrajMeasure under a mixture of environments; and the union-bound reduction from the goal to the pairwise claim.

Selected references

• A. Garivier, E. Kaufmann, Optimal Best Arm Identification with Fixed Confidence, COLT 2016 (JMLR W&CP 49), arXiv:1602.04589v2, 2016. https://arxiv.org/abs/1602.04589
• F. M. J. Willems, Y. M. Shtarkov, T. J. Tjalkens, The context-tree weighting method: basic properties, IEEE Transactions on Information Theory 41(3), 1995. https://doi.org/10.1109/18.382012
• R. Krichevsky, V. Trofimov, The performance of universal encoding, IEEE Transactions on Information Theory 27(2), 1981. https://doi.org/10.1109/TIT.1981.1056331
• H. Chernoff, Sequential design of experiments, Annals of Mathematical Statistics 30(3), 1959. https://doi.org/10.1214/aoms/1177706205
• T. Lattimore, C. Szepesvári, Bandit Algorithms, Cambridge University Press, 2020, Chapter 33. https://doi.org/10.1017/9781108571401

Motivation

A decision maker who chooses a portfolio $x\in\mathbb R^n$ of risky assets with random return vector $R$ receives the scalar return $x'R$ and evaluates it by an expected utility $E[u(x'R)]$. In practice the law of $R$ is not known; what can be estimated with some confidence are its mean $\mu$ and covariance $\Sigma$. The robust mean-covariance objective replaces the unknown law by the worst law consistent with these two moments:

$$U(x)=\inf\{E[u(x'R)] : R \text{ has mean } \mu \text{ and covariance } \Sigma\}.$$

Ioana Popescu (Operations Research 55(1), 2007) showed that $U(x)$ depends on $x$ only through $\mu_x=x'\mu$ and $\sigma_x^2=x'\Sigma x$, and that for large classes of utilities it has a closed form or reduces to a one-dimensional search. This turns a robust stochastic program into a parametric mean-variance program, which is the practical point of the paper. This mission formalizes the closed form for concave utilities with a monotone second derivative (Proposition 7), a class that contains the exponential utility $1-e^{-ay}$ and all concave quadratics. The reduction to $(\mu_x,\sigma_x)$ is the subject of the companion mission Robust Mean-Covariance Solutions for Stochastic Optimization I; the two-point case is mission II.

The underlying univariate question is a moment problem in the tradition of Chebyshev-type bounds: the extremal value of $E[u(r)]$ over all laws with prescribed mean and variance. Scarf (1958) solved an instance for a piecewise-linear inventory cost, and Birge and Dulá (1991) treated two-point extremal laws on bounded domains.

Setting

Fix $m\in\mathbb R$ and $s\ge 0$. The mean-variance class $\mathbb M_{(m,s^2)}$ is the set of Borel probability measures $\nu$ on $\mathbb R$ with $\int r^2\,d\nu<\infty$, $\int r\,d\nu=m$ and $\int (r-m)^2\,d\nu=s^2$ (Lean: MeanVarClass m (s ^ 2)). For a utility $u:\mathbb R\to\mathbb R$ the robust objective is

$$U(m,s)=\inf\Big\{\int u\,d\nu : \nu\in\mathbb M_{(m,s^2)}\Big\},$$

with $\min$ read, as in the paper, "in the wide sense of $\inf$", so the value $-\infty$ is allowed. The paper's $U(x)$ is $U(\mu_x,\sigma_x)$.

For $p\in(0,1)$ the two-point law $r_p$ puts mass $p$ on $b=m+\sqrt{(1-p)/p}\,s$ and mass $1-p$ on $a=m-\sqrt{p/(1-p)}\,s$; these are exactly the two-point laws of $\mathbb M_{(m,s^2)}$. Its expected utility is the two-point objective (8),

$$U(p)=p\,u(b)+(1-p)\,u(a)\qquad(\texttt{twoPointValue u m s p}).$$

A supporting quadratic of $u$ is $q(y)=Ay^2+By+C$ with $q\le u$ on $\mathbb R$; the set of their coefficients is $\mathcal Q$. Since every $r\sim(m,s^2)$ has $E[q(r)]=A(m^2+s^2)+Bm+C$, each element of $\mathcal Q$ gives a lower bound on $U(m,s)$ (Proposition 3). The function $u$ has the one-point support property with respect to $(m,s^2)$ (Definition 3, OnePointSupportWrt u m s) if some supporting quadratic touches $u$ at $m$ and $E[u(r_p)]\to E[q(r_p)]$ as $p\to0^+$ or as $p\to1^-$; it has one-point support (OnePointSupport u) if this holds for every $m$ and every $s\ge0$.

Formalization targets

Goal: Proposition 7

Let $u$ be concave and twice differentiable with monotone $u''$.

(a) If $u'$ is convex, then

$$U(m,s)=u(m)+\lim_{y\to-\infty}u''(y)\,\frac{s^2}{2}\qquad\text{for all } m,\ s.$$

(b) If $u'$ is concave, the same holds with $\lim_{y\to+\infty}u''(y)$.

In each part the limit $L$ of $u''$ exists in $[-\infty,0]$. When $L$ is finite the goal asserts that $u$ is integrable under every law of every class, that $u(m)+Ls^2/2$ is the greatest lower bound of the expected utilities, and that $u$ has one-point support. When $L=-\infty$ and $s>0$ it asserts $U(m,s)=-\infty$.

Milestones

1. Display (9): $\partial U/\partial p=u(b)-u(a)-(b-a)\,\frac{u'(b)+u'(a)}{2}$.
2. For convex $u'$ and $s>0$, $U(p)$ is nonincreasing on $(0,1)$.
3. Appendix (4): $\lim_{p\to1^-}U(p)=u(m)+\frac{s^2}{2}\lim_{y\to-\infty}u''(y)$, including the value $-\infty$.
4. Proposition 3: every $E[u(r)]$ dominates every $A(m^2+s^2)+Bm+C$ with $(A,B,C)\in\mathcal Q$.
5. The function $d(y)=\frac{u(y)-u(m)}{(y-m)^2}-\frac{u'(m)}{y-m}$ is nondecreasing on $\{y \neq m\}$ when $u'$ is convex.
6. Appendix (5): $q(y)=\tfrac12u''(-\infty)(y-m)^2+u'(m)(y-m)+u(m)$ supports $u$, and $\inf_{y\ne m}d(y)=\lim_{y\to-\infty}d(y)=\tfrac12u''(-\infty)$.

An additional item states Proposition 8: a monotone convex $u$ has one-point support and $U(m,s)=u(m)$.

Significance

Proposition 7 turns the worst-case expected utility into a mean-variance criterion with an explicit risk weight: a prudent investor ($u'$ convex) is penalized by $\tfrac12\lim_{y\to-\infty}u''(y)$ per unit of variance, an imprudent one by the limit at $+\infty$. With Proposition 1, the robust portfolio problem becomes $\max_x\ u(\mu_x)+\tfrac12 L\,\sigma_x^2$, a concave mean-variance program. For exponential utility the weight is $-\infty$ (Example 3 of the paper), so the robust investor must eliminate variance entirely; this is a qualitative statement about robustness that follows only from the $-\infty$ case of the theorem.

The result is published with a proof in the appendix of the paper. It has, to the best of our search, no machine-checked version, and the platform currently holds no statement of Propositions 3, 7 or 8 or Definition 3. A formal proof would also check two points where the printed argument is incomplete: the proof's step "$\lim_{y\to-\infty}u'(y)=\infty$" fails for affine $u$, where the theorem nevertheless holds, and the claim that $u$ "has one-point support" fails when $\lim u''=-\infty$ (see the scope section). Related platform work on worst-case expectations over ambiguity sets is the mission Wasserstein Distributionally Robust Optimization II, which uses a different ambiguity set.

Difficulty

The infimum ranges over all laws with two prescribed moments, an infinite-dimensional set with no compactness, and in the relevant cases it is not attained. The natural first idea, restricting to two-point laws and minimizing over $p$, gives only an upper bound (display (7)), and here the minimizing $p$ runs off to the boundary: the extremal law puts vanishing mass on a point escaping to $-\infty$. Identifying the limiting value requires controlling $u(m-sq)/(1+q^2)$ as $q\to\infty$, a second-order asymptotic statement about $u$ at $-\infty$. The matching lower bound requires a supporting quadratic whose curvature is exactly $\tfrac12\lim u''$; showing that it lies below $u$ everywhere, not only near $m$, is a global statement that uses the convexity of $u'$ on the whole line. The finite-limit and infinite-limit cases also behave differently: in the second, no supporting quadratic exists at all.

Formalization scope

Laws are Measure ℝ with IsProbabilityMeasure, finite second moment is MemLp id 2, and the class is parameterized by mean $m$ and variance $s^2$ with $s\ge0$. The statements are univariate: $U(x)$ of the paper is the univariate objective at $(\mu_x,\sigma_x)$ by Proposition 1 of the paper (mission I). Derivatives are deriv u and deriv (deriv u); "twice differentiable" is differentiability of $u$ and of $u'$. Limits of $u''$ are explicit hypotheses (Tendsto … atBot (𝓝 L) or Tendsto … atBot atBot), never limUnder. Limits in $p$ are one-sided inside $(0,1)$. Expectations under two-point laws are written by the explicit formula (8).

"Min" is never a real ⨅, which Lean evaluates to $0$ on sets unbounded below. The finite case uses IsGLB together with integrability of $u$ under every law of the class; the infinite case asserts that integrable laws with arbitrarily small expected utility exist; Proposition 3 is stated as "every value $\ge$ every value". Proposition 3 assumes $u$ integrable under the law; Proposition 8 takes the infimum over laws under which $u$ is integrable, which for convex $u$ loses nothing.

One correction to the printed statement: Proposition 7 opens with "then $u$ has one-point support", which is false when $\lim u''=-\infty$, since no quadratic lies below $1-e^{-ay}$ on $\mathbb R$. The formalization asserts one-point support only in the finite-limit case and the value $-\infty$ in the other. A formalization that took $\min$ as a real infimum, dropped the integrability conjunct, or asserted one-point support unconditionally would be either trivially satisfiable or false; none of these is used.

A complete development needs: moments of two-point laws, a second-order l'Hôpital or Taylor argument at $-\infty$, the trapezoid inequality for convex functions, and Jensen-type integration of quadratic lower bounds. The mean-variance class, the two-point objective and the supporting-quadratic lemmas are reusable for mission II and for other moment-problem bounds. Proofs of any milestone, and of part (b), are welcome.

Selected references

• I. Popescu, Robust Mean-Covariance Solutions for Stochastic Optimization, Operations Research 55(1):98–112, 2007. https://doi.org/10.1287/opre.1060.0353
• H. Scarf, A min-max solution of an inventory problem, in Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, 1958.
• J. R. Birge, J. H. Dulá, Bounding separable recourse functions with limited distribution information, Annals of Operations Research 30, 1991.
• S. Karlin, W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience, 1966.

Why robustify a sampled stochastic program

Many decision problems under uncertainty take the form of a stochastic program: choose a decision $v$ from a feasible set $\mathcal F$ to maximise the expected utility $\mathbb E_{x\sim\mu}[f(v,x)]$, where the distribution $\mu$ of the uncertain parameter $x\in\mathbb R^m$ is known only through i.i.d. samples $x_1,\dots,x_n$. The standard remedy, sample average approximation, maximises $\frac1n\sum_i f(v,x_i)$ instead. Its consistency (convergence of the optimal expected utility of its solutions to the true optimum) is classical, but it needs regularity assumptions of its own, for example those of King and Wets (Stochastics and Stochastic Reports, 1991), cited on p. 98 of the paper; the paper presents its construction as a route to consistency under weaker conditions.

Robust optimization (RO) takes a different route: it protects each sample by an uncertainty set and optimises against the worst point in it. Xu, Caramanis and Mannor (Math. Oper. Res. 2012) show that RO over several overlapping uncertainty sets is equivalent to a distributionally robust stochastic program (their Theorem 2.1, the subject of mission I of this series). Section 3 of the paper uses that equivalence to show that a specific robustification of the sampled problem, with $\ell_\infty$ boxes of shrinking radius around each sample, is consistent under only boundedness and equicontinuity of $f$. This mission formalizes that result, Theorem 3.1.

Setting

Equip $\mathbb R^m$ with the sup norm $\|z\|_\infty=\max_k|z_k|$, its Borel $\sigma$-algebra and Lebesgue measure $dx$. The data are:

• a set of decisions $V$ and a nonempty feasible set $\mathcal F\subseteq V$;
• a utility $f:V\times\mathbb R^m\to\mathbb R$, Borel measurable in $x$ for each $v$;
• a true density $h^*$ on $\mathbb R^m$ (nonnegative, $\int h^*\,dx=1$) and i.i.d. samples $x_1,x_2,\dots$ with distribution $h^*(x)\,dx$;
• radii $\epsilon(n)>0$.

For a sample $x_1,\dots,x_n$ the boxes are $\mathcal Z_i=\{x_i+\delta\mid\|\delta\|_\infty\le\epsilon(n)\}$, and the box-robust sample objective is

$$J_n(v)=\frac1n\sum_{i=1}^n\ \inf_{\|\delta_i\|_\infty\le\epsilon(n)}f(v,x_i+\delta_i)=\sum_{i=1}^n\frac1n\inf_{x_i'\in\mathcal Z_i}f(v,x_i').$$

The RO solution $v(n)$ is a maximiser of $J_n$ over $\mathcal F$. The equicontinuity modulus of $f$ is

$$d(\epsilon)=\sup_{v,\,x,\ \|\delta\|_\infty\le\epsilon}|f(v,x)-f(v,x+\delta)|.$$

The proof works with the distribution set $\mathcal P_n$ of probability measures $\mu$ with $\mu(\bigcup_{i\in S}\mathcal Z_i)\ge|S|/n$ for every $S\subseteq\{1,\dots,n\}$, and with the uniform box kernel density estimator

$$h_n(x)=(n\epsilon(n)^m)^{-1}\sum_{i=1}^nK\Big(\frac{x-x_i}{\epsilon(n)}\Big),\qquad K(z)=\frac{\mathbf 1(\|z\|_\infty\le1)}{2^m}.$$

Formalization targets

Goal: Theorem 3.1 (p. 98)

Assume $|f(v,x)|\le C$ for all $v,x$; $d(\epsilon)\to0$ as $\epsilon\downarrow0$; $\epsilon(n)\downarrow0$ and $n\epsilon(n)^m\uparrow\infty$. Then for every choice of maximisers $v(n)$, with probability one,

$$\lim_{n\to\infty}\int_{\mathbb R^m}f(v(n),x)\,h^*(x)\,dx=\sup_{v\in\mathcal F}\int_{\mathbb R^m}f(v,x)\,h^*(x)\,dx .$$

Milestones (proof of Theorem 3.1, p. 99)

1. $h_n$ is the density of a probability measure in $\mathcal P_n$.
2. $J_n(v)\le\int f(v,x)\,h_n(x)\,dx$ for every $v$.
3. Oscillation over a box: $\sup_{\mathcal Z_i}f(v,\cdot)-\inf_{\mathcal Z_i}f(v,\cdot)\le d(2\epsilon(n))$.
4. Eq. (7): with $M_n=C\int|h_n-h^*|\,dx$, for every $v$,

$$J_n(v)-M_n\le\int f(v,x)h^*(x)\,dx\le J_n(v)+M_n+d(2\epsilon(n)).$$

5. Strong $L^1$ consistency of the box kernel density estimator: if $\epsilon(n)\to0$ and $n\epsilon(n)^m\to\infty$, then $\int|h_n-h^*|\,dx\to0$ almost surely.

Milestones 1–4 are deterministic statements about a fixed sample; milestone 5 is the only probabilistic input.

Significance

Theorem 3.1 gives consistency of a tractable robust reformulation of a sampled stochastic program under conditions the paper notes are weaker than those of King and Wets for sampled stochastic programs: $f$ need only be bounded and equicontinuous in $x$, uniformly in $v$, and the true distribution need only have a density. It also gives an explicit schedule for the size of the uncertainty set, $\epsilon(n)\to0$ with $n\epsilon(n)^m\to\infty$, the bandwidth condition of kernel density estimation. Section 4 of the paper applies the same distributional interpretation to regularised learning methods such as the support vector machine and the Lasso.

The result is proved in the paper, with the $L^1$ consistency of kernel density estimators (Devroye 1983; Devroye and Györfi 1985) cited rather than proved. No part of it is formalized in Lean or on this platform as far as a search of the platform found. A complete development would produce, besides Theorem 3.1, a machine-checked strong $L^1$ consistency theorem for kernel density estimators, which is a basic result of nonparametric statistics in its own right.

Difficulty

The deterministic part (milestones 1–4) is measure-theoretic bookkeeping: the kernel integrates to one only because the box is a sup-norm ball of volume $(2\epsilon)^m$, and every infimum and supremum must be handled with care, since $f$ need not attain them.

The obstacle is milestone 5. Almost-sure $L^1$ convergence of $h_n$ to an arbitrary density $h^*$, with no continuity or support assumption, does not follow from the strong law of large numbers applied pointwise: $h_n(x)$ is an average of $n$ terms whose law changes with $n$ through $\epsilon(n)$, and almost-sure convergence at each fixed $x$ does not give convergence of the integral along a single sample path. The theorem needs both a bias estimate valid for every integrable density and a concentration estimate for the random $L^1$ error. Mathlib has Lebesgue differentiation and the strong law, but no kernel density estimator and no such concentration result.

Formalization scope

• $\mathbb R^m$ is Fin m → ℝ, whose Mathlib norm is the sup norm; boxes are Metric.closedBall. The integrals $\int f(v,x)h^*(x)\,dx$ are Bochner integrals against Lebesgue measure of integrable integrands.
• The samples are a sequence X : ℕ → Ω → Fin m → ℝ on a probability space, independent (iIndepFun) and each with law volume.withDensity h*; $x_1,x_2,\dots$ become X 0, X 1, …, and the $n$-th problem uses the first $n$. "With probability one" is ∀ᵐ ω ∂P.
• The goal quantifies over every selection $v(n)$ of maximisers, with no measurability assumed; a version with one chosen maximiser would be weaker and is ruled out.
• Readings and corrections of the printed text:
  • the kernel argument printed $(x-x_i)/\epsilon$ on p. 98 is read as $(x-x_i)/\epsilon(n)$, as the proof on p. 99 writes it;
  • "$\max_{v,x}|f(v,x)|\le C$" is read as the uniform bound $|f|\le C$ and the "max" in $d(\epsilon)$ as a supremum;
  • "$d(\epsilon)\downarrow0$" is read as $d(\epsilon)\to0$ as $\epsilon\downarrow0$;
  • implicit hypotheses made explicit: $\mathcal F\ne\emptyset$, $\epsilon(n)>0$, measurability of $f(v,\cdot)$, $h^*$ a Lebesgue density;
  • the monotonicity in "$\epsilon(n)\downarrow0$, $n\epsilon(n)^m\uparrow\infty$" is kept in the goal; milestone 5 uses the limits only, as the paper states it;
  • the paper's $M_n$ ("there exists $\{M_n\}\to0$") is made explicit as $M_n=C\int|h_n-h^*|$, so Eq. (7) is stated for every sample.
• Remark 3.2 and Appendix B (an integrable envelope in place of boundedness) are not part of this mission.
• Every real infimum and supremum ranges over a nonempty set of values bounded by $C$ in absolute value, so no statement holds through a junk value; a formalization in which the supremum over $\mathcal F$ or the box infimum could be vacuous is excluded.
• The definitions (boxes, $\mathcal P_n$, the kernel, the estimator, $J_n$, $d$) live in one definition file. $\mathcal P_n$ duplicates, with weights $1/n$, the distribution set of mission I; the duplication is deliberate because draft missions cannot import each other.
• Welcome contributions: the kernel density estimator and its strong $L^1$ consistency as reusable infrastructure, and any of the deterministic milestones.

Selected references

• H. Xu, C. Caramanis, S. Mannor, A Distributional Interpretation of Robust Optimization, Mathematics of Operations Research 37(1):95–110, 2012. https://doi.org/10.1287/moor.1110.0531
• L. Devroye, The equivalence of weak, strong and complete convergence in $L_1$ for kernel density estimates, Annals of Statistics 11(3):896–904, 1983.
• L. Devroye, L. Györfi, Nonparametric Density Estimation: The $L_1$ View, Wiley, 1985.
• A. J. King, R. J.-B. Wets, Epi-consistency of convex stochastic programs, Stochastics and Stochastic Reports 34(1), 1991 (reference [22] of the paper).

Motivation

Linear programs solved in practice carry data that are measured, estimated or rounded. Ben-Tal and Nemirovski (Math. Program. 88, 2000) examined the NETLIB collection of real-world LPs and found that in 13 of them a relative perturbation of only 0.01% in the "ugly" coefficients of the inequality constraints can make the nominal optimal solution more than 50% infeasible (§2.3). Their remedy is the robust counterpart methodology: replace the nominal problem by a deterministic problem whose feasible solutions remain nearly feasible for every, or for all but a small probability of, data realizations.

The paper made the approach concrete for entry-wise uncertainty and gave the probabilistic guarantee that became a standard tool of robust and chance-constrained optimization. The main steps of the history are:

• 1973, A. L. Soyster: the interval (worst-case, "box") counterpart, here called (IRC).
• 1998–1999, Ben-Tal and Nemirovski (Math. Oper. Res. 23; Oper. Res. Lett. 25), and independently El Ghaoui and co-authors: robust optimization with ellipsoidal uncertainty sets.
• 2000, this paper: the ellipsoid-plus-box counterpart (RC[ε, δ, Ω]) and Proposition 1, which bounds each constraint's violation probability by $\exp\{-\Omega^2/2\}$ under independent symmetric perturbations.
• 2004, Bertsimas and Sim (Oper. Res. 52): the budgeted counterpart, with an analogous probability bound.

Setting

An uncertain linear program is

$$\text{minimize } c^Tx\quad\text{s.t.}\quad Ex=e,\qquad Ax\le b,\qquad \ell\le x\le u,\tag{LP}$$

with $x\in\mathbb{R}^n$, $E\in\mathbb{R}^{p\times n}$, $A=(a_{ij})\in\mathbb{R}^{m\times n}$, and bounds $\ell_j\in\mathbb{R}\cup\{-\infty\}$, $u_j\in\mathbb{R}\cup\{+\infty\}$. For each inequality row $i$ a set $J_i\subseteq\{1,\dots,n\}$ lists the uncertain entries $a_{ij}$, $j\in J_i$. Only these entries are uncertain; $E,e,b,\ell,u,c$ are exact.

Given an uncertainty level $\epsilon>0$ and a feasibility tolerance $\delta>0$, write $b_i^+=b_i+\delta\max[1,|b_i|]$.

• $x$ is reliable if it is feasible for (LP) and $\sum_{j\notin J_i}a_{ij}x_j+\sum_{j\in J_i}\tilde a_{ij}x_j\le b_i^+$ for every $i$ and every choice of $\tilde a_{ij}$ with $|\tilde a_{ij}-a_{ij}|\le\epsilon|a_{ij}|$.
• In the random symmetric uncertainty model, the true coefficients are $\tilde a_{ij}=(1+\epsilon\xi_{ij})a_{ij}$, where $\xi_{ij}=0$ for $j\notin J_i$ and, for each row $i$, $\{\xi_{ij}\}_{j\in J_i}$ are independent random variables, each symmetrically distributed in $[-1,1]$.
• $x$ is almost reliable with level $\kappa$ if it is feasible for (LP) and $\Pr\{\sum_j\tilde a_{ij}x_j>b_i^+\}\le\kappa$ for every $i$.

The robust counterpart (RC[ε, δ, Ω]), with a safety parameter $\Omega>0$, has variables $x_j$, $y_{ij}$, $z_{ij}$ and constraints $Ex=e$, $Ax\le b$, $\ell\le x\le u$, $-y_{ij}\le x_j-z_{ij}\le y_{ij}$ for all $i,j$, and

$$\sum_j a_{ij}x_j+\epsilon\Big[\sum_{j\in J_i}|a_{ij}|y_{ij}+\Omega\sqrt{\sum_{j\in J_i}a_{ij}^2z_{ij}^2}\Big]\le b_i^+\qquad\forall i.$$

The interval robust counterpart (IRC[ε, δ]) has variables $x_j,y_j$ and the constraint $\sum_ja_{ij}x_j+\epsilon\sum_{j\in J_i}|a_{ij}|y_j\le b_i^+$ with $-y_j\le x_j\le y_j$, besides the nominal ones. Problem (∗) is the same with $y_j$ replaced by $|x_j|$.

Formalization targets

Goal: Proposition 1 (pp. 418–419)

If $x$ extends to a feasible solution $(x,y,z)$ of (RC[ε, δ, Ω]), then $x$ is feasible for (LP) and, for every $i$,

$$\Pr\Big\{\sum_j(1+\epsilon\xi_{ij})a_{ij}x_j>b_i+\delta\max[1,|b_i|]\Big\}\le\exp\{-\Omega^2/2\}.$$

Milestones

1. The reduction in the proof of Proposition 1 (p. 419), in corrected pointwise form: a violation of row $i$ forces $\sum_{j\in J_i}\xi_{ij}a_{ij}z_{ij}>\Omega\sqrt{\sum_{j\in J_i}a_{ij}^2z_{ij}^2}$.
2. Eq. (1), p. 419: for independent symmetric $\eta_j\in[-1,1]$ and reals $p_j$,

$$\Pr\Big\{\sum_j\eta_jp_j>\Omega\sqrt{\textstyle\sum_jp_j^2}\Big\}\le\exp\{-\Omega^2/2\}.$$

3. $x$ is reliable iff it is feasible for (∗) (p. 417).
4. (∗) is equivalent to (IRC[ε, δ]) (pp. 417–418).
5. Every feasible solution of (IRC) yields one of (RC) with $y_{ij}=y_j$, $z_{ij}=0$ (p. 420).
6. Feasibility for (LP) together with $\sum_ja_{ij}x_j+\epsilon\beta_i(x)\le b_i^+$, $\beta_i(x)=\Omega\sqrt{\sum_{j\in J_i}a_{ij}^2x_j^2}$, suffices to extend $x$ to (RC) (p. 420).
7. The ratio $\alpha_i(x)/\beta_i(x)$, $\alpha_i(x)=\sum_{j\in J_i}|a_{ij}||x_j|$, is at most $\sqrt{\mathrm{card}(J_i)}/\Omega$, with equality attained (p. 420, corrected).

Significance

Proposition 1 turns a probabilistic requirement, which is hard to handle directly, into a single convex (second-order-cone) program. The bound $\exp\{-\Omega^2/2\}$ does not depend on the dimension, on the number of uncertain entries, or on which symmetric distributions the perturbations follow, so $\Omega$ can be chosen from the desired reliability level alone. Together with milestones 3–6, the mission certifies the whole chain: the worst-case notion of reliability is exactly Soyster's linear program (IRC), and (RC) is never more conservative than (IRC), with an advantage that can reach the factor $\sqrt{\mathrm{card}(J_i)}/\Omega$.

The results are proved in the paper; to our knowledge none of them is machine-checked. A formal development produces a reusable model of entry-wise uncertain LPs, the counterparts (∗), (IRC) and (RC) as Lean predicates, and a Hoeffding-type bound for weighted sums of symmetric bounded variables in the exact form (1). The platform's HighDimProb.Concentration.hoeffding_rademacher covers the Rademacher special case only.

Difficulty

The deterministic parts (milestones 1, 3–7) are elementary: worst cases of interval perturbations, and the Cauchy–Schwarz inequality. The obstacle lies in the probabilistic step. The printed proof passes from $\xi_{ij}a_{ij}$ to $\xi_{ij}|a_{ij}|$ with an equality that holds only in distribution, and contains index misprints, so it cannot be transcribed line by line; the reduction has to be restated pointwise. Eq. (1) is a tail bound for general symmetric variables in $[-1,1]$, not only for random signs; the step (c) of the printed proof of (1) is written as an equality that holds only for random signs, so that proof too needs repair. The degenerate case $\sum_jp_j^2=0$ must be handled rather than assumed away.

Formalization scope

• Data are a structure UncertainLP n p m over Fin indices (0-based), with A : Matrix (Fin m) (Fin n) ℝ, J : Fin m → Finset (Fin n) arbitrary, and EReal bounds so that infinite bounds are expressible. The objective $c$ is omitted: no statement involves it.
• The probability space is $(S,\mathbb P)$ with IsProbabilityMeasure; the name $S$ avoids a clash with the safety parameter $\Omega$. Symmetry is equality of the laws of $\xi_{ij}$ and $-\xi_{ij}$; values lie in $[-1,1]$ at every outcome; independence is required within each row only, with no identical distribution (§3.1 says only "independent", which is weaker than the "iid" of §2.2). Probabilities are P.real.
• The hypotheses $\epsilon>0$, $\delta>0$, $\Omega>0$ are the paper's standing assumptions and are carried by every theorem that mentions the parameter.
• Corrections of the printed text: $a_{ij}x_i\to a_{ij}x_j$ in (IRC); $\sum_{j\in J}\to\sum_{j\in J_i}$ in (RC); the reduction of milestone 1 is stated with $a_{ij}$ and $z_{ij}$ in place of the printed $|a_{ij}|$, $x_i-y_{ij}$ and $y_j$, $y_{ij}$; and the ratio of milestone 7 carries the factor $1/\Omega$ that the printed "$\sqrt{\mathrm{card}(J_i)}$" omits.
• Ruling out trivializations: the violation event uses the signed multiplicative model $(1+\epsilon\xi_{ij})a_{ij}$, never $|a_{ij}|$ or an additive perturbation; the goal concludes both nominal feasibility (i) and the probability bound (ii′) for every row; no hypothesis excludes the degenerate case $\sum_{j\in J_i}a_{ij}^2z_{ij}^2=0$; and the probability model is satisfiable (e.g. by $\xi\equiv0$ or by Rademacher signs), so the goal is not vacuous.
• The numerical remarks of the paper (0.92, 5.24, $10^{-6}$, "at least 30") and the NETLIB case study are not formalized.
• Reusable beyond this mission: the uncertain-LP model and the three counterparts, and the tail bound (1). Contributions of general lemmas about symmetric bounded random variables are welcome.

Selected references

• A. Ben-Tal, A. Nemirovski, Robust solutions of Linear Programming problems contaminated with uncertain data, Math. Program. Ser. A 88 (2000) 411–424. https://doi.org/10.1007/s101070000163
• A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res. 21 (1973) 1154–1157. https://doi.org/10.1287/opre.21.5.1154
• A. Ben-Tal, A. Nemirovski, Robust convex optimization, Math. Oper. Res. 23 (1998) 769–805. https://doi.org/10.1287/moor.23.4.769
• A. Ben-Tal, A. Nemirovski, Robust solutions of uncertain linear programs, Oper. Res. Lett. 25 (1999) 1–13. https://doi.org/10.1016/S0167-6377(99)00016-4
• D. Bertsimas, M. Sim, The price of robustness, Oper. Res. 52 (2004) 35–53. https://doi.org/10.1287/opre.1030.0065
• W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963) 13–30. https://doi.org/10.1080/01621459.1963.10500830

Motivation

Algorithmic stability asks how much the output of a learning algorithm changes when its training sample is perturbed. Since Devroye and Wagner (IEEE Trans. Inf. Theory 1979) it has served as a route to generalization bounds that does not go through the complexity of the hypothesis class. Bousquet and Elisseeff (JMLR 2002) popularised uniform stability, and Mukherjee, Niyogi, Poggio and Rifkin (Adv. Comput. Math. 2006) showed that for empirical risk minimisation in supervised learning, a leave-one-out type of stability is necessary and sufficient for consistency.

Shalev-Shwartz, Shamir, Srebro and Sridharan (JMLR 11, 2010) study stability in Vapnik's General Learning Setting, where uniform convergence can fail even though the problem is learnable. In Appendix A.2 they compare replace-one and leave-one-out (LOO) stability. For an empirical risk minimiser they prove that LOO stability is equivalent to consistency and to generalization, provided each property holds with one rate for all distributions (Theorem 31, p. 2667). This mission formalizes that theorem and the lemmas of Section 5.3 on which its proof rests.

Timeline:

• 1979, Devroye–Wagner: leave-one-out estimates for local rules.
• 2002, Bousquet–Elisseeff: uniform stability implies generalization.
• 2002, Kutin–Niyogi (UAI 2002): a taxonomy of stability notions.
• 2006, Mukherjee et al.: LOO stability characterises consistency of ERM in supervised learning.
• 2010, Shalev-Shwartz et al.: in the General Learning Setting, for ERMs, LOO stability, universal consistency and universal generalization are equivalent (Theorem 31). Universally consistent AERMs need not be LOO stable (Example 6).

Setting

A learning problem consists of an instance space $\mathcal Z$ with a $\sigma$-algebra, a nonempty hypothesis class $\mathcal H$, and an objective $f:\mathcal H\times\mathcal Z\to\mathbb R$ with $|f(h;z)|\le B$ for all $h,z$. For a probability measure $\mathcal D$ on $\mathcal Z$:

• the risk is $F(h)=\mathbb E_{z\sim\mathcal D}[f(h;z)]$ and the optimal risk is $F^*=\inf_{h}F(h)$;
• for a sample $S=(z_1,\dots,z_m)\sim\mathcal D^m$ of $m$ i.i.d. draws, the empirical risk is $F_S(h)=\frac1m\sum_{i}f(h;z_i)$, and $F_S(\hat h_S)=\inf_hF_S(h)$ denotes the minimal empirical risk;
• a learning rule $A$ maps each sample of size $m\ge1$ to a hypothesis $A(S)$. It is an ERM if $F_S(A(S))=F_S(\hat h_S)$ for every sample. It is an AERM with rate $\varepsilon_{\mathrm{erm}}$ if $\mathbb E[F_S(A(S))-F_S(\hat h_S)]\le\varepsilon_{\mathrm{erm}}(m)$;
• $A$ is consistent with rate $\varepsilon$ if $\mathbb E_{S\sim\mathcal D^m}[F(A(S))-F^*]\le\varepsilon(m)$. It generalizes with rate $\varepsilon$ if $\mathbb E[|F(A(S))-F_S(A(S))|]\le\varepsilon(m)$, and it on-average generalizes if $|\mathbb E[F(A(S))-F_S(A(S))]|\le\varepsilon(m)$;
• writing $S^{\setminus i}$ for $S$ with $z_i$ removed, $A$ is LOO stable with rate $\varepsilon$ (Definition 29) if

$$\frac1m\sum_{i=1}^m\mathbb E_{S\sim\mathcal D^m}\Big[\big|f(A(S^{\setminus i});z_i)-f(A(S);z_i)\big|\Big]\le\varepsilon(m).$$

A rate is a sequence $\varepsilon(m)$ that is non-increasing and tends to $0$. A property holds universally if it holds under every $\mathcal D$ with one and the same rate.

Formalization targets

Goal: Theorem 31

For an ERM $A$:

$$A\ \text{universally LOO stable}\iff A\ \text{universally consistent}\iff A\ \text{universally generalizes}.$$

The statement has no rates. Each side asserts that some rate exists and serves every distribution.

Milestones

In the order the proof uses them:

1. Utility Lemma 12: $\mathbb E|X-\mathbb EX|\le B/\sqrt m$ for the mean $X$ of $m$ i.i.d. variables bounded by $B$.
2. Utility Lemma 13: $X\le Y$ a.s. implies $\mathbb E|X|\le|\mathbb EX|+2\mathbb E|Y|$.
3. Lemma 14: an AERM that on-average generalizes with rate $\varepsilon_{\mathrm{oag}}$ generalizes with rate $\varepsilon_{\mathrm{oag}}+2\varepsilon_{\mathrm{erm}}+2B/\sqrt m$.
4. Lemma 15: under the same hypotheses the rule is consistent with rate $\varepsilon_{\mathrm{oag}}+\varepsilon_{\mathrm{erm}}$.
5. Lemma 16 (Main Converse Lemma): in a learnable problem, $\mathbb E|F_S(\hat h_S)-F^*|\le2\varepsilon_{\mathrm{cons}}(m')+2B/\sqrt m+2Bm'^2/m$ for $2\le m'\le m/2$.
6. Lemma 17: Eq. (12), together with an AERM that is consistent, gives generalization with rate $\varepsilon_{\mathrm{emp}}+\varepsilon_{\mathrm{erm}}+\varepsilon_{\mathrm{cons}}$.
7. First display of the proof of Theorem 31: a generalizing ERM is LOO stable with rate $\varepsilon_{\mathrm{gen}}(m-1)$.
8. Second display: a LOO stable ERM on-average generalizes on samples of size $m-1$ with rate $\varepsilon_{\mathrm{stable}}(m)+2B/m$.

Significance

Theorem 31 shows that for exact ERMs, LOO stability is not only sufficient but necessary for consistency. It transfers the supervised-learning characterisation of Mukherjee et al. to the General Learning Setting, where uniform convergence is no longer available as an intermediate. The hypothesis is sharp in one direction: Example 6 of the paper gives a universally consistent AERM that is not LOO stable. The equivalence therefore depends on exact minimisation, and an asymptotic minimiser does not suffice.

The lemmas are useful on their own. Lemmas 14 and 15 are the standard bridges between on-average generalization, generalization and consistency. Lemma 16 says that the minimal empirical risk estimates $F^*$ consistently in every learnable problem, even when no ERM learns. Lemma 16 also underlies Theorem 7, the paper's main characterisation of learnability, which is the subject of mission I of this series.

The results are proved in the paper. As far as is known they have not been formalized in any proof assistant. The platform has the textbook side of this framework (Shalev-Shwartz and Ben-David, Understanding Machine Learning, Chapter 13). Those statements are in $\mathbb R^d$ and use a replace-one stability notion; they do not cover leave-one-out stability or the General Learning Setting.

Difficulty

The implications between stability and generalization change the sample size: $S^{\setminus i}$ has $m-1$ points, so a statement about the rule at size $m$ has to be compared with the rule at size $m-1$ under the marginal law of the reduced sample. The step from universal consistency to generalization needs Lemma 16. That lemma estimates $F^*$ from a sample on which the ERM itself may be inconsistent, and it is the only place where universality of the consistency rate is used. Per-distribution consistency of an ERM does not imply generalization (Example 1 of the paper). An argument that fixes $\mathcal D$ throughout therefore cannot succeed.

Formalization scope

Samples are tuples $S:\mathrm{Fin}\,m\to\mathcal Z$ with law Measure.pi (the i.i.d. product), and $S^{\setminus i}$ is Fin.removeNth i S. A learning rule is a family $A_m:\mathcal Z^m\to\mathcal H$. Its value at $m=0$ is never used, and LOO stability is required only for $m\ge2$. $F_S(\hat h_S)$ is the infimum $\inf_hF_S(h)$ and no minimiser is chosen. An ERM is a rule attaining this infimum at every sample. A rate is non-increasing on $m\ge1$ and tends to $0$. Universal properties are stated as "there exists $\varepsilon$ with IsRate ε such that for every probability measure $\mathcal D$ …", with the rate chosen before the distribution. The bound $B$ is any bound on $|f|$; the paper's $B$ is $\sup|f|$, and all its rates increase with $B$.

Measurability is not discussed in the paper. The formalization assumes that each $f(h;\cdot)$ is measurable and that the rule is measurable in the sense that $(S,z)\mapsto f(A_m(S);z)$ is jointly measurable. Lemmas 14–17 also assume that $S\mapsto\inf_hF_S(h)$ is measurable. For a measurable ERM this holds automatically, so Theorem 31 makes no such assumption. Without these assumptions Lean's integral of a non-measurable function is $0$ and every rate bound would hold trivially. For the same reason Utility Lemma 13 assumes $X,Y$ integrable. The ERM hypothesis of the goal must not be weakened to an AERM: the statement would then be false (Example 6).

One statement corrects the printed text. In the second display of the proof of Theorem 31 (p. 2668), the chain adds $2B/m$ and then drops it. The milestone states the bound the argument proves, $\varepsilon_{\mathrm{stable}}(m)+2B/m$. The rate-free Theorem 31 is unaffected.

The development needs product measures, independence and variance bounds, all available in Mathlib, and the marginals of Measure.pi under removal of a coordinate. The definitions of risks, rules and stability notions can be reused by other stability results. Proofs of any milestone are welcome, and so are alternative proofs of the goal.

Selected references

• S. Shalev-Shwartz, O. Shamir, N. Srebro, K. Sridharan, Learnability, Stability and Uniform Convergence, Journal of Machine Learning Research 11 (2010) 2635–2670. https://jmlr.org/papers/v11/shalev-shwartz10a.html
• O. Bousquet, A. Elisseeff, Stability and Generalization, Journal of Machine Learning Research 2 (2002) 499–526. https://www.jmlr.org/papers/v2/bousquet02a.html
• S. Mukherjee, P. Niyogi, T. Poggio, R. Rifkin, Learning theory: stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization, Advances in Computational Mathematics 25 (2006) 161–193. https://doi.org/10.1007/s10444-004-7634-z
• S. Kutin, P. Niyogi, Almost-everywhere algorithmic stability and generalization error, UAI 2002. https://arxiv.org/abs/1301.0579
• L. Devroye, T. Wagner, Distribution-free performance bounds for potential function rules, IEEE Transactions on Information Theory 25(5) (1979) 601–604. https://doi.org/10.1109/TIT.1979.1056087