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Theoretical Computer Science

182 missions · 115 completed

The mathematical foundations of computation: which problems can be solved, by what algorithms, and at what cost in time, space, or communication. Distinguished by its emphasis on rigor and unconditional lower bounds, it spans computational complexity, algorithm design, automata and computability, cryptography, and the analysis of Boolean functions.

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Combinatorics Operations Research Optimization+1·Captain: mikedeng1

Maximizing Non-Monotone Submodular Functions I: A Uniformly Random Set Achieves 1/4 of the Optimum, and 1/2 for Symmetric FunctionsResearch Paper

Motivation

Many combinatorial optimization problems ask for a subset of a finite ground set that maximizes a set function with diminishing returns: Max Cut and Max Directed Cut in graphs, facility location, maximum entropy sampling, and welfare problems in combinatorial auctions all fit this pattern. The common abstraction is the maximization of a submodular function, the discrete analogue of a concave function. Unlike the monotone case, where the objective only grows as elements are added, the non-monotone problem has no constraint at all and is still NP-hard, since Max Cut is a special case.

For Max Cut and Max Directed Cut, the simplest algorithm there is, putting every vertex on a side by an independent fair coin, already cuts half, respectively a quarter, of the optimum in expectation. Feige, Mirrokni and Vondrák (SIAM J. Comput. 40(4), 2011; extended abstract at FOCS 2007) showed that this is not a feature of cut functions: the same random choice achieves the same factors for every nonnegative submodular function, and for every symmetric one. This mission formalizes that result, Theorem 2.1 of the paper, together with the two sampling lemmas on which it rests. The paper's other results (a nonadaptive 1/3-approximation, deterministic and smoothed local search, and query lower bounds) are the subjects of companion missions in the same series.

Setting

Let $X$ be a finite set with $n = |X|$ elements. A set function assigns a real number $f(S)$ to every subset $S \subseteq X$ . It is submodular if

f(S \cup T) + f(S \cap T) \le f(S) + f(T) \qquad \text{for all } S, T \subseteq X,

equivalently if the marginal value $f(B \cup \{x\}) - f(B)$ of an element $x$ does not increase as the set $B$ grows. It is symmetric if $f(X \setminus S) = f(S)$ for every $S \subseteq X$ ; the cut function of an undirected graph is the standard example. The optimum is

OPT = \max_{S \subseteq X} f(S).

For $p \in [0,1]$ , $X(p)$ denotes the random subset of $X$ containing each element independently with probability $p$ ; similarly $A(p)$ is the random subset of a fixed $A \subseteq X$ . The Random Set Algorithm (RS) returns $R = X(1/2)$ , a uniformly random subset of $X$ , without querying $f$ . Its expected value is the average of $f$ over all subsets,

\mathbf{E}[f(R)] = F(\tfrac12, \dots, \tfrac12) = \frac{1}{2^n} \sum_{S \subseteq X} f(S),

where $F(x) = \sum_{S \subseteq X} f(S) \prod_{i \in S} x_i \prod_{i \notin S} (1 - x_i)$ is the multilinear extension of $f$ , the expectation of $f$ on a random set that includes element $i$ independently with probability $x_i$ .

Formalization targets

Goal: Theorem 2.1

For every nonnegative submodular $f : 2^X \to \mathbb{R}_+$ ,

\mathbf{E}[f(X(1/2))] \ge \tfrac14\, OPT,

and if $f$ is in addition symmetric,

\mathbf{E}[f(X(1/2))] \ge \tfrac12\, OPT.

Both parts form the goal, stated as one theorem. The constants $\tfrac14$ and $\tfrac12$ are exact, not asymptotic, and they are tight: the directed cut of a single arc attains $\tfrac14$ , and the cut of a single edge attains $\tfrac12$ .

Milestones

Lemma 2.2. For submodular $g : 2^X \to \mathbb{R}$ , $A \subseteq X$ and $p \in [0,1]$ ,

\mathbf{E}[g(A(p))] \ge (1-p)\, g(\emptyset) + p\, g(A).

Lemma 2.3. For submodular $f : 2^X \to \mathbb{R}$ , sets $A, B \subseteq X$ that need not be disjoint, independent samples $A(p)$ , $B(q)$ , and $p, q \in [0,1]$ ,

\mathbf{E}[f(A(p) \cup B(q))] \ge (1-p)(1-q) f(\emptyset) + p(1-q) f(A) + (1-p)q f(B) + pq f(A \cup B).

The display in the proof of Theorem 2.1. For submodular $f : 2^X \to \mathbb{R}$ and every $S \subseteq X$ , with $\bar S = X \setminus S$ ,

\mathbf{E}[f(X(1/2))] \ge \tfrac14 f(\emptyset) + \tfrac14 f(S) + \tfrac14 f(\bar S) + \tfrac14 f(X).

The milestones need no sign on the function; nonnegativity enters only in the goal.

Significance

The result. Theorem 2.1 gives an algorithm that makes no query at all and is still a constant-factor approximation for unconstrained non-monotone submodular maximization. It sets the baseline that every later algorithm for the problem is measured against: the paper's own nonadaptive $\tfrac13$ -algorithm and its local search algorithms with factors $\tfrac13$ and $\tfrac25$ , followed by later work culminating in the tight $\tfrac12$ -approximation of Buchbinder, Feldman, Naor and Schwartz (FOCS 2012). The paper also shows that $\tfrac14$ is optimal among nonadaptive algorithms required to return one of the queried sets, and that $\tfrac12$ is optimal for symmetric functions among all algorithms using polynomially many value queries, so both factors of Theorem 2.1 have a precise place in the complexity landscape. Lemma 2.3, the probabilistic inequality behind it, is reused in the analyses of the nonadaptive algorithm and of smooth local search.

Formalizing it. The result is proved, with a short proof. What this mission adds is a machine-checked version of the random-set guarantee and of the two sampling lemmas, stated for arbitrary finite ground sets and, for the lemmas, for real-valued submodular functions without a sign. To our knowledge none of these statements has a machine-checked proof; Mathlib has no theory of submodular set functions or of their multilinear extension.

Difficulty

The goal itself is a two-line consequence of the third milestone. The work sits in the lemmas and in one change of viewpoint.

Lemma 2.2 is not a pointwise statement: the random set $A(p)$ can be any subset of $A$ , and $g$ can be smaller on it than both $g(\emptyset)$ and $g(A)$ . The inequality holds only in expectation, and only because submodularity controls the marginal value of each element uniformly across the sets it can be added to. Lemma 2.3 needs a conditioning argument over two independent samples; the sets $A$ and $B$ may overlap, and on $A \cap B$ the union $A(p) \cup B(q)$ contains an element with probability $1 - (1-p)(1-q)$ , so it is not the product distribution with probability $p$ on $A$ and $q$ on $B$ . Finally, the third milestone requires identifying the uniform random subset $X(1/2)$ with the union of independent half-samples of $S$ and of its complement, as a statement about finite sums.

The obvious attempt at the goal, comparing $f(R)$ with $f(S^*)$ for an optimal $S^*$ set by set, fails: $f$ is not monotone, so a random set that contains most of $S^*$ may still have small value, and a random set can pick up elements that hurt.

Formalization scope

The ground set is a Lean type X with [Fintype X] [DecidableEq X]; subsets are Finset X and set functions are f : Finset X → ℝ. Submodularity is the lattice inequality of Definition 1.1, not the decreasing-marginals property. Nonnegativity, the paper's standing assumption $f : 2^X \to \mathbb{R}_+$ , is the hypothesis ∀ S, 0 ≤ f S; it appears only in the goal. Symmetry is ∀ S, f Sᶜ = f S for all subsets, not only for an optimal one. $OPT$ is Finset.univ.sup' Finset.univ_nonempty f, a maximum over the always nonempty family of all subsets, so it is attained. The ground set may be empty; the goal holds there too and no nonemptiness is assumed.

Expectations are written as exact finite sums, not as integrals. $\mathbf{E}[f(X(1/2))]$ is the multilinear extension F f (fun _ => 1/2). $\mathbf{E}[g(A(p))]$ is $\sum_{T \subseteq A} p^{|T|}(1-p)^{|A \setminus T|} g(T)$ , and $\mathbf{E}[f(A(p) \cup B(q))]$ is the double sum over independent samples $S \subseteq A$ , $T \subseteq B$ with the product of the two weights. The ranges $0 \le p \le 1$ and $0 \le q \le 1$ , implied in the paper by the word "probability", are explicit hypotheses; Lemma 2.2 is false without them.

Trivializing formalizations are excluded: the weights are exactly those of the uniform distribution on all $2^n$ subsets, $OPT$ is the true maximum rather than the value at one fixed set, and $f$ is required to be both nonnegative and submodular.

Reusable infrastructure produced by a complete development: the multilinear extension of a set function and its expression as an expectation, product-weight identities for independent sampling of subsets (including the decomposition of $X(1/2)$ along a set and its complement), and Lemmas 2.2 and 2.3, which the companion missions on the nonadaptive algorithm and on smooth local search also need. Proofs of any milestone are welcome independently.

Selected references

U. Feige, V. S. Mirrokni, J. Vondrák, Maximizing Non-Monotone Submodular Functions, SIAM Journal on Computing 40(4):1133–1153, 2011. https://doi.org/10.1137/090779346
U. Feige, V. S. Mirrokni, J. Vondrák, Maximizing non-monotone submodular functions, Proceedings of the 48th IEEE Symposium on Foundations of Computer Science (FOCS), 2007, pp. 461–471. https://doi.org/10.1109/FOCS.2007.29
N. Buchbinder, M. Feldman, J. Naor, R. Schwartz, A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization, SIAM Journal on Computing 44(5):1384–1402, 2015 (FOCS 2012). https://doi.org/10.1137/130929205
G. L. Nemhauser, L. A. Wolsey, M. L. Fisher, An analysis of approximations for maximizing submodular set functions — I, Mathematical Programming 14:265–294, 1978. https://doi.org/10.1007/BF01588971

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Combinatorics Operations Research Optimization·Captain: mikedeng1

Approximation Techniques for Average Completion Time Scheduling IV: List Scheduling from an Optimal One-Machine Schedule Is a 2-Approximation for In-TreesResearch Paper

Motivation

Minimizing the sum of weighted completion times of jobs on identical parallel machines is one of the basic objectives of machine scheduling: it measures the average time a job spends in the system, weighted by its importance. When the jobs are subject to precedence constraints (a job may start only after certain other jobs have finished), the problem is strongly NP-hard already in very restricted cases, and the question becomes how close to optimal a polynomial-time algorithm can guarantee to be.

Chekuri, Motwani, Natarajan and Stein, Approximation Techniques for Average Completion Time Scheduling (SIAM J. Comput. 31(1), 2001, doi:10.1137/S0097539797327180), develop a general way to turn a good schedule for a single machine into a good schedule for $m$ machines. For arbitrary precedence constraints their conversion (Delay List, §4.1–4.3) loses a factor $(1+\beta)\rho+(1+1/\beta)$ over a $\rho$ -approximate one-machine schedule, which is $4$ when the one-machine schedule is optimal. In §4.4 they show that for in-tree precedence without release dates, the plain list-scheduling rule of Graham, fed with an optimal one-machine schedule, already achieves ratio $2$ . In-trees are the precedence structures of assembly processes: every job feeds into at most one later job.

Timeline of the relevant results:

1966–1969: Graham introduces list scheduling on parallel machines and analyzes it for makespan (Graham 1969).
1972: Horn gives a polynomial-time optimal one-machine algorithm for weighted completion time under treelike precedence (Horn 1972).
1977: Adolphson gives $O(n\log n)$ one-machine algorithms for tree and series-parallel precedence (Adolphson 1977, the paper's reference [1]).
2001: Chekuri, Motwani, Natarajan and Stein prove the ratio- $2$ bound for in-trees on $m$ machines (Theorem 4.17).

Setting

There are $n$ jobs $J_0,\dots,J_{n-1}$ and $m\ge 1$ identical machines. Job $J_j$ has a processing time $p_j>0$ and a weight $w_j>0$ ; every job is available at time $0$ (there are no release dates).

The precedence constraints form an in-tree (more generally, an in-forest): every job $j$ has at most one immediate successor $\operatorname{succ}(j)$ , and following successors never returns to the start. Write $i\prec j$ if $j$ is reached from $i$ by following successors one or more times.

A feasible schedule $S^m$ on $m$ machines gives each job a start time $S_j\ge 0$ and a machine; a job runs without interruption for $p_j$ time units; two jobs on the same machine do not overlap; and $i\prec j$ implies that $j$ starts no earlier than $i$ completes. The completion time is $C^m_j=S_j+p_j$ and the value of the schedule is $\sum_j w_jC^m_j$ .

The critical-path length $\kappa_j$ (Definition 4.1 with no release dates) is $\kappa_j=p_j$ if $j$ has no predecessors and $\kappa_j=p_j+\max_{i\prec j}\kappa_i$ otherwise.

A list is an ordering $\pi$ of the jobs that obeys the precedence constraints. It defines the one-machine schedule $S^1$ that runs the jobs in list order without idle time; its completion times are $C^1_j$ , the total processing time of the jobs up to and including $j$ in the list. An optimal one-machine schedule is a list minimizing $C^1=\sum_j w_jC^1_j$ .

List scheduling (Graham's rule, footnote 3 of the paper) on $m$ machines with list $\pi$ : whenever a machine is free, start on it the first job of the list that is ready, i.e. whose predecessors have all completed.

Formalization targets

Goal: Theorem 4.17

Let $\pi$ be an optimal one-machine schedule and $G$ the list schedule on $m$ machines with list $\pi$ . Then for every feasible $m$ -machine schedule $N$ ,

\sum_j w_jC^G_j\ \le\ 2\sum_j w_jC^N_j .

Milestones

Lemma 4.16 (any precedence-respecting list $\pi$ , with its idle-free one-machine schedule $S^1$ ): for every job $i$ ,

C^G_i\ \le\ \kappa_i+\frac{C^1_i}{m}.

Lemma 4.10: $C^m_{\mathrm{OPT}}\ge C^1_{\mathrm{OPT}}/m$ , i.e. $\sum_j w_jC^1_j/m\le\sum_j w_jC^N_j$ for an optimal list and every feasible $N$ .

Lemma 4.11: $C^m_{\mathrm{OPT}}\ge\sum_i w_i\kappa_i=C^\infty_{\mathrm{OPT}}$ , i.e. $\sum_i w_i\kappa_i\le\sum_i w_iC^N_i$ for every feasible $N$ on any number of machines, and the value $\sum_i w_i\kappa_i$ is attained by a feasible schedule on $n$ machines.

Significance

The result. Theorem 4.17 gives a simple, fast algorithm with a guaranteed factor $2$ for a strongly NP-hard problem, halving the factor $4$ that the general Delay List conversion gives for the same class. The per-job bound of Lemma 4.16 is stronger than the aggregate statement: every single job completes within its critical-path length plus a $1/m$ share of its one-machine completion time, so the same bound applies to other objectives built from completion times.

Formalizing it. The paper's proof is complete and short, but it argues about events at a time $t$ (jobs that finish exactly at $t$ , jobs that become ready at $t$ , machines freed at $t$ ) and runs an induction over jobs ordered by start time with an invariant about idle time. A machine-checked version fixes what "list scheduling" means precisely, pins down the counting argument that uses the in-tree structure, and yields reusable definitions of nonpreemptive parallel-machine schedules, critical paths and list schedules. To the knowledge of this mission, none of these results has a machine-checked proof.

Difficulty

List scheduling may start a job that is late in the list before an earlier one, because the earlier job is not yet ready; so the one-machine order is not preserved and the obvious comparison with $S^1$ fails. Idle machines are the other obstacle: a machine can stay idle while a job waits for its predecessors, and a per-job bound of the form $\kappa_i+C^1_i/m$ holds only if such idle time can be accounted for by $J_i$ 's own chain of predecessors. For general precedence constraints, and for out-trees (every job has at most one immediate predecessor), the paper's accounting breaks down, and the paper states the per-job bound only for in-trees; the in-tree structure is essential to the argument. Events with several jobs finishing at the same instant, and ties in start times, have to be handled without loss.

Formalization scope

Jobs are Fin n, machines Fin m, times real numbers. Processing times and weights are strictly positive. There are no release dates: start times are nonnegative. The paper admits $p_j=0$ only in lower-bound instances elsewhere; the bounds here assume $p_j>0$ .
In-trees are encoded by an immediate-successor map succ : Fin n → Option (Fin n) with no cycles; this covers in-forests, the reading of "in-trees" in Theorem 4.17. The precedence relation is its transitive closure.
$\kappa$ is defined by well-founded recursion on the precedence order, exactly as Definition 4.1 with $r\equiv 0$ .
One-machine schedules are represented by their precedence-respecting order and are idle-free; with no release dates and positive processing times idle time only delays jobs, so optimality among orders is optimality among one-machine schedules. The optimal one-machine schedule is a hypothesis of the goal; the paper's $O(n\log n)$ algorithm for computing it (reference [1]) is not formalized, and the running-time claim of Theorem 4.17 is not stated. A separate item asserts that an optimal order exists.
List scheduling is specified by two properties that determine Graham's rule up to machine labels: no machine is idle while a ready job waits, and among jobs ready at a start time the earlier one in the list starts first. A separate item asserts that such a schedule exists for every precedence-respecting list, so the goal is not vacuous.
Optima are never formed as infima: the approximation ratio is stated against every feasible schedule. A statement of the form "there is an algorithm with ratio 2" would be trivial (an optimal schedule exists) and is ruled out: the goal is about the paper's algorithm.
The equality $\sum_i w_i\kappa_i=C^\infty_{\mathrm{OPT}}$ in Lemma 4.11 is stated as attainment on $n$ machines (as many machines as jobs), which together with the lower bound on every number of machines is the optimum with unboundedly many machines.

Welcome contributions: proofs of the two existence items (Graham's list schedule by event-driven construction; an optimal order over the finite set of linear extensions), of Lemmas 4.10 and 4.11, and of Lemma 4.16. The schedule and list-scheduling definitions are reusable for other parallel-machine results with precedence constraints.

Selected references

C. Chekuri, R. Motwani, B. Natarajan, C. Stein, Approximation Techniques for Average Completion Time Scheduling, SIAM J. Comput. 31(1):146–166, 2001. https://doi.org/10.1137/S0097539797327180
R. L. Graham, Bounds on multiprocessing timing anomalies, SIAM J. Appl. Math. 17(2):416–429, 1969. https://doi.org/10.1137/0117039
W. A. Horn, Single-machine job sequencing with treelike precedence ordering and linear delay penalties, SIAM J. Appl. Math. 23(2):189–202, 1972. https://doi.org/10.1137/0123021
D. L. Adolphson, Single machine job sequencing with precedence constraints, SIAM J. Comput. 6(1):40–54, 1977. https://doi.org/10.1137/0206002

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Combinatorics Operations Research Optimization·Captain: mikedeng1

Approximation Techniques for Average Completion Time Scheduling III: From One Machine to Many with Delay ListResearch Paper

Motivation

Minimizing the sum of weighted completion times $\sum_j w_jC_j$ is one of the standard objectives of machine scheduling: it measures the average time a job spends in the system, weighted by its importance. With release dates or precedence constraints the problem is NP-hard already on one machine, and on $m$ identical parallel machines it is harder still, so the literature of the 1990s concentrated on approximation algorithms. Many of these, including LP-based ones, are naturally designed for a single machine, where an order of the jobs determines the schedule.

Chekuri, Motwani, Natarajan and Stein (SIAM J. Comput. 31(1), 2001) gave a generic way to move from one machine to many. Their §4 describes an algorithm, Delay List, that takes any one-machine schedule as a priority list and produces an $m$ -machine schedule, and proves that a $\rho$ -approximate one-machine schedule yields a $\bigl((1+\beta)\rho+1+1/\beta\bigr)$ -approximate $m$ -machine schedule for every $\beta>0$ . The guarantee holds with release dates and arbitrary precedence constraints simultaneously, which at the time gave the best bounds known for several special cases, for example a factor 4 for series-parallel precedence without release dates.

Setting

An instance has $n$ jobs $J_0,\dots,J_{n-1}$ . Job $J_j$ has processing time $p_j>0$ , release date $r_j\ge 0$ and weight $w_j>0$ . Precedence constraints form a strict partial order $\prec$ : $i\prec j$ means that $J_j$ may start only after $J_i$ completes.

A feasible nonpreemptive schedule on $m$ machines assigns each job a start time $S_j$ and a machine; each job runs uninterrupted for $p_j$ time units on its machine, two jobs on one machine do not overlap, $S_j\ge r_j$ , and $S_i+p_i\le S_j$ whenever $i\prec j$ . The completion time is $C_j=S_j+p_j$ and the value of the schedule is $\sum_j w_jC_j$ . A one-machine schedule is the case $m=1$ .

The critical-path length $\kappa_j$ (Definition 4.1) is $p_j+r_j$ for a job without predecessors and $p_j+\max\{\max_{i\prec j}\kappa_i,\,r_j\}$ otherwise; it is the earliest time $J_j$ could complete with unlimited machines.

A list is an ordering $\pi$ of the jobs. Delay List with parameter $\beta>0$ processes time continuously. A job is ready once it is released and all its predecessors have completed; $q^m_j$ is the time it becomes ready. The head is the first unscheduled job of the list. Idle machine-time is recorded as charged to jobs. Whenever a machine is idle:

if the head is ready, it is started, and charged all uncharged idle time in $(q^m_j,s^m_j)$ ;
otherwise the first ready job $J_k$ of the list is started as soon as at least $\beta p_k$ units of uncharged idle time have accumulated, and is charged $\beta p_k$ of it;
otherwise nothing happens.

For a job $J_i$ , $B_i$ is the set of jobs up to and including $J_i$ in the list, $A_i$ the set after it, $O_i\subseteq A_i$ the set of jobs of $A_i$ started before $J_i$ , and $p(A)=\sum_{k\in A}p_k$ . Definition 4.4 builds from the schedule a backward path $P'_i$ ending at $J_i$ , whose length is $\kappa'_i$ .

Formalization targets

Goal: Theorem 4.13

Let $S^1$ be a feasible one-machine schedule of the instance with $\sum_j w_jC^1_j\le\rho\sum_j w_jC'_j$ for every feasible one-machine schedule $C'$ . Let $m\ge 2$ and $\beta>0$ . Every Delay List schedule $S^m$ built on the completion order of $S^1$ satisfies, for every feasible $m$ -machine schedule $N$ ,

\sum_j w_jC^m_j\le\Bigl((1+\beta)\rho+1+\frac1\beta\Bigr)\sum_j w_jC^N_j .

Milestones, in the order the proof uses them

Fact 4.5: $\kappa'_i\le\kappa_i$ .
Fact 4.6: the idle time charged to $J_i$ is at most $\beta p_i$ .
Lemma 4.7: no uncharged idle time remains in $(q^m_i,s^m_i)$ , and that idle time is charged only to jobs in $B_i$ .
Lemma 4.8: the idle time charged to $A_i$ within $(0,s^m_i)$ is at most $m(\kappa'_i-p_i)$ , so $p(O_i)\le m(\kappa'_i-p_i)/\beta\le m(\kappa_i-p_i)/\beta$ .
Theorem 4.9: $C^m_i\le(1+\beta)p(B_i)/m+(1+1/\beta)\kappa'_i-p_i/\beta$ for any list obeying precedence.
Lemma 4.10: $C^m_{\mathrm{OPT}}\ge C^1_{\mathrm{OPT}}/m$ .
Lemma 4.11: $C^m_{\mathrm{OPT}}\ge\sum_i w_i\kappa_i=C^\infty_{\mathrm{OPT}}$ .
Corollary 4.12: $C^m_i\le(1+\beta)C^1_i/m+(1+1/\beta)\kappa_i$ when the list is the completion order of $S^1$ .

A further item states that a Delay List schedule exists for every instance and every list, so that the goal does not hold vacuously.

Significance

The result. Theorem 4.13 turns every one-machine approximation algorithm for weighted completion time with release dates and precedence into an $m$ -machine algorithm at a bounded loss. With an optimal one-machine schedule and $\beta=1$ the factor is $4$ (Corollary 4.14, for series-parallel orders), and the bounds are job-by-job (Theorem 4.9, Corollary 4.12), which the paper uses in Remark 4.15 to extend the method to other metrics and to one-machine schedules that ignore release dates. The same algorithm is the engine of the paper's $2\sqrt2$ -approximation for parallel machines with release dates (§4.5).

Formalizing it. The theorem has been proved since 1997 (SODA) and 2001 (journal). There is no machine-checked version of it or of any of its lemmas, and the platform currently has no model of scheduling with release dates and precedence constraints. A formalization produces a precise specification of Delay List, whose informal description is given in discrete time and repaired in a remark; a checked proof of the charging argument; and reusable lower bounds (Lemmas 4.10 and 4.11) for any later work on parallel-machine scheduling with precedence.

Difficulty

The obvious attempt, list scheduling (start the first available job of the list whenever a machine is free), fails with non-identical processing times: a long job taken out of order can occupy a machine and delay a more valuable job that becomes ready shortly afterwards. Delay List allows out-of-order jobs only against accumulated idle time, and the analysis rests on a charging invariant. Stating it needs care about time (the paper's discrete-time exposition can over-charge by a time unit), about which idle time a charge consumes, and about many jobs being scheduled at one instant. The bound must hold simultaneously for release dates and arbitrary precedence constraints, where idle machines can be forced both by jobs that are not yet released and by chains of predecessors, and it must hold for every tie-breaking choice of the algorithm.

Formalization scope

Jobs are Fin n, machines Fin m, and times are real numbers. Processing times are positive, release dates nonnegative and weights positive, as in §1. Precedence is a strict partial order, the transitive closure of the paper's DAG; $\kappa$ , readiness and feasibility are unchanged by taking the closure. The optimum is never a real infimum: "within a factor $\rho$ of an optimal one-machine schedule" and "within a factor $c$ of an optimal $m$ -machine schedule" are inequalities against every feasible schedule of the same instance, with the same release dates and precedence constraints.

Delay List is formalized in the continuous-time version described in the proof of Fact 4.6, as a predicate on runs that records start times, machines, the order in which jobs are scheduled at equal times, and charge windows. A case-2 charge takes the most recent uncharged idle time, and idle time is charged by whole time slices. Every guarantee is claimed for every run satisfying the predicate. The ties in Definition 4.4 are broken arbitrarily, so statements involving $\kappa'_i$ hold for every admissible path. Lemma 4.10 uses nonpreemptive one-machine schedules. Lemma 4.11's $C^\infty_{\mathrm{OPT}}$ is modelled by $n$ machines.

It would be trivializing to assume the conclusions of Fact 4.6 or Lemma 4.7 as properties of the run, or to measure $\rho$ against a relaxation without release dates or precedence; both are ruled out. The algorithm's rules are the only hypotheses on the run.

Not stated: the running time of Delay List; the discrete-time algorithm; Corollary 4.14 (it needs a formal class of series-parallel orders and the external one-machine algorithm of Adolphson for them); Remark 4.15 (release-date-free one-machine schedules), whose hypotheses the paper does not pin down; and the extension to delays between jobs. Contributions of general infrastructure, such as idle-time accounting for step functions and lemmas about list schedules under precedence, are welcome and reusable beyond this mission.

Selected references

C. Chekuri, R. Motwani, B. Natarajan, C. Stein, Approximation Techniques for Average Completion Time Scheduling, SIAM Journal on Computing 31(1):146–166, 2001. https://doi.org/10.1137/S0097539797327180
R. L. Graham, Bounds for certain multiprocessing anomalies, Bell System Technical Journal 45:1563–1581, 1966. https://doi.org/10.1002/j.1538-7305.1966.tb01709.x
D. Adolphson, Single machine job sequencing with precedence constraints, SIAM Journal on Computing 6(1):40–54, 1977. https://doi.org/10.1137/0206002

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Combinatorics Operations Research Optimization·Captain: mikedeng1

Approximation Techniques for Average Completion Time Scheduling II: A 2.83-Approximation for Parallel Machines with Release DatesResearch Paper

Motivation

Minimizing the average completion time of jobs that arrive over time is a basic objective in machine scheduling. It measures how long a job spends in the system on average. With several identical machines, release dates and no preemption (written $P|r_j|\sum C_j$ ), the problem is strongly NP-hard already on one machine. Research has therefore looked for approximation algorithms: polynomial-time rules whose total completion time is provably within a constant factor of every feasible schedule.

A common approach solves a relaxation that is easy to optimize and converts its solution into a feasible schedule. Chekuri, Motwani, Natarajan and Stein (SIAM J. Comput. 31(1), 2001) use a relaxation that needs neither linear programming nor dynamic programming: pretend that the $m$ machines are one machine that is $m$ times as fast, and allow preemption.

Timeline:

1996, Chakrabarti, Phillips, Schulz, Shmoys, Stein and Wein (ICALP 1996, LNCS 1099, pp. 646–657): a $(2.89+\epsilon)$ -approximation for $P|r_j|\sum C_j$ .
2001, Chekuri, Motwani, Natarajan and Stein (SIAM J. Comput. 31(1), §3 and §4.5). §3 gives a simple $(3-1/m)$ -approximation by list scheduling from the one-machine relaxation. §4.5 combines it with the Delay List conversion to obtain $2\sqrt2\approx2.83$ . This mission's goal is the §4.5 result.
1999, Afrati, Bampis, Chekuri, Karger, Kenyon, Khanna, Milis, Queyranne, Skutella, Stein and Sviridenko (FOCS 1999, pp. 32–43): polynomial-time approximation schemes for $P|r_j|\sum w_jC_j$ . These settle the approximability, but the algorithms are far more involved than the ones formalized here.

Setting

An instance has $n$ jobs $J_0,\dots,J_{n-1}$ and $m\ge1$ identical machines. Job $J_j$ has a processing time $p_j>0$ and a release date $r_j\ge0$ .

A feasible schedule gives each job a start time $S_j\ge r_j$ and a machine. Job $J_j$ runs without interruption on its machine during $[S_j,S_j+p_j)$ , and two jobs on the same machine never overlap. The completion times are $C_j=S_j+p_j$ and the objective is $\sum_j C_j$ . $C^*_j$ denotes the completion times of an arbitrary feasible schedule, against which every bound is stated.

The one-machine relaxation $I1$ has the same jobs and a single machine. Job $J_j$ has processing time $p_j/m$ and release date $r_j$ in $I1$ , and may be preempted. A preemptive schedule $P1$ of $I1$ gives each job a processing rate $\rho_j(t)\ge0$ . The rates sum to at most $1$ at each time, and no job is processed before its release date. Each job receives $p_j/m$ units in total. Its completion time $C^{P1}_j$ is the first time by which all of it has been processed. $P1$ is optimal if $\sum_j C^{P1}_j$ is minimal among all such schedules.

A list is an ordering $\pi$ of the jobs, and the completion order of $P1$ lists the jobs by nondecreasing $C^{P1}_j$ . Two ways of turning a list into an $m$ -machine schedule are compared.

Strict-order list scheduling gives the schedule $N$ . The jobs start in the order of the list. Each job starts at the earliest time that is no earlier than its release date, no earlier than the previous job's start, and at which some machine is free.
Delay List with parameter $\beta>0$ gives the schedule $D$ . When a machine is idle, Delay List starts the first unscheduled job of the list if it has been released. If that job has not been released, the first released job of the list may jump ahead, but only once at least $\beta p_j$ units of idle time (machine × time) have accumulated that no earlier job has charged. The job then charges exactly that amount. A job started in list order charges all uncharged idle time since its release.

Formalization targets

Goal: Lemma 4.19

With $P1$ optimal, $\pi$ its completion order, $N$ the strict-order list schedule of $\pi$ and $D$ a Delay List schedule of $\pi$ with $\beta_0=\sqrt{3-2\sqrt2}$ , every feasible schedule satisfies

\min\Bigl(\sum_j C^N_j,\ \sum_j C^D_j\Bigr)\le 2\sqrt2\,\sum_j C^*_j .

The printed lemma says $2.83$ . Its proof gives $2\sqrt2\approx2.8284$ , which is stated here.

Milestones

In the order the proof uses them:

(4.2): if $\sum_j p_j>\alpha\sum_j C^*_j$ then $\sum_j r_j\le(1-\alpha)\sum_j C^*_j$ .
Lemma 3.1: $\sum_j C^{P1}_j\le\sum_j C^*_j$ for $P1$ optimal.
(3.3): $\sum_j C^N_j\le 2\sum_j C^{P1}_j+(1-1/m)\sum_j p_j$ for any $P1$ .
Lemma 3.2: $\sum_j C^N_j\le(3-1/m)\sum_j C^*_j$ .
Theorem 4.9, specialised to no precedence constraints. With $B_i$ the jobs at or before $J_i$ in the list,

C^D_i\le\frac{(1+\beta)p(B_i)}{m}+\Bigl(1+\frac1\beta\Bigr)(r_i+p_i)-\frac{p_i}{\beta}.

Lemma 4.18: $\sum_j C^D_j\le(2+\beta)\sum_j C^*_j+\frac1\beta\sum_j r_j$ .
The balanced bound: under (4.2)'s hypothesis, $\sum_j C^D_j\le(2+\beta+(1-\alpha)/\beta)\sum_j C^*_j$ .
The constants: at $\alpha=2\sqrt2-2$ and $\beta=\sqrt{3-2\sqrt2}$ , $2+\alpha=2+\beta+(1-\alpha)/\beta=2\sqrt2$ .

Two existence statements accompany them. One says an optimal $P1$ exists. The other says a Delay List schedule exists for every list and every $\beta>0$ .

Significance

The result gives a $2\sqrt2$ -approximation for $P|r_j|\sum C_j$ that is simple to state and runs in $O(n\log n)$ time. It improves the $2.89+\epsilon$ bound of Chakrabarti et al. Neither of its two algorithms achieves the ratio alone. It comes from an analysis in which each algorithm is good exactly when the other is bad. List scheduling is good when processing times are small relative to the optimum. Delay List is good when release dates are small. The inequality (4.2) connects the two cases.

The component results are reusable beyond this paper. The one-machine relaxation lower bound (Lemma 3.1) and the $(3-1/m)$ bound for list scheduling from it (Lemma 3.2) apply to any conversion from a fast single machine. The per-job bound of Theorem 4.9 is the core of the Delay List technique. Its general form, with precedence constraints, drives the paper's results for precedence-constrained scheduling.

All results are proved in the paper. None of them has a machine-checked proof that this mission knows of. A formalization would check the Delay List charging argument, which the paper states only in discrete time and adapts to continuous time in one sentence. It would also produce reusable Lean definitions of parallel-machine schedules with release dates and of list scheduling.

Difficulty

The arithmetic of the goal is routine once the milestones are in place. The substance lies in two places.

The first is Lemma 3.1 together with the "standard makespan argument" behind (3.2). The one-machine relaxation must be related to the $m$ -machine schedule, and to the list schedule, with care about release dates. In particular, in the list schedule every machine is busy between the last release among the first $j$ jobs of the list and the start of the $j$ -th job. Proving this needs the strict order.

The second, and harder, is Theorem 4.9. The obvious argument bounds the waiting time of job $J_i$ by the work of the jobs ahead of it, but Delay List lets later jobs jump ahead. The idle time before $J_i$ starts and the work of the jobs that jump ahead of it must both be controlled, and the paper's charging argument for this depends on where charged idle time lies on the time axis and on which jobs charged it. Making that bookkeeping precise for a continuous-time algorithm is the main formalization cost.

Formalization scope

Jobs are Fin n and machines Fin m with $m\ge1$ . Times are real, processing times are positive and release dates nonnegative. There are no weights and no precedence constraints. "Optimal" is never an infimum. Every bound is stated against every feasible nonpreemptive schedule, and $P1$ 's optimality is the hypothesis that its total completion time is at most that of every preemptive schedule of $I1$ .

Committed conventions:

Preemptive schedules of $I1$ are rate functions, so the machine of $I1$ may be shared. The paper's one-job-at-a-time schedules are a special case.
Lists are bijections Fin n ≃ Fin n. A list of $P1$ may break ties in completion time in any way, and every such list is covered.
$N$ is the strict-order variant of list scheduling, which footnote 3 of the paper contrasts with the greedy variant used in §4. It is a recursive definition over list positions.
Delay List is the continuous-time algorithm, as adopted in the proof of Fact 4.6. It is a predicate on start times, machines, the scheduling order and charge windows. A job scheduled out of order takes its charge from the most recent uncharged idle time; the paper leaves this placement open. Theorem 4.9 and Lemma 4.18 assume $m\ge2$ , the setting of §4.1. The goal assumes only $m\ge1$ .
The printed Lemma 4.18 lacks a $\sum_j$ on the $C^*$ term. The summed form of its proof's last display is stated.

The statement cannot be made easy by the hypotheses. Two existence items show that an optimal $P1$ and a Delay List schedule always exist, so no statement is vacuous. The bound is against every feasible schedule, not against the relaxation's value.

Not stated: the $O(n\log n)$ running time, the on-line version of §3's algorithm, and Delay List with precedence constraints (Theorem 4.9 in general, which is the subject of mission III of this series). Contributions welcome: proofs of the milestones, and reusable lemmas on list scheduling with release dates.

Selected references

C. Chekuri, R. Motwani, B. Natarajan, C. Stein, Approximation Techniques for Average Completion Time Scheduling, SIAM J. Comput. 31(1):146–166, 2001. https://doi.org/10.1137/S0097539797327180
S. Chakrabarti, C. A. Phillips, A. S. Schulz, D. B. Shmoys, C. Stein, J. Wein, Improved scheduling algorithms for minsum criteria, in Proceedings of ICALP 1996, LNCS 1099, Springer, pp. 646–657 (reference [3] of the paper).
F. Afrati et al., Approximation schemes for minimizing average weighted completion time with release dates, in Proceedings of the 40th IEEE FOCS, 1999, pp. 32–43 (reference [2] of the paper).

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Combinatorics Operations Research Optimization+1·Captain: mikedeng1

Approximation Techniques for Average Completion Time Scheduling I: Best-α on One Machine with Release DatesResearch Paper

Motivation

Minimizing the average completion time of jobs that arrive over time is one of the basic objectives of machine scheduling: it measures how long, on average, a job waits in the system. On a single machine with release dates and no preemption (written $1|r_j|\sum C_j$ ), the problem is strongly NP-hard, so research has focused on approximation algorithms whose guarantees are stated against every feasible schedule.

The standard route runs through the preemptive relaxation. When jobs may be interrupted and resumed, the shortest-remaining-processing-time rule (SRPT) produces an optimal schedule, and its value is a lower bound for every nonpreemptive schedule. The question is how to turn that preemptive schedule into a nonpreemptive one without losing too much.

Timeline:

1995, Phillips, Stein and Wein (WADS 1995, pp. 86–97): order the jobs by their SRPT completion times and schedule them nonpreemptively in that order. This gives a 2-approximation. Later 2-approximations are by Hoogeveen and Vestjens (IPCO 1996), Stougie (1995), and Goemans (SODA 1997). Hoogeveen and Vestjens also showed that deterministic on-line algorithms cannot beat 2.
2001, Chekuri, Motwani, Natarajan and Stein (SIAM J. Comput. 31(1)): order by $\alpha$ -points instead of completion times, choose $\alpha$ at random, and take the best $\alpha$ off-line. This gives the $e/(e-1)\approx1.58$ bound for Best- $\alpha$ that is the goal of this mission, and an optimal randomized on-line algorithm.
1999, Afrati et al. (FOCS 1999): a polynomial-time approximation scheme for $1|r_j|\sum w_jC_j$ . This settled the approximability of the problem, but the resulting algorithms are far from simple.

Setting

An instance has $n$ jobs $J_0,\dots,J_{n-1}$ . Job $J_j$ has a processing time $p_j>0$ , a release date $r_j\ge0$ , and, where the objective is weighted, a weight $w_j>0$ . There is one machine.

A nonpreemptive schedule assigns each job a start time $S_j\ge r_j$ such that the intervals $[S_j,S_j+p_j)$ are pairwise disjoint. Its completion times are $C_j=S_j+p_j$ .

A preemptive schedule $P$ specifies, for each time $t$ , which job runs at $t$ , if any. Job $J_j$ runs only at times $t\ge\max(0,r_j)$ , receives exactly $p_j$ units of processing in total, and finishes by some finite time. Its completion time $C^P_j$ is the first time by which all of $J_j$ has been processed. For $\alpha\in(0,1]$ , its $\alpha$ -point $C^P_j(\alpha)$ is the first time by which $\alpha p_j$ units have been processed.

For a job $J_i$ , $T_i$ denotes the idle time of $P$ before $C^P_i$ . $x_{ij}$ denotes the fraction of $J_j$ processed before $C^P_i$ . The paper writes $S^P_i(\beta)$ for the set of jobs with $x_{ij}=\beta$ , and also for their total processing time.

One-machine list scheduling in a given order runs the jobs nonpreemptively in that order. Each job starts at the later of its release date and the completion of the previous job in the list. An $\alpha$ -schedule is list scheduling in nondecreasing order of the $\alpha$ -points $C^P_j(\alpha)$ . $C^\alpha_j$ denotes the completion times of an $\alpha$ -schedule.

Random- $\alpha$ draws $\alpha$ from a distribution on $(0,1]$ and outputs the $\alpha$ -schedule. Best- $\alpha$ outputs the $\alpha$ -schedule of smallest total completion time $\min_\alpha\sum_j C^\alpha_j$ .

Formalization targets

Goal: Corollary 2.7

Let $P$ be optimal among preemptive schedules for $\sum_j C_j$ . Then there is $\alpha\in(0,1]$ such that every $\alpha$ -schedule derived from $P$ satisfies

\sum_j C^\alpha_j\;\le\;\frac{e}{e-1}\sum_j C_j\qquad\text{for every feasible nonpreemptive schedule } (C_j)_j .

Since Best- $\alpha$ returns a schedule no worse than this $\alpha$ -schedule, Best- $\alpha$ is an $e/(e-1)$ -approximation.

Milestones

The calculus behind the constant. For $f(\alpha)=e^\alpha/(e-1)$ and every $\beta\in(0,1]$ ,

\int_0^\beta\frac{1+\alpha-\beta}{\beta}f(\alpha)\,d\alpha=\frac1{e-1}.

Lemma 2.2: $C^P_i=T_i+\sum_{0<\beta\le1}\beta S^P_i(\beta)$ .
Lemma 2.3: $C^\alpha_i\le T_i+(1+\alpha)\sum_{\beta\ge\alpha}S^P_i(\beta)+\sum_{\beta<\alpha}\beta S^P_i(\beta)$ .
Lemma 2.5: if $\alpha$ has density $f$ on $(0,1]$ , then $E[C^\alpha_i]\le(1+\delta)C^P_i$ with $\delta=\max_{0<\beta\le1}\int_0^\beta\frac{1+\alpha-\beta}{\beta}f(\alpha)\,d\alpha$ .
Theorem 2.6, for the weighted objective with $P$ optimal among preemptive schedules: the expected approximation ratio of Random- $\alpha$ is at most $2$ for uniform $\alpha$ , at most $1.8$ for $\alpha=1$ w.p. $3/5$ and $\alpha=1/2$ w.p. $2/5$ , and at most $e/(e-1)$ for the density $e^\alpha/(e-1)$ .

Companion statements, not milestones:

the upper bound of Theorem 2.1, $\sum_jC^\alpha_j\le(1+1/\alpha)\sum_jC^P_j$ ;
the existence of an optimal preemptive schedule.

Significance

The $e/(e-1)$ bound shows that conversion from the preemptive relaxation can beat the factor 2 of the natural ordering. It does so by exploiting that no single instance is bad for many values of $\alpha$ at once. The $\alpha$ -point technique was also used with LP relaxations, for example by Goemans (SODA 1997) and by Schulz and Skutella. The randomized version is an optimal randomized on-line algorithm for $1|r_j|\sum C_j$ . Lemma 2.3 is a statement about any preemptive schedule, so it applies wherever a good preemptive or fractional schedule is available.

All results of the mission are proved in the paper, except that the proof of Theorem 2.6, part 2 is omitted there. No machine-checked proof of them is known. A complete development would give a verified model of preemptive one-machine schedules, $\alpha$ -points and list scheduling, together with the averaging argument over $\alpha$ . These are reusable for the later results of the same paper and for the $\alpha$ -point literature.

Difficulty

The obvious argument bounds each job's $\alpha$ -schedule completion time directly against its preemptive completion time. That argument loses a factor $1+1/\alpha$ (Theorem 2.1), which is at least 2 for every fixed $\alpha$ . The improvement needs Lemma 2.3. There the charge to each job depends on how much of it was done by $C^P_i$ relative to $\alpha$ , and the idle time $T_i$ is not inflated at all. Proving Lemma 2.3 requires reasoning about a preemptive schedule as a measure on time, and about how moving pieces of jobs changes completion times. A proof that treats the preemptive schedule as a finite list of pieces must first show that nothing is lost by this discretization.

The averaging step needs the expectation over $\alpha$ to be an honest integral. The map $\alpha\mapsto C^\alpha_i$ must be shown integrable, which requires a fixed rule for ties between equal $\alpha$ -points.

Formalization scope

Model. Jobs are Fin n, time is real, $p_j>0$ $p_{j} > 0$ and $r_j\ge0$ $r_{j} \geq 0$ . The paper admits $p_j=0$ $p_{j} = 0$ only in its tightness instances.
- A preemptive schedule is a function $\sigma:\mathbb R\to$ Option (Fin n) (none = idle). Each job's run set is measurable, lies in $[\max(0,r_j),\infty)$ , is bounded above, and has Lebesgue measure $p_j$ .
- Completion times and $\alpha$ -points are infima of nonempty sets that are bounded below.
- $T_i$ is the measure of the idle set in $[0,C^P_i)$ .
- The paper's sums over $\beta$ are sums over jobs, weighted by the fraction $x_{ij}$ .
List scheduling is strict: jobs never overtake the list order, and the machine is free from time $0$ $0$ .
- Lemma 2.3, Theorem 2.1, Theorem 2.6.2 and the goal hold for every tie-break among equal $\alpha$ -points.
- The expectations (Lemma 2.5, Theorem 2.6.1 and 2.6.3) use the tie-break by job index. They assert integrability as part of the conclusion.
Optimality. "Approximation ratio $c$ $c$ " is stated as an inequality against every feasible nonpreemptive schedule, never against an infimum.
- The optimality of $P$ among preemptive schedules is the paper's standing assumption for its upper bounds (p. 151). It appears as a hypothesis of Theorem 2.6 and of the goal.
- The lemmas hold for arbitrary $P$ and do not carry it.
- An existence statement shows the hypothesis can be met.
Lemma 2.5's $\delta$ is replaced by any upper bound of the integrals over $\beta\in(0,1]$ . This is equivalent, and it avoids assuming that the maximum is attained.
Not stated:
- the running time $O(n^2)$ of Best- $\alpha$ and the optimality of SRPT;
- the tightness parts of Theorem 2.1 and Corollary 2.4, and the lower bounds of Theorem 2.9, which use zero-length jobs;
- the on-line Theorem 2.8, which needs a model of on-line algorithms.
Trivializing formalization ruled out. Dropping the optimality of $P$ from the goal would turn it into a statement about arbitrary preemptive schedules, which is Lemma 2.5, not Corollary 2.7. Comparing against $\sum_jC^P_j$ instead of every nonpreemptive schedule would likewise remove the content of the corollary.

Contributions are welcome at every level. The calculus milestone and Lemma 2.2 are good first targets.

Selected references

C. Chekuri, R. Motwani, B. Natarajan, C. Stein, Approximation Techniques for Average Completion Time Scheduling, SIAM J. Comput. 31(1):146–166, 2001. https://doi.org/10.1137/S0097539797327180
C. Phillips, C. Stein, J. Wein, Scheduling jobs that arrive over time, Proc. 4th Workshop on Algorithms and Data Structures (WADS), 1995, pp. 86–97 (reference [25] of the paper; no link verified).
J. A. Hoogeveen, A. P. A. Vestjens, Optimal on-line algorithms for single-machine scheduling, Proc. 5th IPCO, 1996, pp. 404–414 (reference [21]; no link verified).
M. X. Goemans, Improved approximation algorithms for scheduling with release dates, Proc. 8th ACM-SIAM SODA, 1997, pp. 591–598 (reference [12]; no link verified).
F. Afrati et al., Approximation schemes for minimizing average weighted completion time with release dates, Proc. 40th FOCS, 1999 (reference [2]; no link verified).

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Combinatorics Operations Research Optimization·Captain: mikedeng1

Local Search Heuristics for k-Median and Facility Location Problems IV: Local Search with Multi-Copy Moves for Capacitated Facility Location Has Locality Gap 4Research Paper

Motivation

Facility location asks where to open service points (warehouses, plants, servers) and how to connect customers to them so that the total opening cost plus the total connection cost is minimum. In the capacitated version each facility can serve only a limited number of customers, which is the situation in most applications: a warehouse has a floor area, a server a bandwidth. The problem is NP-hard, and the algorithms used in practice for it are often simple local search heuristics: start from a solution and repeatedly apply a small change that lowers the cost, until no such change exists.

The quality of such a heuristic is measured by its locality gap: the largest possible ratio between the cost of a solution that no allowed change can improve and the cost of an optimum solution. Arya, Garg, Khandekar, Meyerson, Munagala and Pandit (SIAM J. Comput. 33(3), 2004) gave locality-gap analyses for k-median, uncapacitated facility location, and the capacitated problem in which several copies of a facility may be opened. This mission formalizes their §5, the capacitated case.

Timeline (as surveyed on pp. 545–546 of the paper). For the variant 1-CFL, where at most one facility may be opened at each location, Korupolu, Plaxton and Rajaraman (1998) showed that local search with add, drop and swap moves has locality gap at most 8 when capacities are uniform; Chudak and Williamson (IPCO 1999) refined this to 6, and Pál, Tardos and Wexler gave a local search with gap 9 for nonuniform capacities. For ∞-CFL, the variant with copies studied here, the known algorithms were LP-based: a 3-approximation of Chudak and Shmoys (1999) for uniform capacities, a 4-approximation of Jain and Vazirani for nonuniform capacities, and a 2-approximation of Mahdian, Ye and Zhang. Arya et al. (2004) analysed local search for ∞-CFL with nonuniform capacities: with a new move that drops any set of open copies and opens several copies of one facility, the locality gap is at most 4 (Theorem 5.5), and scaling the facility costs gives $2 + \sqrt3 + \epsilon$ (p. 561). The tight example for uncapacitated facility location (§4.3) also shows a locally optimum solution of cost 3 times the optimum, so the locality gap of the procedure lies between 3 and 4; its exact value was left open (§6).

Setting

An instance consists of a finite set $C$ of clients, a set $F$ of facilities and a distance $c$ on $C \cup F$ that is nonnegative, symmetric and satisfies the triangle inequality; $c_{ji}$ is the cost of serving client $j$ from facility $i$ . Every facility $i$ has an opening cost $f_i \ge 0$ and an integer capacity $u_i > 0$ . Any number of copies of a facility may be opened; each copy of $i$ costs $f_i$ and serves at most $u_i$ clients.

A solution $X$ opens a finite list of copies, copy $s$ being a copy of facility $\mathrm{loc}(s)$ , and assigns every client $j$ to a copy $\sigma(j)$ so that each copy $s$ serves at most $u_{\mathrm{loc}(s)}$ clients. Write $N_X(s)$ for the set of clients served by copy $s$ and $N_X(T)$ for the clients served by a set $T$ of copies. Its costs are

\mathrm{cost}_f(X) = \sum_s f_{\mathrm{loc}(s)}, \qquad \mathrm{cost}_s(X) = \sum_{j\in C} c_{j\,\mathrm{loc}(\sigma(j))}, \qquad \mathrm{cost}(X) = \mathrm{cost}_f(X) + \mathrm{cost}_s(X).

The neighbourhood (9) of a solution whose multiset of open facilities is $S$ consists of

$S + s'$ : one more copy of any facility $s'$ ;
$S - T + l\cdot\{s'\}$ : close any set $T$ of open copies and open $l \ge 1$ copies of a facility $s'$ , provided $l\,u_{s'} \ge |N_S(T)|$ .

$X$ is locally optimum if no neighbour, with any feasible assignment of the clients, has smaller cost.

Formalization targets

Goal: Theorem 5.5

For every instance with at least one client, every locally optimum solution $X$ and every solution $O$ ,

\mathrm{cost}(X) \le 4\,\mathrm{cost}(O).

Milestones

In the order the paper's proof uses them:

Lemma 5.1 (service cost): $\mathrm{cost}_s(X) \le \mathrm{cost}_f(O) + \mathrm{cost}_s(O)$ .
Lemma 5.2: for every set $U$ of copies of $X$ and every facility $s'$ ,

\left\lceil \frac{|N_X(U)|}{u_{s'}}\right\rceil f_{s'} + \sum_{s\in U} |N_X(s)|\, c_{ss'} \ge \sum_{s\in U} f_s .

Lemma 5.4: in the graph with arcs $v_s \to w_o$ of length $c_{so}$ and $w_o \to \mathrm{sink}$ of length $f_o/u_o$ , $|N_X(s)|$ units can be routed from every $v_s$ at cost at most $\mathrm{cost}_s(X) + \mathrm{cost}_s(O) + \mathrm{cost}_f(O)$ .
Inequality (10): the shortest-path flow, with $T_o$ the copies routed through $w_o$ , satisfies $\sum_o\sum_{s\in T_o}|N_X(s)|(c_{so} + f_o/u_o) \le \mathrm{cost}_s(X) + \mathrm{cost}_s(O) + \mathrm{cost}_f(O)$ .
Inequality (11): $\sum_o f_o + \sum_o \sum_{s\in T_o} |N_X(s)|(c_{so} + f_o/u_o) \ge \mathrm{cost}_f(X)$ .
Lemma 5.3 (facility cost): $\mathrm{cost}_f(X) \le 3\,\mathrm{cost}_f(O) + 2\,\mathrm{cost}_s(O)$ .

A companion item states the scaled bound of p. 561: a local optimum for facility costs $(\sqrt3 - 1) f$ has cost at most $(2 + \sqrt3)\,\mathrm{cost}(O)$ in the original instance.

Significance

The result. Theorem 5.5 gives a constant locality gap for a local search procedure for capacitated facility location with copies and nonuniform capacities, a variant previously approached through LP-based algorithms; the drop-add move it analyses is the paper's new operation for this problem. The scaled bound $2 + \sqrt3 \approx 3.732$ gives an approximation algorithm once local search is run to approximate local optimality. The paper also shows (Figure 13, the procedure T-hunt) that the exponentially large neighbourhood can be searched with a knapsack oracle, so the analysis applies to an implementable algorithm.

Formalizing it. The result is proved in the paper; to our knowledge no machine-checked version exists. The mission produces a checked model of capacitated facility location with copies (solutions, costs, the multiset neighbourhood) and of the locality-gap argument. The per-copy model and the flow comparison of Lemma 5.4 are reusable for other capacitated location problems and for local search analyses that compare a local optimum with an optimum through a flow or a matching.

Difficulty

The obvious attempt imitates the uncapacitated analysis: close one copy of $X$ and send its clients to a nearby copy of $O$ . With capacities this fails, since that copy of $O$ may be too small to absorb them, and single-copy moves do not certify a constant bound. With the drop-add move a single copy of $O$ must be charged for a whole group of copies of $X$ , and the groups must be chosen so that the charges add up to a constant times $\mathrm{cost}(O)$ ; the rounding $\lceil |N_X(T)|/u_{s'}\rceil$ of the number of new copies costs an additional $\mathrm{cost}_f(O)$ that has to be absorbed as well.

Formalization scope

Solutions. A solution is a structure CFLSol Cl Fa u: a number n of open copies, a map loc : Fin n → Fa giving the facility of each copy, and an assignment σ : Cl → Fin n with the capacity constraint for every copy. Copies are separate indices because $N_S(s)$ is per copy. Its multiset of facilities is the image multiset of loc.
Costs. A solution's cost is computed under its own assignment. The paper's cost of a multiset is the minimum over feasible assignments. Because every neighbour is compared with every feasible assignment, and $O$ ranges over every assignment, the statements are equivalent to the paper's. The move with $T = \{s\}$ , $s' = \mathrm{loc}(s)$ , $l = 1$ makes every reassignment of $X$ 's clients a neighbour, so a locally optimum $X$ carries a minimum-cost assignment.
Neighbourhood. Local optimality ranges over the whole of (9): every $s'$ , every set $T$ of copies (including $\emptyset$ and all copies) and every $l \ge 1$ with $l u_{s'} \ge |N_X(T)|$ . It is not restricted to what the search procedure T-hunt examines.
Standing assumptions. Distances are nonnegative, symmetric and satisfy the triangle inequality on $C \cup F$ ; $d(x,x) = 0$ is not assumed. Capacities are natural numbers with $u_i > 0$ ; costs are real with $f_i \ge 0$ ; every client has unit demand. Ratios $f_o/u_o$ and the ceiling of Lemma 5.2 are computed in $\mathbb R$ .
Added hypothesis. The goal, Lemmas 5.2 and 5.3 and the companion assume at least one client. Without clients a single idle copy of a facility with $f = 1$ , $u = 1$ is locally optimum at cost 1 while the empty solution costs 0, so these statements fail. The paper's instances implicitly have clients.
No trivialization. $O$ is any solution, not a fixed optimum, and the bounds are multiplied out ( $\mathrm{cost}(X) \le 4\,\mathrm{cost}(O)$ , never a ratio). The empty solution is excluded only by the presence of a client, not by a default cost.
Out of scope. The procedure T-hunt and the knapsack oracle, running time, the $\epsilon$ of approximate local optimality, and arbitrary demands.

Contributions are welcome at every level: proofs of the milestones, alternative proofs of Lemma 5.4 or (10) (for instance through a matching argument instead of flows), and general infrastructure for multiset neighbourhoods and assignment problems.

Selected references

V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, V. Pandit, Local Search Heuristics for k-Median and Facility Location Problems, SIAM J. Comput. 33(3):544–562, 2004. https://doi.org/10.1137/S0097539702416402
M. R. Korupolu, C. G. Plaxton, R. Rajaraman, Analysis of a Local Search Heuristic for Facility Location Problems, J. Algorithms 37(1):146–188, 2000. https://doi.org/10.1006/jagm.2000.1100

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Combinatorics Operations Research Optimization·Captain: mikedeng1

Local Search Heuristics for k-Median and Facility Location Problems I: Single-Swap Local Search for k-Median Has Locality Gap 5Research Paper

Motivation

The k-median problem asks where to open $k$ facilities so that the total distance from a set of clients to their nearest open facility is as small as possible. It is a basic model of facility location in operations research (placing depots, warehouses or servers) and of clustering with representative centres, and it is NP-hard, so the question of interest is how close a polynomial-time method can come to the optimum.

Local search is among the most widely used heuristics for it: start from any $k$ facilities and repeatedly exchange one open facility for a closed one while the cost decreases. Arya, Garg, Khandekar, Meyerson, Munagala and Pandit (SIAM J. Comput. 33(3), 2004) gave the first constant-factor guarantee for this heuristic on metric instances: every local optimum of the single-swap local search costs at most five times any solution with $k$ facilities. This mission formalizes that result.

Timeline of the relevant bounds:

Korupolu, Plaxton and Rajaraman (SODA 1998) analysed a local search for k-median that opens $k(1+\epsilon)$ facilities and costs at most $3 + 5/\epsilon$ times the optimum with $k$ facilities.
Charikar, Guha, Tardos and Shmoys (STOC 1999) gave the first constant-factor approximation for metric k-median, by LP rounding ( $6\tfrac23$ ).
Jain and Vazirani (J. ACM 2001) and Charikar and Guha (FOCS 1999) improved the constant with primal–dual methods to 6 and 4.
Arya et al. (STOC 2001; SIAM J. Comput. 2004) proved the locality gap 5 for single swaps and $3 + 2/p$ for swaps of $p$ facilities at a time, with matching examples.

Setting

A metric instance consists of a finite set $C$ of clients, a finite set $F$ of facilities and a distance $d$ on $C \cup F$ that is nonnegative, symmetric and satisfies the triangle inequality. Write $c_{ji} = d(j,i)$ for the cost of serving client $j$ by facility $i$ .

For a nonempty set $S \subseteq F$ of open facilities every client is served by its nearest open facility, and the cost of $S$ is

\mathrm{cost}(S) = \sum_{j \in C} \min_{i \in S} c_{ji}.

The k-median problem asks for a set $S$ of at most $k$ facilities of minimum cost.

A swap $\langle s, s'\rangle$ closes a facility $s \in S$ and opens a facility $s' \notin S$ , giving $S - s + s' = (S \setminus \{s\}) \cup \{s'\}$ . The neighbourhood of $S$ is

\mathcal B(S) = \{ S - \{s\} + \{s'\} \mid s \in S,\ s' \notin S \},

and $S$ is locally optimum if $\mathrm{cost}(S) \le \mathrm{cost}(S')$ for every $S' \in \mathcal B(S)$ . The local search starts from an arbitrary set of $k$ facilities and applies improving swaps until none exists; swaps preserve the number of facilities, so it stops at a locally optimum set of exactly $k$ facilities. The locality gap is the supremum, over instances, of the ratio between the cost of a worst local optimum and the optimal cost.

The analysis uses the following notation. For a solution $A$ , let $\sigma_A$ assign each client to a nearest facility of $A$ , let $A_j = c_{j\sigma_A(j)}$ be the service cost of client $j$ , and let $N_A(a)$ be the set of clients served by $a \in A$ . For two solutions $S$ and $O$ put $N^o_s = N_O(o) \cap N_S(s)$ . A facility $s \in S$ captures $o \in O$ if $|N^o_s| > \tfrac12 |N_O(o)|$ ; $s$ is bad if it captures some $o \in O$ and good otherwise.

Formalization targets

Goal: Theorem 3.2

For every metric instance, every $k$ , every locally optimum set $S$ of exactly $k$ facilities and every nonempty set $O$ of at most $k$ facilities,

\mathrm{cost}(S) \le 5 \cdot \mathrm{cost}(O).

The comparison solution $O$ is arbitrary, not an optimum; the statement is the locality gap bound in the form the proof gives.

Milestones, in the order the proof uses them

A facility $o$ is captured by at most one facility of $S$ (remark after Definition 3.1).
Property 3.1: for each $o$ there is a bijection $\pi$ of $N_O(o)$ with $\pi(N^o_s) \cap N^o_s = \emptyset$ whenever $s$ does not capture $o$ .
When $|S| = |O|$ there are $|O|$ swaps $\langle s, o\rangle$ , one for each $o \in O$ , such that no facility capturing two or more facilities of $O$ is used, every good facility is used at most twice, and a used $s$ captures no $o' \ne o$ .
Inequality (2): for a locally optimum $S$ and such a swap $\langle s, o\rangle$ ,

\sum_{j \in N_O(o)} (O_j - S_j) + \sum_{\substack{j \in N_S(s)\\ j \notin N_O(o)}} \bigl(O_j + O_{\pi(j)} + S_{\pi(j)} - S_j\bigr) \ge 0.

Significance

Theorem 3.2 shows that the simplest exchange heuristic for k-median is a constant-factor approximation on every metric instance, and the paper states that the analysis is tight: its example of §3.5, given for swaps of two facilities, is said to generalize to swaps of $p \ge 1$ facilities, where the bound $3 + 2/p$ is 5 for $p = 1$ . Combined with the standard device of accepting only swaps that improve the cost by a factor $1 - \epsilon/Q$ , it yields a polynomial-time $5/(1-\epsilon)$ -approximation (p. 548). The same capture-and-reassignment argument is reused for multi-swap k-median, for uncapacitated and capacitated facility location in the same paper, and in later work on k-means and on local search for clustering; its milestones (the capture graph and the mapping $\pi$ ) are the reusable part.

The result has been proved since 2001 and is textbook material (Williamson and Shmoys, The Design of Approximation Algorithms, 2011, Chapter 9). No machine-checked proof of it is known; Mathlib has no k-median problem and no locality-gap result for any clustering objective. The work remaining is to formalize the known proof.

Difficulty

The obvious argument adds up the inequalities $\mathrm{cost}(S - s + o) \ge \mathrm{cost}(S)$ over a pairing of $S$ with $O$ , rerouting the clients of the closed facility $s$ to the nearest remaining facility. This fails when a single facility of $S$ serves most clients of several facilities of $O$ : closing it leaves those clients with no nearby open facility, and no bound in terms of $\mathrm{cost}(O)$ follows. The analysis must choose which swaps to consider so that such facilities are never closed, and must reroute the displaced clients of the facilities it does close to a facility other than the closed one while paying only a constant multiple of their own service costs. Both choices must work for arbitrary ties in the nearest-facility assignments and when $S$ and $O$ share facilities.

Formalization scope

Namespace LocalSearchFL.KMedian. Clients and facilities are types Cl, Fa with Fintype and DecidableEq; the distance is a real-valued function on Cl ⊕ Fa with fields for nonnegativity, symmetry and the triangle inequality, and $d(x,x) = 0$ is not assumed. Solutions are Finset Fa. The cost is defined only for nonempty sets, from a nonemptiness proof, so no value is assigned to the empty solution; the goal takes $S$ nonempty with S.card = k, which is the paper's $k \ge 1$ . Local optimality quantifies over every swap $\langle s, s'\rangle$ with $s \in S$ and $s' \notin S$ , exactly the neighbourhood $\mathcal B(S)$ of Theorem 3.2, and not only over the swaps with $s' \in O$ that the proof uses. The inequality is stated multiplied out, $\mathrm{cost}(S) \le 5 \cdot \mathrm{cost}(O)$ , so it is meaningful when $\mathrm{cost}(O) = 0$ .

The milestones quantify over nearest-facility assignments $\sigma_S$ , $\sigma_O$ with arbitrary ties. Capture is stated in integers as $|N_O(o)| < 2|N^o_s|$ . The bijection $\pi$ of $N_O(o)$ is a permutation of all clients fixing every client outside $N_O(o)$ ; in inequality (2) it is a single permutation preserving every $N_O(o)$ . Milestones 1–3 are purely combinatorial and are stated for arbitrary assignments, which contains the paper's case.

A formalization in which local optimality ranges over the swaps $\langle s, o\rangle$ , $o \in O$ , only, or in which $|O| = |S|$ or $O$ optimal is assumed, or in which the cost of the empty set is $0$ , is a different statement and is ruled out.

A complete development needs the finite-sum and Finset.inf' API of Mathlib, permutations (Equiv.Perm) and finite counting. The capture machinery and the mapping $\pi$ are reusable for the multi-swap and facility location missions of this series. Proofs of individual milestones are welcome independently of the goal.

Selected references

V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, V. Pandit, Local Search Heuristics for k-Median and Facility Location Problems, SIAM J. Comput. 33(3):544–562, 2004. https://doi.org/10.1137/S0097539702416402
M. Charikar, S. Guha, É. Tardos, D. B. Shmoys, A Constant-Factor Approximation Algorithm for the k-Median Problem, J. Comput. System Sci. 65(1):129–149, 2002. https://doi.org/10.1006/jcss.2002.1882
K. Jain, V. V. Vazirani, Approximation Algorithms for Metric Facility Location and k-Median Problems Using the Primal-Dual Schema and Lagrangian Relaxation, J. ACM 48(2):274–296, 2001. https://doi.org/10.1145/375827.375845
M. R. Korupolu, C. G. Plaxton, R. Rajaraman, Analysis of a Local Search Heuristic for Facility Location Problems, J. Algorithms 37(1):146–188, 2000. https://doi.org/10.1006/jagm.2000.1100
D. P. Williamson, D. B. Shmoys, The Design of Approximation Algorithms, Cambridge University Press, 2011. https://doi.org/10.1017/CBO9780511921735

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Functional Analysis Optimization·Captain: Lucas

The Grothendieck Constant: New Upper and Lower BoundsOpen Problem

Motivation

Given a real matrix $A=(a_{ij})\in\mathbb R^{m\times n}$ , consider maximizing the bilinear form $\sum_{i,j}a_{ij}x_iy_j$ over sign vectors $x\in\{\pm1\}^m$ , $y\in\{\pm1\}^n$ . This discrete optimum, written $\mathrm{OPT}(A)$ , is closely tied to the cut norm of a matrix and is NP-hard to compute. Relaxing each sign to a unit vector and each product to an inner product gives the semidefinite value $\mathrm{SDP}(A)$ , computable in polynomial time. Grothendieck's inequality (Grothendieck, 1953) states that the relaxation overshoots by at most a universal factor: there is a finite $K$ , independent of $A$ , of $m,n$ , and of the dimension of the vectors, with $\mathrm{SDP}(A)\le K\cdot\mathrm{OPT}(A)$ for every $A$ . The Grothendieck constant $K_G$ is the least such $K$ — equivalently, the worst-case integrality gap of the canonical semidefinite relaxation of this bilinear problem.

The constant is not a curiosity of one optimization problem. It originated in functional analysis, where it is central to the geometry of Banach spaces and to harmonic analysis; it governs the approximation ratio available for cut norms; and, in quantum information, it measures the maximal advantage of quantum over classical correlations in Bell-type experiments. Its exact value has been open since 1953.

A timeline of the bounds:

1953, Grothendieck. Existence of a finite $K$ , together with the lower bound $K_G\ge\pi/2=1.5707\ldots$
1977, Krivine. $K_G\le\pi/\bigl(2\log(1+\sqrt2)\bigr)=1.7822\ldots$ , obtained by analyzing hyperplane rounding, and conjectured to be optimal.
1984/1991, Davie and Reeds (independently). $K_G\ge1.6769\ldots$ , from an explicit high-dimensional Gaussian hard instance.
2011, Braverman–Makarychev–Makarychev–Naor. Krivine's conjecture is false: $K_G<\pi/(2\log(1+\sqrt2))$ strictly, with no quantitative gap.
2014, Naor–Regev. Mixtures of Krivine schemes are asymptotically optimal: rounding schemes of this one family approach the true value of $K_G$ .
2026, Heilman; Jones–Malavolta. The first improvements on Davie–Reeds, by $10^{-26}$ and $10^{-12}$ respectively; and the first explicit numerical improvements on Krivine's bound, of order $10^{-5}$ (Heilman; Li–Saha–Xue et al.).
2026, Saha–Li–Xue–Chaudhuri–Klivans–Kothari–Meka. The bounds this mission targets:

\frac{6\pi}{11}\ \le\ K_G\ \le\ \frac{\pi}{2\log(1+\sqrt2)}-3.47\times10^{-4},

i.e. $1.7135\ldots\le K_G\le1.7818\ldots$ , which fixes the tenths digit of $K_G$ at $7$ .

Setting

Fix $m,n\in\mathbb N$ and $A\in\mathbb R^{m\times n}$ .

\mathrm{OPT}(A):=\max_{x\in\{\pm1\}^m,\;y\in\{\pm1\}^n}\sum_{i,j}a_{ij}x_iy_j,\qquad \mathrm{SDP}(A):=\sup_{d\in\mathbb N}\ \sup_{u_i,v_j\in S^{d-1}}\sum_{i,j}a_{ij}\langle u_i,v_j\rangle .

Here $u_1,\dots,u_m$ and $v_1,\dots,v_n$ are unit vectors of a common but arbitrary finite dimension $d$ . Since a sign is a unit vector in dimension one, $\mathrm{OPT}(A)\le\mathrm{SDP}(A)$ . Call $K$ a Grothendieck bound if $\mathrm{SDP}(A)\le K\cdot\mathrm{OPT}(A)$ for every $m$ , $n$ and $A$ , and set $K_G:=\inf\{K: K\text{ is a Grothendieck bound}\}$ .

Upper bounds on $K_G$ come from rounding algorithms. A Krivine scheme of dimension $k$ is a pair of partitions of $\mathbb R^k$ into a $+1$ region and a $-1$ region, encoded by measurable odd functions $f,g:\mathbb R^k\to\{\pm1\}$ : the algorithm maps each SDP vector to a Gaussian point in $\mathbb R^k$ , correlated according to the inner products, and reads off the label of the region the point lands in. Taking $f=g=\operatorname{sgn}(z_1)$ recovers random hyperplane rounding. The quality of a scheme is carried by its normalized correlation function

H(t):=\frac{\pi}{2}\,\mathbb E\bigl[f(X)g(Y)\bigr],

where $X,Y$ are standard Gaussian vectors in $\mathbb R^k$ with $\mathbb E[X_iY_i]=t$ for every coordinate $i$ . For the half-space partition $H(t)=\arcsin t$ , whose analysis gives Krivine's bound. Writing the odd expansion $H(t)=b_1t+b_3t^3+\cdots$ , the hyperplane scheme sits at $(b_1,b_3)=(1,\tfrac16)$ .

Formalization targets

Goal

\frac{6\pi}{11}\ \le\ K_G\ \le\ \frac{\pi}{2\log(1+\sqrt2)}-3.47\times10^{-4}

This is the two-sided bound the source paper states as the outcome of its Theorems 2.1 and 2.2. It is the weakest statement that carries both of the paper's contributions at once; each side is also a milestone in its own right, so partial progress is recorded even if only one direction closes.

Milestones

The milestone list runs from the classical background to the two new bounds: $\mathrm{OPT}\le\mathrm{SDP}$ ; the existence of a finite Grothendieck bound; $K_G\ge\pi/2$ ; Krivine's $K_G\le\pi/(2\log(1+\sqrt2))$ ; the affine coefficient constraint $b_3\ge2b_1-\tfrac{11}{6}$ valid for every Krivine scheme (Theorem 2.2, equation (1)); the transfer of a member of the affine family into a lower bound on $K_G$ (Appendix A); the lower bound $K_G\ge6\pi/11$ (Theorem 2.2); and the cubic–quintic upper bound (Theorem 2.1).

Significance

The two target bounds narrow an interval that had been essentially static for four decades: before 2026 the state of the art was $1.6769\ldots\le K_G\le1.7822\ldots$ , wide enough that the tenths digit was unknown. The lower bound is also methodologically new. Every previous lower bound was obtained by exhibiting a hard instance; this one instead proves a ceiling on the performance of every rounding scheme in the Krivine family and converts that ceiling, through the Naor–Regev optimality theorem, into a bound on the constant. The affine constraint $b_3\ge2b_1-\tfrac{11}{6}$ is the transportable core of that argument: being affine in the coefficients, it survives mixing schemes and passing to limits, which is exactly what the reduction to $K_G$ requires.

On the formalization side, nothing here is machine-checked today. The upper bound (Theorem 2.1) is certified by interval arithmetic in the companion paper, and the lower bound's central one-dimensional inequality likewise rests on a computer-assisted certificate; reproducing either inside Lean means building a rigorous numeric layer on top of the analytic argument. Ahead of that, the mission needs a formal definition of $K_G$ itself and of the Krivine-scheme apparatus, neither of which exists in Mathlib — these are reusable well beyond this mission, since Grothendieck's inequality feeds cut-norm approximation and Bell-inequality bounds. Contributions of intermediate lemmas about $\mathrm{OPT}$ , $\mathrm{SDP}$ , Gaussian correlation identities, and Hermite expansions are welcome even when the headline bounds stay open.

Difficulty

The obvious route to a lower bound is to write down a matrix and compute. That route is what Davie and Reeds exhausted; improving it has produced gains of order $10^{-12}$ at best, because the hard instances are high-dimensional Gaussian objects whose $\mathrm{OPT}$ is itself hard to bound tightly. The route taken here avoids instances entirely, and its difficulty lies elsewhere: a constraint on a single scheme is worthless unless it survives averaging over schemes and passing to limits of schemes of growing dimension, since only then does the Naor–Regev optimality theorem convert it into a statement about $K_G$ . Constraints that are nonlinear in the scheme do not survive that passage, which is why the target inequality is affine in $(b_1,b_3)$ . For the upper bound, the difficulty is that the improvement is genuinely asymptotic: it comes from a limit of schemes of growing dimension rather than any fixed low-dimensional partition, and the final margin of $3.47\times10^{-4}$ is certified numerically rather than in closed form.

Formalization scope

$\mathrm{OPT}(A)$ and $\mathrm{SDP}(A)$ are defined as suprema of explicitly described sets of reals, over matrices indexed by Fin m and Fin n with real entries; the sign vectors are real-valued functions constrained to take the values $1$ and $-1$ , and the relaxation quantifies over unit vectors of EuclideanSpace ℝ (Fin d) for an existentially quantified $d$ , so no dimension bound is built in. The empty-index cases $m=0$ or $n=0$ are included and give value $0$ on both sides. $K_G$ is the infimum of the set of Grothendieck bounds; that set is nonempty precisely by Grothendieck's inequality, which is itself a milestone, and it is bounded below, so the infimum is not a junk value.

A Krivine scheme is a structure carrying two measurable $\pm1$ -valued functions on Fin k → ℝ, each odd almost everywhere. Almost-everywhere oddness is forced: no $\pm1$ -valued function satisfies $f(-0)=-f(0)$ at the origin, so a pointwise requirement would make the structure empty and every statement about schemes vacuous. With the null-set relaxation the half-space partition is a scheme in every dimension $k\ge1$ , and the definition file constructs it, pinning down non-vacuity; dimension $k=0$ admits no scheme. The correlation function is the explicit double Gaussian integral against the correlated-pair density, scaled by $\pi/2$ , and the coefficients $b_1,b_3$ are read off as $H'(0)$ and $H'''(0)/6$ — where $H$ fails to be three times differentiable at $0$ these are the ambient junk value $0$ , which a solver should keep in mind when reading the coefficient milestones.

No trivializing reading is available for the goal: it pins $K_G$ between two explicit numerical constants, so it can be satisfied neither vacuously nor by a degenerate convention. Solvers should be aware that the source paper states its two theorems in abridged form and refers to its companion paper for the full proofs, and that the further bounds reported there — the stronger lower rungs $27\pi/49$ and $51\pi/92$ , and the upper values $1.781801841033$ and $1.7813319810625639$ — are explicitly described as system-tested but not author-verified; they are deliberately outside this mission's milestone list.

Selected references

A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79.
J.-L. Krivine, Sur la constante de Grothendieck, C. R. Acad. Sci. Paris (1977).
M. Braverman, K. Makarychev, Y. Makarychev, A. Naor, The Grothendieck constant is strictly smaller than Krivine's bound, FOCS 2011, 453–462. https://doi.org/10.1109/FOCS.2011.77
A. Naor, O. Regev, Krivine schemes are optimal, Proc. Amer. Math. Soc. 142 (2014), 4315–4320. https://doi.org/10.1090/S0002-9939-2014-12145-3
N. Alon, A. Naor, Approximating the cut-norm via Grothendieck's inequality, SIAM J. Comput. 35 (2006), 787–803. https://doi.org/10.1137/S0097539704441629
A. Li, R. Saha, A. Xue, S. Chaudhuri, A. Klivans, P. K. Kothari, R. Meka, Long-Horizon AI Research for Grothendieck Constant: A Case Study in Human–AI Mathematical Collaboration, arXiv:2608.11195v3, 2026. https://arxiv.org/abs/2608.11195
R. Saha, A. Li, A. Xue, S. Chaudhuri, A. Klivans, P. K. Kothari, R. Meka, New upper and lower bounds for the Grothendieck constant, 2026 (companion paper containing the full proofs).

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Machine Learning Operations Research Probability+1·Captain: mikedeng1

Foundations of Machine Learning XIV: Finite Markov Decision Processes and Bellman's EquationsTextbook

Motivation

Reinforcement learning formalizes a scenario supervised learning cannot: an agent that actively interacts with an environment, choosing actions that change both the state it observes next and the reward it receives, rather than passively receiving an i.i.d. labeled sample. Every practical treatment of this scenario — from classical dynamic programming to modern deep reinforcement learning — is built on the Markov decision process (MDP), a model in which the effect of an action depends only on the current state, not on the full history that led to it. Two questions define the theory this mission covers: given a fixed way of acting (a policy), what value does it obtain, and how is that value actually computed rather than merely characterized as the solution of a fixed-point equation? Mohri, Rostamizadeh and Talwalkar's chapter 17 answers both for the stationary, infinite-horizon discounted case, and this mission targets its two central results: that a fixed policy's value is not just characterized but uniquely determined by a linear system with an explicit closed-form solution (Theorem 17.10), and that the optimal value function — obtained instead by choosing the best action at every state — can be computed by an iterative algorithm guaranteed to converge regardless of where it starts (Theorem 17.11).

Setting

A (finite) Markov decision process consists of a finite set of states $S$ , a finite set of actions $A$ , a transition kernel $P[s'\mid s,a]$ giving the distribution over the next state $s'$ after taking action $a$ at state $s$ , and an expected reward $\mathbb E[r(s,a)]$ for that transition. A (stationary) policy $\pi:S\to\Delta(A)$ assigns each state a distribution over actions — possibly, but not necessarily, a point mass on a single action. Fixing $\pi$ turns the MDP into an ordinary Markov chain on $S$ : at each step the agent is at some state $s$ , draws $a\sim\pi(s)$ , receives (expected) reward $\mathbb E[r(s,a)]$ , and moves to a state drawn from $P[\cdot\mid s,a]$ . For a discount factor $\gamma\in[0,1)$ , the value of $\pi$ at $s$ is the expected discounted sum of future rewards starting from $s$ ,

V_\pi(s) = \mathbb E_{a_t\sim\pi(s_t)}\Big[\sum_{t=0}^{+\infty}\gamma^t r(s_t,a_t) \;\Big|\; s_0=s\Big],

and the state-action value function $Q_\pi(s,a)$ is the analogous quantity for taking $a$ at $s$ and then following $\pi$ . Marginalizing the raw kernel and reward over the mixed action $\pi(s)$ gives the induced transition matrix $P_{s,s'}=P[s'\mid s,\pi(s)]=\sum_a \pi(s)(a) P[s'\mid s,a]$ and induced reward vector $R_s=\mathbb E[r(s,\pi(s))]=\sum_a\pi(s)(a)\,\mathbb E[r(s,a)]$ — the objects that turn $\pi$ 's value into a genuinely linear-algebraic quantity. A policy $\pi^*$ is optimal if $V_{\pi^*}(s)\ge V_\pi(s)$ for every policy $\pi$ and every state $s$ ; write $V^*$ for its value function.

Formalization targets

Theorem 17.10 (goal). For a finite MDP and a fixed policy $\pi$ , the matrix $I-\gamma P$ (with $P$ the policy-induced transition matrix) is invertible, and $\pi$ 's value function is the unique solution of the Bellman equations, given in closed form by

V_\pi = (I-\gamma P)^{-1} R.

Proposition 17.9 (milestone). The value function itself satisfies the linear system that Theorem 17.10 solves:

\forall s\in S,\quad V_\pi(s) = \mathbb E_{a\sim\pi(s)}[r(s,a)] + \gamma\sum_{s'} P[s'\mid s,\pi(s)]\,V_\pi(s').

Theorem 17.7 (milestone). A policy $\pi$ is optimal if and only if it places probability only on $Q_\pi$ -maximizing actions: for every $(s,a)$ with $\pi(s)(a)>0$ , $a\in \operatorname{argmax}_{a'} Q_\pi(s,a')$ .

Theorem 17.11 (milestone). The Bellman optimality operator $\Phi$ , $[\Phi(V)](s)=\max_{a} \{\mathbb E[r(s,a)]+\gamma\sum_{s'}P[s'\mid s,a]V(s')\}$ , is a $\gamma$ -contraction for $\lVert\cdot\rVert_\infty$ ; consequently, for any starting vector $V_0$ , the value-iteration sequence $V_{n+1}=\Phi(V_n)$ converges to a fixed point of $\Phi$ .

Significance

Theorem 17.10 is what makes policy evaluation on a finite MDP an exact, finite computation rather than an infinite limit: instead of summing an infinite discounted series or solving an implicit fixed-point equation numerically, a single $|S|\times|S|$ matrix inversion gives the policy's value at every state simultaneously. It is also the base case every planning algorithm in the chapter builds on: policy iteration alternates optimizing a policy with exactly this evaluation step. Theorem 17.11 gives the complementary guarantee for the harder problem of finding the optimal value function directly, without fixing a policy first: value iteration converges from any starting point, with a convergence rate ( $O(\log(1/\epsilon))$ iterations for $\epsilon$ -accuracy) that follows from the same contraction argument. Together, the two results are the mathematical content behind why dynamic-programming planning for finite MDPs is tractable at all — the discount factor $\gamma<1$ , not any structural assumption on rewards or transitions, is what buys both the uniqueness in Theorem 17.10 and the convergence in Theorem 17.11. Formalizing them requires reproducing this linear-algebraic and metric content precisely, not just asserting the conclusions: an invertibility claim asserted without the operator-norm argument, or a convergence claim without the contraction property, would state something true by fiat rather than the book's actual result. No faithful prior art exists on the platform for this exact model (see Formalization scope).

Difficulty

The obvious shortcut for Theorem 17.10 is to assert $I-\gamma P$ is invertible without proof — true, but not what the book does, and not informative about why it holds. The genuine content is that $P$ , being row-stochastic (every row of $P$ sums to exactly $1$ , since $\pi(s)$ and $P[\cdot\mid s,a]$ are both proper distributions), has operator norm $\lVert P\rVert_\infty=1$ exactly, so $\lVert\gamma P\rVert_\infty=\gamma<1$ strictly; this rules out $1$ as an eigenvalue of $\gamma P$ , which is exactly what invertibility of $I-\gamma P$ requires. The same $\gamma<1$ fact, applied differently, drives Theorem 17.11: showing $\Phi$ is $\gamma$ -Lipschitz requires bounding $\Phi(V)(s)-\Phi(U)(s)$ by comparing the maximizing action for $V$ against the same action's value under $U$ (not $U$ 's own maximizer), since the two suprema need not be attained at the same action — a step easy to state incorrectly as a direct comparison of two maxima. Both theorems fail if $\gamma=1$ is allowed: the discounted setting's central asset, a strict contraction, disappears exactly at that boundary.

Formalization scope

States and actions are modeled as finite types (Fintype S, Fintype A); the raw kernel and reward P : S → A → S → ℝ, Er : S → A → ℝ are unconstrained functions, with IsTransitionKernel asserting the required distribution property explicitly rather than assuming it silently. A policy is π : S → A → ℝ with IsPolicy π asserting π s is a distribution over A for every s — deliberately not π : S → A or a PMF-valued function, since Theorem 17.7's own quantifier ("for any pair (s,a) with π(s)(a) > 0") requires treating π(s) as a genuine mixture. PolicyValue is defined as the actual infinite discounted expectation (via an explicit state-occupation-distribution recursion), not as the Bellman fixed point — so that Proposition 17.9 (the value function satisfies the linear system) and Theorem 17.10 (that system has a unique, invertible-matrix solution) are both non-vacuous claims about the same object, rather than one being definitionally true of the other. The trivializing formalization this rules out is asserting IsUnit (1 - γ • P) as a bare hypothesis, or defining V_π as (1-γP)⁻¹R and calling the resulting identity a theorem; both would erase the mission's actual content. Two platform modules model related MDPs (BertsekasSSPModel, a stochastic-shortest-path model with a termination-probability deficit rather than exact row-stochasticity, and FoundationsRL.RLBasics, a finite-horizon episodic model indexed by layer) — neither specializes exactly to this chapter's stationary, always-continuing, infinite-horizon discounted convention, so every definition here is drafted fresh rather than imported. This chunk covers §17.2–17.4.2 (the MDP model, policy value, Bellman's equations, value and policy iteration); §17.4.3 (the linear-programming formulation) and §17.5 (stochastic-approximation learning algorithms — TD(0), Q-learning, SARSA) are out of scope, since they require a stochastic-approximation convergence substrate this mission does not build.

Selected references

Mohri, M., Rostamizadeh, A., and Talwalkar, A. Foundations of Machine Learning, 2nd ed., chapter 17. MIT Press, 2018.
Bellman, R. Dynamic Programming. Princeton University Press, 1957.
Puterman, M. L. Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, 1994.

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning XII: Algorithmic StabilityTextbook

Motivation

Every generalization bound in Chapters 2-11 depends only on the complexity of a fixed hypothesis set $H$ — Rademacher complexity, VC-dimension, growth function — and holds regardless of which algorithm within $H$ actually returns the hypothesis. This is both a strength (broad applicability) and a limitation: it throws away everything specific to how an algorithm searches $H$ , and can be uninformative when $H$ itself is large or unbounded (e.g. a regularized objective that implicitly restricts the search without shrinking $H$ as a set). Chapter 14 introduces a fundamentally different route to a generalization bound — a property of the algorithm rather than the hypothesis class — first used by Devroye, Rogers and Wagner for $k$ -nearest-neighbor rules and given its modern general form by Bousquet and Elisseeff (2002), whose treatment this chapter follows and (for non-differentiable convex losses) extends.

Setting

A labeled example is $z=(x,y)\in X\times Y$ ; for a loss function $L:Y'\times Y\to\mathbb R_+$ (where $Y'$ may differ from $Y$ , e.g. $Y=\{-1,+1\}$ but $Y'=\mathbb R$ for a real-valued hypothesis), the loss of a hypothesis $h$ at $z$ is $L_z(h)=L(h(x),y)$ . Given a learning algorithm $A$ that maps a sample $S$ of size $m$ to a hypothesis $h_S\in H$ , the empirical error and generalization error are $\hat R_S(h)=\frac1m\sum_iL_{z_i}(h)$ and $R(h)=\mathbb E_{z\sim D}[L_z(h)]$ . Uniform $\beta$ -stability (Definition 14.1) says: for any two samples $S$ , $S'$ differing by a single point, the algorithm's returned hypotheses satisfy $|L_z(h_S)-L_z(h_{S'})|\le\beta$ for every $z$ — replacing one training point can change the algorithm's loss on any point by at most $\beta$ . For the regularized algorithms studied in §14.3, a kernel-based regularization algorithm minimizes $F_S(h)=\hat R_S(h)+ \lambda\|h\|_K^2$ over the RKHS $H$ of a positive-definite kernel $K$ , and a loss $L$ is $\sigma$ -admissible (Definition 14.3) if $|L(h'(x),y)-L(h(x),y)|\le\sigma|h'(x)-h(x)|$ for all hypotheses $h,h'$ — a Lipschitz-like smoothness condition satisfied by the standard regression and classification losses.

Formalization targets

Proposition 14.4 (milestone). For a PDS kernel $K$ with $K(x,x)\le r^2$ and a convex, $\sigma$ -admissible loss $L$ , the kernel-based regularization algorithm is $\beta$ -stable with

\beta \le \frac{\sigma^2r^2}{m\lambda}.

Corollary 14.5 (milestone). For SVR (the $\epsilon$ -insensitive loss $L_\epsilon$ , bounded by $M$ ), with probability at least $1-\delta$ :

R(h_S) \le \hat R_S(h_S) + \frac{r^2}{m\lambda} + \Big(\frac{2r^2}\lambda+M\Big)\sqrt{\frac{\log(1/\delta)}{2m}}.

Theorem 14.2 — the mission's goal. For a loss bounded by $M$ and a $\beta$ -stable algorithm $A$ , with probability at least $1-\delta$ over a sample $S$ of size $m$ :

R(h_S) \le \hat R_S(h_S) + \beta + (2m\beta+M)\sqrt{\frac{\log(1/\delta)}{2m}}.

Significance

Theorem 14.2 is the book's demonstration that algorithm-dependent analysis is not merely a special-case curiosity: it is broad enough to cover an entire family (every kernel-based regularization algorithm — KRR, SVR, SVMs, and beyond) uniformly, via a single stability coefficient computation (Proposition 14.4) that is then specialized per algorithm just by plugging in that loss's admissibility constant $\sigma$ . Corollary 14.5's SVR bound is the concrete payoff: a fully explicit, dimension-free generalization guarantee for a widely used regression algorithm, with every constant ( $r$ , $\lambda$ , $m$ ) traceable to the algorithm's own hyperparameters, no VC-dimension or Rademacher-complexity computation required. Unlike Chapters 3-11, whose bounds are oblivious to how $H$ is searched, algorithmic stability is the first tool in the book that can, in principle, certify generalization for a hypothesis class too large or poorly understood for a complexity-based bound to be informative, provided the algorithm itself is stable. No prior art on the Prove2Me platform is faithful: GET /theorems?q=algorithmic+stability, q=uniform+stability return no hits; q=McDiarmid returns only bounded_diff_martingale_two_sided (Boucheron-Lugosi-Massart's own two-sided bounded-differences martingale inequality), which is McDiarmid's inequality's own proof engine (the background result Theorem 14.2's proof applies), not any result of this chapter — a different mathematical object entirely, not reused. All eleven items are drafted fresh.

Not formalized here: Corollary 14.6 (KRR bound), Lemma 14.7 (boundedness of kernel-regularization hypotheses) and Corollary 14.8 (SVM bound). Corollary 14.6 is structurally identical to Corollary 14.5 (a different loss function's admissibility constant plugged into the same Proposition 14.4 + Theorem 14.2 chain) and adds no new formalization content beyond Corollary 14.5, already drafted; Lemma 14.7 and Corollary 14.8 are omitted together, since 14.8's own statement needs 14.7's bound on $|h_S(x)|$ to compute its explicit $M$ (unlike Corollary 14.5, which is given $M$ as a hypothesis) — a genuine additional formalization layer (the reproducing-kernel norm bound $|h_S(x)|\le r\sqrt{B/\lambda}$ ) disproportionate to a single further corollary within this mission's budget.

Difficulty

The chapter's central technical step is recognizing that $\beta$ -stability plus the loss bound $M$ together give exactly the bounded-difference property McDiarmid's inequality needs, applied to $\Phi(S)=R(h_S)-\hat R_S(h_S)$ as a function of the sample: replacing one point of $S$ changes $R(h_S)$ by at most $\beta$ (stability applied to the population loss, an expectation over $z$ ) and changes $\hat R_S(h_S)$ by at most $\beta+M/m$ (stability on the $m-1$ shared points, plus the full loss bound $M/m$ on the one point that actually changed) — two different, asymmetric arguments that must be combined correctly to get $|\Phi(S)-\Phi(S')|\le 2\beta+M/m$ , not merely "stability implies boundedness" asserted directly. Proposition 14.4's own proof (not formalized here beyond its statement) needs a generalized Bregman divergence to handle a possibly non-differentiable convex loss — an extension of Bousquet-Elisseeff's original argument the book credits to itself as novel — via the reproducing-kernel property and Cauchy-Schwarz to convert a divergence bound into a bound on $\|h-h'\|_K$ , then back into a pointwise loss bound.

Formalization scope

IsRKHSOf/IsMinimizer are restated locally in Stability, byte-identical to chunk 06-kernels's own copies (a draft item cannot import another chunk's draft module); H is an abstract real inner-product space with an evaluation map ev : H → X → ℝ standing for "elements of H are functions on X", the same device chunk 06's own RKHS formalization uses, since Mathlib's abstract Hilbert spaces are not themselves spaces of functions. UniformlyStable fixes the sample size m as part of the algorithm's type (A : (Fin m → X × Y) → (X → Y')), matching the book's own standing convention of a fixed sample size m throughout the chapter. Proposition 14.4 is stated pairwise — for any two samples differing by one point and any minimizers of their respective regularized objectives, the pointwise loss bound holds — rather than fixing a global choice-function algorithm A, since the book's own proof picks an arbitrary minimizer of each objective without asserting uniqueness; Corollary 14.5 does fix a choice function A (one minimizer per sample), since Theorem 14.2's own statement needs a single algorithm evaluated across the whole product-measure sample space. No numerical constant in any of the three theorems is altered from the book's own displayed form. A trivializing formalization this mission avoids: stating Theorem 14.2 only for the strict per-hypothesis loss bound (∀ h ∈ H, ∀ z, L_z(h) ≤ M) rather than the book's own weaker, algorithm-specific condition (hbound, ∀ S, ∀ z, L_z(A S) ≤ M) — the weaker hypothesis is kept, exactly matching the book's explicit statement that "a weaker condition suffices."

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 14.
O. Bousquet, A. Elisseeff, "Stability and generalization," Journal of Machine Learning Research 2, 2002, 499-526.
M. Kearns, D. Ron, "Algorithmic stability and sanity-check bounds for leave-one-out cross-validation," Neural Computation 11(6), 1999, 1427-1453.

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning XI: Maximum Entropy Models and DualityTextbook

Motivation

Maximum entropy (Maxent) models are a widely used family of density-estimation algorithms: given a sample and a set of features, they select the distribution that matches the empirical feature averages while being otherwise as "agnostic" (close to a prior, usually uniform) as possible — a principle that, notably, never requires specifying a parametric family of distributions to search over. This mission formalizes the theorem that explains why this works in practice: Maxent's primal optimization (over distributions, subject to feature-matching constraints) is exactly dual to an unconstrained maximum-likelihood problem over a specific, rich parametric family — the Gibbs distributions — even though the Maxent principle never mentions that family at all.

Setting

For a sample $S=(x_1,\dots,x_m)$ drawn i.i.d. from $D$ over a finite set $X$ , and a feature map $\Phi:X\to\mathbb R^N$ with $\|\Phi\|_\infty\le r$ , the Maxent principle seeks $p\in\Delta$ (the simplex of distributions over $X$ ) minimizing the relative entropy $D(p\|p_0)$ to a prior $p_0$ , subject to $\|E_{x\sim p}[\Phi(x)]-E_{x\sim\hat D}[\Phi(x)]\|_\infty\le\lambda$ (problem 12.7). Introducing the indicator function $I_K$ ( $0$ on $K$ , $+\infty$ elsewhere) turns this into the unconstrained primal objective $F(p)=\tilde D(p\|p_0)+I_C(E_p[\Phi])$ (Eq. 12.8), with $C$ the feature-constraint set. A Gibbs distribution with parameter $w\in\mathbb R^N$ is $p_w(x)=p_0(x)e^{w\cdot\Phi(x)}/Z(w)$ , $Z(w)$ the partition function (Eq. 12.9); its associated dual objective is $G(w)=\frac1m\sum_i\log\frac{p_w(x_i)}{p_0(x_i)}-\lambda\|w\|_1$ (Eq. 12.10) — note $-\frac1m\sum_i\log p_w(x_i)$ is exactly the empirical log-loss $L_S(w)$ , so maximizing $G$ is minimizing an L1-regularized log-loss over the Gibbs family.

Formalization targets

Theorem 12.2 — the mission's goal (Maxent duality). $\sup_{w\in\mathbb R^N}G(w)=\min_pF(p)$ . Furthermore, letting $p^*=\arg\min_pF(p)$ and $d^*=\sup_wG(w)$ : for any $\epsilon>0$ and any $w$ with $|G(w)-d^*|<\epsilon$ , $D(p^*\|p_w)\le\epsilon$ .

Theorem 12.3 (Maxent L1-regularization generalization bound, milestone). Fix $\delta>0$ . Let $\hat w$ solve the L1-regularized dual (12.12) with $\lambda=2R_m(H)+r\sqrt{\log(2/\delta)/(2m)}$ . Then, with probability at least $1-\delta$ ,

L_D(\hat w) \le \inf_wL_D(w) + 2\|\hat w\|_1\Big[2R_m(H)+r\sqrt{\log(2/\delta)/(2m)}\Big].

Significance

Theorem 12.2 is one of the most striking dualities in the book: the Maxent principle, phrased purely in terms of closeness to a prior distribution, turns out to always produce a solution in the Gibbs family — not because that family was ever specified, but because relative entropy is the specific measure of closeness whose Fenchel conjugate is the log-partition function. This explains a whole zoo of models (log-linear models, exponential families, Gaussian and bimodal Gibbs distributions from quadratic features) as instances of a single duality theorem, and gives a computationally friendlier route to the (constrained, infinite-if- $X$ -is-large) primal problem via the (unconstrained, $N$ -dimensional) dual. The theorem's proof is a genuine application of conditional (Fenchel) strong duality, not an unconditional fact — this is, per the chapter's own brief, the sharpest trivialization risk in the entire mission series, since "strong duality always holds for convex problems" is false in general, and a formalization skipping the book's own qualification condition ( $\lambda>0$ , placing $u_0$ in the interior of the constraint set) would prove a different, potentially-false statement. No prior art on the platform is faithful: GET /theorems?q=maximum+entropy returns no hits, and Mathlib's generic Fenchel-conjugate machinery (Analysis/Convex/Conjugate) does not package the book's own specific qualification conditions as a single reusable theorem matching Theorem B.39 — reusing it inside a proof (not the audited statement) remains available to whoever proves this theorem later.

Not formalized here: Theorem 12.4 (a Bregman-divergence generalization of Theorem 12.2) and Theorem 12.5 (its L2-regularized concrete special case). BRIEF.md itself flags Theorem 12.4 as possibly too heavy and offers Theorem 12.5 as an easier alternative; this mission omits both, since even Theorem 12.5 requires a second, structurally parallel dual-objective-and-minimizer formalization (for L2 rather than L1 regularization) — disproportionate to this mission's budget once Theorem 12.2's own qualification-condition bookkeeping (the heaviest single item in this mission series) is accounted for. §12.1 (density estimation without features: ML/MAP), §12.7 (coordinate descent), and §12.8-12.9 (Bregman-divergence extensions, L2-regularization in general) are likewise out of scope, per BRIEF.md's own page-range restriction.

Difficulty

Theorem 12.2's proof is the book's own explicit application of the Fenchel duality theorem (Theorem B.39, Appendix B) to the specific triple $f(p)=\tilde D(p\|p_0)$ , $g(u)=I_C(u)$ , $Ap=\sum_xp(x)\Phi(x)$ — every qualification condition (A a bounded linear map, u_0\in A(\mathrm{dom}f)\cap\mathrm{cont}(g), needing \lambda>0 to place u_0 in int(C)) must be checked for this triple, not assumed generically; the conjugate computations themselves (f^*(q)=\log\sum_xp_0(x)e^{q(x)}$ via Lemma B.37, g^(w)=E_{\hat D}[w\cdot\Phi]+\lambda|w|_1 via the dual-norm identity) are specific algebraic derivations, not immediate from abstract duality alone. The second clause's proof needs a further, non-obvious algebraic identity (G(w)-D(p^|p_0)+D(p^|p_w)expanding, via Hölder's inequality applied to the primal feasibility ofp^, to something \le0) that is not a restatement of the first clause but a separate argument built on top of it. Theorem 12.3's proof structurally mirrors chunk 04's SRM bound (bounding L_D(\hat w)-L_S(\hat w)via Hölder's inequality and the Rademacher-complexity feature-concentration bound of Eq. 12.5, then using\hat w`'s optimality twice), but is applied to the log-loss of a Gibbs distribution rather than a generic bounded loss.

Formalization scope

MaxEntPrimalObjective uses EReal (the extended reals) so that the book's own +\infty values (from I_K, \tilde D) are represented exactly, matching the chapter's own explicit use of an extended-real-valued indicator function rather than a soft penalty — a trivializing formalization this mission avoids is silently replacing +\infty with a large real sentinel, which would misstate a convex-analysis object whose entire role in the proof is its infinite value outside the feasible/simplex set. hlam : 0 < lam is a genuine load-bearing hypothesis in the goal theorem, matching the book's own use of \lambda>0 to invoke Theorem B.39's qualification condition — not a free convexity assumption; this is the mission's central faithfulness guard against the chapter's own named trivialization risk. EmpiricalRademacherComplexity/ RademacherComplexity are restated locally, byte-identical to chunks 05-svm/07-boosting's own copies (a draft item cannot import another chunk's draft module). p^* in the goal theorem and \hat w in Theorem 12.3 are both quantified via explicit hypotheses (IsLeast, a minimizer inequality) rather than assumed to exist unconditionally, matching the book's own "let p^*=..."/"let \hat w be a solution of..." phrasing without asserting existence or uniqueness beyond what the book itself asserts. No numerical constant in either theorem is altered from the book's own displayed form.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 12, §12.1-12.6.
E. T. Jaynes, "Information theory and statistical mechanics," Physical Review 106(4), 1957, 620-630.
S. Della Pietra, V. Della Pietra, J. Lafferty, "Inducing features of random fields," IEEE Transactions on Pattern Analysis and Machine Intelligence 19(4), 1997, 380-393.

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning X: Regression and Rademacher Complexity BoundsTextbook

Motivation

Every generalization bound presented so far in this series is for classification, where the error of a prediction is binary (correct or not). Regression asks a different question: predictions are real-valued, and error is measured by the magnitude of the deviation from the true label, via a loss function L. Chapter 11 develops generalization theory for bounded regression, showing that the same two complexity measures used for classification — Rademacher complexity and a VC-dimension analogue — extend naturally, once the loss function itself is folded into the machinery via a Lipschitz-contraction argument (Rademacher route) or a reduction to classification via level-set thresholding (pseudo-dimension route).

Setting

A regression hypothesis h:X→ℝ is scored by a loss L:ℝ×ℝ→ℝ against a joint distribution D on X×ℝ (the stochastic scenario, since regression labels are rarely exactly reproducible); R(h) = E_{(x,y)~D}[L(h(x),y)] (Eq. 11.1) and R̂_S(h) = (1/m)∑L(h(x_i),y_i) (Eq. 11.2). For a finite hypothesis set, Theorem 11.1 gives a Hoeffding/union-bound guarantee directly, the regression analogue of chunk 02-pac's finite-hypothesis bound. For infinite H, §11.2.2 develops a Rademacher-complexity route: Proposition 11.2 shows that if L is µ-Lipschitz in its first (predicted-value) argument, the Rademacher complexity of the loss-composed family G = {(x,y)↦L(h(x),y) : h∈H} is controlled by µ times H's own Rademacher complexity, via Talagrand's contraction lemma (chunk 05-svm's Lemma 5.7); Theorem 11.3 combines this with chunk 03's Theorem 3.3 to give the chapter's headline bound. §11.2.3 develops an independent, purely combinatorial route: pseudo-dimension (Definition 11.5), a real-valued analogue of VC-dimension defined via threshold-witnessed shattering (Definition 11.4, restated via its own Eq. 11.3 as the VC-dimension of a thresholded indicator family); Theorem 11.8 gives a pseudo-dimension generalization bound by reducing regression to a family of classification problems (one per threshold t), using the tail-integral identity Eq. 11.5.

Formalization targets

Theorem 11.1 (milestone). For L bounded by M and H finite: for any δ>0, with probability at least 1-δ, for all h∈H: R(h) ≤ R̂_S(h) + M√((log|H|+log(1/δ))/(2m)).

Proposition 11.2 (milestone). For L non-negative, bounded by M, µ-Lipschitz in its first argument: for any sample S, R̂_S(G) ≤ µR̂_S(H).

Theorem 11.3 — the mission's goal. Under Proposition 11.2's hypotheses on L: for any δ>0, with probability at least 1-δ, for all h∈H: E[L(h(x),y)] ≤ (1/m)∑L(h(x_i),y_i) + 2µR_m(H) + M√(log(1/δ)/(2m)), and also with 2µR̂_S(H) + 3M√(log(2/δ)/(2m)).

Theorem 11.8 (milestone). For Pdim(G)=d, L non-negative bounded by M: for any δ>0, with probability at least 1-δ over a sample of size m, for all h∈H: R(h) ≤ R̂_S(h) + M√(2d log(em/d)/m) + M√(log(1/δ)/(2m)).

Significance

Theorem 11.3 is the chapter's own choice of headline result (§11.2's stated goal is to show "how the Rademacher complexity bounds of theorem 3.3 can be used to derive generalization bounds for regression"), and its proof genuinely reuses two pieces of prior machinery from this series — chunk 03's Theorem 3.3 and chunk 05's Talagrand's-lemma-style contraction — combined via a new observation (Proposition 11.2) specific to loss-composed families, not a restatement of either. Theorem 11.8 is the chapter's second, structurally independent technique: its em/d bound parallels chunk 03's Corollary 3.19 (both ultimately reduce to a VC-dimension-style growth-function argument), but the reduction itself — regression to a continuum of threshold classification problems, via the Lebesgue-integral tail identity Eq. (11.5) applied to |R(h)-R̂_S(h)| — is genuinely new content for this book, and pseudo-dimension has no prior art on the platform or in Mathlib. No prior art exists for this chapter's overall content either: GET /theorems?q=generalization%20bound%20regression and GET /theorems?q=pseudo-dimension both return zero hits.

Difficulty

Proposition 11.2's proof needs Talagrand's contraction lemma applied with the Lipschitz constant taken in the first argument of L only — the predicted value h(x_i), holding the true label y_i fixed — exactly the pitfall BRIEF.md names: a loss Lipschitz in the wrong argument, or in both arguments jointly, would not license this step. Theorem 11.8's proof is the chapter's most involved: it defines, for every h∈H and threshold t≥0, a classifier c(h,t):(x,y)↦1_{L(h(x),y)>t}, bounds |R(h)-R̂_S(h)| by M·sup_{t∈[0,M]}|R(c(h,t))- R̂_S(c(h,t))| via the tail-integral identity, and then applies a VC-dimension-style classification bound (Corollary 3.19) to the family of thresholded classifiers — whose VC-dimension is, by Eq. (11.3), exactly Pdim(G) by construction. A formalization that conflated pseudo-dimension with ordinary VC-dimension, or reused chunk 03's HasVCDim definition by relabeling, would misrepresent this chapter's genuinely different (real-valued, threshold-witnessed) combinatorial notion — precisely the pitfall BRIEF.md flags.

Formalization scope

Y := ℝ throughout (the book's own "Y a measurable subset of ℝ"), a harmless simplification consistent with every hypothesis, loss and Lipschitz condition in this chapter being stated for real-valued scores and labels. EmpiricalRademacherComplexity/ RademacherComplexity restate chunk 03-rademacher-vc's Definitions 3.1/3.2 locally, since a draft item cannot import another chunk's draft module. Shatters/PseudoDim are formalized via the book's own equivalent reformulation (Eq. 11.3, the thresholded-indicator form), rather than the sign-function form of Definition 11.4 directly, since the two coincide except at a measure-zero boundary the book itself does not address; PseudoDim mirrors chunk 03's HasVCDim Prop-valued pattern (does not cover Pdim(G)=+∞; every consuming theorem takes it as an explicit hypothesis) but is a structurally distinct definition built on Shatters, never a relabeling of HasVCDim, per BRIEF.md's pitfall note. Proposition 11.2's and Theorem 11.3's Lipschitz hypothesis (hLlip) is stated with the true label y' universally quantified outside the two-point comparison y1, y2 (the predicted values), matching "for any fixed y' ∈ Y, y ↦ L(y,y') is µ-Lipschitz" exactly — Lipschitzness in the first argument only, per BRIEF.md's pitfall note. RademacherComplexity (Measure.map Prod.fst D) H m gives the book's R_m(H) (H's Rademacher complexity under the marginal sampling distribution of the inputs x, i.e. D's first marginal). No numerical constant is altered from the book in any of the four theorems.

Not formalized: the L_p-loss worked example following Theorem 11.3's proof (an instantiation of the general theorem for a specific loss family, not a separate numbered theorem); Theorem 11.6 and Theorem 11.7 (worked pseudo-dimension examples for hyperplanes and vector spaces, background/illustration rather than the chapter's general machinery — drafting only these examples instead of the general Theorem 11.8 would be this chapter's trivializing formalization); the two-sided variant of Theorem 11.1 mentioned immediately after its proof (an unnumbered remark, not a separately displayed/numbered theorem); and all of §11.3 (linear regression, kernel ridge regression, SVR, Lasso and their online variants), which is applications-heavy per BRIEF.md's chapter restriction to §11.1-11.2.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 11 (§11.1-11.2).
D. Haussler, "Decision theoretic generalizations of the PAC model for neural net and other learning applications," Information and Computation 100(1), 1992 (pseudo-dimension's origin).
D. Pollard, Convergence of Stochastic Processes, Springer, 1984 (the tail-integral identity Eq. 11.5's classical antecedent).

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning VIII: Multi-Class Classification and the Margin BoundTextbook

Motivation

Every generalization bound in chapters 2-5 is for binary classification. Most real-world classification problems have more than two classes, and the number of classes can itself be in the hundreds or thousands (topic classification, speech recognition). Chapter 9 extends the margin-based generalization theory of chapter 5 (SVMs) to this multi-class, mono-label setting, using the same Rademacher-complexity machinery as chunk 03-rademacher-vc, but with a new combinatorial ingredient — bounding the Rademacher complexity of a family built by taking a pointwise maximum over several hypothesis sets — needed because a multi-class prediction is itself an argmax over per-class scores.

Setting

A multi-class hypothesis is a scoring function $h:X\times Y\to\mathbb R$ with $Y=\{1,\dots,k\}$ (mono-label case); the predicted label is $\arg\max_y h(x,y)$ , and the margin $\rho_h(x,y)=h(x,y)-\max_{y'\ne y}h(x,y')$ (p. 215) is negative exactly when $h$ misclassifies $(x,y)$ . The empirical margin loss $\hat R_{S,\rho}(h)$ (Eq. 9.5) uses the same margin-loss function $\Phi_\rho$ (Definition 5.5) as chunk 05-svm, restated locally here. $\Pi_1(H) = \{x\mapsto h(x,y):y\in Y,h\in H\}$ (p. 217) projects a multi-class hypothesis set onto ordinary real-valued functions on $X$ — the object the chapter's Rademacher-complexity bound actually controls, since $H\subseteq\mathbb R^{X\times Y}$ has no norm of its own without such a projection. Lemma 9.1 is a purely combinatorial tool: the empirical Rademacher complexity of a family built by taking the pointwise max over $l$ hypothesis sets is bounded by the sum of their individual empirical Rademacher complexities — used to control the argmax structure of a multi-class prediction. Theorem 9.2 combines this with chunk 03's Rademacher-complexity generalization machinery (Theorem 3.3) to give the chapter's margin bound. Proposition 9.3 and Corollary 9.4 specialize this to kernel-based hypotheses, where each class has its own weight vector in a reproducing kernel Hilbert space and the $k$ weight vectors are jointly constrained by an $L^p$ -type group norm $\|W\|_{H,p}\le\Lambda$ .

Formalization targets

Lemma 9.1 (milestone). For $F_1,\dots,F_l$ hypothesis sets in $\mathbb R^X$ , $l\ge1$ , and $G=\{\max\{h_1,\dots,h_l\}:h_i\in F_i\}$ : $\hat R_S(G)\le\sum_{j=1}^l\hat R_S(F_j)$ .

Theorem 9.2 — the mission's goal. For $H\subseteq\mathbb R^{X\times Y}$ , $Y=\{1,\dots,k\}$ , fix $\rho>0$ . For any $\delta>0$ , with probability at least $1-\delta$ , for all $h\in H$ :

R(h) \le \hat R_{S,\rho}(h) + \tfrac{4k}\rho R_m(\Pi_1(H)) + \sqrt{\tfrac{\log(1/\delta)} {2m}}.

Proposition 9.3 (milestone). For a PDS kernel $K$ with feature map $\Phi$ and $K(x,x)\le r^2$ : $R_m(\Pi_1(H_{K,p})) \le \sqrt{r^2\Lambda^2/m}$ .

Corollary 9.4 (milestone). Under Proposition 9.3's hypotheses, fix $\rho>0$ . For any $\delta>0$ , with probability at least $1-\delta$ , for all $h\in H_{K,p}$ : $R(h) \le \hat R_{S,\rho}(h) + 4k\sqrt{r^2\Lambda^2/\rho^2/m} + \sqrt{\log(1/\delta)/(2m)}$ .

Significance

Theorem 9.2 is the multi-class generalization of chunk 05-svm's Theorem 5.8, and its proof is the chapter's genuine new technique rather than a restatement: it needs a $k$ -way application of Lemma 9.1 (once for the argmax structure of the margin, once summing over the $k$ possible labels), which is exactly where the $4k$ factor comes from. Corollary 9.4 is the direct theoretical basis for the multi-class SVM algorithm the chapter derives next (§9.3.1): the displayed dual optimization problem literally minimizes the right-hand side of the corollary's bound. No prior art exists on the platform: GET /theorems?q=multi-class%20classification returns zero hits, and chunk 03's Rademacher-complexity machinery (needed by the proof route) is a draft, not reusable, per the "drafts cannot import drafts" rule.

Difficulty

Lemma 9.1's proof is a genuine two-function argument (max as $\tfrac12(h_1+h_2+|h_1-h_2|)$ , Talagrand's lemma applied to $|\cdot|$ ) generalized to $l$ functions by induction, not a one-line consequence of chunk 03's single-hypothesis-set bound. Theorem 9.2's own proof (PDF pp. 234-236) is the chapter's most involved: it introduces an auxiliary margin function $\rho_{\theta,h}$ with a free parameter $\theta$ later fixed to $2\rho$ , splits the resulting Rademacher complexity into a "diagonal" term (bounded via a further one-hot decomposition across the $k$ classes, giving the first factor of $k$ ) and a "off-diagonal" term bounded via Lemma 9.1 (giving the second factor, folded into the same $4k$ constant). A formalization that stated Theorem 9.2 for $H$ itself rather than $\Pi_1(H)$ , or that treated $k$ as an unrelated free constant rather than the actual number of classes, would misstate the theorem — precisely the pitfall BRIEF.md names for this chapter. Proposition 9.3's proof is a clean Cauchy-Schwarz/Jensen argument in the RKHS but needs the $L^p$ -group-norm hypothesis class $H_{K,p}$ stated with its exact footnote definition (PDF p. 236), not a simplified $p=2$ special case.

Formalization scope

GeneralizationError, EmpiricalRademacherComplexity and RademacherComplexity are restated locally in this chunk's MultiClass namespace (the last two identical in content to chunk 03-rademacher-vc's own copies); MarginLossFunction restates chunk 05-svm's Definition 5.5 (the same function, needed here for this chapter's own EmpiricalMarginLoss); IsPDS restates chunk 06-kernels's PDS-kernel definition. All are duplicated rather than imported since a draft item cannot import another chunk's draft module, and none of 03, 05, 06 is listed as reusable in missions/README.md's "Published definitions" table at the time of this session. GeneralizationError is formalized via the book's own established equivalence " $h$ misclassifies $(x,y)$ iff $\rho_h(x,y)\le0$ " (the form Theorem 9.2's own proof displays and works with), rather than via an explicit argmax classifier construction — checked as faithful, not a weakening, since it is exactly the quantity the chapter's proof bounds. MarginFunction's ⨆_{y'≠y} is a real supremum rather than a Finset.sup', avoiding a nonempty-finset side proof at definition time; every consuming theorem supplies 2 ≤ k (Y = Fin k) to guard it against trap 5. MaxFamily's index type is Fintype+Nonempty rather than a Finset-cardinality parameter l, a harmless generalization matching "l ≥ 1 hypothesis sets" via Nonempty. IsPDS's feature map Φ and its defining property K(x,y) = ⟪Φ(x),Φ(y)⟫ are supplied as hypotheses to the two kernel theorems rather than as a separate "feature mapping associated to a kernel" definition — the book itself treats this as a given correspondence, not a construction. No numerical constant is altered: 4k/ρ and log(1/δ) in Theorem 9.2, r²Λ²/m in Proposition 9.3, and 4k and r²Λ²/ρ²/m in Corollary 9.4 are exactly as displayed.

Not formalized: §9.1's discussion of the multi-label case (Eq. 9.2/9.3, the Hamming-distance risk) and Eq. 9.4 (empirical Hamming error) — background for a case this chapter's own generalization-bound section (§9.2) does not cover (the mono-label case only); the multi-class SVM primal/dual optimization problems (§9.3.1, an algorithm derived from Corollary 9.4, not a generalization-theoretic theorem); AdaBoost.MH (§9.3.2, a boosting algorithm, analyzed via a convex-surrogate argument rather than the Rademacher-complexity route this mission formalizes); and the uniform-over- $\rho$ extension mentioned at the end of the Theorem 9.2 proof (an unnumbered remark referencing Theorem 5.9's technique from a different chapter, not restated here). Drafting only the algorithmic consequences (the multi-class SVM's optimization problem) in place of the generalization bounds themselves would be this chapter's trivializing formalization.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 9.
V. Koltchinskii, D. Panchenko, "Empirical margin distributions and bounding the generalization error of combined classifiers," Annals of Statistics 30(1), 2002 (Lemma 9.1's technique).
K. Crammer, Y. Singer, "On the algorithmic implementation of multiclass kernel-based vector machines," JMLR 2, 2001 (the multi-class SVM algorithm §9.3.1 derives).

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning VII: On-Line Learning and On-Line-to-Batch ConversionTextbook

Motivation

Every guarantee in the preceding chapters assumes a fixed distribution and i.i.d. sampling. On-line learning drops both assumptions: an algorithm processes one example at a time, in an adversarial (worst-case) sequence, and is judged by regret against the best fixed comparator in hindsight rather than by generalization error. This chapter develops the theory for this setting — mistake bounds and regret bounds for prediction with expert advice, a margin-based mistake bound for the Perceptron — and then closes a conceptual gap: since on-line algorithms need no distributional assumption, can their guarantees be converted into ordinary distributional (batch) generalization guarantees when the data does happen to be i.i.d.? The on-line-to-batch conversion theorem answers yes, using nothing but an Azuma's-inequality martingale argument on the sequence of hypotheses the algorithm actually produces.

Setting

At round t, an on-line algorithm receives x_t, predicts ŷ_t, receives the true label y_t, and incurs loss L(ŷ_t,y_t); its regret R_T (Eq. 8.1) compares its cumulative loss to the best fixed action's in hindsight. §8.2 develops this for prediction with expert advice: the Halving algorithm (realizable case), Weighted Majority and its randomized version RWM (zero-one loss, Theorem 8.4's L_T ≤ log(N)/(1-β) + (2-β)L_T^min, proved by the chapter's recurring potential-function technique applied to W_t = ∑_i w_{t,i}), and the Exponential Weighted Average algorithm (convex losses). §8.3.1 analyzes the Perceptron, a linear classification algorithm whose margin-based mistake bound (Theorem 8.8, separable case; the non-separable Theorem 8.11, restated here, in terms of an arbitrary comparator v's hinge losses) depends only on the normalized margin, not the ambient dimension. §8.4 shows that averaging the hypotheses h_1,…,h_T an on-line algorithm produces while processing an i.i.d. sample S yields a hypothesis with controlled true risk: Lemma 8.14 bounds the average of the per-round risks R(h_t) by the average on-line loss via a martingale argument on V_t = R(h_t) - L(h_t(x_t),y_t), and Theorem 8.15 upgrades this, via the loss's convexity, to a bound on the risk of the averaged hypothesis (1/T)∑h_t.

Formalization targets

Theorem 8.4 (milestone). Fix β∈[1/2,1). For any T≥1: L_T ≤ log(N)/(1-β) + (2-β)L_T^min; for β=max{1/2,1-√(log(N)/T)}: L_T ≤ L_T^min + 2√(T log N).

Theorem 8.11 (milestone). M ≤ inf_{ρ>0,‖v‖₂≤1}[(r/ρ+√(r²/ρ²+4‖l_ρ‖₁))/2]², where l_ρ=(l_t)_{t∈I}, l_t=max{0,1-y_t(v·x_t)/ρ}.

Lemma 8.14 (milestone). For any δ>0, with probability at least 1-δ: (1/T)∑_tR(h_t) ≤ (1/T)∑_tL(h_t(x_t),y_t) + M√(2log(1/δ)/T).

Theorem 8.15 — the mission's goal (first inequality). Under Lemma 8.14's hypotheses, with L additionally convex in its first argument: for any δ>0, with probability at least 1-δ: R((1/T)∑_th_t) ≤ (1/T)∑_tL(h_t(x_t),y_t) + M√(2log(1/δ)/T).

Significance

Theorem 8.15 is the chapter's conceptual capstone: it is the only bridge in the whole book between the adversarial on-line-learning framework and the distributional PAC/statistical framework every other chapter develops, and its proof needs nothing beyond Lemma 8.14 plus convexity — no new machinery, just the right observation about the loss's structure. Theorem 8.4 is the chapter's cleanest instance of its recurring potential-function proof technique (reused, with variations, for Theorems 8.3, 8.6 and 8.7), and — checked against the platform's existing OnlineConvexOpt.Introduction.randomized_weighted_majority_mistake_bound (Hazan series) — a genuinely different result from what is already on the platform: that lemma bounds a mistake count with a (1+ε) multiplier, this bounds the RWM algorithm's own weighted-mixture loss with a 1/(1-β) term and a distinct optimal-β substitution, confirming BRIEF.md's assessment that the two are close but not interchangeable. Theorem 8.11 is the non-realizable generalization of the separable-case Perceptron bound (Theorem 8.8) that motivates soft-margin algorithms generally, expressed via an arbitrary comparator's hinge loss rather than assuming perfect separability. No prior art exists for the chapter's other content: GET /theorems?q=online%20to%20batch returns zero hits, and GET /theorems?q=perceptron returns only an unrelated neural-network topology result.

Difficulty

Theorem 8.4's proof (mirrored by Theorem 8.3's WM analogue) derives matching upper and lower bounds on the potential W_t, combines them via a logarithm, and substitutes a specific optimal β found by differentiating the resulting bound — a genuine two-step optimization argument, not a direct algebraic identity. Theorem 8.11's proof solves a quadratic inequality in √M after summing the hinge-loss-defining inequalities over the update set I and invoking the Cauchy-Schwarz step already used in Theorem 8.8's proof; keeping the inf over both ρ and v in the statement (not fixing them, per BRIEF.md's pitfall note) is what makes this a genuine bound rather than a bound for one arbitrary choice. Lemma 8.14's proof is an application of Azuma's inequality (the book's own Theorem D.7) to the martingale difference sequence V_t = R(h_t) - L(h_t(x_t),y_t), which requires h_t to be measurable with respect to the history strictly before round t — the on-line algorithm's hypothesis at round t must not depend on the pair drawn at that same round, per BRIEF.md's pitfall note. Theorem 8.15's step beyond Lemma 8.14 is the passage from the average of T individual risks to the risk of the averaged hypothesis, licensed by Jensen's inequality under the loss's convexity in its first argument — dropping convexity breaks exactly this step, not merely weakening a constant.

Formalization scope

GeneralizationError restates chunk 11-regression's Eq. (11.1) convention locally (Y := ℝ, consistent with that chunk's own harmless simplification), needed here since Theorem 8.15 requires averaging hypotheses into a single real-valued function. OnlineHypothesis A S t is formalized so that its type signature itself enforces history-adaptedness: the on-line algorithm A : (n:ℕ) → (Fin n → X × ℝ) → (X → ℝ) is a function of the prefix of the sample seen so far, and OnlineHypothesis A S t applies it only to S's first t pairs — this is what licenses Azuma's inequality's martingale-difference argument (the conditional-mean-zero property of V_t), per BRIEF.md's pitfall note. Revision (2026-09-19), correcting an earlier claim in this section: history-adaptedness does not by itself guard against GeneralizationError's Bochner integral silently junking to 0 for a non-measurable hypothesis (a distinct property — whether h_t, as a function of x, is Measurable — from whether h_t depends on round t's own draw). Moderation found this a live gap in both Lemma 8.14 and Theorem 8.15's drafted statements; both now carry an explicit hAmeas/hLmeas hypothesis in addition to the history-adapted type signature. RWM's w_{t,i}, W_t, p_{t,i}, L_t, L_T, L_{T,i}, L_T^min are modeled as their own recursively-defined algorithm state (mirroring, but never substituting into, chunk 07-boosting's AdaBoost pattern), matching this chapter's own loss-based (not mistake-count) quantities, per BRIEF.md's pitfall note distinguishing them from AdaBoost's and RWM-mistake variants. The Perceptron's w_t, update-index set I, and M = |I| are modeled the same way, using Eq. (8.23)'s equivalent sign-agreement update rule (the book's own reformulation of Figure 8.6's sgn-based rule). Theorem 8.11's inf_{ρ>0,‖v‖₂≤1} is a genuine nested restricted infimum (⨅ ρ ∈ Set.Ioi 0, ⨅ v ∈ Metric.closedBall 0 1, …), not a bound instantiated at fixed ρ, v, per BRIEF.md's explicit pitfall note. No numerical constant is altered from the book in any of the four theorems.

Not formalized: Theorems 8.1-8.3 (Halving and WM mistake bounds — the chapter's warm-up results, superseded in content by the more general RWM/EWA theorems that follow), Theorem 8.5 (a matching lower bound, a distinct impossibility result rather than an algorithm's guarantee), Theorems 8.6-8.7 (Exponential Weighted Average regret bounds — a third algorithm with its own potential-function proof, out of scope per BRIEF.md's restriction to §8.2's Halving/WM/RWM), Theorems 8.8-8.10 (the Perceptron's separable-case bound and its leave-one-out-based expected generalization bounds, both superseded in generality by Theorem 8.11 for this mission's purposes), Theorem 8.12 (Perceptron's L²-norm hinge-loss bound, the book's own note that it is implied by, and looser than, Theorem 8.11's L¹-norm bound), the dual/kernel Perceptron (an equivalent reformulation, not new generalization content), and Theorem 8.15's second displayed inequality (a regret-form corollary depending on the regret decomposition of the surrounding discussion, not drafted per BRIEF.md's own recommendation to commit to the first inequality as the goal). §8.3.2 (Winnow) and §8.5 (the game-theoretic connection) are out of scope per BRIEF.md's chapter restriction.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 8 (§8.2, §8.3.1, §8.4).
N. Littlestone, M. K. Warmuth, "The weighted majority algorithm," Information and Computation 108(2), 1994 (WM/RWM's origin).
F. Rosenblatt, "The perceptron: a probabilistic model for information storage and organization in the brain," Psychological Review 65(6), 1958 (the Perceptron algorithm).
Y. Freund, R. E. Schapire, "Large margin classification using the perceptron algorithm," Machine Learning 37(3), 1999 (Theorem 8.11's hinge-loss mistake bound).

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning VI: AdaBoost and Margin TheoryTextbook

Motivation

Weak learning — a base classifier only slightly better than random guessing — is easy to come by; strong learning, in the PAC sense of Chapter 2, is not. Boosting is the technique that turns the first into the second: combine many weak classifiers, each trained on a reweighted version of the sample that emphasizes previously misclassified points, into a single strong ensemble. AdaBoost, the algorithm this chapter studies, does this with a specific, closed-form weighting rule, and comes with two distinct theoretical guarantees: its training error decreases exponentially fast in the number of rounds (Theorem 7.2), and — more surprisingly — its test error can keep improving even after the training error has already reached zero, an empirical phenomenon that Chapter 3's VC-dimension bound cannot explain at all (it predicts overfitting for large numbers of rounds) but that a margin-based analysis, structurally identical to Chapter 5's SVM theory, does (Theorem 7.7). This mission formalizes both routes.

Setting

AdaBoost (Figure 7.1) takes a labeled sample $S=((x_1,y_1),\dots,(x_m,y_m))$ with $y_i\in\{-1,+1\}$ and a base classifier set $H\subseteq\{-1,+1\}^X$ , and runs for $T$ rounds. It maintains a distribution $D_t$ over the sample indices, starting uniform ( $D_1(i)=1/m$ ); at round $t$ it selects a base classifier $h_t$ with small $D_t$ -weighted error $\varepsilon_t=\Pr_{i\sim D_t}[h_t(x_i)\ne y_i]$ , sets $\alpha_t=\frac12\log\frac{1-\varepsilon_t} {\varepsilon_t}$ and $Z_t=2\sqrt{\varepsilon_t(1-\varepsilon_t)}$ , and reweights: $D_{t+1}(i)=D_t(i)\exp(-\alpha_ty_ih_t(x_i))/Z_t$ . After $T$ rounds it returns $f=\sum_{t=1}^T\alpha_th_t$ ; its normalized version is $\bar f=f/\sum_t\alpha_t$ . Since $\varepsilon_t<1/2$ makes $\alpha_t>0$ , $\bar f$ is a genuine convex combination of base classifiers, i.e. a member of the convex hull $\mathrm{conv}(H)=\{\sum_k\mu_kh_k:\mu_k\ge0, h_k\in H,\sum_k\mu_k\le1\}$ (Eq. 7.12). The chapter reuses Chapter 5's confidence-margin apparatus (empirical margin loss $\hat R_{S,\rho}$ , Rademacher complexity $\hat R_S$ / $R_m$ ) to analyze $\bar f$ 's generalization.

Formalization targets

Theorem 7.2 (AdaBoost empirical error bound, milestone). The empirical (zero-one) error of $f$ satisfies $\hat R_S(f) \le \exp(-2\sum_{t=1}^T(1/2-\varepsilon_t)^2)$ , and, if $\gamma\le1/2-\varepsilon_t$ for all $t$ , $\hat R_S(f)\le\exp(-2\gamma^2T)$ : training error decays exponentially in $T$ whenever every round beats random guessing by a fixed margin (the "edge" $\gamma$ ).

Lemma 7.4 (milestone). $\hat R_S(\mathrm{conv}(H))=\hat R_S(H)$ : the convex hull of a hypothesis set, though generally much larger, has exactly the same empirical Rademacher complexity as the set itself.

Corollary 7.5 (Ensemble Rademacher margin bound, milestone). For $H$ a set of real-valued functions and $\rho>0$ , with probability at least $1-\delta$ , every $h\in\mathrm{conv}(H)$ satisfies $R(h)\le\hat R_{S,\rho}(h)+\frac2\rho R_m(H)+\sqrt{\log(1/\delta)/(2m)}$ (and the empirical-complexity analogue with an extra additive $3\sqrt{\log(2/\delta)/(2m)}$ term) — this is Theorem 5.8's margin bound applied to $\mathrm{conv}(H)$ , then rewritten via Lemma 7.4 so its complexity term is $H$ 's own, not the (much larger) convex hull's.

Theorem 7.7 — the mission's goal. Assume $\varepsilon_t<1/2$ for every $t\in[T]$ (so $\alpha_t>0$ ). Then for any $\rho>0$ ,

\hat R_{S,\rho}(\bar f) \le 2^T\prod_{t=1}^T\sqrt{\varepsilon_t^{1-\rho}(1-\varepsilon_t)^{1+\rho}}.

Significance

Theorem 7.7's bound is what makes margin theory a genuine explanation of AdaBoost's empirical behavior: combined with Corollary 7.5 (applied to $\bar f\in\mathrm{conv}(H)$ ), it shows that if AdaBoost's edge stays bounded away from zero, the empirical margin loss at a fixed $\rho$ decreases exponentially in $T$ while the generalization bound's complexity term does not depend on $T$ at all — so continuing to boost past zero training error can still shrink the true risk, by growing the margin on the training points that are already correctly classified. This resolves the puzzle that opens §7.3.1: AdaBoost's test error is empirically observed to keep decreasing well after its training error hits zero, which the chapter's own earlier VC-dimension bound on $\mathcal F_T$ (Eq. 7.9, growing as $O(dT\log T)$ ) predicts should eventually overfit, not improve. No prior art on the Prove2Me platform is faithful: GET /theorems?q=boosting and q=AdaBoost return no hits; this chunk's Rademacher-complexity apparatus is restated locally (a draft item cannot import chunk 05-svm's or 03-rademacher-vc's own draft copies) rather than reused, matching the precedent those chunks' own STATUS.md records recommend for every later chunk needing the same machinery.

Not formalized here: Theorem 7.6 (the VC-dimension-based ensemble margin bound, a direct corollary of Corollary 7.5 via chunk 03's VC-dimension apparatus) — restating 03's own machinery a second time for a single further corollary is disproportionate within this mission's budget, and the chapter's actual capstone targets the sharper, dimension-free Rademacher-complexity route (Theorem 7.7) instead. Also out of scope: §7.2.2's coordinate- descent equivalence, §7.2.3's practical (decision-stump) use, and §7.3.4-7.3.5's margin- maximization LP and game-theoretic interpretation — discussion sections with no numbered result feeding the goal's proof.

Difficulty

Theorem 7.2's proof needs the telescoping identity $D_{T+1}(i) = e^{-y_if(x_i)}/(m\prod_tZ_t)$ (Eq. 7.2), obtained by repeatedly unfolding the recursive weight update — a genuine induction on $t$ , not a one-line algebraic manipulation — before the elementary inequality $1_{u\le0}\le e^{-u}$ turns the empirical error into a telescoping product of the $Z_t$ 's, each of which is then re-expressed in closed form via a case split on $y_ih_t(x_i)=\pm1$ . Theorem 7.7's proof reuses the same identity but with an added margin-shift term $\rho\|\alpha\|_1$ inside the exponential, requiring the same telescoping machinery plus a separate accounting of $e^{\rho\sum_t\alpha_t}$ against the product of $[\sqrt{(1-\varepsilon_t)/\varepsilon_t}]^\rho$ factors coming from each $\alpha_t$ 's own closed form — a proof that shares its main structural step with Theorem 7.2 but is not a trivial corollary of it. Corollary 7.5's proof is Lemma 7.4 (itself a careful supremum-exchange argument using the dual-norm characterization of $\ell^1$ , not a routine calculation) composed with Theorem 5.8, applied to the specific set $\mathrm{conv}(H)$ rather than a generic hypothesis class — a formalization that stated the corollary only for a "sufficiently nice" abstract class, without deriving it from Lemma 7.4's convex-hull identity, would be proving a different, weaker-provenance statement.

Formalization scope

WeightedError, AdaBoostAlpha, AdaBoostNormalizer, AdaBoostDist, AdaBoostEpsilon, AdaBoostEnsemble, AdaBoostNormalizedEnsemble, EmpiricalError and ConvHull are new, capturing AdaBoost as an actual algorithm (a genuine recursion on the round index, closed under Definitions.Def_FoundationsML_Boosting_AdaBoostDist's own recursive equation) rather than an unspecified "boosting procedure" — the trivialization trap BRIEF.md names for this chapter. AdaBoostDist takes the sequence of base classifiers actually selected at each round, h : ℕ → X → ℝ, as external data rather than deriving it via an argmin over H; this is checked in SELF_REVIEW.md to drop no content either milestone or the goal theorem's statement actually needs, since neither invokes h_t's optimality, only the weighted error ε_t it produces under AdaBoost's own distribution D_t. PhiRho, EmpiricalMarginLoss, MarginGeneralizationError, EmpiricalRademacherComplexity and RademacherComplexity are restated locally, byte-identical to chunk 05-svm's own copies of Definitions 5.5, 5.6, 2.1 (specialized), 3.1, 3.2 (a draft item cannot import another chunk's draft module); this duplication collapses once 05-svm and 03-rademacher-vc are uploaded and listed in missions/README.md's "Published definitions" table. No numerical constant in any of the four theorems is altered from the book's own displayed form. A trivializing formalization this mission avoids: stating Theorem 7.2/7.7 for an arbitrary sequence of error rates $\varepsilon_1,\dots,\varepsilon_T$ satisfying $\varepsilon_t<1/2$ , disconnected from any actual algorithm — AdaBoostEpsilon instead ties every ε_t to the weighted error AdaBoost's own recursively defined D_t assigns to its own selected h_t, so the bound is provably about this algorithm's error trajectory, not an arbitrary one.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 7.
Y. Freund, R. E. Schapire, "A decision-theoretic generalization of on-line learning and an application to boosting," Journal of Computer and System Sciences 55(1), 1997, 119-139.
R. E. Schapire, Y. Freund, P. Bartlett, W. S. Lee, "Boosting the margin: a new explanation for the effectiveness of voting methods," The Annals of Statistics 26(5), 1998, 1651-1686.

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Machine Learning Probability Random Matrix Theory+1·Captain: mikedeng1

High-Dimensional Probability VIII: Dudley's Integral InequalityTextbook

Motivation

Many questions in high-dimensional probability reduce to bounding the expected supremum of a random process $(X_t)_{t\in T}$ — the maximum, over an entire indexed family of random variables, of how large any one of them can get. When $T$ is finite this is routine (a union bound over $|T|$ terms suffices), but the interesting cases have $T$ infinite, even uncountable: a supremum over a continuum of test functions, a norm expressed as a supremum over a sphere, or an empirical process indexed by a whole class of functions. A naive union bound is unusable here, since $|T|$ is infinite.

R. M. Dudley's 1967 entropy bound (R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, Journal of Functional Analysis 1 (1967), 290–330) resolved this for Gaussian processes, controlling the expected supremum purely in terms of the metric entropy of $T$ — how many balls of radius $\varepsilon$ are needed to cover $T$ , at every scale $\varepsilon$ . The technique behind the proof, chaining, builds a sequence of increasingly fine finite approximations to $T$ and telescopes the resulting bounds; it is one of the central tools of the field, reused throughout empirical process theory, statistical learning theory (via Vapnik-Chervonenkis theory), and non-asymptotic random matrix theory. This mission formalizes the chapter's generalization of Dudley's bound beyond Gaussian processes, to any process with sub-gaussian increments, together with the purely combinatorial Sauer-Shelah lemma that the chapter's applications to statistical learning theory build on.

Setting

Fix a probability space $(\Omega,\mathcal F,P)$ . A random process is a family $(X_t)_{t\in T}$ of real random variables on $(\Omega,\mathcal F,P)$ indexed by an arbitrary set $T$ , with no independence or measurability-of-the-supremum assumed between different $t$ 's. Since $\sup_{t\in T}X_t(\omega)$ need not be measurable in $\omega$ for a general index set $T$ , its expectation is understood — following the book's own convention, set once in Chapter 7 and reused throughout — through the process's finite-dimensional marginals:

\mathbb E\sup_{t\in T}X_t \;:=\; \sup_{T_0\subseteq T\text{ finite, nonempty}}\ \mathbb E\max_{t\in T_0}X_t.

Now fix a metric $d$ on $T$ , making $(T,d)$ a metric space. The covering number $N(T,d,\varepsilon)$ , for $\varepsilon>0$ , is the smallest cardinality of a finite $\varepsilon$ -net of $T$ : a finite set $N\subseteq T$ such that every point of $T$ lies within distance $\varepsilon$ of some point of $N$ (or $N(T,d,\varepsilon):=\infty$ if no finite $\varepsilon$ -net exists). The quantity $\log N(T,d,\varepsilon)$ is the metric entropy of $T$ at scale $\varepsilon$ : it measures how large $T$ looks when resolved only down to scale $\varepsilon$ .

A process $(X_t)_{t\in T}$ has sub-gaussian increments with parameter $K\ge0$ if

\|X_t-X_s\|_{\psi_2}\;\le\;K\,d(t,s)\qquad\text{for all }t,s\in T,

where $\|\cdot\|_{\psi_2}$ is the sub-gaussian (Orlicz) norm of Chapter 2: the smallest $u>0$ with $\mathbb E\exp((X_t-X_s)^2/u^2)\le2$ . This says the increments of the process are controlled by the metric $d$ the way a Gaussian process's increments are controlled by its own canonical metric $d(t,s):=\|X_t-X_s\|_{L^2}$ — but without assuming $(X_t)_{t\in T}$ is Gaussian.

A class of Boolean functions $F$ on a set $\Omega$ shatters a subset $\Lambda\subseteq\Omega$ if every function $g:\Lambda\to\{0,1\}$ arises as the restriction to $\Lambda$ of some $f\in F$ . The VC (Vapnik-Chervonenkis) dimension $\mathrm{vc}(F)$ is the largest cardinality of a subset of $\Omega$ shattered by $F$ (or $\infty$ if arbitrarily large finite subsets, or an infinite one, are shattered) — a purely combinatorial measure of how rich the class $F$ is.

Formalization targets

Goal (Theorem 8.1.3, Dudley's integral inequality)

\exists\,C>0:\quad\mathbb E\sup_{t\in T}X_t\;\le\;CK\int_0^\infty\sqrt{\log N(T,d,\varepsilon)}\;d\varepsilon

for every mean-zero random process $(X_t)_{t\in T}$ on a metric space $(T,d)$ with sub-gaussian increments parameter $K\ge0$ , whenever the integral is finite. $C$ is the book's own unnamed absolute constant, never depending on $T$ , $K$ , or the process. This is the weakest stable form of the claim: no numeral is hard-coded for $C$ , and the statement asks only for the shape of the bound, matching what the book actually proves.

Milestone (Theorem 8.3.16, Sauer-Shelah lemma)

|F|\;\le\;\sum_{k=0}^{d}\binom nk\;\le\;\left(\frac{en}{d}\right)^{d},\qquad d:=\mathrm{vc}(F),

for every class $F$ of Boolean functions on a finite $n$ -point set $\Omega$ . This is a purely combinatorial fact, with no probability involved, but it is the bridge (via the covering-number bound Theorem 8.3.18, outside this mission's scope) between the chapter's Dudley-inequality engine and its statistical-learning applications — a bound on how large a finite class of Boolean functions can be, in terms of a single combinatorial complexity parameter.

Significance

Dudley's inequality is, in the book's own words, "the main result" of the chaining chapter: it converts a purely geometric quantity — the metric entropy of an index set, computable in many cases from covering-number estimates already available for balls, ellipsoids, and other convex bodies — into a probabilistic control on the size of a random process indexed by that set. This is what lets later chapters (uniform laws of large numbers over function classes, the matrix deviation inequality, the Dvoretzky-Milman theorem on almost-spherical sections of convex bodies) bound suprema over infinite, even uncountable, index sets without ever performing a union bound. The bound is also known to be tight only up to a logarithmic factor in general — Sudakov's minoration inequality (Chapter 7) gives a matching lower bound for Gaussian processes, and the book's own Exercise 8.1.12 exhibits a set where the two bounds genuinely diverge — so the constant $C$ here cannot in general be sharpened away.

The Sauer-Shelah lemma is one of the two founding results of VC theory (together with the Glivenko-Cantelli-type uniform convergence it feeds into), independently discovered by Vapnik and Chervonenkis, Sauer, and Shelah in the early 1970s; it underlies the sample-complexity bounds of statistical learning theory (a hypothesis class with finite VC dimension is PAC-learnable) and, through Theorem 8.3.18, gives one of the two standard routes (the other being direct combinatorial counting) to bounding covering numbers of infinite function classes.

Both results are decades old and have long-established, standard proofs; no open mathematical question is being formalized. What this mission contributes is the machine-checked statement infrastructure — the goal and the Sauer-Shelah milestone, together with the definitions (CoveringNumber, ProcessESup, Shatters, VcDim) a faithful Lean rendering of either result needs — for a solver to close with a proof. No formalization of Dudley's inequality or the Sauer-Shelah lemma is known to exist on the platform prior to this mission.

Difficulty

The natural first idea for bounding $\mathbb E\sup_{t\in T}X_t$ is a single-scale $\varepsilon$ -net argument: replace $T$ by a finite $\varepsilon$ -net, bound the maximum over the (finite) net by a union bound using the sub-gaussian tail, and separately bound the error of replacing $T$ by the net using the Lipschitz-in-probability control the sub-gaussian-increments hypothesis gives. This works, but it only ever sees $T$ at one fixed resolution $\varepsilon$ , and optimizing over $\varepsilon$ afterward gives a bound with an extra $\sqrt{\log(1/\varepsilon)}$ -type loss that does not match Dudley's inequality. The actual difficulty is genuinely multi-scale: chaining builds a whole sequence of nets at dyadic scales $\varepsilon=2^{-k}$ simultaneously, connects each point of $T$ to its nearest net point at every scale to form a "chain" of successive approximations back to a single fixed basepoint, and telescopes the resulting sum of increments — turning $X_t$ itself into a sum of differences between successive links of the chain, each individually well controlled by the sub-gaussian hypothesis at its own scale. Passing from the resulting discrete sum over dyadic scales (Theorem 8.1.4) to the continuous integral of the goal is a further, separate technical step.

For the Sauer-Shelah lemma, the natural first idea — bound $|F|$ directly by counting — has no obvious purchase on an arbitrary class of Boolean functions. The actual argument goes through Pajor's lemma, which reduces bounding $|F|$ to counting the shattered subsets of $\Omega$ instead of the functions in $F$ themselves; only then does the cardinality bound $d=\mathrm{vc}(F)$ on shattered sets become directly usable, via a binomial-sum estimate.

Formalization scope

CoveringNumber T ε is ℕ∞-valued (ℕ∞ = WithTop ℕ), defined as the infimum, over the subtype of finite $\varepsilon$ -nets of the whole type T (an instance of MetricSpace T), of their cardinality; the infimum of the empty family in this complete lattice is ⊤, reproducing " $N:= \infty$ if no finite net exists" with no case split. ProcessESup is EReal-valued, defined as the supremum over finite nonempty $T_0\subseteq T$ of the Bochner integral of the finite max — EReal, not ℝ, because a real-valued supremum would silently return the junk value $0$ if the family of marginal expectations were unbounded above. Shatters and VcDim are direct transcriptions of Definition 8.3.1, with VcDim valued in ℕ∞ via a supremum of Set.encard over the (always-nonempty, since $\varnothing$ is trivially shattered) subtype of shattered subsets. The goal's mean-zero hypothesis is stated as Integrable (X t) P ∧ ∫ X t = 0 rather than the bare equation, since a non-integrable variable's Bochner integral is 0 in Mathlib by convention regardless of its true mean — a bare-equation hypothesis would let a non-mean-zero, non-integrable process satisfy the theorem vacuously. Two further hypotheses make explicit what the book's own displayed statement treats as understood without spelling out: that $N(T,d,\varepsilon)$ is finite for every $\varepsilon>0$ (total boundedness of $T$ ), and that the resulting integrand is integrable on $(0,\infty)$ — both hold whenever $T$ is totally bounded, since the integrand vanishes once $\varepsilon\ge\mathrm{diam}(T)$ , so neither hypothesis excludes any case the book's own proof does not also need. [Nonempty T] excludes the degenerate empty index set. The absolute constant $C$ is existentially quantified ahead of every type, instance, and hypothesis it is uniform over, and pinned to no numeral, matching " $C$ is an absolute constant" — a formalization hard-coding a specific numeral for $C$ would be invalidated by the next sharper constant in the literature and would not match what the book proves.

A trivializing formalization of the Sauer-Shelah lemma would fix $\mathrm{vc}(F)$ at a hard-coded small value, or drop the second (exponential) inequality in favor of the weaker first one; this mission's statement keeps both inequalities, with $d$ genuinely computed from VcDim, and handles the $d=0$ boundary (where the exponential bound's base involves a division by zero under Lean's x/0=0 convention) explicitly rather than excluding it, since x^0=1 still recovers the book's correct bound $|F|\le1$ there.

This mission covers Theorem 8.1.3 and Theorem 8.3.16 only; Theorem 8.3.18 (covering numbers via VC dimension) and Theorem 8.2.3 (the uniform law of large numbers, the chapter's direct application of Dudley's inequality) are left out, not approximated, for lack of the additional empirical- process measurability machinery — the class of Lipschitz functions of Eq. (8.22), measurability of the resulting empirical process — that a faithful statement of either would need beyond what this mission's items already provide. CoveringNumber and ProcessESup are reusable by any later chapter needing a metric space's covering numbers or a general random process's expected supremum (this book's own Chapters 7, 9, and 11 all use one or both); Shatters and VcDim are reusable by any later development of VC theory or statistical learning theory. Solvers' contributions are welcome on: the chaining argument itself (the mission's hardest open leaf, via the discrete dyadic form of Theorem 8.1.4), Pajor's lemma underlying Sauer-Shelah, and the binomial-sum estimate closing its second inequality.

Selected references

R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, Journal of Functional Analysis 1 (1967), 290–330. https://doi.org/10.1016/0022-1236(67)90017-1
N. Sauer, On the density of families of sets, Journal of Combinatorial Theory, Series A 13 (1972), 145–147. https://doi.org/10.1016/0097-3165(72)90019-2
V. N. Vapnik, A. Ya. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability & Its Applications 16 (1971), 264–280. https://doi.org/10.1137/1116025
R. Vershynin, High-Dimensional Probability: An Introduction with Applications in Data Science, Cambridge University Press, 2018, Chapter 8. https://doi.org/10.1017/9781108231596

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning V: Kernel Methods and the Representer TheoremTextbook

Motivation

Linear methods like SVMs work only when the classes are linearly separable, but most real data is not. Chapter 6 shows how to get non-linear decision boundaries for free: replace the input space's inner product with a kernel $K$ that implicitly computes an inner product in a (possibly very high- or infinite-dimensional) feature space, without ever explicitly computing the feature mapping. This works for any positive definite symmetric (PDS) kernel — and the chapter's central theorem shows that such a kernel always induces a genuine Hilbert space (the reproducing kernel Hilbert space, RKHS) in which the kernel is literally an inner product. The chapter's capstone, the representer theorem, then shows that a broad class of optimization problems over this (possibly infinite-dimensional) Hilbert space always has a solution expressible as a finite linear combination of kernel evaluations at the training points — turning an infinite-dimensional problem into a finite, $m$ -dimensional one.

Setting

A kernel $K:X\times X\to\mathbb R$ is PDS (Definition 6.3) if for every finite sample $\{x_1,\dots,x_m\}\subseteq X$ , the Gram matrix $[K(x_i,x_j)]$ is symmetric positive semidefinite. Theorem 6.8 shows every PDS kernel is an inner product $K(x,x')=\langle \Phi(x),\Phi(x')\rangle$ in some Hilbert space $H$ (the RKHS), which further has the reproducing property $h(x)=\langle h,K(x,\cdot)\rangle$ for every $h\in H$ — evaluating $h$ at a point is itself an inner product with the kernel section at that point. Theorem 6.10 shows PDS kernels are closed under sum, product, tensor product, pointwise limit, and power-series composition, letting complex kernels (Gaussian, and many others) be built from simple ones (polynomial kernels) without re-verifying positive-semidefiniteness from scratch. Section 6.3's representer theorem (Theorem 6.11) then considers minimizing, over $h\in H$ , an objective $F(h)=G(\|h\|_H)+L(h(x_1),\dots,h(x_m))$ that depends on $h$ only through its norm and its values at $m$ fixed points.

Formalization targets

Theorem 6.8 (RKHS existence, milestone). For a PDS kernel $K$ , there exist a Hilbert space $H$ and $\Phi:X\to H$ with $K(x,x')=\langle\Phi(x),\Phi(x')\rangle$ , and $H$ has the reproducing property $h(x)=\langle h,K(x,\cdot)\rangle$ for all $h\in H$ , $x\in X$ .

Theorem 6.10 (closure properties, milestone). PDS kernels are closed under sum, product, tensor product, pointwise limit, and power-series composition with non-negative coefficients.

Theorem 6.11 — the mission's goal. For any non-decreasing $G:\mathbb R\to\mathbb R$ and any loss $L:\mathbb R^m\to\mathbb R\cup\{+\infty\}$ , $\mathrm{argmin}_{h\in H} G(\|h\|_H)+ L(h(x_1),\dots,h(x_m))$ admits a solution $h^\star=\sum_{i=1}^m\alpha_i K(x_i,\cdot)$ ; if $G$ is increasing, every solution has this form.

Significance

Theorem 6.11 is the chapter's payoff and one of the most widely used structural results in kernel methods: it explains, in one general statement covering SVMs, kernel ridge regression, Gaussian process MAP estimation and many other algorithms simultaneously, why the dual (finite, $m$ -coefficient) formulation always suffices — the RKHS's infinite dimensionality never has to be confronted directly. Theorem 6.8 is the structural fact the whole chapter (and every later kernelized algorithm in the book, chapters 9-11, 15) depends on: without it, "PDS kernel" would be a purely combinatorial condition on Gram matrices with no guarantee it corresponds to any actual inner product. No prior art on the Prove2Me platform is faithful to any of this chapter's content: GET /theorems?q=Representer theorem and q=reproducing kernel return no faithful match (one unrelated hit concerns a Gaussian-measure reproducing kernel in a different, probabilistic context, not this chapter's PDS-kernel/RKHS construction). All six items are drafted fresh.

Difficulty

Theorem 6.8's proof is a genuine construction: define $H_0$ as finite linear combinations of kernel sections $\Phi(x)=K(x,\cdot)$ , define an inner product on $H_0$ using $K$ itself, verify it is well-defined (independent of the representation), positive semidefinite (via the PDS hypothesis), and — via the Cauchy-Schwarz-for-PDS-kernels lemma (Lemma 6.7) — actually positive definite, then complete $H_0$ to a genuine Hilbert space $H$ in which it is dense, and finally extend the reproducing property from the dense subspace $H_0$ to all of $H$ by a continuity argument. This is substantial analysis, not a restatement. Theorem 6.11's proof uses the orthogonal decomposition $H=H_1\oplus H_1^\perp$ (where $H_1=\mathrm{span}\{K(x_i, \cdot)\}$ ) and the reproducing property to show the orthogonal component $h^\perp$ never helps and, when $G$ is strictly increasing, strictly hurts — a short argument, but one that depends essentially on Theorem 6.8's reproducing property holding for the specific $H$ constructed, not just any Hilbert space with the kernel as its inner product.

Formalization scope

IsPDS uses the book's own second SPSD characterization (c^T K c ≥ 0 for every finite sample and coefficient vector c) rather than the non-negative-eigenvalues characterization, avoiding spectral theory for a Prop-valued definition; the book states the two are equivalent. IsRKHSOf and IsMinimizer are formalization scaffolding, not book-numbered definitions: IsRKHSOf packages Theorem 6.8's own two displayed equations (6.8, 6.9) as a reusable predicate shared between Theorem 6.8 (its conclusion) and Theorem 6.11 (its "H its corresponding RKHS" hypothesis), using an explicit evaluation map ev : H → X → ℝ to stand in for "elements of H are functions on X," since Mathlib's abstract Hilbert spaces are not themselves spaces of functions; IsMinimizer packages argmin. Both X in Theorem 6.8's existential and Theorem 6.11's ambient type are plain Type rather than Type*, avoiding universe-polymorphic quantification over the constructed Hilbert space's own type — a harmless simplification, since every application in this book instantiates X at a concrete, small type (typically $\mathbb R^N$ or a finite set). Theorem 6.11's loss codomain ℝ ∪ {+∞} is WithTop ℝ, not EReal (which would also admit -∞, an unstated generalization the book's own display does not license, since EReal's ⊤+⊥=⊥ collapse is a genuine faithfulness risk the book's own L:\mathbb R^m\to\mathbb R\cup\{+\infty\} avoids by construction). WithTop ℝ on its own does not avoid every collapse, though: an unconstrained L may be the constant function ⊤ (a legal instance of ℝ∪\{+\infty\}), forcing the objective identically ⊤ and every point to vacuously minimize it, which would make the theorem's second conjunct false. An added hypothesis, ∃ h₀, F h₀ ≠ ⊤, makes explicit the book's own implicit assumption that the objective is finite somewhere — see the Formalization scope note below. Theorem 6.11's two clauses are otherwise kept exactly as distinct as the book states them: existence needs only Monotone G (non-decreasing); "any solution has this form" needs StrictMono G (increasing) as an added hypothesis on the second conjunct only — per this chunk's own BRIEF.md, the crux of the theorem, and the trap this mission is most careful to avoid collapsing. Theorem 6.10's five closure clauses are stated as one conjunction (matching the book's single theorem, not five separate items); the power-series clause keeps the book's own radius-of-convergence domain restriction and adds an explicit summability hypothesis guarding the ∑' term.

Not formalized: Theorem 6.2 (Mercer's condition) — not needed by the goal's own proof chain (it is an equivalent characterization of PDS mentioned before the RKHS construction, not a premise Theorem 6.8's or 6.11's proof invokes) and its own hypotheses (compact $X\subset \mathbb R^N$ , continuous $K$ , an eigenfunction expansion of a compact self-adjoint integral operator) are real analytic content this mission's budget does not include; Lemma 6.7 (Cauchy-Schwarz for PDS kernels) and Lemma 6.9 (normalized PDS kernels) — supporting lemmas for Theorem 6.8's proof, not independently numbered results the goal cites; Theorem 6.12/Corollary 6.13 (Rademacher complexity/margin bounds for kernel-based hypotheses) — the chapter's optional further milestone, connecting to chunks 03/05's machinery, cut for budget; §6.5-6.8 (sequence kernels, weighted transducers, rational kernels, Bochner's theorem, approximate feature maps) — explicitly out of scope per this chunk's own BRIEF.md, a distinct, applications-heavy topic.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 6, §6.1-6.4.
B. Schölkopf, R. Herbrich, A. J. Smola, "A generalized representer theorem," COLT 2001, Lecture Notes in Computer Science 2111, 2001, 416-426.
N. Aronszajn, "Theory of reproducing kernels," Transactions of the American Mathematical Society 68(3), 1950, 337-404.

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Machine Learning Probability Statistics·Captain: mikedeng1

Foundations of Machine Learning III: Structural Risk Minimization and Model SelectionTextbook

Motivation

Chapters 2 and 3 bound the estimation error of a hypothesis chosen from a fixed hypothesis set $H$ , but the choice of $H$ itself is left open: a richer $H$ lowers the approximation error (how close $H$ comes to the Bayes classifier) at the price of a looser generalization bound, and a poorer $H$ does the reverse. Chapter 4 is the book's answer to this trade-off. It first shows that Empirical Risk Minimization (ERM) alone cannot resolve it — ERM ignores the complexity of $H$ entirely — and then develops Structural Risk Minimization (SRM): decompose a rich hypothesis set into a nested countable union $H=\bigcup_{k\ge1}H_k$ of increasingly complex pieces, and let the learning algorithm balance empirical fit against a complexity penalty for each $H_k$ automatically. The chapter closes by showing how the same balance can be achieved computationally through convex surrogate losses, whose minimization is tractable where minimizing the zero-one loss directly is not.

Setting

For a hypothesis $h$ chosen from $H$ , the excess error $R(h)-R^*$ decomposes into an estimation term $R(h)-\inf_{h\in H}R(h)$ and an approximation term $\inf_{h\in H}R(h)-R^*$ (Eq. 4.1). Proposition 4.1 bounds ERM's estimation error by twice the uniform deviation $\sup_{h\in H}|R(h)-\hat R_S(h)|$ . For a nested family $(H_k)_{k\ge1}$ and $h\in H$ , $k(h)$ denotes the least index with $h\in H_{k(h)}$ ; SRM selects $h_S^{SRM}$ by minimizing $F_k(h)=\hat R_S(h)+R_m(H_k)+\sqrt{\log k/m}$ jointly over $k\ge1$ and $h\in H_k$ , where $R_m(H_k)$ is $H_k$ 's Rademacher complexity (Definitions 3.1/3.2, restated locally in this chunk's ModelSelection namespace). Theorem 4.2 is the resulting learning guarantee. Section 4.4 develops a competing model-selection procedure, cross-validation, and Theorem 4.4 directly compares its guarantee to SRM's on a held-out split of the sample. Section 4.7 turns to real-valued scoring functions $h:X\to\mathbb R$ with sign convention $f_h(x)=\mathrm{sign}(h(x))$ and a convex non-decreasing surrogate $\Phi$ of the zero-one loss; the Bayes scoring function $h^*(x)=\eta(x)-\tfrac12$ (Eq. 4.9) and the $\Phi$ -loss $L_\Phi$ (Eq. 4.10) let Theorem 4.7 bound the true excess error by a power of the surrogate's own excess loss.

Formalization targets

Proposition 4.1 (ERM bound, milestone). For any sample $S$ , $\Pr[R(h_S^{ERM}) - \inf_{h\in H}R(h) > \epsilon] \le \Pr[\sup_{h\in H}|R(h)-\hat R_S(h)| > \epsilon/2]$ .

Theorem 4.2 — the mission's goal. For a nested countable union $H=\bigcup_{k\ge1}H_k$ and $h_S^{SRM}$ minimizing $F_k(h)$ over the whole union, for any $\delta>0$ , with probability at least $1-\delta$ :

R(h_S^{SRM}) \le \inf_{h\in H}\Big[R(h)+2R_m(H_{k(h)})+\sqrt{\tfrac{\log k(h)}m}\Big] + \sqrt{\tfrac{2\log(3/\delta)}m}.

Theorem 4.4 (Cross-validation versus SRM, milestone). Splitting a sample of size $m$ into $S_1$ (size $(1-\alpha)m$ , training) and $S_2$ (size $\alpha m$ , validation), for any $\delta>0$ , with probability at least $1-\delta$ :

R(h_S^{CV}) - R(h_{S_1}^{SRM}) \le 2\sqrt{\tfrac{\log\max(k(h_S^{CV}),k(h_{S_1}^{SRM}))}{\alpha m}} + 2\sqrt{\tfrac{\log(4/\delta)}{2\alpha m}}.

Theorem 4.7 (Convex-surrogate excess-error bound, milestone). For $\Phi$ convex and non-decreasing with $s\ge1,c>0$ satisfying $|h^*(x)|^s \le c^s(L_\Phi(x,0)-L_\Phi(x,h^*_\Phi(x)))$ for all $x$ : $R(h)-R^* \le 2c(L_\Phi(h)-L^*_\Phi)^{1/s}$ .

Significance

Theorem 4.2 is the chapter's headline result and the theoretical justification for regularization-based learning: it shows that a single algorithm, without knowing in advance which $H_k$ contains a good hypothesis, achieves a guarantee that is — up to the $\sqrt{\log k(h)/m}$ penalty — as favorable as if an oracle had revealed the best-in-class index in advance (Eq. 4.6). It is also the chapter's genuine new content beyond chunk 03-rademacher-vc's single-hypothesis-set bound: the countable union bound (a 1/k²-weighted union over $k\ge1$ converging to $\pi^2/6$ , hence the $\log 3$ appearing in place of $\log 2$ ) is not a restatement of Theorem 3.3 but a distinct argument, and the goal's inf over the whole nested family is what makes SRM a model-selection method rather than a bound for one fixed $k$ . Theorem 4.4 is the chapter's only head-to-head comparison between two competing model-selection procedures, on two genuinely different samples. Theorem 4.7 is the bridge between the learning-theoretic guarantees of chapters 2-4 and the actually-implemented convex optimization problems of chapters 5 (SVM), 6 (kernels) and beyond, all of which minimize a convex surrogate rather than the zero-one loss directly. No prior art exists on the platform: GET /theorems?q=structural%20risk%20minimization and GET /theorems?q=model%20selection both return zero hits.

Difficulty

Theorem 4.2's proof genuinely uses the union bound over a countably infinite family indexed by $k\ge1$ with weight $1/k^2$ converging to $\pi^2/6 < 2$ (Eq. 4.5) — this is the chapter's distinct new technique, not an application of chunk 03's finite/VC-dimension machinery to a single $H_k$ ; a formalization that stated the bound only for one fixed $k$ , or dropped the inf over the whole union in favor of a single best-in-class $h^*$ , would be Theorem 4.2's named trivializing formalization (BRIEF.md's pitfall note) rather than the theorem itself. Theorem 4.4 requires keeping two distinct samples ( $S_1$ , $S_2$ , of different, precisely related sizes) and two distinct hypotheses ( $h_S^{CV}$ , $h_{S_1}^{SRM}$ ) apart throughout; conflating them collapses the comparison to a tautology. Theorem 4.7's difficulty is in its setup, not its statement: the Bayes scoring function, the $\Phi$ -loss, and the pointwise $\Phi$ -minimizer $h^*_\Phi$ (which the book allows to take the extended values $\pm\infty$ at the degenerate points $\eta(x)\in\{0,1\}$ ) all need care to state without silently altering the theorem's content.

Formalization scope

GeneralizationError, EmpiricalError, EmpiricalRademacherComplexity and RademacherComplexity are restated locally in this chunk's ModelSelection namespace (identical in content to chunk 03-rademacher-vc's own copies), since a draft item cannot import another chunk's draft module. LeastIndex H h (k(h)) is Nat.sInf {k | 1 ≤ k ∧ h ∈ H k}; every theorem using it carries the standing hypothesis that h lies in the relevant union, guarding against trap 5 (Nat.sInf of an empty set). Theorem 4.2's hSRM and Proposition 4.1's hERM are hypothesis-supplied functions satisfying the book's optimality property, not constructed via choice over an unconstrained H; H.Nonempty (Proposition 4.1) and (⋃ k ≥ 1, Hk k).Nonempty (Theorem 4.2) guard the outer sInf/inf terms against trap 5. Theorem 4.7's $h^*_\Phi$ is formalized as a real-valued function satisfying the pointwise minimization property for all $x$ ; the book's own extended-real convention ( $h^*_\Phi(x)=\pm\infty$ exactly where $\eta(x)\in\{0,1\}$ ) is outside this formalization — disclosed here and in MODERATION_NOTES.md — since no real number satisfies the minimizing property at those degenerate points, the theorem as stated applies precisely to the case a real-valued $h^*_\Phi$ can be supplied, which is the book's own generic case. No numerical constant is altered from the book in any of the four theorems: 2 and log(3/δ) in Theorem 4.2, 2 (twice) and log(4/δ) in Theorem 4.4, and 2c and the exponent 1/s in Theorem 4.7 are exactly as displayed.

Not formalized: the discussion of computing $k^*$ via binary search (a computational, not a statistical, result); $n$ -fold and leave-one-out cross-validation (Section 4.5, a practical variant of Theorem 4.4's two-sample cross-validation without its own numbered generalization bound); regularization-based algorithms (Section 4.6, the uncountable-union extension of SRM, which the book itself only sketches without a numbered theorem); Lemma 4.5 and Proposition 4.6 (intermediate results establishing that $h^*_\Phi$ induces the same classifier as $h^*$ , needed for Theorem 4.7's proof but not part of its statement); and the worked examples for the hinge, exponential and logistic losses (instantiations of Theorem 4.7's $s,c$ , not separate theorems). Drafting only these worked instantiations in place of Theorem 4.7's general statement would be a trivializing formalization for this chapter.

Selected references

M. Mohri, A. Rostamizadeh, A. Talwalkar, Foundations of Machine Learning, 2nd ed., MIT Press, 2018, Chapter 4.
V. Vapnik, Statistical Learning Theory, Wiley-Interscience, 1998 (structural risk minimization).
T. Zhang, "Statistical behavior and consistency of classification methods based on convex risk minimization," Annals of Statistics 32(1), 2003 (Theorem 4.7's origin).

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Mathematical Physics Quantum Information·Captain: Lucas

Undecidability of the Spectral GapResearch Paper

Motivation

The spectral gap of a quantum many-body Hamiltonian is the difference between the energy of its ground state and the energy of its first excited state, in the limit of infinitely many particles. Whether a given microscopic interaction produces a gapped or a gapless system decides much of the macroscopic physics: gapped systems have exponentially decaying correlations and well-defined quantum phases, gapless systems sit at critical points and can display algebraically decaying correlations. Several long-standing questions — the Haldane conjecture for antiferromagnetic spin chains, the existence of gapped topological spin liquids, and the Yang–Mills mass gap — are instances of the question "given the interaction, is the system gapped?".

Cubitt, Pérez-García and Wolf proved that this question, posed for families of two-dimensional translationally invariant nearest-neighbour spin models, admits no algorithmic answer: the spectral gap problem is undecidable (Nature 528, 207–211 (2015); full version: Forum of Mathematics, Pi 10:e14 (2022), also arXiv:1502.04573).

Timeline of the ingredients the proof rests on: Turing's undecidability of the halting problem (1936); Berger's undecidability of the domino problem (1966) and Robinson's aperiodic tile set (Inventiones 12, 177–209 (1971)); Feynman's and Kitaev's circuit-to-Hamiltonian constructions, which turn a computation into a ground state; Gottesman and Irani's translationally invariant one-dimensional Hamiltonians encoding computation (FOCS 2009); and Bitansky–Vadhan-style quantum Turing machine engineering from Bernstein and Vazirani (SIAM J. Comput. 26, 1411–1473 (1997)). The 2015 result was later sharpened to one-dimensional chains by Bausch, Cubitt, Lucia and Pérez-García (PRX 10, 031038 (2020)).

Setting

Fix a local dimension $d$ and, for each side length $L$ , the square lattice $\Lambda(L)=\{1,\dots,L\}^2$ with open boundary conditions. Each site carries a copy of $\mathbb{C}^d$ , so the state space of the lattice has the standard product basis indexed by assignments of a level in $\{1,\dots,d\}$ to each site. A model is specified by three Hermitian matrices: an on-site term $h_1$ of size $d\times d$ , and two interactions $h_{\mathrm{row}},h_{\mathrm{col}}$ of size $d^2\times d^2$ acting on horizontally and vertically adjacent pairs. The Hamiltonian of the finite lattice is

H^{\Lambda(L)} \;=\; \sum_{\text{horizontal edges}} h_{\mathrm{row}}^{(i,j)} \;+\; \sum_{\text{vertical edges}} h_{\mathrm{col}}^{(i,j)} \;+\; \sum_{k\in\Lambda(L)} h_1^{(k)},

the same three matrices being used at every edge and every site, which is what translational invariance means here. The quantity $\max\{\|h_1\|,\|h_{\mathrm{row}}\|,\|h_{\mathrm{col}}\|\}$ is the local interaction strength.

Write $\lambda_0(H^{\Lambda(L)})\le\lambda_1(H^{\Lambda(L)})\le\cdots$ for the eigenvalues and $\Delta(H^{\Lambda(L)})=\lambda_1-\lambda_0$ for the finite-size gap. The family $\{H^{\Lambda(L)}\}_L$ is

gapped (Definition 1 of the source) if there are $\gamma>0$ and $L_0$ such that for all $L>L_0$ the ground state of $H^{\Lambda(L)}$ is non-degenerate and $\Delta(H^{\Lambda(L)})\ge\gamma$ ;
gapless (Definition 2 of the source) if there is $c>0$ such that for every $\varepsilon>0$ there is an $L_0$ with: for all $L>L_0$ , every point of $[\lambda_0,\lambda_0+c]$ lies within $\varepsilon$ of the spectrum of $H^{\Lambda(L)}$ .

These two conditions are not negations of each other; the construction guarantees that every instance falls into one of them. The ground state energy density is $E_\rho=\lim_{L\to\infty}\lambda_0(H^{\Lambda(L)})/L^2$ .

Formalization targets

Goal — Theorem 3 of the source

For a fixed universal machine and every $n$ , one explicit family of interactions, built from fixed integer-valued matrices $A,A',B,C,D,D'$ , a diagonal projector $\Pi$ , a rational $\beta>0$ that may be taken arbitrarily small, and an algebraic $\alpha(n)\le 2\beta$ ,

h_1(n)=\alpha(n)\Pi,\qquad h_{\mathrm{col}}(n)=D+\beta D',

h_{\mathrm{row}}(n)=A+\beta\Bigl(A'+e^{i\pi\varphi}B+e^{-i\pi\varphi}B^{\dagger}+e^{i\pi 2^{-|\varphi|}}C+e^{-i\pi 2^{-|\varphi|}}C^{\dagger}\Bigr),

with $\varphi=\varphi(n)$ the rational whose binary expansion after the point is the binary expansion of $n$ reversed, satisfies: the local interaction strength is at most $1$ ; if the machine halts on input $n$ the family is gapped with gap at least $1$ ; and if it does not halt the family is gapless. Since halting is undecidable, no algorithm decides gappedness, even with the promise that exactly one of the two alternatives holds and even at fixed local dimension $d$ .

Milestones

The milestone list follows the numbering of the full version: Lemma 8 and Theorem 9 (reduction of halting to ground state energy and to arbitrary low-energy properties), Corollary 7 (the same undecidability for unconstrained local dimension, with rational interactions), Proposition 53 and Corollary 54 (the diverging ground state energy and its promise version), and Theorem 5 (undecidability of the ground state energy density).

Significance

The result rules out a general algorithm — and therefore any complete general method — for deciding gappedness from the interaction matrices, however much computing power is available; the property genuinely depends on arbitrarily large system sizes. It also implies, via the standard link between undecidability and independence, that there are concrete finite-dimensional models whose gap is independent of the axioms of any consistent recursively axiomatized formal system (Corollary 4 of the source), and it transfers to other low-energy properties such as the existence of algebraically decaying ground-state correlations.

The theorem is proved; none of it is formalized. This mission produces the machine-checked version. The reusable infrastructure it forces into existence is substantial on its own: a formal model of translationally invariant lattice Hamiltonians and their thermodynamic-limit spectral behaviour, the tiling layer, and computational-history-state Hamiltonians. Each milestone is a self-contained statement that can be attacked without the others.

Difficulty

The obvious approach — encode a halting computation as an energy penalty — gives the ground state energy of a finite lattice, not a property of the limit; this is exactly what Lemma 8 achieves, and it is not enough, because a gap is a statement about the sequence of spectra as $L\to\infty$ and is insensitive to any single lattice size. The construction must make the halting information visible at all sufficiently large sizes at once while a fixed finite local dimension carries every instance $n$ . That forces three separate difficulties: an aperiodic (Robinson) tiling to create squares of every size $2^n$ inside one translationally invariant model; a quantum phase-estimation Turing machine whose transition amplitudes encode $n$ in a single phase $e^{i\pi\varphi(n)}$ , so that the instance index does not inflate the local dimension; and a history-state Hamiltonian whose low-energy spectrum can be controlled well enough that a positive energy density in the halting case turns into a genuine spectral gap, and a vanishing one into a dense spectrum above the ground state.

Formalization scope

The development commits to the following conventions, all of which are visible in the definition items of this mission.

Lattices are finite: sites are pairs of indices in $\{0,\dots,L-1\}$ , edges are consecutive pairs within a row or a column (open boundary conditions; the periodic case of Section 6.3 of the source is out of scope).
Operators are complex matrices indexed by product-basis configurations; the interactions are embedded by acting as the given matrix on the two sites of an edge and as the identity elsewhere.
The spectrum is taken as the set of real numbers in the matrix spectrum, and $\lambda_0$ is its infimum; every statement carries the Hermiticity hypotheses that make this the usual spectrum. Multiplicities are dimensions of eigenspaces, which is how the "identity of spectra as multisets" of Theorem 9 is expressed.
Gapped, gapless and the energy density are properties of the whole family $\{H^{\Lambda(L)}\}_L$ generated by a fixed triple of matrices, exactly as in Definitions 1 and 2.
Operator norms are $\ell_2$ operator norms; the local interaction strength is the maximum of the three.
Machines are represented by partial recursive codes: "halts on input $n$ " is definedness of the evaluation, and "has not halted after $L$ steps" is the step-bounded evaluation returning nothing. The explicit local-dimension bounds of Lemma 8 and Theorem 9, which are stated in the source in terms of the number of internal states and the alphabet size of a Turing machine, are replaced by the existence of a finite local dimension.

Degenerate readings are excluded: a zero local dimension satisfies none of the statements, since a non-degenerate ground state requires a one-dimensional eigenspace and the gapless condition requires a non-empty spectrum; and every existential statement fixes the matrices before quantifying over all instances $n$ and all lattice sizes $L$ .

Contributions are welcome at any milestone, and also on the infrastructure the milestones need — Wang tilings and the Robinson tile set, Gottesman–Irani history-state Hamiltonians, and quantum Turing machines in the Bernstein–Vazirani sense — which are needed for Theorem 6 and Lemma 47 of the source and are not yet part of this mission's item list.

Selected references

T. S. Cubitt, D. Pérez-García, M. M. Wolf, Undecidability of the Spectral Gap (full version), Forum of Mathematics, Pi 10:e14, 1–102 (2022). https://doi.org/10.1017/fmp.2021.15 — the version all statements of this mission are formalized against; preprint: https://arxiv.org/abs/1502.04573
T. S. Cubitt, D. Pérez-García, M. M. Wolf, Undecidability of the spectral gap, Nature 528, 207–211 (2015). https://doi.org/10.1038/nature16059
R. M. Robinson, Undecidability and nonperiodicity for tilings of the plane, Inventiones Mathematicae 12, 177–209 (1971). https://doi.org/10.1007/BF01418780
D. Gottesman, S. Irani, The quantum and classical complexity of translationally invariant tiling and Hamiltonian problems, FOCS 2009. https://arxiv.org/abs/0905.2419
E. Bernstein, U. Vazirani, Quantum complexity theory, SIAM J. Comput. 26, 1411–1473 (1997). https://doi.org/10.1137/S0097539796300921
J. Bausch, T. S. Cubitt, A. Lucia, D. Pérez-García, Undecidability of the spectral gap in one dimension, Phys. Rev. X 10, 031038 (2020). https://doi.org/10.1103/PhysRevX.10.031038

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Mechanism Design·Captain: Shuze Chen

Algorithmic Game Theory IV: VCG and the Limits of TruthfulnessTextbook

Motivation

Mission III of this series ends at an impossibility: without money, incentive compatibility over three or more alternatives means dictatorship. This mission formalizes the classical escape route — quasilinear utilities and payments — and the exact price of it. Vickrey (1961) discovered that a second-price auction makes truth-telling dominant; Clarke (1971) and Groves (1973) generalized the idea to arbitrary social choice: welfare-maximizing rules can always be made truthful by the right payments. The converse program — which choice rules are implementable at all — runs through Rochet (1987) and Myerson (1981) to Saks–Yu (2005): weak monotonicity characterizes implementability on convex domains, and on single-parameter domains the characterization is complete and elementary — monotone rules with critical-value payments. Chapter 9, §§9.3 and 9.5 of Nisan–Roughgarden–Tardos–Vazirani (eds.), Algorithmic Game Theory (Cambridge, 2007), written by Nisan, is the source text.

Setting

A set $A$ of alternatives and a finite set $\iota$ of players. Player $i$ holds a private valuation $v_i : A \to \mathbb{R}$ from a publicly known domain $V_i \subseteq \mathbb{R}^A$ ; utilities are quasilinear: choosing $a$ and charging $p_i$ gives $i$ utility $v_i(a) - p_i$ . A (direct revelation) mechanism is a social choice function $f$ from valuation profiles to $A$ together with payment functions $p_i$ (Definition 9.14). The mechanism is incentive compatible if no unilateral misreport from the domain ever beats the truth (Definition 9.15).

A VCG mechanism (Definition 9.16) has $f$ maximizing social welfare $\sum_i v_i(a)$ and payments of the Groves form $p_i = h_i(v_{-i}) - \sum_{j\ne i} v_j(f(v))$ ; the Clarke pivot rule takes $h_i(v_{-i}) = \max_b \sum_{j \ne i} v_j(b)$ . A rule is weakly monotone (Definition 9.28) if a unilateral change of valuation that moves the outcome from $a$ to $b$ satisfies $v_i'(b) - v_i'(a) \ge v_i(b) - v_i(a)$ . A single-parameter domain (Definition 9.33) is given by a win set $W_i \subseteq A$ per player and bids $t \in [t_0, t_1]$ : the valuation is $t$ on $W_i$ and $0$ elsewhere.

Formalization targets

Goal (capstone) — Theorem 9.36

A normalized mechanism (losers pay 0) on a single-parameter domain is incentive compatible iff the rule is monotone and every winning bid pays the critical value — the threshold below which the bid loses.

Theorem 9.17 — VCG is truthful

Every VCG mechanism is incentive compatible.

Lemma 9.20 — Clarke pivot

With Clarke pivot payments, a welfare-maximizing rule makes no positive transfers, and is individually rational when valuations are nonnegative.

Theorem 9.29 — weak monotonicity

Necessity: incentive compatibility forces WMON, on any domain. Sufficiency: on convex domains, WMON rules admit implementing payments (Saks–Yu).

Significance

These are the working theorems of every later mechanism-design mission: the approximation mechanisms of Chapter 12, the profit-maximization results of Chapter 13, and the sponsored-search analysis of Chapter 28 all argue through Theorem 9.36's monotonicity-plus-critical-value normal form, and VCG is the benchmark they approximate. Formalizing the cluster produces the platform's quasilinear-mechanism vocabulary — domains, truthfulness, Groves payments, weak monotonicity, single-parameter settings — on top of the social-choice layer of Mission III.

The capstone and Theorem 9.17 are textbook results with complete proofs in the source; the Saks–Yu half of Theorem 9.29 is stated but not proved in the book ("quite involved"), so that milestone carries a genuinely hard formalization with a published paper proof. None have prior Lean formalizations.

Difficulty

Theorem 9.17 is a three-line inequality chase once the Groves form is unfolded — a deliberate warm-up. Lemma 9.20 adds the attained maximum over a finite alternative set. The necessity half of 9.29 is a two-application argument; the sufficiency half is the hard point of the mission: the known proofs walk two-cycle inequalities into a path-integral construction of payments on a convex domain, and nothing of the kind exists in Mathlib. For the capstone, the delicate part is the critical value: the book defines it as a supremum that "is undefined" when the player always wins, and the honest formal rendering — a constant payment c that is a least upper bound of the losing bids whenever losing bids exist — makes the case split explicit; the equivalence proof must thread monotonicity, the threshold structure of the winning set, and normalization through both directions.

Formalization scope

Valuations are functions A → ℝ; domains are sets V i : Set (A → ℝ); mechanisms are total functions with every property quantified only over profiles from the domain, so behavior on invalid inputs carries no content. The Groves term hᵢ is a function of the full profile constrained to be invariant under changes of coordinate i — the standard rendering of "depends only on $v_{-i}$ ". The Clarke payment uses a Finset.sup' over a finite nonempty A, so no junk supremum arises. In the single-parameter setting the valuation induced by a bid is Set.indicator, bids live in Set.Icc t0 t1 with t0 ≤ t1, and the critical value is characterized by IsLUB guarded by nonemptiness of the losing set — the book's "undefined" caveat made precise without a junk sSup. Weak monotonicity's sufficiency half carries Convex ℝ (V i) and finite A (the Saks–Yu setting); the necessity half deliberately carries no hypotheses beyond incentive compatibility itself.

Selected references

W. Vickrey, Counterspeculation, auctions, and competitive sealed tenders, J. Finance 16 (1961), 8–37. DOI
E. H. Clarke, Multipart pricing of public goods, Public Choice 11 (1971), 17–33. DOI
T. Groves, Incentives in teams, Econometrica 41 (1973), 617–631. DOI
M. Saks, L. Yu, Weak monotonicity suffices for truthfulness on convex domains, Proc. 6th ACM EC (2005), 286–293. DOI
N. Nisan, T. Roughgarden, É. Tardos, V. V. Vazirani (eds.), Algorithmic Game Theory, Cambridge University Press, 2007, Chapter 9, §§9.3, 9.5. DOI

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Captain: marwahaha

Asymmetric Hashing Square Bound: omega < 2.3747Research Paper

AI generated, I think it's correct

Motivation

The matrix-multiplication exponent measures the asymptotic arithmetic cost of multiplying square matrices. A bound $\omega<c$ means that, over the field under consideration, $n\times n$ matrices can be multiplied in $O(n^{c+\varepsilon})$ field operations for every $\varepsilon>0$ . Matrix multiplication is a central benchmark in algebraic complexity and a basic subroutine in linear algebra, graph algorithms, and symbolic computation.

The Coppersmith--Winograd tensor and the laser method produced the strongest bounds on $\omega$ for several decades. The 1990 tensor-square analysis gave $\omega<2.375477$ . Later analyses of larger powers improved the numerical bound, but they organized their recursion through values assigned independently to constituent tensors. Duan, Wu, and Zhou identified a loss in that organization: several fine constituents that can coexist inside one coarse block may be counted as though they had to be selected independently. Their asymmetric-hashing framework partially compensates for this combination loss. The paper's full second-power specialization improves the best bound obtainable from the square of the Coppersmith--Winograd tensor to $\omega<2.374631$ ; see Section 6.3 and its parameter Table 2 in Duan--Wu--Zhou.

This mission isolates that second-power result. It is smaller than the paper's record-setting eighth-power calculation, but it contains the genuinely new asymmetric-hashing and hole-repair mechanisms in their first complete form. It therefore provides a focused bridge from the existing formalization of the classical $2.375477$ square analysis to later combination-loss methods.

Setting

For a field $K$ , the matrix-multiplication tensor

\langle a,b,c\rangle_K =\sum_{i<a}\sum_{j<b}\sum_{k<c} x_{ij}\otimes y_{jk}\otimes z_{ki}

encodes multiplication of an $a\times b$ matrix by a $b\times c$ matrix. A restriction applies one linear map to each tensor leg, while a degeneration permits polynomial families of such maps and takes their first nonzero coefficient. A degeneration from the diagonal tensor $I_r$ gives a border-rank upper bound of $r$ .

The Coppersmith--Winograd tensor with parameter $q$ is

CW_q= \sum_{i=1}^{q} (x_i y_i z_0+x_i y_0z_i+x_0y_i z_i) +x_0y_0z_{q+1}+x_0y_{q+1}z_0+x_{q+1}y_0z_0.

It has border rank at most $q+2$ . Its coordinate partition has six supported types, and the square $CW_q^{\otimes2}$ has fifteen coarse constituent types $(i,j,k)$ with $i+j+k=4$ . A large tensor power contains many blocks with prescribed joint and marginal type distributions. The laser method retains blocks whose variables are disjoint and interprets their direct sum through Schönhage's asymptotic sum inequality.

Duan--Wu--Zhou refine this organization by also retaining a split distribution for the fine indices inside each coarse constituent. Coarse $X$ - and $Y$ -blocks are made unique, while compatible coarse triples may initially share a $Z$ -block. The resulting partially damaged constituent tensors are described as broken copies of a standard-form tensor. The formal target uses $q=6$ , the full Section 6 construction, and the paper's released second-power parameters.

Formalization targets

Goal: the full second-power asymmetric-hashing bound

For every field $K$ ,

\operatorname{matMulExp}(K)<\frac{23747}{10000}=2.3747.

The source reports the stronger numerical endpoint $2.374631$ , so the displayed rational inequality has strict slack. The Lean declaration has exactly the same field quantification and uses exactly the same matMulExp definition as the existing Coppersmith--Winograd $2.376$ mission; only the theorem name and rational endpoint change.

Source-level milestones

The mission first isolates the available-block shuffling interface extracted from Definitions 5.3--5.5 and Claims 5.8--5.10, then formalizes the finite covering core of the Hole Lemma 5.6. The subsequent tensor realization by zeroing and identification, the multiple-copy Corollary 5.11, the compatibility-rate identity of Lemma 6.7, the probabilistic part of Claim 6.8, and the global restricted-splitting value inequality in Equation (25) remain visible structural leaves rather than being hidden inside scalar assumptions. The numerical milestone instantiates Equation (25) with the exact $q=6$ data of Section 6.3 and Table 2 and checks a strict value surplus at $\tau=23747/30000$ . The structural proof must also make explicit the conversion from the paper's six-symmetrized value to a direct HasTauValueAtLeast witness for the mode-symmetric CW square. The final bridge applies the existing tau-value/rank machinery and transfers the Strassen-preorder exponent bound to matMulExp.

Significance

The mathematical result gives the first improvement over the classical Coppersmith--Winograd number while continuing to use only the tensor square. It separates improvement of the tensor analysis from improvement obtained merely by moving to a much higher tensor power. The same standard-form and restricted-splitting language is then reused by the paper's higher-power algorithm, which reports $\omega<2.371866$ .

For formalization, the mission adds reusable infrastructure for nested tensor partitions. Existing CW-square work records coarse support types and actual matrix-multiplication restrictions. This mission extends that layer with fine split distributions, compatibility between levels, broken-block bookkeeping, and repair of holes without replacing tensor statements by unverified scalar values. Those definitions are prerequisites for later asymmetric-hashing, complete-split, and more-asymmetry analyses.

The bound is known mathematically and was published at FOCS 2023. The open work is a machine-checked reconstruction. The underlying CW tensor, border-rank certificate, canonical tensor-square grading, Salem--Spencer sets, direct-sum tau-value notion, asymptotic sum inequality, and exponent equivalence already exist on Prove2Me. The new frontier is the cross-level combination-loss analysis and its exact numerical specialization.

Difficulty

The central difficulty is that coarse and fine decompositions cannot be optimized independently. Two coarse triples may share a $Z$ -block, and a fine $Z$ -block can be useful for one triple, compatible with several, or removed by a collision. Counting all locally valuable fine constituents therefore does not certify a direct sum. Conversely, requiring every coarse $Z$ -block to be unique discards precisely the combinations that produce the improvement.

The Hole Lemma must also preserve the actual tensor. A broken copy lacks some fine variable blocks; combining several such copies is useful only when a degeneration covers every required block with controlled loss and does not duplicate monomials. On the numerical side, the same-marginal maximum-entropy term and restricted-splitting values must be bounded with certified real inequalities. Floating-point output from MATLAB is evidence for a witness, not a Lean proof.

Formalization scope

The mission uses the existing TensorObj, MMObj, restriction, degeneration, asymptotic-rank, HasTauValueAtLeast, matMulExp_strassen, and matMulExp declarations in environment 777aaa61dcd2a1258d2b4962dbe983ede4d23b2e. Top-level results quantify over an arbitrary field. Finite supports and block indices are represented by finite types; probability and split distributions are nonnegative real functions of total mass one; entropy and numerical optimization live in the reals.

The formalization is restricted to $CW_6^{\otimes2}$ for the capstone, although generic definitions and source lemmas may quantify over levels and finite index types. A valid proof must connect scalar rate inequalities to witnessed restrictions or degenerations yielding direct sums of concrete matrix-multiplication tensors. A constant-valued surrogate for the restricted-splitting value, a hypothesis that already assumes the desired exponent bound, or a certificate definition containing its own conclusion is outside scope.

Contributions are welcome for standard-form tensor encodings, finite permutation arguments, hole repair, type and split counting, entropy maximization certificates, certified logarithm and power inequalities, and the final tau-value/rank assembly. Statements should identify the corresponding definition, lemma, claim, equation, or table in the source.

Selected references

Ran Duan, Hongxun Wu, and Renfei Zhou, Faster Matrix Multiplication via Asymmetric Hashing, 64th IEEE Symposium on Foundations of Computer Science (FOCS), 2023. arXiv:2210.10173 and released verification code.
Don Cop persmith and Shmuel Winograd, Matrix Multiplication via Arithmetic Progressions, Journal of Symbolic Computation 9, 1990, pp. 251--280. DOI 10.1016/S0747-7171(08)80013-2.
Arnold Schönhage, Partial and Total Matrix Multiplication, SIAM Journal on Computing 10(3), 1981, pp. 434--455. DOI 10.1137/0210032.

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Computational Geometry·Captain: wurtle

Generalization of Hinging PlanesResearch Paper

A continuous piecewise linear (CPWL) function is one assembled from finitely many flat pieces glued along flat seams. Every ReLU network computes such a function, and every such function is computed by some ReLU network. Questions about how deep a network must be are therefore questions about the internal structure of CPWL functions.

In 1993 Breiman built such functions from hinges: maxima of two affine maps. Sums of hinges approximate anything, but from two dimensions up they fail to represent most CPWL functions exactly. Wang and Sun (2005) widened the maxima, proving that every CPWL function on ℝⁿ is a signed sum of maxima of at most n+1 affine maps. Twenty years on it remains the workhorse structural fact, reducing any question about a network to a question about a single max gate and underpinning every known upper bound on the depth of exact representation.

That includes the newest one: at STOC 2026, Bakaev et al disproved the short standing conjecture that ⌈log₂(n+1)⌉ hidden layers are necessary, showing ⌈log₃(n−1)⌉+1 suffice. In this mission we deliver a machine-checked proof of the Wang and Sun theorem so future formalizations of network expressivity can invoke it rather than reprove it. Note that we take as given the lattice representation of Tarela and Martínez, independently proved by Ovchinnikov, which writes any CPWL function as a max of mins of its affine pieces. That is the one external ingredient the argument consumes, and our definition of CPWL builds it in.

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Captain: marwahaha

Sensitivity ConjectureResearch Paper

Nearly every measure of Boolean function complexity was known to be equivalent — except sensitivity. Proving the conjecture unified the whole picture.

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Captain: wurtle

k-SAT is NP-HardResearch Paper

Prove that $k$ -SAT is NP-complete for every fixed $k\ge3$ , with at most $k$ literal occurrences per clause.

Construct the reductions and certificate verifier using the existing WordRAM definitions. We reuse proved polynomial backend and Cook–Levin SAT completeness.

References:

Richard M. Karp. Reducibility Among Combinatorial Problems. Complexity of Computer Computations, 85–103, 1972. Main Theorem, problem 11; reduction, p. 98.
Stephen A. Cook and Robert A. Reckhow. Time Bounded Random Access Machines. Journal of Computer and System Sciences 7(4), 354–375, 1973.
Torben Hagerup. Sorting and Searching on the Word RAM. STACS 1998, 366–398.

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Combinatorics Operations Research Optimization+1·Captain: mikedeng1

A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization 2: Randomized Double Greedy Achieves 1/2 of the Optimum in ExpectationResearch Paper

Motivation

Many selection problems assign a value to each subset of a finite collection: the coverage supplied by chosen facilities, the influence reached by chosen seeds, or the value of a coalition. A submodular set function has diminishing returns in the precise sense that the combined value of two sets, counting their overlap once, does not exceed the sum of their separate values. When the function is also monotone, taking more elements never hurts. The unconstrained problem studied here permits nonmonotone functions, so both accepting and rejecting an element can matter. The question is what a single pass through the elements can guarantee when the function is available through value queries. Buchbinder et al., FOCS 2012

The randomized algorithm in this mission attains an expected one-half approximation for every nonnegative submodular function. The paper presents this as tight in the value-oracle setting: it recalls the earlier result of Feige, Mirrokni and Vondrák that a fixed improvement beyond one-half requires exponentially many queries. The contribution here is therefore both the guarantee and a short adaptive rule that attains it in a linear number of iterations. The local proposal follows the FOCS 2012 version of the paper; its theorem numbering differs from the later SIAM Journal on Computing article. Buchbinder et al., §I.A and Theorem I.2

Setting

Let $\mathcal N$ be a finite ground set, and let $f:2^{\mathcal N}\to\mathbb R_{\ge0}$ assign a nonnegative real value to every subset. The unconstrained submodular maximization problem asks for the largest value $f(S)$ among all $S\subseteq\mathcal N$ . Write $OPT$ for that value when no confusion arises, and $O$ for a set attaining it. Submodularity means

f(A\cup B)+f(A\cap B)\le f(A)+f(B)\qquad(A,B\subseteq\mathcal N).

There is no monotonicity or normalization assumption: $f(\varnothing)$ and $f(\mathcal N)$ may both be positive. A value oracle returns $f(S)$ for a requested subset $S$ . The paper's complexity claim counts such queries, assuming a query takes constant time. Buchbinder et al., §I and footnotes 1–2

Algorithm 2 visits the elements once in an arbitrary order $u_1,\ldots,u_n$ . It keeps two sets, starting at $X_0=\varnothing$ and $Y_0=\mathcal N$ . At step $i$ , it measures the gain $a_i$ from adding $u_i$ to $X_{i-1}$ and the gain $b_i$ from removing $u_i$ from $Y_{i-1}$ . It clips each gain at zero, giving $a'_i=\max(a_i,0)$ and $b'_i=\max(b_i,0)$ . It adds $u_i$ to $X$ with probability $a'_i/(a'_i+b'_i)$ and otherwise removes it from $Y$ . When both clipped gains vanish, the paper defines the add probability as one. After all elements have been processed, the two sets coincide, and the algorithm returns their common value. The state law is adaptive: its probability at step $i$ depends on the actual pair of sets produced by earlier choices. Buchbinder et al., Algorithm 2

Formalization targets

The main target is Theorem I.2 for this exact algorithm and for every enumeration of the ground set:

\max_{S\subseteq\mathcal N}f(S)\le 2\,\mathbb E[f(X_n)].

The milestone statements retain the paper's key local quantities. For a comparison optimum $O$ , set $OPT_i=(O\cup X_i)\cap Y_i$ . Lemma II.1 asserts $a_i+b_i\ge0$ . The endpoint statement identifies $OPT_0=O$ and $OPT_n=X_n=Y_n$ . Inequality (3) bounds the conditional loss in the positive-gain case; Lemma III.1 compares the expected change of $OPT_i$ with the expected combined change of $X_i$ and $Y_i$ . The telescoped display keeps the initial endpoint values $f(\varnothing)$ and $f(\mathcal N)$ before using nonnegativity. Buchbinder et al., Lemmas II.1 and III.1, inequality (3), proof of Theorem I.2

A companion target is Theorem I.4 via its second proof. For two normalized monotone submodular utilities $f_1,f_2$ , let $g(S)=f_1(S)+f_2(\mathcal N\setminus S)$ . The maximum of $g$ is exactly the optimal welfare of a two-player partition. Algorithm 2 on $g$ is asked to satisfy

3\max_{S\subseteq\mathcal N}g(S)\le4\,\mathbb E[g(X_n)].

This is the paper's three-quarter guarantee in its welfare application. Buchbinder et al., Theorem I.4 and Proof (2)

Significance

The main theorem gives a specific randomized rule whose expected value is at least half the best subset value, even when accepting an element can lower the objective. It applies without restricting the cardinality or shape of the chosen subset. The welfare corollary shows that keeping the initial endpoint values in the analysis yields a stronger guarantee for the objective formed from two monotone players. Buchbinder et al., Theorems I.2 and I.4

This mission formalizes the statement of the algorithm, its intermediate state laws, its comparison set, and the paper's numbered proof targets. The algorithmic guarantee is proved in the source paper; the local Lean theorem files are open statements with sorry and do not yet give machine-checked proofs of these results. A completed development would supply a reusable formal model of an adaptive finite random process over pairs of subsets, as well as the specific submodular inequalities. The published Submodular and OPT definitions from the earlier Feige–Mirrokni–Vondrák formalization are reused here.

Difficulty

The two possible updates cannot be assessed independently. The probability of each choice depends on the current state, and the comparison set $OPT_i$ can gain or lose the processed element in a way that differs from the two algorithm sets. A bound on the expected value of $X_i$ alone does not control the movement of $OPT_i$ . The proof must handle the clipped gains, including the case when both are zero, while preserving the exact joint law of $(X_i,Y_i)$ . Buchbinder et al., proof of Lemma III.1

Formalization scope

The ground set is a finite Lean type; subsets are Finset X, and values are real numbers. An order is a list with no repeated elements that covers the type, including the empty type. The run is an explicit finite mass function on pairs of subsets after every prefix of the list. Expectation is a finite weighted sum, so it has no integrability exception. The transition clips the two real marginal gains and handles $0/0$ by assigning probability one to the add branch, exactly as Algorithm 2 specifies. The optimum is the published maximum over all subsets. No ratio divides by a possibly zero optimum.

The theorem fixes Algorithm 2 itself; an arbitrary process with nested sets or a process defined by its desired approximation property does not satisfy this scope. The Lean goal states the value bound and leaves the paper's linear-time claim outside the formal theorem. The algorithm uses four value evaluations per processed element in its printed rule; the Lean development represents those evaluations, not an implementation cost model. The statement that its two final sets coincide is a separate milestone.

The source's main-text decreasing-returns definition has an overbroad quantifier on the added element. This development uses the equivalent lattice inequality given in the paper's footnote, which permits nonmonotone functions. The proof of Lemma II.1 also has a set-index slip, and the proof of Theorem I.2 prints $F$ for $f$ in one display; neither slip is copied into a formal statement. The one-step inequality (3) is stated for any nested pair with the processed element in $Y\setminus X$ , a generalization of the conditioned reachable states in the paper. Contributions proving the endpoint invariant, conditional inequality, one-step expected estimate, and final bound are all within scope.

Selected references

Niv Buchbinder, Moran Feldman, Joseph Naor and Roy Schwartz, A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization, Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science, 2012. FOCS version used here.
Niv Buchbinder, Moran Feldman, Joseph Naor and Roy Schwartz, A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization, SIAM Journal on Computing 44(5), 2015. DOI: 10.1137/130929205. The cited statement indices above refer to the FOCS version.

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