Title: Learning Regularized Monotone Graphon Mean-Field Games
Venue: NeurIPS 2023
Accepted: yes
Keywords: mean-field approximation

graphon games

multi-agent reinforcement learning
Abstract: This paper studies two fundamental problems in regularized Graphon Mean-Field Games (GMFGs). First, we establish the existence of a Nash Equilibrium (NE)  of any $\lambda$-regularized GMFG   (for  $\lambda\geq 0$). This result relies on weaker conditions than previous works analyzing both unregularized GMFGs ($\lambda=0$) and $\lambda$-regularized MFGs, which are special cases of GMFGs. Second, we propose provably efficient algorithms to learn the NE in  weakly monotone GMFGs, motivated by Lasry and Lions (2007). Previous literature either only analyzed continuous-time algorithms or required extra conditions to analyze discrete-time algorithms. In contrast, we design a discrete-time algorithm and derive its convergence rate  solely under  weakly monotone conditions. Furthermore, we develop and analyze the action-value function estimation procedure during the online learning process, which is absent from algorithms for monotone GMFGs. This serves as a sub-module in our optimization algorithm. The efficiency of the designed algorithm is corroborated by empirical evaluations.

Learning Regularized Monotone Graphon Mean-Field Games

Abstract

This paper studies two fundamental problems in regularized Graphon Mean-Field Games (GMFGs). First, we establish the existence of a Nash Equilibrium (NE) of any λ -regularized GMFG (for λ ≥ 0 ). This result relies on weaker conditions than those in previous works for analyzing both unregularized GMFGs ( λ = 0 ) and λ -regularized MFGs, which are special cases of GMFGs. Second, we propose provably efficient algorithms to learn the NE in weakly monotone GMFGs, motivated by Lasry and Lions [2007]. Previous literature either only analyzed continuous-time algorithms or required extra conditions to analyze discrete-time algorithms. In contrast, we design a discrete-time algorithm and derive its convergence rate solely under weakly monotone conditions. Furthermore, we develop and analyze the action-value function estimation procedure during the online learning process, which is absent from algorithms for monotone GMFGs. This serves as a sub-module in our optimization algorithm. The efficiency of the designed algorithm is corroborated by empirical evaluations.

1 Introduction

In Multi-Agent Reinforcement Learning (MARL), the sizes of state and action spaces grow exponentially in the number of agents, which is known as the 'curse of many agents' [Sonu et al., 2017, Wang et al., 2020] and restrict its applicability to large-scale scenarios. The Mean-Field Game (MFG) has thus been proposed to mitigate this problem by exploiting the homogeneity assumption [Huang et al., 2006, Lasry and Lions, 2007], and it has achieved tremendous successes in many real-world applications [Cousin et al., 2011, Achdou et al., 2020]. However, the homogeneity assumption is an impediment when modeling scenarios in which the agents are heterogeneous. GMFGs, as extensions of MFGs, have thus been proposed to model the behaviors of heterogeneous agents and ameliorate the 'curse of many agents' at the same time [Parise and Ozdaglar, 2019, Carmona et al., 2022].

Despite the empirical successes of the Graphon Mean-Field Game (GMFG) [Aurell et al., 2022a], its theoretical understanding is lacking. First, sufficient conditions for Nash Equilibrium (NE) existence in regularized GMFG have not been established. Most works only address the existence of the NE in unregularized GMFGs. However, regularization is employed in practical implementations for improved exploration and robustness [Geist et al., 2019]. Moreover, previous works prove the existence of NE in regularized MFGs, a special case of GMFGs, only under the contraction condition , which is overly restrictive for real-world applications. Second, the analysis of discretetime algorithms for monotone GMFGs is lacking. Most existing works design provably efficient discrete-time algorithms only under contraction conditions, as shown in Table 1. Complementarily, previous works on monotone GMFGs either only derive the convergence rate for continuous-time algorithms, which ignores the discretization error, or require extra conditions, (e.g., potential games) to analyze discrete-time algorithms.

In this paper, we first consider GMFGs in full generality, i.e., without any contraction or monotone conditions. The goal is to establish the existence of NE in the regularized GMFG in this general setting. Then we focus on monotone GMFGs motivated by Lasry and Lions [2007]. We aim to learn the unique NE from the online interactions of all agents with and without the action-value function oracle. When the oracle is absent, the action-value functions should be estimated from the data of sampled agents generated in the online game.

In the analysis, we have to overcome difficulties that arise from both the existence of the NE problem and the learning of the NE. First, the proof of the existence of NE in regularized GMFG involves establishing some topological spaces and operators related to NE on which fixed point theorems are applicable. However, the direct construction of the space and the operators for GMFGs with uncountably infinite agents is challenging. Second, the design and analysis of the discrete-time NE learning algorithm require subtle exploitation of the monotone condition. Unlike continuous-time algorithms with infinitesimal step sizes, the design of appropriate step sizes is additionally required for the discrete-time algorithm to guarantee that iterates evolve appropriately. This guarantee originates from the delicate utilization of the monotone condition in the optimization procedures.

To address the difficulty of the existence problem, we construct a regularized MFG from the regularized GMFG and show that the NE of the constructed game can be converted into the NE of the original game, thus mitigating the difficulty of having an uncountable number of agents. To handle the difficulty in the NE learning problem, we design the Monotone GMFG Policy Mirror Descent (MonoGMFG-PMD) algorithm, which iteratively implements policy mirror descent for each player. We show that this procedure results in a decrease of the KL divergence between the iterate and the NE, and this decrease is related to the gap appearing in the weakly monotone condition. When the action-value function oracle is absent, we also design and analyze action-value functions estimation algorithms to serve as a submodule of the optimization procedures.

Main Contributions: We first establish the existence of the NE in the λ -regularized GMFG with λ ≥ 0 assuming Lipschitzness of graphons and continuity of transition kernels and reward functions. Our result relaxes the assumption of the Lipschitzness of transition kernels and rewards required in previous works on unregularized GMFGs and the contraction condition in the literature on regularized MFG [Cui and Koeppl, 2021a]. Then we design and analyze the MonoGMFG-PMD algorithm. Using an action-value function oracle, the convergence rate for MonoGMFG-PMD is proved to be ˜ O ( T -1 / 2 ) after T iterations. Without the oracle, the convergence rate includes an additional ˜ O ( K -1 / 2 + N -1 ) term that arises from sampling N agents and collecting data from K episodes, reflecting the generalization error and the approximation error of the estimation algorithm. As shown in Table 1 , our algorithm can be implemented from the online interaction of agents and does not require the distribution flow manipulation. Detailed explanations of the properties stated in the columns of Table 1 are provided in Appendix A. Our result for MonoGMFG-PMD provides the first convergence rate for discrete-time algorithms in monotone GMFGs .

2 Related Works

MFGs were proposed by Huang et al. [2006] and Lasry and Lions [2007] to model the interactions among a set of homogeneous agents. In recent years, learning the NE of the MFGs formulated by

discrete-time Markov Decision Process (MDP)s has attracted a lot of interest. There is a large body of works that design and analyze algorithms for the MFGs under contraction conditions [Anahtarci et al., 2019, 2022, Cui and Koeppl, 2021a, Xie et al., 2021a, Zaman et al., 2022, Yardim et al., 2022]. Typically, these works design reinforcement learning algorithms to approximate the contraction operators in MFGs, and the NE is learned by iteratively applying this operator. In contrast, another line of works focuses on the MFGs under monotone conditions. Motivated by Lasry and Lions [2007], the transition kernels in these works are independent of the mean fields. Perrin et al. [2020] proposed and analyzed the continuous-time fictitious play algorithm, which dynamically weighs the past mean fields and the best response policies to derive new mean fields and policies. With the additional potential assumption, Geist et al. [2021] derived the convergence rate for the discrete-time fictitious play algorithm. Perolat et al. [2021] then proposed the continuous-time policy mirror descent algorithm but only proved the asymptotic convergence, i.e., the consistency. In summary, there is no convergence rate result for any discrete-time algorithm for MFGs under the monotone condition. In addition, the relationship between the contraction conditions and the monotone conditions is not clear from existing works, but they complement each other.

To capture the potential heterogeneity among agents, GMFGs have been proposed by Parise and Ozdaglar [2019] in static settings as an extension of MFGs. The heterogeneous interactions among agents are represented by graphons. Aurell et al. [2022b], Caines and Huang [2019] then extended the GMFGs to the continuous-time setting, where the existence and the uniqueness of NE were established. Vasal et al. [2020] formulated discrete-time GMFGs and provided way to compute the NE with master equations. With the precise underlying graphons values, Cui and Koeppl [2021b] proposed algorithms to learn the NE of GMFGs with the contraction condition by modifying MFGs learning algorithms. Fabian et al. [2022] considered the monotone GMFG and proposed the continuous-time policy mirror descent algorithm to learn the NE. However, only consistency was provided in the latter two works.

Notation Let [ N ] := { 1 , · · · , N } . Given a measurable space (Ω , F ) , we denote the collection of all the measures and the probability measures on (Ω , F ) as M (Ω) and ∆(Ω) , respectively. For a metric space ( X , ∥ · ∥ ) , we use C ( X ) and C b ( X ) to denote the set of all continuous functions and the set of all bounded continuous functions on X , respectively. For a measurable space ( X , F ) and two distributions P, Q ∈ ∆( X ) , the total variation distance between them is defined as TV( P, Q ) := sup A ∈F | P ( A ) -Q ( A ) | . Asequence of measures { µ n } on X is said to converge weakly to a measure µ if ∫ X g ( x ) µ n (d x ) → ∫ X g ( x ) µ (d x ) for all g ∈ C b ( X ) .

3 Preliminaries

3.1 Graphon Mean-Field Games

We consider a GMFG that is defined through a tuple ( I , S , A , H, P, r, W, µ 1 ) . The horizon (or length) of the game is denoted as H . In GMFG, we have infinite players, each corresponding to a point α ∈ I = [0 , 1] . The state and action space of them are the same, denoted as S ⊆ R d s and A ⊆ R d a respectively. The interaction among players is captured by graphons . Graphons are symmetric functions that map [0 , 1] 2 to [0 , 1] . Symmetry here refers to that W ( α, β ) = W ( β, α ) for all α, β ∈ [0 , 1] . We denote the set of graphons as W = { W : [0 , 1] 2 → [0 , 1] | W is symmetric. } . The set of graphons of the game is W = { W h } H h =1 with W h ∈ W . The state transition and reward of each player are influenced by the collective behavior of all the other players through an aggregate z ∈ M ( S ) . At time h ∈ [ H ] , we denote the state distribution of player β ∈ I as µ β h ∈ ∆( S ) . The aggregate for player α ∈ I with the underlying graphon W h ∈ W is then defined as

The transition kernels P = { P h } H h =1 of the game are functions P h : S × A × M ( S ) →S for all h ∈ [ H ] . At time h , if player α takes action a α h ∈ A at state s α h ∈ S , her state will transition according to s α h +1 ∼ P h ( · | s α h , a α h , z α h ) . The reward functions are denoted as r h : S × A × M ( S ) → [0 , 1] for all h ∈ [ H ] . We note that the players in GMFG are heterogeneous. This means that different players will, in general, receive different aggregates from other players. The distribution µ 1 ∈ ∆( S ) is the initial state distribution for all the players. A policy for an player α is π α = { π α h } H h =1 ∈ Π H , where π α h : S → ∆( A ) takes action based only on the current state, and Π is the set of all these policies.

A policy for all the players π I ∈ ˜ Π = Π H ×I is the collection of the policies of each player, i.e, π I = { π α } α ∈I . In the following, we denote the state distributions of all the players at time h and the state distributions of all the players at any time (distribution flow) respectively as µ I h = { µ α h } α ∈I and µ I = { µ I h } H h =1 ∈ ˜ ∆ = ∆( S ) H ×I . Eqn. (1) shows that the aggregate z α h is a function of µ I h and W h , so to make this dependence explicit, we also write it as z α h ( µ I h , W h ) .

We consider the entropy-regularized GMFG. It has been shown that the regularization results in policy gradient algorithms converging faster [Shani et al., 2020, Cen et al., 2022]. In this game, the rewards of each player are the sum of the original rewards and the negative entropy of the policy multiplied by a constant. In a λ -regularized GMFG ( λ ≥ 0 ), the value function and the action-value function of player α with policy π α on the MDP induced by the distribution flow µ I are defined as

for all h ∈ [ H ] , where the expectation E π α is taken with respect to the stochastic process induced by implementing policy π α on the MDP induced by µ I . The cumulative reward function of player α is defined as J λ,α ( π α , µ I ) = E µ 1 [ V λ,α 1 ( s, π α , µ I )] . Then the notion of an NE is defined as follows.

Definition 1. An NE of the λ -regularized GMFG is a pair ( π ∗ , I , µ ∗ , I ) ∈ ˜ Π × ˜ ∆ that satisfies: (i) (player rationality) J λ,α ( π ∗ ,α , µ ∗ , I ) = max ˜ π α ∈ Π H J λ,α (˜ π α , µ ∗ , I ) for all α ∈ I up to a zeromeasure set on I with respect to the Lebesgue measure. (ii) (Distribution consistency) The distribution flow µ ∗ , I is equal to the distribution flow induced by implementing the policy π ∗ , I .

Similar to the NE for the finite-player games, the NE of the λ -regularized GMFG requires that the policy of each player is optimal. However, in GMFGs, the optimality is with respect to µ ∗ , I .

3.2 Mean-Field Games

As an important subclass of GMFG, MFG corresponds to the GMFG with constant graphons, i.e, W ( α, β ) = p for all α, β ∈ I . MFGs involve infinite homogeneous players. All the players employ the same policy and thus share the same distribution flow. The aggregate in Eqn. 1 degenerates to z α h = ∫ 1 0 p · µ β h d β = p · µ h for all α ∈ I . Here µ h is the state distribution of a representative player. Thus, an MFG is denoted by a tuple ( ¯ S , ¯ A , ¯ H, ¯ P, ¯ r, µ 1 ) . The state space, the action space, and the horizon are respectively denoted as ¯ S , ¯ A , and ¯ H . Here, the transition kernels ¯ P = { ¯ P h } H h =1 are functions ¯ P h : S × A × ∆( S ) → S , and reward functions ¯ r h : S × A × ∆( S ) → [0 , 1] for all h ∈ [ H ] . In the MFG, all the players adopt the same policy π = { π h } H h =1 where π h ∈ Π . The value function and the action-value function in the λ -regularized MFG with the underlying distribution flow µ = { µ h } H h =1 ∈ ∆( S ) H can be similarly defined as Eqn. (2) and (3) respectively as follows

for all h ∈ [ H ] . The cumulative reward is ¯ J λ ( π, µ ) = E µ 1 [ ¯ V λ 1 ( s, π, µ )] . The notion of NE can be similarly derived as follows.

Definition 2. An NE of the λ -regularized MFG is a pair ( π ∗ , µ ∗ ) ∈ Π H × ∆( S ) H that satisfies: (i) (player rationality) J λ ( π ∗ , µ ∗ ) = max ˜ π ∈ Π H J λ (˜ π, µ ∗ ) . (ii) (Distribution consistency) The distribution flow µ ∗ is equal to the distribution flow induced by the policy π ∗ .

Remark 1. Compared with Definition 1, the definition of NE in MFG only involves the policy and the distribution flow of a single representative player, since the agents are homogeneous in MFGs.

4 Existence of the NEs in Regularized GMFGs and MFGs

We now state some assumptions to demonstrate the existence of a NE for λ -regularized GMFGs.

Assumption 1. The state space S is compact, and the action space A is finite.

This assumption imposes rather mild constraints on S and A . In real-world applications, the states are usually physical quantities and thus reside in a compact set. For the action space, many deep reinforcement learning algorithms discretize the potential continuous action sets into finite sets [Lowe et al., 2017, Mordatch and Abbeel, 2018].

Assumption 2. The graphons W h for h ∈ [ H ] are continuous functions.

The stronger version of this assumption (Lipschitz continuity) is widely adopted in GMFG works [Cui and Koeppl, 2021b, Fabian et al., 2022]. It helps us to build the continuity of the transition kernels and rewards with respect to players.

Assumption 3. For all h ∈ [ H ] , the reward function r h ( s, a, z ) is continuous on S × A × M ( S ) , that is if ( s n , a n , z n ) → ( s, a, z ) as n → ∞ , then r h ( s n , a n , z n ) → r h ( s, a, z ) . The transition kernel P h ( · | s, a, z ) is weakly continuous in S × A × M ( S ) , that is if ( s n , a n , z n ) → ( s, a, z ) as n →∞ , P h ( · | s n , a n , z n ) → P h ( · | s, a, z ) weakly.

This assumption states the continuity of the models, i.e., the transition kernels and the rewards, as functions of the state, action, and aggregate. We note that the Lipschitz continuity assumption of the model in the previous works implies that our assumption is satisfied [Cui and Koeppl, 2021b, Fabian et al., 2022]. Next, we state the existence result of regularized GMFG.

Theorem 1. Under Assumptions 1, 2 and 3, for all λ ≥ 0 , the λ -regularized GMFG ( I , S , A , H, P, r, W, µ 1 ) admits an NE ( π I , µ I ) ∈ ˜ Π × ˜ ∆ .

This theorem strengthens previous existence results in Cui and Koeppl [2021b] and Fabian et al. [2022] in two aspects. First, our assumptions are weaker. These two existing works require a finite state space and the Lipschitz continuity of the models. In contrast, we only need a compact state space and the model continuity. Second, their results only hold for the unregularized case ( λ = 0 ), whereas ours holds for any λ ≥ 0 . In the proof of Theorem 1, we construct a MFG from the given GMFG and show that an NE of the constructed MFG can be converted to an NE of the GMFG. Then we prove the existence of NE in the constructed regularized MFG. Our existence result for the regularized MFG is also a significant addition to the MFG literature.

Remark 2. Although we show that an NE of the constructed MFG can be converted to an NE of GMFG, this proof does not imply that GMFG forms a subclass of or is equivalent to MFG. This is because we have only demonstrated the relationship between the NEs of these two games, but the exact realizations of the GMFG and the conceptually constructed MFG may differ. It means that the sample paths of these two games may not be the same, which include the realizations of the states, actions, and rewards of all the players.

We next state the assumption needed for the MFG.

Assumption 4. The MFG ( ¯ S , ¯ A , ¯ H, ¯ P, ¯ r, µ 1 ) satisfies that: (i) The state space ¯ S is compact, and the action space ¯ A is finite. (ii) The reward functions ¯ r h ( s, a, µ ) for h ∈ [ H ] are continuous on S × A × ∆( S ) . The transition kernels are weakly continuous on S × A × ∆( S ) ; that is if ( s n , a n , µ n ) → ( s, a, µ ) as n →∞ , ¯ P h ( · | s n , a n , µ n ) → ¯ P h ( · | s, a, µ ) weakly.

Then the existence of the NE is stated as follows.

Theorem 2. Under Assumption 4, the λ -regularzied MFG ( ¯ S , ¯ A , ¯ H, ¯ P, ¯ r, µ 1 ) admits an NE ( π, µ ) ∈ Π H × ∆( S ) H for any λ ≥ 0 .

Our result in Theorem 2 imposes weaker conditions than previous works [Cui and Koeppl, 2021a, Anahtarci et al., 2022] to guarantee the existence of an NE. These existing works prove the existence of NE by assuming a contractive property and the finiteness of the state space. They also require a strong Lispchitz assumption [Anahtarci et al., 2022], where the Lipschitz constants of the models should be small enough. In contrast, we only require the continuity assumption in Theorem 2. This is due to our analysis of the operator for the regularized MFG and the use of Kakutani fixed point theorem Guide [2006].

5 Learning NE of Monotone GMFGs

In this section, we focus on GMFGs with transition kernels that are independent of the aggregate z , i.e., P h : S × A → S for h ∈ [ H ] . This model is motivated by the seminal work Lasry and Lions [2007], where the state evolution in continuous-time MFG is characterized by the Fokker-Plank equation. However, the form of the Fokker-Plank equation results in the state transition of each player being independent of other players. This model is also widely accepted in the discrete-time GMFG and MFG literature [Fabian et al., 2022, Perrin et al., 2020, Perolat et al., 2021].

5.1 Monotone GMFG

We first generalize the notion of monotonicity from multi-population MFGs in Perolat et al. [2021] to GMFGs.

Definition 3 (Weakly Monotone Condition) . A GMFG is said to be weakly monotone if for any ρ I , ˜ ρ I ∈ ∆( S × A ) I and their marginalizations on the states µ I , ˜ µ I ∈ ∆( S ) I , we have

̸

for all h ∈ [ H ] , where W h is the underlying graphon. It is strictly weakly monotone if the inequality is strict when ρ I = ˜ ρ I .

In two MDPs induced by the distribution flows µ I of π I and ˜ µ I of ˜ π I , the weakly monotone condition states that we can achieve higher rewards at stage h ∈ [ H ] by swapping the policies. This condition has two important implications. The first is the uniqueness of the NE.

Proposition 1. Under Assumptions 1, 2, and 3, a strictly weakly monotone λ -regularized GMFG has a unique NE for any λ ≥ 0 up to a zero-measure set on I with respect to the Lebesgue measure.

In the following, we denote this unique NE as ( π ∗ , I , µ ∗ , I ) , and we aim to learn this NE. The second implication concerns the relationship between the cumulative rewards of two policies.

Proposition 2. If a λ -regularized GMFG satisfies the weakly monotone condition, then for any two policies π I , ˜ π I ∈ ˜ Π and their induced distribution flows µ I , ˜ µ I ∈ ˜ ∆ , we have

̸

If the λ -regularized GMFG satisfies the strictly weakly monotone condition, then the inequality is strict when π I = ˜ π I .

Proposition 2 shows that if we have two policies, we can improve the cumulative rewards on the MDP induced by these policies by swapping the policies. This implies an important property of the NE ( π ∗ , I , µ ∗ , I ) . Since π ∗ , I is optimal on the MDP induced by µ ∗ , I , we have ∫ 1 0 J λ,α ( π ∗ ,α , µ ∗ , I ) d α ≥ ∫ 1 0 J λ,α ( π α , µ ∗ , I ) d α for any π I ∈ ˜ Π . Then Proposition 2 shows that ∫ 1 0 J λ,α ( π ∗ ,α , µ I ) d α ≥ ∫ 1 0 J λ,α ( π α , µ I ) d α for any policy π I and the distribution flow µ I it induces. This means that the NE policy gains cumulative rewards not less than any policy π I on the MDP induced by π I . This motivates the design of our NE learning algorithm.

5.2 Policy Mirror Descent Algorithm for Monotone GMFG

In this section, we introduce the algorithm to learn the NE, which is called MonoGMFG-PMD and whose pseudo-code is outlined in Algorithm 1. It consists of three main steps. The first step estimates the action-value function (Line 3). We need to evaluate the action-value function of a policy on the MDP induced by itself. This estimate can be obtained for each player independently by playing the π I t several times. We assume access to a sub-module for this and quantify the estimation error in our analysis. The second step is the policy mirror descent (Line 4). Given λη t < 1 , This step can be equivalently formulated as

Algorithm 1 MonoGMFG-PMD

Procedure:

1: Initialize π α 1 ,h ( · | s ) = Unif ( A ) for all s ∈ S , h ∈ [ H ] and α ∈ I .

2: for t = 1 , 2 , · · · , T do

3: Compute the action-value function ˆ Q λ,α h ( s, a, π α t , µ I t ) for all α ∈ I and h ∈ [ H ] , where µ I t is the distribution flow induced by π I t .

5: π α t +1 ,h ( · | s ) = (1 -β t )ˆ π α t +1 ,h ( · | s ) + β t Unif ( A )

4: ˆ π α t +1 ,h ( · | s ) ∝ ( π α t,h ( · | s ) ) 1 -λη t exp ( η t ˆ Q λ,α h ( s, a, π α t , µ I t ) ) for all α ∈ I and h ∈ [ H ]

6: end for

7: Output ¯ π I = Unif ( π I [1: T ] )

where R ( p ) = ⟨ p, log p ⟩ is the negative entropy function. This step aims to improve the performance of the policy π I t on its own induced MDP. Intuitively, since the policy π ∗ , I in NE performs better than π I t on the MDP induced by µ I t as shown in Section 5.1, the improved policy π I t +1 should be closer to π ∗ , I than π I t . The third step mixes the current policy with the uniform policy (Line 5) to encourage exploration.

MonoGMFG-PMDis different from previous NE learning algorithms for monotone GMFG in Perolat et al. [2021], Fabian et al. [2022] in three different ways. First, MonoGMFG-PMD is designed to learn the NE of the λ -regularized GMFG with λ > 0 , whereas other algorithms learn the NE of the unregularized GMFGs. As a result, our policy improvement (Line 4) discounts the previous policy as ( π α t,h ) 1 -λη t , but other algorithms retain π α t,h . Second, our algorithm is discrete-time and thus amenable for practical implementation. However, other provably efficient algorithms evolve in continuous time. Finally, MonoGMFG-PMD encourages exploration in Line 5, which is important for the theoretical analysis. Such a step is missing in other algorithms.

5.3 Theoretical Analysis for MonoGMFG-PMD with Estimation Oracle

This section provides theoretical analysis for the MonoGMFG-PMD algorithm given an action-value function oracle in Line 3, i.e., ˆ Q λ,α h = Q λ,α h . We denote the unique NE of the λ -regularized GMFG as ( π ∗ , I , µ ∗ , I ) . For any policy π I , we measure the distance to the policy π ∗ , I of NE using

This metric measures the weighted KL divergence between policy π I and the NE policy, and the weights are the NE distribution flow µ ∗ , I .

Theorem 3. Assume that the GMFG is strictly weakly monotone and we have an action-value function oracle. Let η t = η = O ( T -1 / 2 ) and β t = β = O ( T -1 ) in MonoGMFG-PMD. For any λ > 0 we have

Theorem 3 provides the first convergence rate result for a discrete-time algorithm on strictly weakly monotone GMFGs under mild assumptions. In contrast, Perolat et al. [2021], Fabian et al. [2022] only consider the continuous-time policy mirror descent, which is difficult for the practical implementation, and only provide the asymptotic consistency results. Geist et al. [2021] derive exploitability results for a fictitious play algorithm but require the potential structure and the Lipschitzness of the reward function. Our proof for Theorem 3 mainly exploits properties of NE discussed in Section 5.1. Concretely, we use the fact that the policy mirror descent procedure reduces the distance between the policy iterate and the NE as D ( π I t +1 ) -D ( π I t ) ≈ ∫ 1 0 J λ,α ( π α t , µ I t ) -J λ,α ( π ∗ ,α , µ I t ) d α . Thus, the policy iterate becomes closer to the NE. However, the discretization error and the exploration influence (Line 5) also appear, requiring additional care to show that D ( π I t ) , in general, decreases.

Algorithm 2 Estimation of Action-value Function

Procedure:

1: Sample N players { i/N } N i =1 ⊆ [0 , 1] .

3: Collect data { ( s i τ,h , a i τ,h , r i τ,h ) } N,K,H i,τ,h =1 of sampled players from K independent episodes.

2: The i th player implements π b ,i t for i ∈ [ N ] , and the other players implement π I t .

4: Initialize ˆ V λ,i H +1 ( s, a ) = 0 for all s ∈ S , a ∈ A and i ∈ [ N ] .

5: for time h = H, · · · , 1 do

6: for Player i = 1 , · · · , N (in parallel) do

7: ˆ Q λ,i h = argmin f ∈F h ∑ K τ =1 ( f ( s i τ,h , a i τ,h ) -r i τ,h -ˆ V λ,i h +1 ( s i τ,h +1 ) ) 2 .

8: ˆ V λ,i h ( s ) = ⟨ ˆ Q λ,i h ( s, · ) , π i/N t,h ( · , s ) ⟩ -λR ( π i/N t,h ( · , s ) ) .

9: end for

10: end for

11: Output { ˆ Q λ,i h } N,H i,h =1 .

5.4 Theoretical Analysis for MonoGMFG-PMD with General Function Classes

In this section, we remove the requirement that one is given oracle access to an action-value function and we estimate it in Line 3 of MonoGMFG-PMD using general function classes. We consider the action-value function class F = F 1 ×··· × F H , where F h ⊆ { f : S × A → [0 , ( H -h +1)(1 + λ log |A| )] } is the class of action-value functions at time h ∈ [ H ] . Then we estimate the action-value functions using Algorithm 2.

This algorithm mainly involves two steps. The first is involves data collection (Line 3). Here we assign policies to players and collect data from their interactions. We let the sampled N players implement behavior policies { π b ,i t } N i =1 , which can be different from { π i/N t } N i =1 . This will not change the aggregate z α h ( µ I h , W h ) for any α ∈ I , since only a zero-measure set of players change their policies. The second is the action-value function estimation (Lines 7 and 8). The action-value function is selected based on the previous value function, and the value function is updated from the derived estimate. This can be implemented in parallel for all players. We highlight that the estimation analysis cannot leverage results from general non-parametric regression [Wainwright, 2019], since the response variable ˆ V λ,i h +1 is not independent of s i τ,h +1 in our setting.

Assumption 5 (Realizability) . For any policy π I ∈ ˜ Π and the induced distribution flow µ I ∈ ˜ ∆ , we have Q λ,α h ( · , · , π α , µ I ) ∈ F h for h ∈ [ H ] .

This assumption ensures that we can find the nominal action-value function in the function class. For a policy π ∈ Π H and a function f : S × A → R , we define the operator ( T π h f )( s, a ) = E s ′ ∼ P h ( ·| s,a ) [ ⟨ f ( s ′ , · ) , π h +1 ( ·| s ′ ) ⟩ -λR ( π h +1 ( ·| s ′ ))] . For a policy π I and the induced distribution flow µ I , we have Q λ,α h ( s, a, π α , µ I ) = r h ( s, a, z α h ) + ( T π α h Q λ,α h +1 )( s, a ) .

Assumption 6 (Completeness) . For any policy π I ∈ ˜ Π and the induced distribution flow µ I ∈ ˜ ∆ , we have that for all f ∈ F h +1 , r h ( · , · , z α h ( µ I h , W h )) + ( T π α h f ) ∈ F h for all α ∈ I , h ∈ [ H -1] .

This completeness assumption ensures that the estimates from F also satisfy the relationship between nominal action-value functions through T π α h . These realizability and completeness assumptions are widely adopted in the off-policy evaluation and offline reinforcement learning literature [Uehara et al., 2022, Xie et al., 2021b].

Assumption 7. The reward functions { r h } H h =1 are Lipschitz in z , i.e., | r h ( s, a, z ) -r h ( s, a, z ′ ) | ≤ L r ∥ z -z ′ ∥ 1 for all s ∈ S , a ∈ A , h ∈ [ H ] . The graphons W h for h ∈ [ H ] are Lipschitz continuous functions, i.e., there exists a constant L = L W > 0 (depending only on W = { W h } H h =1 ) such that | W h ( α, β ) -W h ( α ′ , β ′ ) | ≤ L W ( | α -α ′ | + | β -β ′ | ) for all α, α ′ , β, β ′ ∈ [0 , 1] , and h ∈ [ H ] .

The Lipschitzness assumption is common in the GMFG works [Parise and Ozdaglar, 2019, Carmona et al., 2022, Cui and Koeppl, 2021b]. It helps us to approximate the action-value function of a player by that of sampled players. We denote the state distributions of player i induced by policy π b ,i t as µ b ,i t . Then we require the behavior policies { π b ,i t } N i =1 to satisfy the following requirements.

BeachBarProblemwithSBM-graphons

100

Exploitability

10-1

10-2

N=10,K=300

p=1

N=10,K=100

p=0.5

N= 5,K = 300

p=0

Unreg PMD

1

5

10

15

20

25

OptimizationIteration

(a) Beach Bar problem with SBM graphons

BeachBarProblemwithexp-graphons

Exploitability

100

10-1

10-2

1

5

10

15

20

25

OptimizationIteration

Figure 1: Simulation results for Beach Bar problem with SBM and exp-graphons. Assumption 8. For any t ∈ [ T ] , the behavior policies explore sufficiently. More precisely, for any policy π ∈ Π H and induced distributions µ ∈ ∆( S ) H , we have sup s ∈S ,a ∈A π h ( a | s ) /π b ,i t,h ( a | s ) ≤ C 1 and sup s ∈S d µ h ( s ) / d µ b ,i t,h ( s ) ≤ C 2 for h ∈ [ H ] and i ∈ [ N ] where C 1 , C 2 > 0 are constants.

This assumption guarantees that the behavior policies explore the actions that may be adopted by π I t and π ∗ , I . Such an assumption is widely adopted in offline reinforcement learning and off-policy evaluation works [Uehara et al., 2022, Xie et al., 2021b].

Theorem 4. Assume that the GMFG is weakly monotone and that Assumptions 5, 6, 7, and 8 hold. Let η t = η = O ( T -1 / 2 ) and β t = β = O ( T -1 ) in Algorithm 1 (MonoGMFG-PMD). Then with probability at least 1 -δ , Algorithms 1 and 2 yield

where B H = H (1 + λ log |A| ) , and N ∞ (5 B H /K, F [ H ] ) = max h ∈ [ H ] N ∞ (5 B H /K, F h ) is the ℓ ∞ -covering number of the function class.

The error in Theorem 4 consists of both the optimization and estimation errors. The optimization error corresponds to the first term, which also appears in Theorem 3. The estimation error consists of the generalization error and the approximation error, in the second and third terms respectively. When the function class F is finite, this term scales as O ( K -1 / 2 ) , which originates from the fact that we estimate the action-value function from the empirical error instead of its population counterpart. The approximation error scales as O ( N -1 ) . This term originates from the fact that the action-value function of player α is approximated by that of the sampled player near α . To learn a policy that is at most ε > 0 far from the NE, we can set T = ˜ O ( ε -2 ) , K = ˜ O ( ε -2 ) , and N = ˜ O ( ε -1 ) , which in total results in TK = ˜ O ( ε -4 ) episodes of online plays.

6 Experiments

In this section, we conduct experiments to corroborate our theoretical findings. We run different algorithms on the Beach Bar problem Perrin et al. [2020], Fabian et al. [2022]. The underlying graphons are set to Stochastic Block Model (SBM) and exp-graphons. The details of experiments are deferred to Appendix B. Since the NEs of the games are not available, we adopt the exploitability to measure the proximity between a policy and the NE. For a policy π I and its induced distribution flow µ I , the exploitability for the λ -regularized GMFG is defined as

First, the experimental results demonstrate the necessity of modelling the heterogeneity of agents. Figure 1 demonstrates the performance degradation of approximating GMFG by MFG. Here we let the agents play in the GMFG with constant graphons W h ( α, β ) = p for p ∈ { 0 , 0 . 5 , 1 } . The agents have oracle access to the action-value function. We observe that this approximation results in gross errors for learning the NEs of GMFGs with non-constant graphons.

Second, the experiments show that the algorithms designed for unregularized GMFG cannot learn the NE of regularized GMFG. We implement the discrete-time version of the algorithm in Fabian et al.

[2022]; results marked 'Unreg PMD' show that the exploitability first decreases and then increases. In line with the discussion in Section 5.2, this originates from keeping too much gradient knowledge in previous iterates π I t . The gradient of the policy is largely correct in the several initial iterations, but a large amount of past knowledge results in it deviating in later iterations, since the past knowledge accumulates. In contrast, our algorithm discounts the past knowledge as ( π I t ) 1 -λη t .

Finally, the results indicate the influence of action-value function estimation. In the experiments, we run our algorithm when N = 5 , K = 300 , N = 10 , K = 100 , and N = 10 , K = 300 . Figure 1 shows that the algorithm with N = 10 , K = 300 can achieve a smaller error than the algorithms both with N = 10 , K = 100 and N = 5 , K = 300 . This is in agreement with Theorem 4.

7 Conclusion

In this paper, we focused on two fundamental problems of λ -regularized GMFG. Firstly, we established the existence of NE. This result greatly weakened the conditions in the previous works. Secondly, the provably efficient NE learning algorithms were proposed and analyzed in the weakly monotone GMFG motivated by Lasry and Lions [2007]. The convergence rate of MonoGMFG-PMD features the first performance guarantee of discrete-time algorithm without extra conditions in monotone GMFGs. We leave the lower bound of this problem to the future works.

Acknowledgments and Disclosure of Funding

Fengzhuo Zhang and Vincent Tan acknowledge funding by the Singapore Data Science Consortium (SDSC) Dissertation Research Fellowship, the Singapore Ministry of Education Academic Research Fund (AcRF) Tier 2 under grant number A-8000423-00-00, and AcRF Tier 1 under grant numbers A-8000980-00-00 and A-8000189-01-00. Zhaoran Wang acknowledges National Science Foundation (Awards 2048075, 2008827, 2015568, 1934931), Simons Institute (Theory of Reinforcement Learning), Amazon, J.P. Morgan, and Two Sigma for their support.

References

Achdou, P. Cardaliaguet, F. Delarue, A. Porretta, F. Santambrogio, Y. Achdou, and M. Laurière. Mean field games and applications: Numerical aspects. Mean Field Games: Cetraro, Italy 2019 , pages 249-307, 2020.

Anahtarci, C. D. Kariksiz, and N. Saldi. Fitted q-learning in mean-field games. arXiv preprint arXiv:1912.13309 , 2019.

Anahtarci, C. D. Kariksiz, and N. Saldi. Q-learning in regularized mean-field games. Dynamic Games and Applications , pages 1-29, 2022.

Aurell, R. Carmona, G. Dayanıklı, and M. Laurière. Finite state graphon games with applications to epidemics. Dynamic Games and Applications , 12(1):49-81, 2022a.

Aurell, R. Carmona, and M. Lauriere. Stochastic graphon games: Ii. the linear-quadratic case. Applied Mathematics & Optimization , 85(3):1-33, 2022b.

Bertsekas and S. E. Shreve. Stochastic optimal control: the discrete-time case , volume 5. Athena Scientific, 1996.

Cai, Z. Yang, C. Jin, and Z. Wang. Provably efficient exploration in policy optimization. In International Conference on Machine Learning , pages 1283-1294. PMLR, 2020.

E. Caines and M. Huang. Graphon mean field games and the gmfg equations: ε -nash equilibria. In 2019 IEEE 58th conference on decision and control (CDC) , pages 286-292. IEEE, 2019.

Carmona, D. B. Cooney, C. V. Graves, and M. Lauriere. Stochastic graphon games: I. the static case. Mathematics of Operations Research , 47(1):750-778, 2022.

Cen, C. Cheng, Y. Chen, Y. Wei, and Y. Chi. Fast global convergence of natural policy gradient methods with entropy regularization. Operations Research , 70(4):2563-2578, 2022.

Cousin, S. Crépey, O. Guéant, D. Hobson, M. Jeanblanc, J. Lasry, J. Laurent, P. Lions, P. Tankov, and O. Guéant. Mean field games and applications. Paris-Princeton lectures on mathematical finance 2010 , pages 205-266, 2011.

Cui and H. Koeppl. Approximately solving mean field games via entropy-regularized deep reinforcement learning. In International Conference on Artificial Intelligence and Statistics , pages 1909-1917. PMLR, 2021a.

Kai Cui and Heinz Koeppl. Learning graphon mean field games and approximate nash equilibria. In International Conference on Learning Representations , 2021b.

Fabian, . Cui, and H. Koeppl. Learning sparse graphon mean field games. arXiv preprint arXiv:2209.03880 , 2022.

Geist, B. Scherrer, and O. Pietquin. A theory of regularized markov decision processes. In International Conference on Machine Learning , pages 2160-2169. PMLR, 2019.

Geist, J. Pérolat, M. Laurière, R. Elie, S. Perrin, O. Bachem, R. Munos, and O. Pietquin. Concave utility reinforcement learning: the mean-field game viewpoint. arXiv preprint arXiv:2106.03787 , 2021.

H. Guide. Infinite dimensional analysis . Springer, 2006.

Györfi, M. Kohler, A. Krzyzak, and H. Walk. A distribution-free theory of nonparametric regression , volume 1. Springer, 2002.

Hinderer. Foundations of non-stationary dynamic programming with discrete time parameter, 1970.

Huang, R. P. Malhamé, and P. E. Caines. Large population stochastic dynamic games: closed-loop mckean-vlasov systems and the nash certainty equivalence principle. 2006.

H.-J. Langen. Convergence of dynamic programming models. Mathematics of Operations Research , 6(4):493-512, 1981.

Lasry and P. Lions. Mean field games. Japanese journal of mathematics , 2(1):229-260, 2007.

Lowe, Y. I. Wu, A. Tamar, J. Harb, OpenAI Pieter A., and I. Mordatch. Multi-agent actor-critic for mixed cooperative-competitive environments. Advances in neural information processing systems , 30, 2017.

Mordatch and P. Abbeel. Emergence of grounded compositional language in multi-agent populations. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 32, 2018.

Francesca Parise and Asuman Ozdaglar. Graphon games. In Proceedings of the 2019 ACM Conference on Economics and Computation , pages 457-458, 2019.

Perolat, S. Perrin, R. Elie, M. Laurière, G. Piliouras, M. Geist, K. Tuyls, and O. Pietquin. Scaling up mean field games with online mirror descent. arXiv preprint arXiv:2103.00623 , 2021.

Perrin, J. Pérolat, M. Laurière, M. Geist, R. Elie, and O. Pietquin. Fictitious play for mean field games: Continuous time analysis and applications. Advances in Neural Information Processing Systems , 33:13199-13213, 2020.

Shani, Y. Efroni, and S. Mannor. Adaptive trust region policy optimization: Global convergence and faster rates for regularized mdps. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 34, pages 5668-5675, 2020.

Sonu, Y. Chen, and P. Doshi. Decision-theoretic planning under anonymity in agent populations. Journal of Artificial Intelligence Research , 59:725-770, 2017.

Uehara, C. Shi, and N. Kallus. A review of off-policy evaluation in reinforcement learning. arXiv preprint arXiv:2212.06355 , 2022.

Vasal, R. K. Mishra, and S. Vishwanath. Master equation of discrete time graphon mean field games and teams. arXiv preprint arXiv:2001.05633 , 2020.

J. Wainwright. High-dimensional statistics: A non-asymptotic viewpoint , volume 48. Cambridge university press, 2019.

Wang, Z. Yang, and Z. Wang. Breaking the curse of many agents: Provable mean embedding Qiteration for mean-field reinforcement learning. In International Conference on Machine Learning , pages 10092-10103. PMLR, 2020.

Xie, Z. Yang, Z. Wang, and A. Minca. Learning while playing in mean-field games: Convergence and optimality. In International Conference on Machine Learning , pages 11436-11447. PMLR, 2021a.

Xie, C. Cheng, N. Jiang, P. Mineiro, and A. Agarwal. Bellman-consistent pessimism for offline reinforcement learning. Advances in neural information processing systems , 34:6683-6694, 2021b.

Yardim, S. Cayci, M. Geist, and N. He. Policy mirror ascent for efficient and independent learning in mean field games. arXiv preprint arXiv:2212.14449 , 2022.

A. uz Zaman, A. Koppel, S. Bhatt, and T. Basar. Oracle-free reinforcement learning in mean-field games along a single sample path. arXiv preprint arXiv:2208.11639 , 2022.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper focuses on two fundamental problems in regularized Graphon Mean-Field Games (GMFGs). The first problem is to establish the existence of a Nash Equilibrium (NE) of any $\\lambda$-regularized GMFG (for $\\lambda \\geq 0$). The second problem is to propose provably efficient algorithms to learn the NE in weakly monotone GMFGs. Regarding the first problem,  this paper used weaker conditions than previous works analyzing unregularized GMFGs ($\\lambda = 0$) or $\\lambda$-regularized MFGs, which are special cases of $\\lambda$-regularized GMFGs. To address the second problem, the paper proposes a discrete-time algorithm and derives its convergence rate solely under weakly monotone conditions. Furthermore, the paper develops and analyzes the action-value function estimation procedure during the online learning process, which is absent from algorithms for monotone GMFGs. The efficiency of the designed algorithm is corroborated by empirical evaluations.\n\nSTRENGTHS:\n1. It's quite fascinating to uncover the link between MFG and GMFG as explored in the paper. Notably, the use of lambda-regularized MFG in Proof 1 to find the NE of regularized GMFG provides an intuitive understanding.\n\n2. PMD for the general function approximation is impressive. The decision to employ policy mirror descent adds an interesting dimension to the methodology.\n\nThe paper appears to be well-rounded and articulately composed. It has a clear presentation of its findings.\n\nWEAKNESSES:\nThe paper does have quite a few notations, but it's understandable considering the complexity of GMFG. I've got a few questions, which might also point out some areas in the paper that could be improved. I've put these questions and potential weaknesses together in the question section for easy reference.\n\nQUESTIONS:\nI mainly work on Stochastic game theory, not mean field game, so some of my questions might be quite basic.\n\n1. From the first strength, I'm curious about how an NE of the $\\lambda$-regularized GMFG is built from an NE of the made-up $\\lambda$-regularized MFG. What makes this possible? This is really interesting to me. Is this a common method for MFG and GMFG? Also, is step 3 in the proof of theorem 1, something new compared to the previous literature?\n\n2. In stochastic game theory, I know that $\\lambda$-regularizing usually gives a better landscape for optimization, making it easier to find (and define) NE in the regularized setup. Is it the same in the MFG setting?\n\n3. Is the analysis linked to this paper? It'd be helpful to know the technical differences between [1] and this paper.\n[1] Zhan, Wenhao, et al. \"Policy mirror descent for regularized reinforcement learning: A generalized framework with linear convergence.\" SIAM Journal on Optimization 33.2 (2023): 1061-1091.\n\nIf this is well-addressed, I am down to re-evaluate this paper.\n\nLIMITATIONS:\nThe authors adequately addressed the limitation and potential negative societal impact on their work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper analyzes policy mirror descent for solving regularized GMFG. The results provide new guarantees for learning GMFG without stringent oracle assumptions, and unlike some past works, it does not restrict the results to continuous time analysis. Furthermore, the paper provides an analysis of the case of function approximation. Experimental results are also presented for certain GMFG.\n\nSTRENGTHS:\nThe paper provides convincing theoretical guarantees for the proposed algorithm. Table 1 is in general convincing of the theoretical contributions of the work. The analysed setting is novel and removes theoretical oracle assumptions in past work. Furthermore, the analysis is in discrete time (i.e., purely algorithmic) and not in continuous time dynamics.\n\nThe theoretical results are very clearly presented, and the assumptions are explicit. There is no ambiguity, and the proofs seem correct (although I might have missed details).\n\nFurthermore, the incorporation of a function class in the analysis as opposed to oracle access is new in MFG/GMFG to the best of my knowledge.\n\nWEAKNESSES:\nWhile the provided result for offline RL-based approximation of value functions is interesting theoretically, it might be prohibitive in practice as the results stand.\n\nTable 1 seems to indicate a large variety of assumptions employed in literature. It is not directly clear how the setting compares to other alternative settings, for instance, if related, a comparison of weak monotonicity with other definitions of monotonicity as well as contraction. This makes it difficult to compare the theory.\n\nExperimental results are restricted to toy problems, however, it is possible that no alternative benchmarks exist for GMFG.\n\nQUESTIONS:\nHow does the function class complexity $\\mathcal{N}_\\infty$ scale in general with various schemes of approximation? Several examples could illuminate the reader. Similarly, making the definition of the covering number in the main body explicit could help reading easier.\n\nLine 230: The statement regarding switching policies is not clear to me. In general, is there an intuitive explanation of the weak monotonicity assumption?\n\nHow do the results compare to the specific case of MFG? The graphon structure seems to be more general than MFG, admitting it as a special GMFG with a particular graphon. For a clear comparison, it could be interesting to state the implied bounds (if any) for the special case of monotone MFG with the corresponding implication made clear. Otherwise a direct comparison might be difficult with continuous time results in MFG.\n\nLIMITATIONS:\nPotential additional limitations to be discussed were mentioned in the weaknesses and limitations.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIntuitively, a \"graphon mean field game\" (GMFG) describes the large-$N$ limit of a game with $N$ players, where the payoff of a player $i$ depends on a weighted average of the states of other players $j\\in [N]$. The graphon aspect comes from the fact that players have \"identities\" given by numbers $U_i\\in [0,1]$, and the weights in the averages are of the form $W(U_i,U_j)/N$, where $W$ is a suitable function (if $W$ is constant, we have a regular mean field game MFG). GMFGs are potentially useful in multiagent reinforcement learning whenever agent interactions are not too strong. \n\nThe present paper does not deal with the finite-$N$ problem, but rather with its continuous limit. One important point is that it considers regularized versions of GMFGs, with an added penalization term. The main results are as follows.\n\n* Theorem 1 is a new result on the existence of equilibria for (potentially regularized) GMFGs. The main attraction of this result, in comparison with previous work, is that it only makes continuity assumptions on the reward function and transition probabilities (the function $W$ is still assumed Lipschitz). Moreover, the regularization had not been considered previously. Theorem 1 is obtained via a careful reduction to a non-graphon Mean Field Game, for which the authors also prove a new existence result (Theorem 2). \n\n* The paper then considers algorithms for approximating Nash equilibria for GMFGs that satisfy a monotonicity condition. Theorem 3 obtains a result under the existence of an \"action function oracle\". When that is not available, one can resort to function approximation: this leads to Theorem 4, which works under additional assumptions (and require the regularization). \n\nA small set of experiments suggests that the authors' method performs well in practice, and also that regularization is important to achieve good performance.\n\nSTRENGTHS:\nThe paper is original and significant. It is also quite clear. The following two points stand out. \n\n* The existence results work under weak conditions because of the clever use of \"soft\" techniques. \n* The algorithms work under relatively natural conditions.\n\nWEAKNESSES:\n* It is not clear to me if the Lipschitz assumption on $W$ is really needed. \n* The proof of Theorem 4 is not particularly surprising. (I hesitate to call this a \"weakness\", but it is true that this part of the analysis is not too surprising.)\n* In certain settings the user may be interested in the unregularized GMFG; however, Theorem 4 requires nonzero regularization. The paper does not provide bounds on how close a (near-)NE for the regularized case is to being a NE for the unregularized game.\n\nQUESTIONS:\nThese are simply rewordings of my comments on weaknesses. \n\nQ1: Regarding Theorems 1 and 2, it looks like the only place where the Lipschitz assumption on $W$ is used is (D.5). However, it looks that all that is needed is uniform continuity, which is equivalent to continuity.\n\nQ2: In certain settings the user may be interested in the unregularized GMFG; however, Theorem 4 requires nonzero regularization. Can one prove bounds on how close a (near-)NE for the regularized case is to being a NE for the unregularized game?\n\nLIMITATIONS:\nNone were discussed, but I don't think there was any need to do so.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-08"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies regularized Graphon Mean-Field Games (GMFGs). They make two theoretical improvements over previous works on this topic:\n* They prove existence of Nash equilibrium under weaker assumptions (e.g., weaker requirement on the continuity of the game) than previous works.\n* For the special case of monotone regularized GMFGs, they give an mirror-descent algorithm that learns the unique Nash equilibrium. Compared to previous works, the novelty here is that the algorithm works for regularized games in discrete time (as opposed to unregularized games and continuous time).\n\nIn terms of techniques, their first result follows the proof plan of Cui and Koeppl, [2021] that reduces the problem to proving existence of Nash equilibrium for a subclass of GMFGs called MFGs. Their main technical contribution is proving an equilibrium existence result (Theorem 2) for MFGs under weaker assumptions using a different approach than previous works. Their second result essentially adapts the algorithm from Perolat et al. [2021] to their setting.\n\nSTRENGTHS:\nThis paper builds upon previous works and makes reasonable improvements. Most interestingly, the condition of their equilibrium existence result (Theorem 2) seems to be significantly weaker than previous works. The paper is well-written. They did a great job introducing the problem and the results and explaining the techniques and the difference from previous works.\n\nWEAKNESSES:\nAs someone who is not closely following this line of work, it is hard for me to gauge the significance of the new equilibrium existence result, i.e., whether the weakened assumption is significantly more applicable than the assumptions made in previous works. The paper briefly mentions the assumptions in previous works are ``overly restrictive for real-world applications'' but did not provide any concrete example.\n\nThe algorithm for learning Nash equilibrium in their setting seems to be rather straightforward adaptation (with more careful analysis) from previous work of Perolat et al. [2021].\n\nQUESTIONS:\nCould you elaborate why the new equilibrium existence results are more applicable than previous works with concrete examples?\n\nWhat are the concrete benefits of the exploration step (Line 5 of Algorithm 1) in your learning algorithm? Is it necessary for proving your result?\n\nLIMITATIONS:\nn/a", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-25"}