Title: Revisiting Area Convexity: Faster Box-Simplex Games and Spectrahedral Generalizations
Venue: NeurIPS 2023
Accepted: yes
Keywords: Optimization

optimal transport

linear programming

semidefinite programming
Abstract: We investigate area convexity [Sherman17], a mysterious tool introduced to tackle optimization problems under the challenging $\ell_\infty$ geometry. We develop a deeper understanding of its relationship with conventional analyses of extragradient methods [Nemirovski04, Nesterov07]. We also give improved solvers for the subproblems required by variants of the [Sherman17] algorithm, designed through the lens of relative smoothness [BBT17, LFN18}.

Leveraging these new tools, we give a state-of-the-art first-order algorithm for solving box-simplex games (a primal-dual formulation of $\ell_\infty$ regression) in a $d \times n$ matrix with bounded rows, using $O(\log d \cdot \epsilon^{-1})$ matrix-vector queries. As a consequence, we obtain improved complexities for approximate maximum flow, optimal transport, min-mean-cycle, and other basic combinatorial optimization problems. We also develop a near-linear time algorithm for a matrix generalization of box-simplex games, capturing a family of problems closely related to semidefinite programs recently used as subroutines in robust statistics and numerical linear algebra.

Revisiting Area Convexity: Faster Box-Simplex Games and Spectrahedral Generalizations

Abstract

We investigate different aspects of area convexity [She17], a mysterious tool introduced to tackle optimization problems under the challenging glyph[lscript] ∞ geometry. We develop a deeper understanding of its relationship with more conventional analyses of extragradient methods [Nem04, Nes07]. We also give improved solvers for the subproblems required by variants of the [She17] algorithm, designed through the lens of relative smoothness [BBT17, LFN18].

Leveraging these new tools, we give a state-of-the-art first-order algorithm for solving box-simplex games (a primal-dual formulation of glyph[lscript] ∞ regression) in a d × n matrix with bounded rows, using O (log d · glyph[epsilon1] -1 ) matrix-vector queries. Our solver yields improved runtimes for approximate maximum flow, optimal transport, min-mean-cycle, and other basic combinatorial optimization problems. We also develop a near-linear time algorithm for a matrix generalization of box-simplex games, capturing a family of problems closely related to semidefinite programs recently used as subroutines in robust statistics and numerical linear algebra.

1 Introduction

Box-simplex, or glyph[lscript] ∞ -glyph[lscript] 1 , games are a family of optimization problems of the form

These problems are highly expressive, as high-accuracy solutions capture linear programs (LPs) as a special case [LS15]. 3 More recently, approximation algorithms for the family (1) have been applied to improve runtimes for applications captured by LPs naturally bounded in the glyph[lscript] ∞ geometry. The current state-of-the-art first-order method for (1) is by Sherman [She17], who used the resulting solver to obtain a faster algorithm for the maximum flow problem. As further consequences, solvers for (1) have sped up runtimes for optimal transport [JST19], a common task in modern computer vision [KPT + 17], and min-mean-cycle [AP20], a basic subroutine in solving Markov decision processes [ZP96], among other combinatorial optimization problems [AJJ + 22, JJST22].

The runtimes of first-order methods for approximating (1) are parameterized by an additive accuracy glyph[epsilon1] > 0 and a Lipschitz constant 4 L := ‖ A ‖ 1 → 1 ; in several prominent applications of (1) where the relevant A has favorable structure, e.g. the column-sparse edge-vertex incidence matrix of a graph, L is naturally small. Standard non-Euclidean variants of gradient descent solve a smoothed variant of (1) using ˜ O ( L 2 glyph[epsilon1] -2 ) matrix-vector products with A and A glyph[latticetop] [Nes05, KLOS14]. Until

recently however, the lack of accelerated algorithms for (1) (using a quadratically smaller ˜ O ( Lglyph[epsilon1] -1 ) matrix-vector products) was a notorious barrier in optimization theory. This barrier arises due to the provable lack of a strongly convex regularizer in the glyph[lscript] ∞ geometry with dimension-independent additive range [ST18], 5 a necessary component of standard non-Euclidean acceleration schemes.

Sherman's breakthrough work [She17] overcame this obstacle and gave an algorithm for solving (1) using roughly O ( L log( d ) log( Lglyph[epsilon1] -1 ) · glyph[epsilon1] -1 ) matrix-vector products. 6 Roughly speaking, Sherman exploited the primal-dual nature of the problem (1) to design a smaller regularizer over the glyph[lscript] ∞ ball [ -1 , 1] n , which satisfied a weaker condition known as area convexity (rather than strong convexity). The algorithm in [She17] combined an extragradient method 'outer loop' requiring O ( L log d · glyph[epsilon1] -1 ) iterations, and an alternating minimization subroutine 'inner loop' solving the subproblems required by the outer loop in O (log( Lglyph[epsilon1] -1 )) steps. However, Sherman's analysis in [She17] was quite ad hoc, and its relationship to more standard optimization techniques has remained mysterious. These mysteries have made further progress on (1) and related problems challenging in several regards.

Sherman's algorithm pays a logarithmic overhead in the complexity of its inner loop to solve subproblems to high accuracy. Does it tolerate more approximate subproblem solutions?

The analysis of Sherman's alternating minimization scheme relies on multiplicative stability properties of the dual variable y . In generalizations of (1) where the dual variable is unstable, how do we design solvers handling instability to maintain fast subproblem solutions?

The relationship between the area convexity definition [She17] and more standard Bregman divergence domination conditions used in classical analyses of extragradient methods [Nem04, Nes07] is unclear. Can we unify these extragradient convergence analyses?

A central goal of this work is to answer Question 3 by revisiting the area convexity technique of [She17], and developing a deeper understanding of its relationship to more standard tools in optimization theory such as relative smoothness [BBT17, LFN18] and the classical extragradient methods of [Nem04, Nes07]. We summarize these insights in Section 2 and give self-contained expositions in Appendices A and B of the supplement. As byproducts, our improved understanding of area convexity results in new state-of-the-art runtimes for (1) by affirmatively answering Question 1, and the first accelerated solver for a matrix generalization by affirmatively answering Question 2.

1.1 Our results

Box-simplex games. We give an algorithm for solving (1) improving the runtime of [She17] by a logarithmic factor, by removing the need for high-accuracy inner loop solutions. 7

Theorem 1. There is an algorithm (stated in full as Algorithm 4 in the supplement) deterministically computing an glyph[epsilon1] -approximate saddle point to (1) in time

We define nnz and ‖ A ‖ 1 → 1 in Section 1.3. The most direct comparison to Theorem 1 is the work [She17], which we improve. All other first-order methods for solving (1) we are aware of are slower than [She17] by at least an ≈ min( glyph[epsilon1] -1 , √ n ) factor. Higher-order methods for (1) (e.g. interior point methods) obtain improved dependences on glyph[epsilon1] at the cost of polynomial overhead in the dimension.

As a byproduct of Theorem 1, we improve all the recent applications of (1), including approximate maximum flow [She17], optimal transport [JST19], min-mean-cycle [AP20], and semi-streaming variants of bipartite matching and transshipment [AJJ + 22]. These are summarized in Section 5 of the supplement. The design of efficient approximate solvers for optimal transport in particular is a problem which has received a substantial amount of attention from the learning theory community

(see Table 1 and the survey [PC19]). The runtime of Theorem 1 represents a natural conclusion to this line of work. 8 We believe our improvement in Theorem 1 (avoiding a high-accuracy subproblem solve) takes an important step towards bridging the state-of-the-art in the theory and practice of optimal transport, a well-motivated undertaking due to its widespread use in applications.

Box-spectraplex games. We initiate the algorithmic study of box-spectraplex games of the form

where { A i } i ∈ [ n ] , B are d × d symmetric matrices and Y glyph[followsequal] 0 satisfies Tr ( Y ) = 1 (i.e. Y ∈ ∆ d × d , cf. Section 1.3). This family of problems is a natural formulation of semidefinite programming (SDP), and generalizes (1), the special case of (2) where all of the matrices { A i } i ∈ [ n ] , B , Y are diagonal.

Our main result on solving (2), stated below as Theorem 2, has already found use in [JRT23] to obtain faster solvers for spectral sparsification and related discrepancy-theoretic primitives, where existing approximate SDP solvers do not apply. Beyond the fundamental nature of the problem solved by Theorem 2, we are optimistic that it will find use in other applications of structured SDPs.

To motivate our investigation of (2), we first describe some history of the study of the simpler problem (1). Many applications captured by (1) are structured settings where optimization is at a relative scale , i.e. we wish to obtain an (1 + glyph[epsilon1] ) -multiplicative approximation to the value of a LP, OPT := min A x ≤ b c glyph[latticetop] x . We use bipartite matching as an example: x is a fractional matching and A is a nonnegative edge-vertex incidence matrix. For LPs with a nonnegative constraint matrix A , custom positive LP solvers achieve (1 + glyph[epsilon1] ) -multiplicative approximations at accelerated rates scaling as ˜ O ( glyph[epsilon1] -1 ) . This result then implies a ˜ O ( glyph[epsilon1] -1 ) -rate algorithm for approximating bipartite matching and its generalization, optimal transport (where the first such algorithm used positive LP solvers [BJKS18, Qua20]). However, existing accelerated positive LP solvers [ZO15] are both sequential and randomized, and whether this is necessary has persisted as a challenging open question.

In several applications with nonnegative constraint matrices [JST19, AP20, AJJ + 22], this obstacle was circumvented by recasting the problem as a box-simplex game (1), for which faster, deterministic, and efficiently-parallelizable solvers exist (see e.g. Section 4.1 of [AJJ + 22] for the bipartite matching reduction). 9 In these cases, careful use of box-simplex game solvers (and binary searching for the problem scale) match the guarantees of positive LP solvers, with improved parallelism or determinism. Notably, simpler primitives such as simplex-simplex game solvers do not apply in these settings.

The current state-of-affairs in applications of fast SDP solvers is very similar. For various problems in robust statistics (where the goal is to approximate a covariance matrix or detect a corruption

spectrally) [CDG19, CMY20, JLT20] and numerical linear algebra (where the goal is to reweight or sparsify a matrix sum) [LS17, CG18, JLM + 21], positive SDP solvers have found utility. However, these uses appear quite brittle: current positive SDP solvers [ZLO16, PTZ16, JLT20] only handle the special case of packing SDPs (a special class of positive SDP with one-sided constraints), preventing their application in more challenging settings (including pure covering SDPs).

In an effort to bypass this potential obstacle when relying on positive SDP solvers more broadly, we therefore develop a nearly-linear time solver for box-spectraplex games in Theorem 2 (we define T mv, the time required to perform a matrix-vector product, in Section 1.3).

Theorem 2. There is an algorithm which computes an glyph[epsilon1] -approximate saddle point to (2) in time

with probability ≥ 1 -δ , for L := ‖ ∑ i ∈ [ n ] | A i |‖ op + ‖ B ‖ op . A deterministic variant uses time

Theorem 2 follows as a special case of Theorem 3 in the supplement, which has refined guarantees stated in terms of the cost to perform certain queries of matrix exponentials required by our algorithm, typical of first-order methods over ∆ d × d . The first runtime in Theorem 2 implements these queries using randomized sketching tools, and the second uses an exact implementation. Beyond its generic utility, we find Theorem 2 conceptually interesting, as its development used our relative smoothness viewpoint on area convexity in Section 2 (for overcoming instability of dual matrix variables), and new techniques for proving area convexity of a matrix variant of the [She17] regularizer.

To put Theorem 2 in context, we compare it to existing solvers for simplex-spectraplex games, a relative of (2) where both players are glyph[lscript] 1 constrained. Simplex-spectraplex games are more wellstudied [KV05, WK06, AK07, BBN13, AL17, CDST19], and the state-of-the-art algorithms [BBN13, AL17, CDST19] query ˜ O ( L 2 . 5 glyph[epsilon1] -2 . 5 ) vector products in { A i } i ∈ [ n ] and B . A slight variant of these algorithms (using a separable regularizer in place of area convex techniques) gives a complexity of ˜ O ( n · L 2 . 5 glyph[epsilon1] -2 . 5 ) such queries for the more challenging box-spectraplex games (2), which is the baseline in the literature. In comparison, Theorem 2 requires ˜ O ( L 3 . 5 glyph[epsilon1] -3 . 5 ) products, but assumes access to {| A i |} i ∈ [ n ] . This is not without loss of generality, as | A | can be dense even when A is sparse, though in important robust statistics or spectral sparsification applications (where { A i } i ∈ [ n ] are rank-one), it is not restrictive. We find it interesting to understand whether access to {| A i |} i ∈ [ n ] and the ˜ O ( Lglyph[epsilon1] -1 ) overhead in Theorem 2 (due to higher-rank randomized sketches) can be avoided.

1.2 Related work

The family of box-simplex games (1) is captured by linear programming [LS15], where state-of-theart solvers [CLS21, vdBLL + 21] run in time ˜ O (( n + d ) ω ) or ˜ O ( nd +min( n, d ) 2 . 5 ) , where ω ≈ 2 . 37 is the current matrix multiplication constant [AW21, DWZ22]. These LP solvers run in superlinear time, and practical implementations do not currently exist; on the other hand, the convergence rates depend polylogarithmically on the accuracy parameter Lglyph[epsilon1] -1 . The state-of-the-art approximate solver for (1) (with runtime depending linearly on the input size nnz ( A ) and polynomially on Lglyph[epsilon1] -1 ) is the accelerated algorithm of [She17], which is improved by our Theorem 1 by a logarithmic factor. Finally, we mention [BSW19] as another recent work which relies on area convexity techniques to design an algorithm in a different problem setting; their inner loop also requires high-precision solves (as in [She17]), and this similarly results in a logarithmic overhead in the runtime.

For the specific application of optimal transport, a problem captured by (1) receiving significant recent attention from the learning theory community, several alternative solvers have been developed beyond Table 1. By exploiting relationships between Sinkhorn regularization and faster solvers for matrix scaling [CMTV17, ALdOW17], [BJKS18] gave an alternative solver obtaining a ˜ O ( n 2 · glyph[epsilon1] -1 ) rate. These algorithms call specialized graph Laplacian system solvers as a subroutine, which are currently

less practical than our methods based on matrix-vector queries. Finally, the recent breakthrough maximum flow algorithm of [CKL + 22] extends to solve optimal transport in O ( n 2+ o (1) ) time (though its practicality is unclear). For moderate glyph[epsilon1] ≥ n -o (1) , this is slower than Theorem 1.

To our knowledge, there have been no solvers developed tailoring specifically to the family (2). These problems are solved by general-purpose SDP solvers, where the state-of-the-art runtimes of ˜ O ( n ω √ d + nd 2 . 5 ) or ˜ O ( n ω + d 4 . 5 + n 2 √ d ) [JKL + 20, HJS + 22] are highly superlinear (though again depend polylogarithmically on the inverse accuracy). We believe our techniques in proving Theorem 2 extend to show that a variant of gradient descent in the glyph[lscript] ∞ geometry [KLOS14] solves (2) in an unaccelerated ˜ O ( L 2 glyph[epsilon1] -2 ) iterations, with the same per-iteration complexity as Theorem 2.

1.3 Notation

General notation. We use ˜ O to suppress polylogarithmic factors in problem parameters for brevity. Throughout [ n ] := { i ∈ N | 1 ≤ i ≤ n } and ‖·‖ p is the glyph[lscript] p norm. We refer to the dual space (bounded linear operators) of a set X by X ∗ . The all-zeroes and all-ones vectors in dimension d are denoted 0 d and ✶ d . When u, v are vectors of equal dimension, u ◦ v denotes their coordinatewise multiplication. For A ∈ R n × d we denote its i th row (for i ∈ [ n ] ) by A i : and j th column (for j ∈ [ d ] ) by A : j . For p, q ≥ 1 we define ‖ A ‖ p → q := sup ‖ v ‖ p =1 ‖ A v ‖ q . The number of nonzero entries in A is denoted nnz ( A ) , and T mv ( A ) is the time it takes to multiply a vector by A . We use med ( x, -1 , 1) to project x onto [ -1 , 1] , where med means median. We define the d -dimensional simplex and the d × d spectraplex ∆ d := { y ∈ R d ≥ 0 | ‖ y ‖ 1 = 1 } and ∆ d × d := { Y ∈ S d × d glyph[followsequal] 0 | Tr Y = 1 } . Throughout we reserve h ( y ) := ∑ j ∈ [ d ] y j log y j (for y ∈ ∆ d ) for negated (vector) entropy.

Optimization. For convex function f , ∂f ( x ) refers to the subgradient set at x ; we sometimes use ∂f to denote any (consistently chosen) subgradient. Following [LFN18], we say f is L -relatively smooth with respect to convex function r if Lr -f is convex, and we say f is m -strongly convex with respect to r if f -mr is convex. For a differentiable convex function f we define the associated Bregman divergence V f x ( x ′ ) := f ( x ′ ) -f ( x ) -〈∇ f ( x ) , x ′ -x 〉 , which satisfies the identity

For convex function f on two variables ( x, y ) , we use ∂ y f ( x, y ) to denote the subgradient set at y of the restricted function f ( x, · ) . We call a function f of two variables ( x, y ) ∈ X × Y convexconcave if its restrictions to the first and second block are respectively convex and concave. We call ( x, y ) ∈ X×Y an glyph[epsilon1] -approximate saddle point if its duality gap , max y ′ ∈Y f ( x, y ′ ) -min x ′ ∈X f ( x ′ , y ) , is at most glyph[epsilon1] . For differentiable convex-concave function f clear from context, its subgradient operator is g ( x, y ) := ( ∂ x f ( x, y ) , -∂ y f ( x, y )) ; when f is differentiable, we will call this a gradient operator. We say operator g is monotone if for all w,z in its domain, 〈 g ( w ) -g ( z ) , w -z 〉 ≥ 0 : examples are the subgradient of a convex function or subgradient operator of a convex-concave function.

1.4 Organization

We summarize our improved area convexity insights in Section 2 (with extended expositions in the supplement), and showcase how they prove Theorem 1 in Section 3. Due to space constraints, we defer proving Theorem 2 and our applications to the supplement (Sections 4 and 5, respectively).

2 Revisiting area convexity

In this section and Section 3, we consider box-simplex games of the form (1), for A ∈ R n × d , b ∈ R d , and c ∈ R n . For simplicity we assume ‖ A ‖ 1 → 1 ≤ 1 , lifting this assumption (by scale invariance) in the proof of Theorem 1. We approximate saddle points to (1) by using the family of regularizers:

where the absolute value is applied to A entrywise, and squaring is applied to x entrywise. This family was introduced by [JST19], a minor modification to a similar family given by [She17], and has favorable properties for the geometry present in the problem (1). In the remainder of the paper,

we specialize the notation X := [ -1 , 1] n , and for any v ∈ R n we let Π X ( v ) := med ( v, -1 , 1) be truncation applied entrywise. We also use Y := ∆ d and for any v ∈ R d ≥ 0 we let Π Y ( v ) := v/ ‖ v ‖ 1 normalize onto Y . We let Z := X × Y , and define the gradient operator of (1):

We refer to the x and y components of g ( x, y ) by g x ( x, y ) := A y + c and g y ( x, y ) := b -A glyph[latticetop] x . Finally, to minimize clutter we denote the Bregman divergence in r ( α ) A by V ( α ) := V r ( α ) A .

Section 2.1 presents our unification of the convergence analysis in [She17] with more standard analyses, by way of interpreting area convexity as Bregman domination (6).

Section 2.2 presents several helper results from convex analysis which demonstrates how area convexity implies relative smoothness of the subproblems encountered by [She17], leading to simpler subproblem solvers. This also gives way to developments which generalize to the matrix setting (and are crucial for our proof of Theorem 2).

Ultimately, our proof of Theorem 1 in Section 3 will build upon both of these sets of insights.

2.1 Area convexity as Bregman domination

In Appendix B of the supplement, we analyze extragradient algorithms for approximately solving variational inequalities in an operator g , i.e. which find z with small 〈 g ( z ) , z -u 〉 for all u in the domain (in the minimax optimization setting where g is taken to be the gradient operator, e.g. the operator in (5), this corresponds to duality gap). Specifically, we analyze convergence under the following condition, weaker than previous notions in [She17, CST21].

Definition 1 (Relaxed relative Lipschitzness) . We say an operator g : Z → Z ∗ is 1 η -relaxed relatively Lipschitz with respect to r : Z → R if for all ( z, z ′ , z + ) ∈ Z × Z × Z ,

This notion is related to and subsumes the notions of relative Lipschitzness [CST21] and area convexity [She17] which have recently been proposed to analyze extragradient methods, which we define below. In particular, Definitions 2 and 3 were motivated by designing solvers for (1).

Definition 2 (Relative Lipschitzness [CST21]) . We say an operator g : Z → Z ∗ is 1 η -relatively Lipschitz with respect to r : Z → R if for all ( z, z ′ , z + ) ∈ Z × Z × Z ,

Definition 3 (Area convexity [She17]) . We say convex r : Z → R is η -area convex with respect to an operator g : Z → Z ∗ if for all ( z, z ′ , z + ) ∈ Z × Z × Z , defining c := 1 3 ( z + z ′ + z + ) ,

Area convexity is monotone in η as the right-hand side is nonnegative for any z, z ′ , z + .

Relaxed relative Lipschitzness simultaneously generalizes relative Lipschitzness and area convexity. The relationship is obvious for Definition 2, but the generalization of Definition 3 relies on the following simple observation, which has not previously appeared explicitly to our knowledge:

In Appendix B of the supplement we give a framework unifying previous analyses of [Nem04, Nes07, She17, CST21], possibly of further utility. The realization (6) that area convexity implies a Bregman divergence bound lets us reinterpret the [She17] outer loop to tolerate larger amounts of inexactness in the subproblems, where inexactness is measured in Bregman divergence. We combine our improved outer loop analysis with tools leveraging relative smoothness (to be discussed) to remove alternating minimization from [She17]. We will use the following known fact.

Fact 1 (Lemma 2, [CST21]) . For convex functions f, r , if f is L -relatively smooth with respect to r , then ∂f is L -relatively Lipschitz with respect to r .

We also summarize some useful properties of r ( α ) A adapted from [JST19].

Lemma 1. For α ≥ 1 2 , r ( α ) A is jointly convex; for α ≥ 2 , r ( α ) A is 1 3 -area convex with respect to g (5) .

2.2 Area convexity as relative smoothness

In [She17], the second property in Lemma 1 was used with an ad hoc outer loop analysis (unified with existing analyses in Appendix B of the supplement) to prove convergence. However, it remains to discuss how to implement steps of the outer loop method, each of which solves a regularized subproblem of the form, for ( g x , g y ) ∈ X ∗ ×Y ∗ (and recalling our regularizer (4))

These subproblem forms are standard for mirror descent-based methods (including extragradient methods). In contrast to schemes using separable regularizers (admitting closed-form solutions), the subproblems induced by (4) themselves require an iterative method to solve. In [She17], a linearly-convergent minimization subroutine for F was given based on multiplicative stability of simplex variables, using a fairly nonstandard analysis (see Lemma 6, [JST19]). Our next observation is that a linearly-convergent subproblem solver follows off-the-shelf from [LFN18] and viewing area convexity under the lens of relative smoothness , a more standard condition in optimization theory. We use the following convex analysis facts, deferring proofs to Section 3.2 of the supplement.

Lemma 2. Let X ⊆ R m and Y ⊆ R n be convex compact subsets. Suppose F : X × Y → R is jointly convex over its argument ( x, y ) ∈ X × Y . For y ∈ Y , define x br ( y ) := argmin x ∈X F ( x, y ) and f ( y ) := F ( x br ( y ) , y ) . Then for all y ∈ Y , ∂ y F ( x br ( y ) , y ) ⊂ ∂f ( y ) .

Lemma 3. In the setting of Lemma 2, suppose for any x ∈ X , F ( x, · ) (as a function over Y ) always is r : Y → R plus a linear function (where the linear function may depend on x ). Then r -f is convex, and f -q is convex for any q : Y → R such that F -q : X × Y → R is jointly convex.

Recall from Lemma 1 that F is jointly convex over ( x, y ) for any α ≥ 1 2 . Hence, for α = 2 , we may apply Lemma 3 with r ( y ) := 2 h ( y ) to conclude that f ( y ) := min x ∈ [ -1 , 1] n F ( x, y ) is 2 -relatively smooth with respect to h , as a function over Y = ∆ d . Moreover, applying Lemma 3 with q ( y ) := h ( y ) , and again using the joint convexity fact in Lemma 1, shows that f is further 1 -relatively strongly convex with respect to h . At this point, a direct application of Theorem 3.1 in [LFN18] (which gives an algorithm for optimization under relative smoothness and strong convexity) yields a linearly-convergent algorithm for minimizing F . We remark that in light of Lemma 2, we can implement gradient queries to f by computing the best response argument for a given y .

Interestingly, this argument used nothing more than Lemma 3, joint convexity of r ( α ) A , and the ability to tune α to induce relative strong convexity (in contrast to [She17], which requires multiplicative stability properties). An important consequence is that the same technique generalizes to the matrix setting via new joint convexity facts we prove in Section 4 of the supplement, where multiplicative stability breaks due to non-monotonicity of matrix exponentials; the corresponding subproblem analysis in [She17] hence does not apply. This gives a simple proof-of-concept matching Theorem 2 up to logarithmic factors, which we improve via our approximation-tolerant extragradient methods.

3 Box-simplex games without alternating minimization

In this section, we show how to combine the insights regarding area convexity from Section 2, along with a careful implementation of our inner loop subproblem solvers, to prove Theorem 1.

3.1 Approximation-tolerant extragradient method

We begin by stating two oracles whose guarantees, when combined with Definition 3, give our conceptual algorithm for (1). These oracles can be viewed as approximately implementing steps of the extragradient method framework in Appendix B of the supplement. We develop subroutines which satisfy these relaxed definitions in the following Section 3.2. In the following definitions (and throughout), we let V ( α ) := V r ( α ) A be the divergence associated with r ( α ) A for notational convenience.

Definition 4 (Gradient step oracle) . For a problem (1) , we say O grad : Z × Z ∗ →Z is an ( α, β ) -gradient step oracle if on input ( z, v ) , it returns z ′ such that

Definition 5 (Extragradient step oracle) . For a problem (1) , we say O xgrad : Z × Z ∗ ×Y → Z is an ( α, β ) -extragradient step oracle if on input ( z, v, ¯ y ) , it returns ( z + , ¯ y + ) such that

When β = 0 , Definitions 4 and 5 reduce to the conventional proximal oracle steps used by the extragradient method of [Nem04]. In our solver for (1), we use β > 0 to compensate for our inexact subproblem solves. The asymmetry in Definitions 4 and 5 reflect an asymmetry in the analyses of extragradient methods. In typical analyses, the regret is bounded for the 'gradient oracle' points, but the regret upper bound is stated in terms of the divergences of the 'extragradient oracle' points (which our inexact oracles need to compensate for).

The utility of our Definitions 3, 4, and 5 reveals itself through the following lemma.

Lemma 4. Let z ∈ Z , ¯ y ∈ Y , α ≥ 2 , β, γ ≥ 0 and 0 ≤ η ≤ 1 3 . Let z ′ ← O grad ( z, ηg ( z )) and ( z + , ¯ y + ) ← O xgrad ( z, η 2 g ( z ′ ) , ¯ y ) , where O grad is an ( α, β ) -gradient step oracle and O xgrad is an ( α + β, γ ) -extragradient step oracle. Then for all u ∈ Z ,

Proof. By definition of O grad (with u ← z + ) and O xgrad, we have

Combining yields

where in the second inequality we used that V ( α + β ) dominates V ( α ) , and V ( α ) z ′ ( z + ) ≥ 0 by Lemma 1. The conclusion follows by applying Definition 3 (see (6)) and the second fact in Lemma 1.

When β = γ = 0 , Lemma 4 is the same as Appendix B of the supplement, and hence yields similar implications as standard extragradient methods: a scaling of the left-hand side upper bounds duality gap of z ′ , and the right-hand side telescopes (and is bounded using the following standard fact).

Lemma 5. Let α, γ ≥ 0 , and let z 0 = ( x 0 , y 0 ) where x 0 = 0 n and y 0 = 1 d ✶ d . Then z 0 is the minimizer of r ( α ) A over Z , and V ( α ) z 0 ( u ) ≤ 1 + α log d, V γh y 0 ( u y ) ≤ γ log d for all u = ( u x , u y ) ∈ Z .

Finally, for convenience to the reader, we put together Lemma 4 and 5 to obtain an analysis of the following conceptual 'outer loop' extragradient algorithm (subject to the implementation of gradient and extragradient step oracles), Algorithm 1. Our end-to-end algorithm will be an explicit implementation of the framework in Algorithm 1; we provide a runtime analysis and error guarantee for our complete algorithm in Theorem 1, as well as pseudocode in Algorithm 4 of the supplement.

Algorithm 1: ConceptualBoxSimplex ( A , b, c, O grad , O xgrad ) 1 Input: A ∈ R n × d with L := ‖ A ‖ 1 → 1 , desired accuracy glyph[epsilon1] ∈ (0 , L ) , O grad a (2 , 2) -gradient step oracle, O xgrad a (4 , 4) -extragradient step oracle 2 Initialize x 0 ← 0 n , y 0 ← 1 d ✶ d , ¯ y 0 ← 1 d ✶ d , ˆ x ← 0 n , ˆ y ← 0 d , T ←glyph[ceilingleft] 6(8 log d +1) L glyph[epsilon1] glyph[ceilingright] , η ← 1 3 3 Rescale A ← 1 L A , b ← 1 L b , c ← 1 L c 4 for t = 0 to T -1 do 5 g t ← ( A y t + c, b -A glyph[latticetop] x t ) 6 z ′ t := ( x ′ t , y ′ t ) ←O grad ( z t , ηg t ) 7 g ′ t ← ( A y ′ t + c, b -A glyph[latticetop] x ′ t ) 8 ( z t +1 , ¯ y t +1 ) := ( x t +1 , y t +1 , ¯ y t +1 ) ←O xgrad ( z t , η 2 g ′ t , ¯ y t ) 9 end 10 Return: (ˆ x, ˆ y ) ← 1 T ∑ T -1 t =0 ( x ′ t , y ′ t )

Proof. First, clearly the rescaling in Line 3 multiplies the entire problem (1) by 1 L , so an glyph[epsilon1] L -approximate saddle point to the new problem becomes an glyph[epsilon1] -approximate saddle point for the original. Throughout the rest of the proof it suffices to treat L = 1 . Next, by telescoping and averaging Lemma 4 with α = β = 2 , γ = 4 , and η = 1 3 , we have for z 0 = ( x 0 , y 0 ) and any u = ( u x , u y ) ∈ Z

The last inequality used the bounds in Lemma 5 and the definition of T . Moreover since g is bilinear, and ˆ z := (ˆ x, ˆ y ) is the average of the z ′ t iterates, we have 〈 g (ˆ z ) , ˆ z -u 〉 ≤ glyph[epsilon1] . Taking the supremum over u ∈ Z bounds the duality gap of ˆ z and gives the conclusion.

3.2 Implementing oracles

In this section, we give generic constructions of gradient and extragradient step oracles. 10

Algorithm 2: GradStepOracle ( z, v, α, β )

1 Input: z = ( x, y ) ∈ Z , v = ( v x , v y ) ∈ Z ∗ , α, β ≥ 0

3 x ′ ← x br ( y ′ )

4 Return: ( x ′ , y ′ )

Lemma 6. For β ≥ α ≥ 1 2 , Algorithm 2 is an ( α, β ) -gradient step oracle.

Proof. By (3) and the first-order optimality condition for y ′ , we have for any u = ( u x , u y ) ∈ Z ,

Further, define for any ˆ y ∈ Y ,

Note that f is a partial minimization of a function on two variables which is a linear term plus r ( α ) A , which is convex by Lemma 1. For any fixed x , r ( α ) A ( x, y ) is itself αh ( y ) plus a linear function.

Lemma 3 then shows f is α -relatively smooth with respect to h . We then have for all ˆ y, u y ∈ Y ,

Here the equality used Lemma 2 where the linear shift between r ( α ) A and the minimization problem inducing f cancels in the expression ∂f (ˆ y ) -∂f ( y ) , and the inequality used Fact 1. Combining (9) and (11) (with ˆ y ← y ′ ) and using V βh dominates V αh yields

Finally, first-order optimality of x ′ with respect to the objective induced by y ′ implies for all u x ∈ X ,

so combining with (12) we have for u = ( u x , u y ) , 〈 v + ∇ r ( α ) A ( z ′ ) -∇ r ( α ) A ( z ) , z ′ -u 〉 ≤ V βh y ( u y ) . The conclusion follows by using the identity (3) to rewrite 〈∇ r ( α ) A ( z ′ ) -∇ r ( α ) A ( z ) , u -z ′ 〉 .

Algorithm 3: XGradStepOracle ( z, v, ¯ y, α, β )

1 Input: z = ( x, y ) ∈ Z , v = ( v x , v y ) ∈ Z ∗ , ¯ y ∈ Y , α, β ≥ 0

3 x + ← x br ( y + )

5 Return: ( x + , y + , ¯ y + )

We defer a proof of the following to Section 3.2 of the supplement, as it is similar to Lemma 6. We remark that the auxiliary sequence ¯ y, ¯ y + used in the extragradient step oracle definition is itself inspired by running a few steps of a descent method on the subproblem, inducing a telescoping divergence sequence used by the outer loop to pay for inaccuracy in the subproblem solution.

Lemma 7. For β ≥ α ≥ 1 2 , Algorithm 3 is an ( α, β ) -extragradient step oracle.

3.3 Proof of Theorem 1

Theorem 1. There is an algorithm (stated in full as Algorithm 4 in the supplement) deterministically computing an glyph[epsilon1] -approximate saddle point to (1) in time

Proof. We briefly describe the algorithm from the supplement for completeness here, as correctness follows straightforwardly from combining Corollary 1 with Lemmas 6 and 7. The algorithm is an instance of the conceptual Algorithm 1. Each step of Algorithm 1 then first implements Algorithm 2 (used in Lemma 6), with the parameters required by Lemma 4, to obtain a gradient step oracle. Similarly, it next implements Algorithm 3 (used in Lemma 7), with the parameters required by Lemma 4, to obtain an extragradient step oracle. Correctness follows from Corollary 1. For the runtime, without loss of generality nnz ( A ) ≥ max( n, d ) (else we may drop columns or rows appropriately), and each of T iterations is dominated by a constant number of matrix-vector multiplications.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focuses on box-simplex games that are min max of degree 2 polynomials being bilinear in the inner and outer optimization variables. The inner problem is over the simplex and the outer problem is over the unit hypercube. The goal of the paper is to unify the previous framework derived in [She17] with standard Bregman divergence domination conditions. \nThis latter framework [She17] consists on exploiting the primal-dual nature of the min-max problem to setup a regularizer over the unit hypercube. This satisfies the so-called \"area convexity\" condition instead of strong convexity. The outer loop method relies on an extragradient method and the inner loop relies on an alternating minimization sub-procedure.\nIn order to analyze this framework, the authors explain the connections with relative smoothness and extragradient methods.\nThis yields an algorithm to solve box-simplex games that improves the one of [She17] by a log factor.\n\nSTRENGTHS:\nThe paper is well-written and I believe quite clear for non-experts in this field. \nThe authors are willing to bridge the gap between area convexity and more standard convexity tools, that makes the contribution original. \nThe proof sketches sound correct.\n\nWEAKNESSES:\nI am not a specialist in this field and with a very limited amount of time (too short to read the 39 pages of supplementary material), it is completely impossible to judge whether the framework is correct or not. The authors stated the main theorems and did provide proof sketches and hints that could possibly convince some readers. I have doubts that this work, possibly sound and surely interesting, could be a good fit with NeurIPS.\n\nThere are neither concrete algorithmic implementation nor benchmarking in the main submission document.\n\nQUESTIONS:\n- after l72 (2): could you please define Delta^(dxd) (the set of trace 1 PSD matrices)? \n\n- In (4) what is h?\n\nLIMITATIONS:\nThere is no limitation section provided by the authors in the main submission. \nThe authors do not address any limitation and do not provide any further research directions.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study first-order algorithms for box-simplex games, a special kind of two-player zero-sum bi-affine constrained games. The constraints of these games dictate that the first player selects an action represented by a vector within the n-dimensional box ($[-1,1]^n$), while the second player chooses an action defined by a vector from the d-dimensional simplex ($\\Delta_d$).\n\nThe authors show an algorithm for the computation of an $\\epsilon$-approximate saddle point of a box-simplex bi-affine game, expressed as $min_{x\\in[-1,1]^n,y\\in\\Delta_d} x^T A y^T - <b,x> + <c,y>$, in time $O(nnz(A)\\cdot L \\cdot\\log(d)/\\epsilon)$, where L is the maximum 1-norm of any column of A, and nnz(A) denotes the number of non-zero entries in matrix A. They advance over the previously best-known first-order method by Sherman, removing logarithmic terms.\n\nBox-simplex games has some important applications, and the authors' algorithm introduces a new state-of-the-art, including applications like approximate max flow and optimal transport.\n\nIn the latter part of the paper, the authors broaden their focus to include a new class of two-player games: box-spectraplex games. This is a generalization of box-simplex games where one player still picks an action in a n-dimensional box ($[-1,1]^n$), but the other player chooses as an action, a positive semi-definite matrix Y with trace equals to 1. An applications of these kind of games include spectral sparcification. The authors present a first-order algorithm to computing an approximate saddle point of a box-spectraplex game.\n\n\nAfter rebuttal: The author's rebuttal successfully addressed my concerns, and as a result, I have chosen to maintain my original score.\n\nSTRENGTHS:\nThis paper studies an interesting class of games with important applications, thereby contributing to the current state-of-the-art algorithm. Moreover, the authors novel reinterpretation of the area-convexity as the more standard relative smoothness allows for the use of more conventional analysis methods, which is interesting on its own. I looked at the statement of the lemmas and they seem reasonable. I only verified the proofs in the main body.\n\nWEAKNESSES:\nIn the spirit of constructive criticism, I believe that the writing could be improved for better readability. Currently, the appendix appears more like a journal version of the paper rather than serving its intended purpose. A few recommendations to enhance the paper include:\n- On line 63, defining the notation nnz(A) and $||A||_{1\\rightarrow 1}$ earlier might improve clarity.\n- On lines 105-16, it could be beneficial to define the operator $T_{mv}$ sooner.\n- Several references to 'outer' and 'inner loop' throughout the paper lacked clarity. Lines 38, 215, and 224 are examples of this.\n\nFinally, the move towards box-spectraplex games seems to be more theoretical than practical, with limited application examples. Please see the question section.\n\nQUESTIONS:\nCould the authors provide more examples of problems that could be simplified to box-spectraplex games?\n\nI was a bit confused in the proof sketch of Theorem 1. For the telescoping sum to be valid, it appears necessary that $\\bar{y}_{t+1}$ must equal $\\bar{y}^+_t$ (where $\\bar{y}_t$ represents the input to the extra gradient step-oracle at step t, and $\\bar{y}^+_t$ indicates the output of the extra gradient oracle at step t). However, upon reviewing Algorithm 3 in the supplementary material, different notation seems to be used. Could the authors provide clarification on this?\n\nLIMITATIONS:\nThe authors list any assumptions for their theorems, and I do not believe this work can have negative societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper looks at the bilinear min-max problem Area Convexity framework proposed by Sherman through the lens of Relative Lipschitzness condition of Cohen et al. which relates to the standard Bregman analysis in mirror-type algorithms in first-order optimization. Leveraging this connection, they provide the standard analysis of two-timescale Extragradient method for area convex regularizer based Bregman divergence. Tuning a parameter in the analysis and using the structure of the area convex regularizer, they implement an approximate version of Extragradient where each approximate iteration requires O(nnz(A)) computations. This allows them to remove the log(1/\\eps) factor Sherman incurred in the analysis of inner loop. The extend similar result to the case where\n\nSTRENGTHS:\nThe paper had a new ideas. E.g., tuning parameter alpha/beta in the regularizer was critical step that allowed for approximate implementation as done in Line 5-9 and Line 12-17 of Algorithm 3.\n\nWEAKNESSES:\nThe main weakness of the paper is in its presentation. The way main paper differs from the supplementary almost feels like reading a new paper with different orders, equation numbers etc. Also, there are concepts introduced which may not be directly relevant to the problem at hand: E.g. Definition 1 is not used throughout the main paper. I had to dig until Appendix B to understand why there was a fuss about it on the very first technical section, i.e., Section 2.1.\nThe paper borrows key ideas developed by Sherman (which are popular by now) and some previous developments such as Lu et al, and Cohen et al. In particular, the idea of Area Convexity is very much exclusively used for bilinear games and there has been no major movement towards general convex-concave problems. In that sense, this is an incremental work in my opinion.\n\nQUESTIONS:\nMy questions below are referring to the supplementary material (or actual paper).\n\n1. It would be best if you can explain how steps 7,8,9 relate to steps in Algorithm 1 with exact parameters used for it. Leaving it as \"By observation\" may not work for large fraction of readers. Same goes for steps 14-17. It would also be worthwhile to remind the reader that h is the entropy divergence function used here.\n2. The proof of Lemma 6 needs to deliberate more on what F to use define f(y-hat) in (13) such that Lemma 4 can be applied.\n3. Use x_br for two different purposes in the same section is a bad notation. In particular, Lemma 4 defines x_br as argmin F(x,y) for F defined Section 2.2 of the main paper (again, I could not find it in the supplementary paper). Is this the same used in Lemma 6?\n4. Remind the reader about H at the beginning of Section 4.\n5. In the Proof of Lemma 8, you use Theorem 3.3 of LS01. It would be better if you provide the result as there is already quite a bit of jumping around while reading the paper. I haven't been able to verify the claims in the proof of this lemma.\n6. I have read some of the references in the previous paper of this area. Can you also comment BSW19 paper? As I understand, they also implement Shermans MWU algorithm for inner subproblems and hence incur log(1/\\eps) additional factor similar to Sherman or JST19. \n7. Can you also compare the result for box-spectraplex case with BBN13, AL17, CDST19? In particular, it will be better to state clearly in what regimes Algorithm in Section 4.4 would be better than box-spectraplex variations of the above papers.\n\n\nBSW19: Faster width-dependent algorithm for mixed packing and covering LPs\n\nLIMITATIONS:\nI am giving this paper borderline accept since it was not very well written and almost seems like reading two papers at a time which borrows/uses existing literature heavily (sometimes without giving context). If properly written, this paper can be improved substantially. But that will require significant rewriting and hence, I don't see how it will be justified in this conference set up.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors consider box-simplex games, a bilinear min-max optimization problem with box and simplex constraints. The current best-known method for solving such problems is Sherman's algorithm, which is based on the concept of \"area convexity\". The key insight of this work is to reinterpret area convexity as a more general notion of relative Lipschitzness in the optimization literature. Using this new perspective, they streamline the proof of Sherman's algorithm, propose an improved subproblem solver that eliminates the additional log overhead, and further extend the algorithm to box-spectraplex games.\n\nSTRENGTHS:\n- The authors show that area convexity implies a Bregman divergence bound similar to relative Lipschitzness, which is a simple yet insightful observation. I appreciate that it better clarifies the connection between Sherman's algorithm and the classical extragradient methods, making it conceptually easier to understand and extend. \n- The proposed subproblem solver improves a $\\log(1/\\epsilon)$ factor from the original Sherman's algorithm. As a corollary, it leads to the state-of-the-art runtime complexity for optimal transport among first-order algorithms.\n\nWEAKNESSES:\nI only wish the presentation of the paper could be more clear. In particular, the proposed extragradient algorithm is not fully described in the main text, but only implicitly in the proof of Theorem 1. It would be better if the authors can present the pseudocode and better explain the steps in the algorithm. Also, it is a bit confusing that the supplement is an extended version of the paper rather than the appendix, which makes it a bit hard to track down the proof.\n\nQUESTIONS:\n- In Definition 4, the notation $V_z^{\\alpha}$ is undefined. \n- If I understand correctly, both the GradStepOracle and XGradStepOracle are designed for solving the same kind of subproblem (at the bottom of page 6). Could you explain why they are implemented differently? And how are they related to the algorithm in [LFN18]?\n\nLIMITATIONS:\nThe authors addressed the limitations by specifying the considered problem class.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper explores the relationship between area convexity and extragradient methods, and provides improved solvers for subproblems required by variants of the algorithm. The paper also presents a state-of-the-art first-order algorithm for solving box-simplex games and a near-linear time algorithm for a matrix generalization of box-simplex games. The authors demonstrate that their algorithms improve runtimes for combinatorial optimization problems and provide new insights into numerical linear algebra. The contributions of the paper include the development of efficient algorithms with improved convergence rates and computational efficiency, as well as the application of these algorithms to various optimization problems in fields such as optimal transport and min-mean-cycle.\n\nSTRENGTHS:\nThrough a deeper understanding of the relationship between box-simplex games and matrix generalization problems, the authors propose improved solvers that leverage the lens of relative smoothness and convex analysis to design efficient algorithms with faster runtimes. These proposed algorithms not only improve runtimes for combinatorial optimization problems but also provide new insights into numerical linear algebra and have applications in various combinatorial optimization problems such as approximate maximum flow, optimal transport, and min-mean-cycle.\n\nIn addition to its technical contributions, the paper is well-written, well-structured, and effectively communicates its main ideas and contributions. The authors present a rigorous analysis of the proposed algorithms, complete with detailed proofs, explanations, and appropriate use of mathematical notation. The inclusion of appendix sections, examples, illustrations, and a comprehensive literature review further enhance the quality, readability, and accessibility of the work.\n\nOverall, the strengths of this paper lie in its thorough exploration of area convexity, its innovative use of relative smoothness, and the development of improved algorithms for solving optimization problems.\n\nWEAKNESSES:\n1.\tThe author claims to have improved the runtime for the Box-simplex problem, resulting in increased performance of related applications. However, a detailed comparison with previous results is not provided. Providing such a comparison would help readers better understand the level of improvement and contribution of the paper.\n2.\tThe paper mentions the use of a proximal oracle in Algorithm 6, but more details on its implementation and computational complexity would be helpful. It would be valuable for the authors to discuss how the proximal oracle contributes to the overall efficiency of the algorithm and whether there are any limitations or challenges in its practical implementation.\n3.\tThe paper could benefit from experimental evaluation to validate the proposed approach. Conducting experiments on real-world or synthetic datasets would provide empirical evidence of the effectiveness and efficiency of the alternating minimization scheme.\n\nQUESTIONS:\nRegarding Weakness 1: The author claims to have improved the runtime of multiple applications by solving the Box-simplex problem. However, a detailed comparison with previous results is not provided. Providing such a comparison would help readers better understand the level of improvement and contribution of the paper.\n\nRegarding Weakness 2: The paper mentions the use of a proximal oracle in Algorithm 6. Could the authors provide more details on the implementation and computational complexity of this oracle? It would be helpful to understand how the proximal oracle contributes to the overall efficiency of the algorithm and whether there are any limitations or challenges in its practical implementation.\n\nLIMITATIONS:\nThe paper does not discuss the limitations and any potential negative consequences of their work. However, since their algorithm is focused on optimization problems, the impact on society would largely depend on the specific applications and use cases. A brief discussion on the potential real-world implications and ethical considerations would provide valuable context for readers to better understand the broader implications of the research.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-24"}