Game-theoretic concepts permeate a multitude of machine learning (ML) problem domains, ranging from conventional constrained optimization [7,69,91,95] and feature selection tasks [4,34,53] to burgeoning fields such as adversarial training [45,68,81] and multi-agent system dynamics [18,117]. In these research thrusts, the prevailing approach involves formulating the optimal solution of the pertinent task as a Nash equilibrium [89,103] of some carefully constructed zero-sum, cooperative, or general-sum N -player game. A deeper exploration of the optimization literature, however, reveals that to secure tight convergence guarantees, the prevailing methods tend to subtly posit the assumption of a "nearly" concave/monotone game structure whose strategy space is constrained to be some convex set [20,31,32,96]. This is problematic, because in many real-world applications, spanning from optimal transport [78,79,97] to bounded value problems in environmental engineering [64,92] and robotics especially those involving complex interactions among multiple agents [9,10,99,107], the game can exhibit nonconvexities both in the players' utilities and their strategy spaces, posing significant challenges for conventional approaches [23,27,84,118], and often compelling recourse to ad hoc heuristics [17,33,51,114]. Thus, a key objective is to devise algorithms that systematically exploit the unique geometric structure of the feasible set [80,82,98]. Attempts at providing such geometric foundations can be found in a number of recent paperes [see, e.g., 3,5,25,35,41,42,48,54,55,83,91,101,108,119]. A notable outcome of this line of work is the interpretation of constraints in game theory and equilibrium computation through the lens of Riemannian manifolds.
The Riemannian framework, commonly utilizing Hadamard and Stiefel manifolds, has been productive in a wide range of other statistical tasks, from online principal component analysis [71] to diffusion tensor data processing [74], and maximum likelihood estimation for certain heavy-tailed non-Gaussian distributions [125]. It remains a significant challenge, however, to bring Riemannian methods to bear in game-theoretic scenarios, especially those characterized by nonconvex strategy spaces, including multi-agent robotic systems where the feasible joint angles of robotic arms use a SO(3) rotation manifold [47,113,121], environmental pollution control games where the physical transmission of pollutants along bounded surface areas represents a manifold constraint [88,106], and optimal transport problems via robust Wasserstein barycenters [24,63,109]. Some of the specific challenges arising in such domains include:
1. Existence of Solution Concept. One fundamental hurdle is that the problem may not possess solutions. This contrasts starkly with minimization tasks where, given a bounded domain, the existence of an optimal solution is always assured [36]. The establishment of Nash equilibria, on the other hand, typically calls for the application of topological fixed-point theorems [15,58], whose core requirement is the convexity of the strategy space. Formally, the Nash Equilibrium problem in an N -player game with (non-Euclidean) strategy spaces
where we employ the standard shorthand (y i , y -i ) to denote the action of the i-th player and the actions of all other players in the game. A guarantee for the existence of Nash equilibria in geodesically concave games over Hadamard manifolds was introduced by Németh [90] and extended by Li [76], employing the framework of variational inequalities. Following this, [100] further broadened the scope by introducing the notion of local Nash equilibria in games situated on either finite-dimensional differentiable manifolds or infinite-dimensional Hilbert spaces. In a more recent advance, [130] proved a strong duality theorem for min-max optimization, applicable across a variety of Riemannian manifolds that are equipped with unique geodesics.
A general understanding of the efficiency and convergence properties of game dynamics over the Riemannian manifolds remains elusive. To fill this lacuna, an emerging body of research is devoted to extending standard game dynamics and optimization algorithms such as Gradient Descent Ascent (GDA) [49] and Extragradient (EG) [56,130] to variational inequalities, and tailoring the resulting algorithms to the Riemannian setting. Most of the aforementioned research focuses on min-max scenarios. For instance, [49] examined the Riemannian Gradient Descent Ascent (RGDA) for nonconvex-nonconcave settings, but their findings only offer sub-optimal rates and merely best/average-iterate convergence. [46] analyzed the Riemannian Hamiltonian Method (RHM), equating its performance with second-order methods in Euclidean settings, but the practical application of such methods is limited due to the computational expense of handling second-order derivatives in large-scale optimization problems. On a promising note, [57] demonstrated that the Riemannian Corrected Extragradient (RCEG) method achieves last-iterate convergence at a linear rate in the geodesically strongly-convexconcave case, thereby matching the results in the Euclidean regime. Their findings also extend to the stochastic or non-smooth case, where RCEG and RGDA achieve near-optimal convergence rates, albeit with factors dependent on the curvature of the manifold. For a more comprehensive discussion of related work in both Euclidean and Riemannian settings, we refer the interested reader to our supplementary material.
When investigating multi-player games where strategy spaces are represented as manifolds, several significant challenges arise. Firstly, we need to understand the appropriate solution concept in this nonconvex regime. Secondly, it is crucial to identify meaningful structure in the game so that the solution concepts are tractable using computationally lightweight methods-e.g., first-order methods-in the face of nonconvexity. 1 Lastly, from both the optimization and the equilibrium perspective, it is desirable to construct algorithms or game dynamics that remain uncoupled and more importantly model-free; that is, they remain unaffected by transformations that alter the curvature of the underlying manifold.
Our work aims to shed new light on the problem of finding game-theoretic equilibria on Riemannian manifolds. Specifically, we consider finding Nash equilibria in geodesically monotone Riemannian games (Definition 2)-a new family of games proposed in this paper that captures both the wellstudied monotone games [102] and geodesically convex-concave min-max optimization as special cases. In a geodesically monotone Riemannian game, players choose their actions from a Riemannian manifold, and their loss functions jointly satisfy the geodesic monotonicity condition.
We revisit the Riemannian Gradient Descent (RGD) Dynamics, the leading method in Riemannian optimization, in geodesically monotone Riemannian games.
Main Contribution: For geodesically strongly monotone Riemannian games, we show that the RGD exhibits curvature-independent and linear last-iterate convergence rate to the Nash equilibrium. Additionally, our instantiation of RGD (Algorithm 1) is completely agnostic to the curvature of the manifold. To the best of our knowledge, this is the first curvature-independent linear convergence rate for even the special case of min-max manifold optimization, and hence answers an open question raised in [57]. We showcase our finding in both the deterministic and stochastic/mini-batch setting (Theorem 1).
Organization. In Section 2, we provide some necessary background on manifold theory. In Section 3, we supply the definition of geodesically monotone Riemannian games and present our performance function. Section 4 is devoted to the analysis of RGD in geodesically strongly monotone Riemannian games. Finally, Section 5 concludes our work with a discussion of future directions and challenges. We defer the full proofs of our results as well as further discussion on applications to the Appendix. In Appendix A we describe some limitations of our work.
Focusing on min-max problems on Riemannian manifolds, existing research has mainly been devoted to: (1) studying the existence and uniqueness of equilibria; (2) computation of equilibrium concepts. The former results have been established for the special case of Hadamard manifolds [65,67,93] and then generalized to finite-dimensional manifolds with unique geodesics [130]. Similarly, algorithms for computing the equilibrium were first developed for the special case of hyperbolic Hadamard manifolds with negative curvature [38,75,120]. However, only asymptotic convergence guarantees were provided. It was not until recently that a nonasymptotic convergence rate guarantee has been derived for Riemannian gradient-based algorithms when applied to min-max optimization problems. In the geodesically smooth and strongly-convex-strongly-concave setting, the best known method achieves a last-iterate and curvature-dependent global rate [57]. In the geodesically convex-concave setting, the best known method only achieves a time-average global rate [57,130]. To the best of our knowledge, we provide the first set of results that fill in a gap of the equilibrium computation for games on Riemannian manifolds, where the possibility of curvatureindependent and/or last-iterate convergence rates for gradient-based algorithms has been unexplored before.
In the supplementary material we provide a further discussion of related theoretical work and applications.
The topological definitions of manifolds and their analytic constructions have given rise to a vast literature which we cannot hope to review here; for a basic introduction, see [16,70] and references therein. Here, we review the key definitions and tools needed for the study of optimization on manifolds:
A n-dimensional smooth manifold M is a topological space that is locally homeomorphic to R n and whose transition maps-the local homeomorphisms that cover the manifold-are smooth functions, meaning they are infinitely differentiable. A Riemannian manifold (M, g) is a smooth manifold M equipped with a Riemannian metric g, which is a smoothly varying positive-definite symmetric bilinear form on the tangent spaces of M. Intuitively, for each point y ∈ M, the Riemannian metric g assigns an inner product •, • y and correspondingly a n-dimensional real vector space T y M, called the tangent space at y. The tangent space T y M at a point y ∈ M is a real vector space consisting of all tangent vectors to smooth curves in M passing through y. The dimension of the tangent space T y M is equal to the dimension of the manifold M. We denote it by T M = ∪ y∈M T y M. Tangent spaces provide a means to analyze the local geometry of a manifold near a given point.
A geodesic is a smooth curve γ : [0, 1] → M in a Riemannian manifold M that locally minimizes distance and has the property that its tangent vector remains parallel along the curve with respect to the Riemannian connection.
Parallel transport is a means of transporting vectors along curves while preserving their length and direction with respect to the Riemannian metric g. More formally, given a smooth curve γ : [0, 1] → M, starting at γ(0) = y, ending at γ(1) = y ′ and a tangent vector v ∈ T y M at the starting point y, parallel transport defines a smooth vector field Γ
The parallel transport of v along the curve γ to the endpoint γ( 1) is then given by the vector Γ y ′ y (v) ∈ T y ′ M. Parallel transport is dependent on both the curve γ and the Riemannian metric g, and it provides a natural way to compare tangent vectors at different points in the manifold, while taking the manifold's geometry into account.
The exponential map EXP y (•) : T y M → M and the logarithmic map LOG y (•) : M → T y M are essential tools in connecting the local geometry of a Riemannian manifold (M, g) with the tangent space T y M. The exponential map maps a tangent vector v ∈ T y M to a point in the manifold along the geodesic starting at y. Mathematically, if γ v : [0, 1] → M is the geodesic with γ v (0) = y and γv (0) = v, then EXP y (v) = γ v (1). The logarithmic map LOG y (•) is the inverse operation, mapping a point y ′ ∈ M to the tangent vector v ∈ T y M such that EXP y (v) = y ′ . Note that the logarithmic map is well-defined when the manifold is equipped with a unique geodesic. The norm of LOG y (y ′ ) corresponds to the geodesic distance between points y and y:
Throughout this paper, we only consider Riemannian manifolds M, such that for any pair of points y, y ′ in M, there exists a unique minimizing geodesic within M that connects them. We also make use of the following properties of the parallel transport and logarithmic map:
• For any triplet of points y, y ′ , y ′′ ∈ M that lie in the same geodesic, and any vector u in the tangent space T y of y, we have
• For any pair of points y, y ′ ∈ M, we have Γ y ′ y LOG y (y ′ ) = -LOG y ′ (y). Moreover, for a point y ∈ M and a radius D > 0, we use the notation d M (y, D) = {y ′ ∈ M : d M (y, y ′ ) ≤ D}.
We begin with the definition of a geodesically convex set and function, respectively:
• The function f is called geodesically strongly convex with the modulus µ > 0 if the following statement holds:
where grad y f (y) ∈ T y M is a Riemannian gradient of f at a point y.
Finally, in our work we adopt the following smoothness conditions for the utilities of our games, conventionally used in Riemannian optimization [2,129,130]:
• Additionally, if function f is also differentiable, it is called geodesically L-smooth if for ∀y,
where grad y f (y ′ ), ∈ T y ′ M is the Riemannian gradient of f at y ′ .
In this section, we consider games where players' actions lie on a Riemannian manifold, and we extend Rosen's monotone games to this setting. Rosen's original formulation considers players whose actions lie in Euclidean spaces [102]. This new formulation allows us to study games where the action space for player i has a geometric structure, represented by the Riemannian manifold M i . We extend this concept to accommodate more diverse geometries, broadening the range of potential applications, as illustrated in Section 3.2. The missing proofs in this section are presented in Appendix D.
Definition 1 (Games on Riemannian Manifolds). A game on Riemannian manifold consists of the following components:
1. A finite set of players N = 1, 2, . . . , N . 2. For each player i ∈ N , there is a Riemannian manifold M i . Player i's action set is M i , and we denote the joint action profile by M = Π i∈N M i ,. 3. For each player i ∈ M, a loss function l i : M → R is defined, which assigns a real-valued loss to every combination of strategies in the joint strategy space M. We let L = {l i } i∈N .
A Riemannian game is smooth if all loss functions l i 's are smooth with respect to the Riemannian metric on the manifold M. We denote by F (y 1 , y 2 , . . . , y N ) = (grad y1 l 1 (y), grad y2 l 2 (y), . . . , grad yN l N (y)) : M → T M the vector of concatenated gradients of the players' loss functions with respect to the Riemannian manifold at the joint strategy profile y ∈ M.
The notion of a Nash equilibrium in a Riemannian game remains similar to the one in the Euclidean space: a strategy profile y * ∈ M is a Nash Equilibrium if no player can unilaterally decrease their loss by deviating from their current strategy, given the strategies of the other players. Formally, y * is an equilibrium if
for all players i ∈ N , all strategies y ′ i ∈ M i . In the following lemma, we provide a necessery condition for a strategy profile to be a Nash equilibrium of a arbitrary Riemannian games. Lemma 1. A necessery condition for the strategy profile y * ∈ M to be a Nash equilibrium of a smooth Riemannian game, is that F (y * ) y * = 0. Definition 2 (Geodesically Monotone Riemannian Games). A geodesically µ-strongly monotone Riemannian game satisfies the following monotonicity condition: for any two action profiles y, y
, where d M is the Riemannian distance on manifold M. When µ = 0, we refer to the game as a geodesically monotone Riemannian game.
In the next lemma, we show that geodesically monotone Riemannian games are a generalization of geodesically convex, geodesic concave min-max optimization.
Lemma 2. Consider a two player Riemannian game, where y 1 ∈ M 1 and y 2 ∈ M 2 are the actions of player 1 and 2 respectively. The loss of player 1 is u(y 1 , y 2 ) : M 1 × M 2 → R and the loss of player 2 is -u(y 1 , y 2 ). If u is µ-strongly geodesically convex in y 1 , and µ-strongly geodesically concave in y 2 , then the Riemannian game is µ-strongly geodesically monotone.
Consider a strategy profile y ∈ M and a radius D > 0. The standard measures of proximity to a Nash equilibrium in games in Euclidean spaces are the gap function and the total gap function. The gap function is inspired by the variational characterization of the game's solution, as presented in Corollary 1, and the total gap function can be interpreted as the cumulative suboptimality gaps of the players. Note that a total gap of value zero indicates that the action profile is a Nash equilibrium. Definition 3 (Duality Gap and Total Gap in Riemannian Games). The duality gap and total gap for a radius D > 0 are defined as follows:
In the subsequent lemma, we establish that the duality gap provides an upper bound for the total gap. One notable consequence of this lemma is the characterization of solutions for monotone Riemannian games.
Lemma 3 (Adapted from [43,44]). For any monotone Riemannian game, the following inequality hold for all D > 0 and y ∈ M:
T gap D (y) ≤gap √ N•D (y). Corollary 1. A strategy profile y * ∈ M is a Nash equilibrium of a monotone Riemannian game, if and only if F (y * ) y * = 0.
We measure the proximity of a strategy profile y to a Nash equilibrium by evaluating the norm of the loss function's gradient F (y) y . In the following lemma, we show that the norm of the gradient bounds the gap function, which therefore also bounds the total gap due to Lemma 3.
We provide two examples of Riemmanian games to give a sense of their expressivity. One of the examples is a generic model from the optimization and game theory literature [87,104] and the other is a formalization of a class of applied problems in economics and statistical machine learning [39,60,94].
Example 3.1 (Potential game on Riemannian manifolds). We refer to a Riemmanian game (N , S = N i=1 S i , {u i } N i=1 ) as a potential game if there exists a potential function f :
for all i ∈ N , all y ∈ S and all ỹi ∈ S i . If the potential function f is geodesically strongly concave, we have (N , S, {u i } N i=1 ) is a geodesically strongly monotone game. Example 3.2 (Robust matrix Karcher mean problem). We consider a robust version of classical matrix Karcher mean problem. More specifically, the Karcher mean of N symmetric positive definite matrices {A i } N i=1 is defined as the matrix X ∈ M = {X ∈ R n×n : X ≻ 0, X = X ⊤ } that minimizes the sum of squared distance induced by the Riemannian metric:
The loss function is thus defined by
which is known to be nonconvex in Euclidean spaces but is geodesically strongly convex. Then, the robust version of classical matrix Karcher mean problem aims at solving the following problem:
where γ > 0 encodes the tradeoff between the computation of Karcher mean over a set of {Y i } N i=1 and the difference between the observed samples {A i } N i=1 and {Y i } N i=1 . It is clear that the above problem is a geodesically strongly-convex-strongly-concave min-max optimization problem.
In addition to these examples, it is worth mentioning that Riemmanian games contain all general games in Euclidean spaces and both minimization and maximization optimization problems in geodesic spaces. A typical class of examples consists in min-max optimization in geodesic spaces which abstracts many machine learning problems, e.g,. principal component analysis [13], dictionary learning [110,111], deep neural networks (DNNs) [50] and low-rank matrix learning [52,116]. In particular, the problem of principal component analysis can be formulated as optimization on a Grassmann manifold.
In this section, we show that the Riemannian Gradient Descent, formally defined in Algorithm 1, attains a curvature-independent linear convergence rate for strongly monotone and smooth Riemannian games. This holds true in two cases: (i) when we have exact access to the gradients of the loss functions, and (ii) when our access to the gradients is only stochastic but we can use mini-batches to reduce the variance. 2 The assumptions we place on the Riemannian game under consideration in this section are summarized in Assumption 1. Note that we do not need to assume that a Nash equilibrium exists, because the existence is already implied by Assumption 1, as shown in Lemma 6. We postpone the proofs in this section to Appendix E. Assumption 1. Throughout this section we make the following assumptions:
• The Riemannian game is µ-strongly monotone and L-smooth, i.e., each player i's loss function l i is L-smooth for any pair of action profiles y, y ′ in M, there exists a unique minimizing geodesic within M that connects them. We denote the condition number of the game by κ = µ L . • The Riemannian manifold and the metric (d M (•, •) : M → R, M) form a complete metric space.
• In the stochastic setting, we have an upper bound B ≥ d M (y 0 , y * ) for the initial strategy y 0 .
Require:
1: • Riemannian game (N , M, L).
• Access to an unbiased estimator of the gradient F : M → T M with variance σ 2 .
• Initial strategy profile = y 0 ∈ M, maximum number of iterations K, step size schedule
Query an unbiased estimator of the gradient at y k for m k times, and let g (k) ∈ T y k M be the average of the output of the m k queries.
Perform Riemannian gradient descent with step size η k , y k+1 ← EXP y k -η k • g (k) ∈ M. 5: end for 6: return the final strategy profile y K .
The core of our proof rests on a descent inequality, detailed in Lemma 5. The descent inequality shows that the norm of the gradient contracts in both the deterministic case and the stochastic setting 2 Consider m independent and identically distributed (i.i.d.) samples drawn from an unbiased estimator of F , with a variance denoted by σ 2 . The empirical average of these samples has variance σ 2 m .
with a carefully tuned batch-size schedule. For clarity of exposition, we only include the proof for the deterministic case here and defer the proof for the stochastic case to the Appendix.
The transition to a stochastic setting presents an additional layer of complexity: bounding the term
. We employ Young's inequality to completely avoid the distance distortion. Contrasted with conventional approaches for Euclidean space, our methodology exacerbates the dependence on the condition number, introducing an additional factor of 1 κ 2 as seen in Theorem 1, but allows our convergence rate to be curvature-independent. 3 A slightly sharper bound can be derived for the deterministic case, as evidenced in the proof of Lemma 5. However, for the sake of clarity and coherence, we opted to present a unified bound applicable to both deterministic and stochastic settings.
L 2 , conditioning on y k , the iterates of Algorithm 1 satisfy the following descent inequality for every k < K:
Proof. As the game is µ-strongly monotone, we have
By the smoothness of the loss functions in L we have
Combining the two inequalities we have
Next, we show that we can prove the existence of a Nash equilibrium by combining Lemma 5 and Lemma 3. In other words, the existence of a Nash equilibrium is guaranteed by Assumption 1.
Lemma 6. A Riemannian game that satisfies Assumption 1 has a Nash equilibrium.
3 A way to bound this term is to introduce an additional point
) y k = 0, we can use Cauchy-Schwartz and smoothness of the loss functions to bound g
However this approach depends on the term dM(y k+1 , y k+1 ) which suffers from distance distortion.
We summarize the convergence rates in Theorem 1.
Theorem 1. Under Assumption 1, the RGD (Algorithm 1) with constant step size
has the following convergence rate with respect to any target ǫ ≥ 0.
• In the deterministic case (i.e., σ 2 = 0) for
• In the stochastic setting, for
Moreover, the total number of times a stochastic gradient is queried is at most 109σ 2 κ 6 ǫ 2 .
Comparison with the Riemannian Corrected Extragradient method [130]. Geodesically strongly convex-concave min-max optimization is a special instance of strongly monotone Riemannian games (see Lemma 2). In this setting, [57] studied the Riemannian Corrected Extragradient method (RCEG) [130] and used the distance to equilibrium, d M (y K * , y * ), as the proximity measure. By the smoothness of the loss functions, F (y K * ) y K * ≤ Ld M (y K * , y * ), and thus we can directly compare our bounds using the norm of the gradient. While our analysis and choice of step size is independent of the curvature of the manifold, for their analysis [57], they require a step size that depends on the curvature of the manifold.
To output a strategy with gradient norm at most ǫ in the deterministic case, both algorithms require the number of iterations that scales logarithmically in d M (y 0 , y * ), L, ǫ. The main difference is that for the analysis of RCEG, the number of iterations scales with the product of 1 κ and a metric that quantifies the distance distortion caused by the manifold, while our bound depends on 1 κ 2 but is independent of the curvature of the manifold. We observe a similar tradeoff in the stochastic case.
We have introduced the Riemannian game framework, a generalization of min-max optimization over Riemannian manifolds. We show that the Riemannian Gradient Descent, arguably the simplest first-order method in manifold optimization, can achieve curvature-independent and linear convergence rate with a step size that is agnostic to the manifold's curvature. We have also extended the result to the stochastic setting.
Our work raises several open questions. The first is whether we can obtain best-of-both-worlds convergence rate in geodesically strongly monotone Riemannian games, achieving both curvature independence and linear dependence on the condition number. Another intriguing challenge is to obtain curvature-independent and last-iterate convergence rate for monotone Riemannian games. Finally, can we design first-order methods that converge in structural but geodesically nonmonotone Riemannian games?
As a theoretical contribution, our results are valid provided that the assumptions we have made, summarized in Assumption 1, are met. Please note that this paper offers no guarantees if any of these conditions are violated.
We cannot survey all the increasing stream of works on the geometric properties of Riemannian manifolds but refer the reader to the books [16,70] and the references therein.
The minimization on Riemannian manifolds. Riemannian optimization [1,12] is an extensive and active area of research, for which one aspires to develop Riemannian gradient-based algorithms that share analogous properties to the more broadly studied Euclidean methods, including deterministic [8,14,26,37,62,129], projection-free [122,123], stochastic and adaptive [6,11,59], variance-reduced [105,115,128], and saddle-point-escaping [25,112], among others. In this context, the curvature-independent and last-iterate convergence rate can be achieved by Riemannian gradient descent method [129].
The last-iterate convergence for games in Euclidean spaces. In the unconstrained setting, if we further assume that either the game is strongly monotone or the payoff matrix in a bilinear game has all singular values bounded away from zero, the linear convergence rate is known for gradient-based algorithms [28,40,77,85,127]. The results for the constrained setting are relatively scare. Indeed, most of convergence results for monotone games are asymptotic [29,73] and we are aware of [124] which provided a linear convergence rate of optimistic gradient method for bilinear games when the constraint set is a polytope. However, the bound depends on a problem-dependent parameter that can be arbitrarily close to zero, making their results invalid for all smooth and monotone games. Recently, [20] has shown the first last-iterate convergence rates for all smooth and monotone games, matching the lower bounds [43,44].
From a technical viewpoint, the choice of performance measures is crucial to proving last-iterate convergence. It is known that classical gradient-based algorithms have a time-average convergence rate of O(1/t) in terms of the gap function for smooth monotone games [66,86,91]. Other than the gap function, one can measure the convergence using the norm of the operator if the setting is unconstrained [61,72,126], or the natural residual or similar notions if the setting is constrained and further satisfies the additional cocoercive condition [31]. More recently, a line of work [19,21,22] show how to obtain the optimal O(1/t) last-iterate convergence rate in constrained settings without relying on the cocoercive assumption, and these results can be further extended to a structural nonmonotone setting known as the comonotone setting.
Proof of Lemma 1: The proof follows from the descent property of Riemannian gradient descent (A4.2 in [12]). Assume, for sake of contradiction, that F (y * ) y * = 0. Then there exists player i such that grad yi l i (y * ) y * i = 0, and consider the following deviation for player i parameterized by η > 0,
Assume that the loss function l i is L-smooth, then for sufficiently small η we have the sufficient decrease property (Proposition 4.8 in [12]),
Proof of Lemma 6: Consider an exectuion of Algorithm 1 with exact access to the gradient (e.g., σ 2 = 0 and g (k) = F (y k ) ), and constant step-size η k = µ L 2 initialized at an arbitrary strategy profile y 0 ∈ M. Lemma 5 implies that for k ≥ 1,
Observe that d M (y k+1 , y k ) = η F (y k ) y k , which further implies that Proof of Theorem 1: The following is true for both the deterministic and stochastic setting. By Corollary 1 F (y * ) = 0, and by smoothness of the loss functions in L we have that,
First, we prove the deterministic case. By the descent inequality in Lemma 5 and the chosen stepsize η k = µ L 2 , we know that for k ≥ 1,
The deterministic case then follows from Lemma 3, Lemma 4 and our choice of K * =
The proof in the stochastic setting follows by Lemma 3, Lemma 4, Jensen's inequality, and noting that for
Moreover, the total number of stochastic queries is at most
where in the last inequality we used that e x -1 ≥ x, and that κ ≤ 1 ⇒ e κ 2 2 ≤ 1.7.
37th Conference on Neural Information Processing Systems (NeurIPS 2023).
Note that the problem is intractable without any structural assumption on the game[30].