Sample Efficient Stochastic Policy Extragradient Algorithm for Zero-Sum Markov Game

Two-player zero-sum Markov game is a fundamental problem in reinforcement learning and game theory . Although many algorithms have been proposed for solving zero-sum Markov games in the existing literature , many of them either require a full knowledge of the environment or are not sample-efficient . In this paper , we develop a fully decentralized and sample-efficient stochastic policy extragradient algorithm for solving tabular zero-sum Markov games . In particular , our algorithm utilizes multiple stochastic estimators to accurately estimate the value functions involved in the stochastic updates , and leverages entropy regularization to accelerate the convergence . Specifically , with a proper entropy-regularization parameter , we prove that the stochastic policy extragradient algorithm has a sample complexity of the order Õ ( Amaxµmin 5.5 ( 1−γ ) 13.5 ) for finding a solution that achieves -Nash equilibrium duality gap , where Amax is the maximum number of actions between the players , µmin is the lower bound of state stationary distribution , and γ is the discount factor . Such a sample complexity result substantially improves the state-of-the-art complexity result . 1 INTRODUCTION Competitive reinforcement learning ( RL ) is an emerging and popular framework that has broad applications in various areas , including market pricing applications ( Kononen and Oja , 2004 ) , realtime strategy-making ( Vinyals et al. , 2019 ) , board games ( Silver et al. , 2017 ; Moerland et al. , 2018 ) and inverse RL ( Zhang et al. , 2019 ) . In particular , an important and fundamental formulation of competitive RL is the two-player zero-sum Markov game , which involves two competing players that interact with a common environment and receive zero-sum rewards . Both players aim to learn the optimal policy that achieves the Nash equilibrium of accumulated rewards . Algorithms for solving Markov games are very different from conventional single-agent RL algorithms . In particular , both players must learn to improve their policies based on feedback from the opponent and the environment , but usually the opponent will not reveal any sensitive information ( e.g. , actions or policy ) or cooperate with each other . In the existing literature , numerous algorithms have been developed for solving zero-sum Markov games , including Q-learning ( Fan et al. , 2020 ; Zhu and Zhao , 2020 ) , fitted Q iteration ( Zhang et al. , 2021 ) , policy gradient ( Daskalakis et al. , 2020 ; Zhao et al. , 2021 ) , policy extragradient ( Cen et al. , 2021 ) , model-based Monte Carlo estimation ( Zhang et al. , 2020 ) , optimistic gradient descent ascent ( Wei et al. , 2021 ) , etc . However , many of these algorithms require both players to access their opponent ’ s actions ( Wei et al. , 2017 ; Sidford et al. , 2020 ; Huang et al. , 2021 ; Jafarnia-Jahromi et al. , 2021 ) , which causes privacy issues . On the other hand , some algorithms need to know about the environment transition kernel and reward mapping ( Cen et al. , 2021 ) , which are usually unknown a priori in practice . Moreover , other algorithms require both players to perform asymmetric policy updates using different numbers of iterations , learning rates or exploration probabilities ( Zhao et al. , 2021 ; Daskalakis et al. , 2020 ) , which are generally hard to coordinate in advance between two competing players . Therefore , it is desired to develop an algorithm for solving Markov games that avoids all the aforementioned issues . Moreover , from a theoretical perspective , the convergence and sample complexity of these existing algorithms for solving Markov games have not been comprehensively studied . Specifically , some studies established the convergence of the algorithms with i.i.d . samples ( Zhao et al. , 2021 ; Guo et al. , 2021 ) , which violates the dependent nature of samples collected from the dynamic Markov decision process . Also , other algorithms suffer from an extremely high sample complexity to achieve an approximate Nash equilibrium solution ( Wei et al. , 2021 ) . Hence , we are motivated to answer the following fundamental question . • Q : Can we develop a fully decentralized algorithm that is model-free and takes symmetric and private policy updates for solving zero-sum Markov games , with provable convergence guarantee and an improved sample complexity for achieving a Nash equilibrium solution ? In this paper , we provide positive and comprehensive answers to this question by developing a fully decentralized stochastic policy extragradient algorithm . In Table 1 of Appendix F , we compare the key properties and sample complexity of our algorithm with those of the existing algorithms . 1.1 OUR CONTRIBUTIONS We consider a standard zero-sum Markov game with discounted reward over infinite horizon . To solve such a Markov game , we propose a stochastic variant of the policy extragradient algorithm ( Cen et al. , 2021 ) that satisfies the following amenable properties . • Our algorithm uses multiple stochastic estimators to estimate the value functions involved in the predictive updates for solving entropy-regularized matrix games , and therefore the algorithm does not rely on any prior knowledge of the environment transition kernel ( model-free ) . Moreover , the resulting stochastic policy updates of our algorithm for both players are symmetric and do not involve any sensitive information of the opponent ( private ) . • Compared with the stochastic estimators used in ( Wei et al. , 2021 ) , our estimators have much smaller variance that helps improve the estimation accuracy . Specifically , by developing new techniques ( explained in the next bullet ) , we establish tight high-probability estimation error bounds for our stochastic estimators with Markovian samples . Then , with a proper entropyregularization parameter , we show that our stochastic policy extragradient algorithm requires a sample complexity of the order O ( Amaxµmin 5.5 ( 1−γ ) 13.5 ) to achieve an -Nash equilibrium duality gap , which substantially improves the state-of-the-art complexity result of ( Wei et al. , 2021 ) . • We develop novel techniques to bound the estimation error of the proposed stochastic estimators , whose numerator and denominator involve sample average approximations . First , we propose a special estimation error decomposition that avoids divergence of the bound caused by possibly small numerical values of the sample average involved in the denominator of the stochastic estimators . Second , we leverage this error decomposition and the recursive structure of the stochastic estimators to derive a contraction property of the estimation errors , which eventually leads to tight bounds for the estimation error . We refer to Section 4 for more elaboration on our technical novelties . 1.2 OTHER RELATED WORK Other settings of two-player zero-sum Markov games . In this paper , we focus on a standard setting of two-player zero-sum Markov game with discount and infinite horizon in the discrete time domain . There are other settings of two-player zero-sum Markov games . For example , Bai et al . ( 2020 ) ; Huang et al . ( 2021 ) studied a two-player zero-sum Markov game with finite horizon and without discount , whereas Daskalakis et al . ( 2020 ) considered finite random horizon without discount . Jafarnia-Jahromi et al . ( 2021 ) also considered the setting without discount , and it allows one of the players to constantly adjust its policy based on the entire history of states and actions . Ghosh et al . ( 2021 ) studied a two-player zero-sum Markov game in the continuous time domain . Multi-agent general-sum Markov game . Some works studied multi-agent Markov games , which extend the two-player zero-sum Markov games to multiple players without the zero-sum constraint ( Wang and Sandholm , 2002 ; Hu and Wellman , 2003 ; Deng et al. , 2021 ; Leonardos et al. , 2021 ) . More specifically , Leonardos et al . ( 2021 ) defined and studied Markov potential game which has a potential function assumption . Guo et al . ( 2019 ) ; Elie et al . ( 2020 ) ; Gu et al . ( 2021 ) studied mean-field games with a large number of players . Entropy regularization and value iteration Our algorithm leverages entropy regularization and value iteration to accelerate convergence . Entropy regularization is a popular technique that has been widely used in reinforcement learning ( Neu et al. , 2017 ; Geist et al. , 2019 ; Mei et al. , 2020 ; Cen et al. , 2020 ) and Markov game ( Mertikopoulos and Sandholm , 2016 ; Savas et al. , 2019 ; Cen et al. , 2021 ) to encourage environment exploration and accelerate algorithm convergence . Value iteration is also a classical method that is widely used in both single-agent reinforcement learning ( Ernst et al. , 2005 ; Tamar et al. , 2016 ; Farahmand and Ghavamzadeh , 2021 ) and Markov games ( Zhu and Zhao , 2020 ; Cen et al. , 2021 ) . With full knowledge of the environment , it exponentially converges to the fixed point of Bellman operator ( Cen et al. , 2021 ) . Compared to another similar classical method called policy iteration , value iteration does not need policy evaluation which involves additional computation ( Sutton and Barto , 2018 ) . 2 BACKGROUND OF MARKOV GAME AND ENTROPY REGULARIZATION 2.1 TWO-PLAYER ZERO-SUM MARKOV GAME In a zero-sum Markov game , two players compete with each other in a common environment . Throughout , the state space is denoted as S. The action spaces and policies of both players are denoted as A ( 1 ) , π ( 1 ) and A ( 2 ) , π ( 2 ) , respectively . Here , π ( 1 ) ∈ ∆ ( |A ( 1 ) | ) , π ( 2 ) ∈ ∆ ( |A ( 2 ) | ) are random policies defined over the corresponding simplex sets . The reward function is denoted as R : S ×A ( 1 ) ×A ( 2 ) → [ 0 , 1 ] , and the discount factor is denoted as γ ∈ ( 0 , 1 ) . At any time t , both players observe state st ∈ S of the environment . Then , both players respectively select their actions following their own policies , i.e. , a ( 1 ) t ∼ π ( 1 ) ( ·|st ) and a ( 2 ) t ∼ π ( 2 ) ( ·|st ) . After that , the environment state transfers to a new state st+1 following the underlying transition kernel P ( ·|st , a ( 1 ) t , a ( 2 ) t ) , and both players receive zero-sum rewards , i.e. , R ( 1 ) t = −R ( 2 ) t = Rt , where Rt : = R ( st , a ( 1 ) t , a ( 2 ) t ) . With this Markov decision process , we can define the following state value function associated with the players ’ policies π ( 1 ) and π ( 2 ) for any environment state s ∈ S. Vπ ( 1 ) , π ( 2 ) ( s ) = E [ ∞∑ t=0 γtRt ∣∣∣s0 = s ] . ( 1 ) The goal of both players is to compete via the following minimax game in all states s. min π ( 2 ) max π ( 1 ) Vπ ( 1 ) , π ( 2 ) ( s ) . ( 2 ) In particular , it has been shown in ( Shapley , 1953 ) that there exists a Nash equilibrium policy pair π ( 1 ) ∗ , π ( 2 ) ∗ for zero-sum Markov games , i.e. , Vπ ( 1 ) , π ( 2 ) ∗ ( s ) ≤ Vπ ( 1 ) ∗ , π ( 2 ) ∗ ( s ) ≤ Vπ ( 1 ) ∗ , π ( 2 ) ( s ) holds for any other policies π ( 1 ) , π ( 2 ) and for all states s. 2.2 ENTROPY-REGULARIZED MARKOV GAME Entropy regularization is a popular technique that has been widely used in reinforcement learning ( Neu et al. , 2017 ; Geist et al. , 2019 ; Mei et al. , 2020 ; Cen et al. , 2020 ) and Markov game ( Mertikopoulos and Sandholm , 2016 ; Savas et al. , 2019 ; Cen et al. , 2021 ) to encourage environment exploration and accelerate algorithm convergence . Specifically , for the zero-sum Markov game , we can define an entropy-regularized state value function by adding entropy regularization to the state value function in ( 1 ) as follows ( Cen et al. , 2021 ) . V ( τ ) π ( 1 ) , π ( 2 ) ( s ) : =E [ ∞∑ t=0 γt [ Rt − τ lnπ ( 1 ) ( a ( 1 ) t |st ) + τ lnπ ( 2 ) ( a ( 2 ) t |st ) ] ∣∣∣s0 = s ] , ( 3 ) where τ > 0 is called the regularization parameter . With the above definition , we further define the following entropy-regularized state-action value function ( also called Q-function ) ( Cen et al. , 2021 ) . Q ( τ ) π ( 1 ) , π ( 2 ) ( s , a ( 1 ) , a ( 2 ) ) : =R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) [ V ( τ ) π ( 1 ) , π ( 2 ) ( s′ ) ] . ( 4 ) In particular , V ( τ ) π ( 1 ) , π ( 2 ) can be obtained from Q ( τ ) π ( 1 ) , π ( 2 ) as follows . V ( τ ) π ( 1 ) , π ( 2 ) ( s ) = [ π ( 1 ) ( s ) ] > Q ( τ ) π ( 1 ) , π ( 2 ) ( s ) π ( 2 ) ( s ) + τH ( π ( 1 ) ( s ) ) − τH ( π ( 2 ) ( s ) ) : =fτ ( Q ( τ ) π ( 1 ) , π ( 2 ) ( s ) , π ( 1 ) ( s ) , π ( 2 ) ( s ) ) , ( 5 ) whereH ( π ) denotes the entropy of policy π , and we define this mapping as fτ for convenience . For the entropy regularized Markov game , it has an equilibrium policy pair that solves the minimax optimization problem minπ ( 2 ) maxπ ( 1 ) V ( τ ) π ( 1 ) , π ( 2 ) ( s ) . Such a policy pair is called the quantal response equilibrium ( QRE ) . Our goal is to find the equilibrium policy pair of the original Markov game in ( 2 ) by solving the entropy-regularized Markov game with a proper regularization parameter τ . In particular , compared with the equilibrium policy of the Markov game , the QRE tends to have a larger entropy due to the entropy regularization , which encourages the players to explore and obtain a better understanding of the environment . Another advantage of considering the entropy-regularized Markov game is that the entropy regularization makes the minimax problem have a better optimization geometry that accelerates the convergence of the optimization process . 3 STOCHASTIC POLICY EXTRAGRADIENT ALGORITHM FOR ENTROPY-REGULARIZED MARKOV GAME In this section , we develop a stochastic policy extragradient ( SPE ) algorithm for solving entropyregularized Markov games . First , we recap the policy extragradient ( PE ) algorithm , which is introduced in ( Cen et al. , 2021 ) to solve entropy-regularized Markov games with full knowledge of the environment transition kernel and reward mapping . Then , we propose the model-free SPE algorithm that solves entropy-regularized Markov games using only stochastic samples . 3.1 REVIEW OF POLICY EXTRAGRADIENT ALGORITHM Value iteration is a classical reinforcement learning algorithm that requires full knowledge of the environment and achieves an exponential convergence rate . In particular , for the entropy-regularized Markov game , the k-th value iteration update is defined as follows . Qk ( s , a ( 1 ) , a ( 2 ) ) =R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) [ Vk ( s ′ ) ] , ∀s , a ( 1 ) , a ( 2 ) , ( 6 ) Vk+1 ( s ) = min π ( 2 ) ( s ) max π ( 1 ) ( s ) fτ ( Qk ( s ) ; π ( 1 ) ( s ) , π ( 2 ) ( s ) ) , ∀s , ( 7 ) where we define Qk ( s ) : = Qk ( s , · , · ) ∈ R|A ( 1 ) |×|A ( 2 ) | . This algorithm alternatively updates all the entries of the value functions Qk and Vk . Thanks to the entropy regularization in the function fτ ( see ( 5 ) for the definition ) , the minimax matrix game in ( 7 ) is τ -strongly concave in π ( 1 ) and τ -strongly convex in π ( 2 ) , and therefore it has a unique solution . To solve the entropy-regularized minimax matrix game in ( 7 ) , Cen et al . ( 2021 ) proposed a predictive update ( PU ) algorithm . Specifically , with uniform policy initialization , i.e. , π ( m ) k,0 ( s ) = 1 |A ( m ) | , ∀m ∈ { 1 , 2 } , ∀s ∈ S , the PU algorithm performs the following policy updates : for t = 0 , 1 , 2 , ... ( PU ) : π ( 1 ) k , t+1 ( a ( 1 ) |s ) ∝ π ( 1 ) k , t ( a ( 1 ) |s ) 1−ητ exp ( ηQ ( 1 ) k , t ( s , a ( 1 ) ) ) π ( 2 ) k , t+1 ( a ( 2 ) |s ) ∝ π ( 2 ) k , t ( a ( 2 ) |s ) 1−ητ exp ( − ηQ ( 2 ) k , t ( s , a ( 2 ) ) ) π ( 1 ) k , t+1 ( a ( 1 ) |s ) ∝ π ( 1 ) k , t ( a ( 1 ) |s ) 1−ητ exp ( ηQ ( 1 ) k , t+1 ( s , a ( 1 ) ) ) π ( 2 ) k , t+1 ( a ( 2 ) |s ) ∝ π ( 2 ) k , t ( a ( 2 ) |s ) 1−ητ exp ( − ηQ ( 2 ) k , t+1 ( s , a ( 2 ) ) ) , ( 8 ) where we use the following notations ( superscript ( \m ) denotes the opponent of the m-th player. ) . Q ( m ) k , t ( s , a ( m ) ) : = E a ( \m ) ∼π ( \m ) k , t ( s ) [ Qk ( s , a ( 1 ) , a ( 2 ) ) ] , m ∈ { 1 , 2 } ( 9 ) Q ( m ) k , t+1 ( s , a ( m ) ) : = E a ( \m ) ∼π ( \m ) k , t+1 ( s ) [ Qk ( s , a ( 1 ) , a ( 2 ) ) ] , m ∈ { 1 , 2 } . ( 10 ) Once we obtain the output policy pair ( π ( 1 ) k , π ( 2 ) k ) of the PU algorithm , we can obtain an approximation of Vk+1 ( s ) as V ′k+1 ( s ) = fτ ( Qk ( s ) ; π ( 1 ) k ( s ) , π ( 2 ) k ( s ) ) , which will be further used in the next Q-value function update ( 6 ) to replace Vk+1 ( s′ ) . The updates ( 6 ) , ( 7 ) & ( 8 ) are referred to as policy extragradient ( PE ) algorithm . In the PE algorithm , the PU update in ( 8 ) allows both players to take symmetric updates without revealing their private actions , and has been shown to converge to the unique solution of the entropyregularized matrix game ( 7 ) exponentially fast ( Cen et al. , 2021 ) . However , the PE algorithm has several limitations . First , in the PU update , each player m ∈ { 1 , 2 } needs to query the quantities Q ( m ) k , t ( s , a ( m ) ) , Q ( m ) k , t ( s , a ( m ) ) from its opponent . To compute these quantities , the opponent needs to multiply the entire Q-table by its own policy vector . This requires both players to coordinate with each other and share aQ-table . Second , the update of theQ-table in ( 6 ) requires full knowledge of the environment transition kernel P and the reward mapping R , which are unknown a priori in practice . To overcome these limitations , we develop a fully stochastic PE algorithm in the next subsection . 3.2 STOCHASTIC POLICY EXTRAGRADIENT ALGORITHM The major challenge of the PE algorithm is computing the quantities Q ( m ) k , t , Q ( m ) k , t and V ′ k+1 , which requires coordinating with the opponent and involves the environment information . Here , we develop a model-free and fully stochastic variant of PE that estimates these key quantities using Markovian stochastic samples . We refer to this algorithm as stochastic policy extragradient ( SPE ) . Specifically , we first estimate the quantity V ′k+1 ( s ) = fτ ( Qk ( s ) ; π ( 1 ) k ( s ) , π ( 2 ) k ( s ) ) . By definition of fτ in ( 5 ) and the update of Qk in ( 6 ) ( now we use V ′k ( s ′ ) instead of Vk ( s′ ) ) and using some standard tricks on random variables ( see Lemma 2 in Appendix B for a full proof ) , we can rewrite V ′k+1 ( s ) as V ′k+1 ( s ) = E [ ( R ( s̃ , ã ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ) 1 { s̃ = s } ] µk ( s ) + τH ( π ( 1 ) k ( s ) ) − τH ( π ( 2 ) k ( s ) ) , ( 11 ) where µk ( s ) denotes the stationary state distribution associated with the policy pair ( π ( 1 ) k , π ( 2 ) k ) , and the expectation is taken over s̃ ∼ µk , ã ( 1 ) ∼ π ( 1 ) k ( s ) , ã ( 2 ) ∼ π ( 2 ) k ( s ) , s ′ ∼ P ( ·|s̃ , ã ( 1 ) , ã ( 2 ) ) . To estimate this quantity , we query a set Nk+1 ( with cardinality Nk+1 ) of samples from the Markov decision process following the pair of policies ( π ( 1 ) k , π ( 2 ) k ) . Then , we estimate V ′ k+1 ( s ) as V̂k+1 ( s ) = 1 Nk+1 ∑ i∈Nk+1 ( Ri + γV̂k ( si+1 ) ) 1 { si = s } 1 Nk+1 ∑ i∈Nk+1 1 { si = s } + τH ( π ( 1 ) k ( s ) ) − τH ( π ( 2 ) k ( s ) ) . ( 12 ) Intuitively , we use the sample average of Markovian samples to estimate the expectation terms in ( 11 ) . Thanks to the concentration phenomenon of dependent samples ( Paulin , 2015 ) , these sample averages converge to the desired expected values provided that the sample size is sufficiently large . Next , we estimate Q ( m ) k , t , m ∈ 1 , 2 . Leveraging ( 9 ) and ( 6 ) , we obtain the following equivalent characterization for both players m ∈ 1 , 2 ( see Lemma 2 in Appendix B for the proof of equivalence ) . Q ( m ) k , t ( s , a ( m ) ) = E [ ( R ( s̃ , ã ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ) 1 { s̃ = s , ã ( m ) = a ( m ) } ] µk , t ( s ) π ( m ) k , t ( a ( m ) |s ) , ( 13 ) where 1 { · } denotes the indicator function , µk , t denotes the stationary state distribution associated with the policy pair ( π ( 1 ) k , t , π ( 2 ) k , t ) , and the expectation is taken over s̃ ∼ µk , t , ã ( 1 ) ∼ π ( 1 ) k , t ( s ) , ã ( 2 ) ∼ π ( 2 ) k , t ( s ) , s ′ ∼ P ( ·|s̃ , ã ( 1 ) , ã ( 2 ) ) . To estimate this quantity , we query a set Nk , t ( with cardinality Nk , t ) of samples from the Markov decision process following a pair of smoothed policies π′ ( m ) k , t ( s ) = ( 1− ′ ) π ( m ) k , t ( s ) + ′ |A ( m ) |1 , where ′ ∈ [ 0 , 1 ] is a small smoothing constant that will be theoretically determined later . Then , we estimate Q ( m ) k , t ( s , a ( m ) ) as follows . Q̂ ( m ) k , t ( s , a ( m ) ) = 1 Nk , t ∑ i∈Nk , t ( Ri + γV̂k ( si+1 ) ) 1 { si = s , a ( m ) i = a ( m ) } ( 1 Nk , t ∑ i∈Nk , t 1 { si = s } ) π ′ ( m ) k , t ( a ( m ) |s ) , ( 14 ) where we have replaced the expectations with sample averages , and replaced V ′k ( s ′ ) with V̂k ( s′ ) . Here , the Markovian samples are queried following the ′-smoothed policies ( π′ ( 1 ) k , t , π ′ ( 2 ) k , t ) . On one hand , ′ should not be too small so that it keeps the denominator of the above estimation away from zero . On the other hand , ′ should not be too large so that it is sufficiently close to the original policy . Similarly , to estimate Q ( m ) k , t , we query another set N k , t ( with cardinality Nk , t ) of samples from the Markov decision process following a pair of smoothed policies π′ ( m ) k , t ( s ) = ( 1− ′ ) π ( m ) k , t ( s ) + ′ |A ( m ) |1 , where ′ ∈ [ 0 , 1 ] will be theoretically determined later . Then , we estimate Q ( m ) k , t as follows . Q̂ ( m ) k , t ( s , a ( m ) ) = 1 Nk , t ∑ i∈Nk , t ( Ri + γV̂k ( si+1 ) ) 1 { si = s , a ( m ) i = a ( m ) } ( 1 Nk , t ∑ i∈Nk , t 1 { si = s } ) π ′ ( m ) k , t ( a ( m ) |s ) . ( 15 ) Remark 1 . The estimators ( 12 ) , ( 14 ) & ( 15 ) essentially use ratio of sample averages to approximate ratio of expectations . Estimators with similar structure have been used in ( Ortner and Auer , 2007 ; Xia et al. , 2016 ; Wei et al. , 2021 ) . However , these works analyze the estimation error of their estimators with independence samples . As a comparison , our analysis of the estimation error bounds the additional bias induced by the correlated Markovian samples , and achieves an improved sample complexity in the main Theorem 2 . We summarize our stochastic policy extragradient ( SPE ) algorithm in Algorithm 1 . Specifically , in SPE , we estimate the quantities Q ( m ) k , t , Q ( m ) k , t , V ′ k using their corresponding stochastic estimators . As a result , the SPE algorithm is model-free , and the updates for both players are symmetric and private . Algorithm 1 Stochastic policy extragradient ( SPE ) for entropy-regularized Markov game Initialize : V ′0 ( s ) for all s ∈ S. for value iterations k = 0 , 1 , . . . , K − 1 do Initialize π ( 1 ) k,0 , π ( 2 ) k,0 with uniform distribution . for PU iterations t = 0 , 1 , . . . , Tk − 1 do Players 1,2 sample Nk , t Markovian samples following smoothed policies π ′ ( 1 ) k , t , π ′ ( 2 ) k , t . Every player m ∈ { 1 , 2 } computes Q̂ ( m ) k , t ( s , a ( m ) ) for all s , a ( m ) using ( 14 ) . Players 1,2 sample Nk , t Markovian samples following smoothed policies π ′ ( 1 ) k , t , π ′ ( 2 ) k , t . Every player m ∈ { 1 , 2 } computes Q̂ ( m ) k , t ( s , a ( m ) ) for all s , a ( m ) using ( 15 ) . Implement the t-th PU iteration for all s , a ( 1 ) , a ( 2 ) using ( 8 ) with estimations ( 14 ) & ( 15 ) . end Let π ( m ) k = π ( m ) k , Tk , m = 1 , 2 . Players sample Nk Markovian samples following π ( 1 ) k , π ( 2 ) k . Compute V̂k+1 ( s ) for all s using ( 12 ) . end Output : π ( 1 ) K−1 , π ( 2 ) K−1 . 4 FINITE-TIME CONVERGENCE ANALYSIS OF SPE Throughout our convergence analysis , we adopt the following two standard assumptions . Assumption 1 . Denote Tπ ( 1 ) , π ( 2 ) ( s , s′ ) : = inf { t ≥ 1 : st = s′|s0 = s } as the first-visit time under the policy pair π ( 1 ) , π ( 2 ) . We assume that sup s , s′∈S sup π ( 1 ) , π ( 2 ) Eπ ( 1 ) , π ( 2 ) [ Tπ ( 1 ) , π ( 2 ) ( s , s ′ ) ] < +∞ . ( 16 ) Assumption 1 is widely used in the reinforcement learning literature ( Ortner and Auer , 2007 ; Ortner , 2020 ; Wei et al. , 2021 ; Jafarnia-Jahromi et al. , 2021 ) . It ensures that every state will be visited at least once within a finite duration of time , thus ensuring that all the states will be visited infinitely often . This guarantees sufficient exploration . In our analysis , we use the following equivalent statement of Assumption 1 for convenience , which means that the stationary state distribution µπ ( 1 ) , π ( 2 ) has a uniform lower bound µmin > 0 . Their equivalence is based on Theorem 5.5.11 of ( Durrett , 2019 ) . µmin : = inf s∈S inf π ( 1 ) , π ( 2 ) µπ ( 1 ) , π ( 2 ) ( s ) = [ sup s∈S sup π ( 1 ) , π ( 2 ) Eπ ( 1 ) , π ( 2 ) Tπ ( 1 ) , π ( 2 ) ( s , s ) ] −1 > 0 . ( 17 ) Assumption 2 . There exists a mixing time tmix ∈ N such that for any policy pair π ( 1 ) , π ( 2 ) and its corresponding stationary state distribution µπ ( 1 ) , π ( 2 ) , we have dTV ( Pπ ( 1 ) , π ( 2 ) ( stmix ) , µπ ( 1 ) , π ( 2 ) ) ≤ 1 4 . where Pπ ( 1 ) , π ( 2 ) ( stmix ) denotes the state distribution under the policy pair π ( 1 ) , π ( 2 ) at time tmix , and dTV denotes the total variation distance between two probability distributions . In this subsection , we analyze the finite-time convergence of Algorithm 1 for solving the entropyregularized Markov game ( 5 ) . We focus on the convergence rate of the following Nash equilibrium duality gap , which is a standard optimality metric widely adopted in the existing literature ( Xu et al. , 2020 ; Jin and Sidford , 2021 ; Wei et al. , 2021 ) . D ( τ ) ( π ( 1 ) , π ( 2 ) ) : = max s ( max π V ( τ ) π , π ( 2 ) ( s ) −min π′ V ( τ ) π ( 1 ) , π′ ( s ) ) . In particular , when τ = 0 , D ( 0 ) ( π ( 1 ) , π ( 2 ) ) corresponds to the duality gap of the original Markov game . Throughout , we define Amax : = max { |A ( 1 ) | , |A ( 2 ) | } , Qmax : = 1+γτ lnAmax1−γ and Vmax : = 1+τ lnAmax 1−γ . We also require that the batch sizes of Algorithm 1 satisfy the following conditions . Nk , t , Nk , t ≥ 650tmixAmax µmin ln ( 20Tsum|S|Amax δ √ µmin ) , ∀k , t , ( 18 ) Nk+1 ≥ 650tmix µmin ( 1− γ ) 2 ln ( 4 δ √ µmin ) , ∀k . ( 19 ) Then , we obtain the following convergence result of Algorithm 1 , where Tsum : = ∑K−1 k=0 Tk . Theorem 1 ( Finite-time convergence rate ) . Apply Algorithm 1 to solve the entropy-regularized Markov game with τ ∈ ( 0 , 1 ] . Choose learning rate η = [ 2 ( τ +Qmax ) ] −1 , initialization ‖V̂0‖∞ ≤ Vmax and batch sizes Nk , t , Nk , t , Nk+1 that satisfy ( 18 ) & ( 19 ) . Then , the Nash equilibrium duality gap converges at the following rate with probability at least 1− δ . D ( τ ) ( π ( 1 ) K−1 , π ( 2 ) K−1 ) ≤ O ( Vmax lnAmax 1− γ K−1∑ k=0 γK−k ( 1− ητ ) Tk−1 + Vmax 1− γ [ tmixAmax µmin ln ( Tsum|S|Amax δµmin ) ] 2/3 K−1∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−2−t ( 1 N 2/3 k , t + Vmax τN 2/3 k , t+1 ) + V 3maxγ K τ2 ( 1− γ ) 2 + tmixV 3 max τ2µmin ( 1− γ ) 3 ln ( K|S| δµmin ) K−1∑ k=0 γK−k−1 Nk+1 ) . ( 20 ) Remark 2 . In the proof of Theorem 1 , we also prove that the convergence rate of the Q-function estimation error ‖QK−Q ( τ ) ∗ ‖∞ is ( 1−γ ) times the convergence rate in ( 20 ) . Here , QK corresponds to the Q-function associated with the policy pair ( π ( 1 ) K , π ( 2 ) K ) produced by Algorithm 1 in the K-th iteration , and Q ( τ ) ∗ corresponds to the optimal Q-function associated with the Nash equilibrium policy pair π ( 1 ) ∗τ , π ( 2 ) ∗τ of the entropy-regularized Markov game . Theorem 1 characterizes the convergence of duality gap of the SPE algorithm under general hyperparameter scheduling of the batch sizes Nk , t , Nk , t , Nk+1 and number of inner iterations Tk . Specifically , it can be seen that as the number of outer iterations K and inner iterations Tk increase , the duality gap converges to an exponentially weighted average of N−2/3k , t , N −2/3 k , t+1 and N −1 k+1 , and the gap can be made arbitrarily small by choosing sufficiently large batch sizes . To provide an intuitive understanding , we can set these hyper-parameters as constants , i.e. , Tk ≡ T , Nk , t ≡ N , Nk , t ≡ N , Nk+1 ≡ N ′ , and then the convergence rate in ( 20 ) simplifies to O ( V 3maxγ K τ2 ( 1− γ ) 2 + γVmax lnAmax ( 1− γ ) 2 ( 1− ητ ) T−1 + tmixV 3 max τ2µmin ( 1− γ ) 4 ln ( K|S| δµmin ) 1 N + Vmax ητ ( 1− ητ ) ( 1− γ ) 2 [ tmixAmax µmin ln ( Tsum|S|Amax δµmin ) ] 2/3 ( 1 N2/3 + Vmax τN 2/3 ) ) . ( 21 ) To elaborate , the first term characterizes the exponential convergence of the K outer value iterations . This convergence is faster than the sublinear convergence O ( √ lnK/K ) established in ( Wei et al. , 2021 ) . The second term characterizes the exponential convergence of the T inner PU iterations . Moreover , the last two terms characterize the estimation errorsO ( N−1 ) andO ( N−2/3 ) +O ( N−2/3 ) of the estimators ( 12 ) and ( 14,15 ) , respectively . As a comparison , Wei et al . ( 2021 ) established a much larger estimation error O ( N−1/3 ) . These improvements , as we show in Theorem 2 later , lead to a substantially improved overall sample complexity over the state-of-the-art result . In particular , due to the exponentially weighted average structure in ( 20 ) , the sample complexity can be further optimized by an adaptive scheduling of the batch sizes . Technical Novelty . We further comment on the proof of Theorem 1 . Note that the analysis of the PE algorithm in ( Cen et al. , 2021 ) requires full knowledge of the environment and does not characterize the convergence of duality gap . As a comparison , to establish the duality gap convergence rate ( 20 ) of the model-free SPE , we need to make substantial new developments to tightly bound the estimation errors of the proposed stochastic estimators . We elaborate our technical contributions below . • As we explained in Remark 1 , the sample averages involved in our estimators are correlated Markovian samples . To bound the additional bias induced by these correlated samples , we apply the concentration inequalities developed in ( Paulin , 2015 ) for dependent samples to establish tight high-probability estimation error bounds . • We develop a much refined analysis of the state value function estimation error ‖V̂k+1 − V ′k+1‖∞ , which is the key to develop tight bounds for all the other estimation errors . Specifically , we first propose the following error decomposition for any state s |V̂k+1 ( s ) −V ′k+1 ( s ) |= ∣∣∣ v̂k+1 ( s ) µ̂k ( s ) − vk+1 ( s ) µk ( s ) ∣∣∣≤ |v̂k+1 ( s ) −vk+1 ( s ) | µk ( s ) + ∣∣v̂k+1 ( s ) ∣∣∣∣∣µk ( s ) −µ̂k ( s ) µk ( s ) µ̂k ( s ) ∣∣∣ , where v̂k+1 ( s ) , µ̂k+1 ( s ) are sample average estimators of vk+1 ( s ) , µk+1 ( s ) , respectively , and we refer to Appendix B for the definitions of these terms . The motivation is that the ∣∣v̂k+1 ( s ) ∣∣ in the second term helps cancel out the estimator µ̂k ( s ) in the denominator , and then all the denominators do not involve any sample average estimators , which may take a small numerical value that causes divergence and a loose concentration bound . By leveraging this special decomposition and the recursive structure of the stochastic estimator ( 12 ) , we are able to establish the following key contraction property of the estimation error ( see ( 85 ) in Appendix B ) . ‖V̂k+1 − V ′k+1‖∞ ≤ γ‖V̂k − V ′k‖∞ +O ( N −1/2 k+1 ) . By telescoping the above contraction bound , we obtain tight estimation error bounds for all the proposed stochastic estimators . As a comparison , Wei et al . ( 2021 ) directly applied the AzumaHoeffding inequality with independent samples to bound the entire estimator and obtain a loose error bound , and Liu et al . ( 2021 ) simply assumed a small upper bound for the estimation error . • We develop a stochastic predictive update ( SPU ) algorithm with general inexact value function estimations and a finite-time convergence analysis of its duality gap ( see Lemma 1 for the SPU algorithm and its convergence proof ) . This generalizes the convergence result of the PU algorithm established in ( Cen et al. , 2021 ) , which uses exact value functions based on full knowledge of the environment . Finally , by incorporating our developed tight estimation error bounds into the finite-time duality gap bound of SPU , we obtain the desired convergence rate in Theorem 1 . Based on Theorem 1 , we obtain the following sample complexity of SPE for achieving an -Nash equilibrium duality gap of the original Markov game , i.e. , D ( 0 ) ( π ( 1 ) K−1 , π ( 2 ) K−1 ) ≤ . Here , we adopt an adaptive batch size scheduling scheme to optimize the complexity order . The overall sample complexity is given by ∑K−1 k=0 [ Nk+1 + 2 ∑Tk−1 t=0 ( Nk , t +Nk , t+1 ) ] . Theorem 2 ( Sample complexity ) . Implement Algorithm 1 with η = O ( 1 − γ ) , τ = O ( ( 1−γ ) lnAmax ) , K = O [ 1 1−γ ln ( lnAmax ( 1−γ ) ) ] and Tk = 1 + k ln γ −1 ln ( 1−ητ ) −1 . Choose the following adaptive batch sizes . Nk+1 = Õ ( tmix ( ln2Amax ) γ− k2 2µmin ( 1− γ ) 8 ) , Nk , t = Nk , t 3 2 ( 1− γ ) 3 = Õ ( tmixAmax ( 1− ητ ) −3 ( t+1 ) 5 µmin ( 1− γ ) 3 ) . Then , for any ≤ lnAmax1−γ , the overall sample complexity to achieve D ( 0 ) ( π ( 1 ) K−1 , π ( 2 ) K−1 ) ≤ is Õ ( tmixAmax µmin 5.5 ( 1−γ ) 13.5 ) . Please refer to ( 118 ) in Appendix E for a complete expression . The above complexity result is obtained by choosing a small τ = O ( ( 1−γ ) lnAmax ) for the convergence rate result in Theorem 1 . Specifically , we show in Lemma 6 that the duality gap is Lipschitz continuous with regard to the entropy regularization parameter , i.e. , ∣∣D ( τ ) ( π ( 1 ) , π ( 2 ) ) − D ( 0 ) ( π ( 1 ) , π ( 2 ) ) ∣∣ ≤ 2τ lnAmax 1−γ . Therefore , by choosing a proper small τ , convergence of the duality gap D ( τ ) of the entropy-regularized Markov game implies the convergence of the duality gap D ( 0 ) of the original Markov game . Remark on Improvement of Sample Complexity . We elaborate on the improvement of sample complexity in two different levels : population level ( with known environment ) and stochastic level ( using stochastic samples ) . First , in the population level , the original policy extragradient ( PE ) algorithm in ( Cen et al. , 2021 ) already achieves a faster convergence rate than the OGDA-based algorithm proposed in ( Wei et al. , 2021 ) . Specifically , to achieve an -Nash equilibrium point of the original Markov game , PE requires O ( Amax ( 1−γ ) ln 2 −1 ) iterations by choosing a proper entropy regularization parameter τ = O ( ( 1−γ ) lnAmax ) . As a comparison , the OGDA-based algorithm requires Õ ( |S|3 ( 1−γ ) 9 2 ) iterations by substituting = 0 ( no stochastic error ) and their choice of step size into the Theorem 1 of Wei et al . ( 2021 ) . This improvement is in the order of Õ ( −1 ) and is due to the use of entropy regularization in the PE algorithm , which improves the geometry of the bilinear matrix game . Second , the rest of the improvement of sample complexity ( about Õ ( −1.5 ) ) comes from the stochastic level . Specifically , our stochastic PE ( SPE ) allows to use a large constant learning rate η = O ( 1− γ ) , whereas the OGDA-based algorithm in ( Wei et al. , 2021 ) needs to use a substantially smaller learning rate η = O ( √ ( 1− γ ) 5|S|−1 ) , which significantly slows down its convergence in both the population and stochastic levels . Moreover , both the PE and our SPE take O ( ln −1 ) inner updates to achieve an accurate solution of the matrix game ( 7 ) , whereas the OGDA-based algorithm uses only one inner update to solve the matrix game and hence suffers from a larger optimization error . Finally , as we explained in the previous technical novelty paragraph , we improve the techniques used in bounding the estimation errors . Specifically , Wei et al . ( 2021 ) bounds the estimation errors in all the iterations by a small constant using independent samples . As a comparison , we establish a key contraction property of the estimation error over the iterations with correlated Markovian samples . Such a property allows us to use a growing batch size that bounds the errors loosely in the initial iterations and achieves tight error bounds in the end . 5 CONCLUSION In this paper , we developed a model-free , provably convergent , sample efficient , symmetric and private stochastic policy extra gradient algorithm for solving two-player zero-sum Markov games . Our algorithm leverages entropy regularization to facilitate the algorithm convergence and develops new stochastic estimators to accurately estimate the value functions . We proved that our SPE algorithm achieved a fast convergence rate in terms of the Nash equilibrium duality gap and moreover , achieves a substantially improved sample complexity over the state-of-the-art result . We believe our algorithm deepens the understanding of Markov games from a computation complexity perspective . In the future study , it is interesting to extend SPE algorithm to the multi-agent setting for solving general-sum Markov games and competitive games that involve multiple cooperative teams . Another interesting direction is to improve the algorithm to further reduce the sample complexity to approach the theoretical lower bound established in ( Zhang et al. , 2020 ) . ACKNOWLEDGMENTS The work of Ziyi Chen , Shaocong Ma and Yi Zhou was supported in part by U.S. National Science Foundation under the Grants CCF-2106216 and DMS-2134223 . REFERENCES Bai , Y. , Jin , C. , and Yu , T. ( 2020 ) . Near-optimal reinforcement learning with self-play . Proc . Advances in Neural Information Processing Systems ( Neurips ) , 33 . Cen , S. , Cheng , C. , Chen , Y. , Wei , Y. , and Chi , Y . ( 2020 ) . Fast global convergence of natural policy gradient methods with entropy regularization . ArXiv:2007.06558 . Cen , S. , Wei , Y. , and Chi , Y . ( 2021 ) . Fast policy extragradient methods for competitive games with entropy regularization . ArXiv:2105.15186 . Daskalakis , C. , Foster , D. J. , and Golowich , N. ( 2020 ) . Independent policy gradient methods for competitive reinforcement learning . In Proc . Advances in Neural Information Processing Systems ( Neurips ) , volume 33 . Deng , X. , Li , Y. , Mguni , D. H. , Wang , J. , and Yang , Y . ( 2021 ) . On the complexity of computing markov perfect equilibrium in general-sum stochastic games . ArXiv:2109.01795 . Durrett , R. ( 2019 ) . Probability : theory and examples , volume 49 . Cambridge university press . Elie , R. , Perolat , J. , Laurière , M. , Geist , M. , and Pietquin , O . ( 2020 ) . On the convergence of model free learning in mean field games . In Proc . the AAAI Conference on Artificial Intelligence ( AAAI ) , volume 34 , pages 7143–7150 . Ernst , D. , Geurts , P. , and Wehenkel , L. ( 2005 ) . Tree-based batch mode reinforcement learning . Journal of Machine Learning Research , 6:503–556 . Fan , J. , Wang , Z. , Xie , Y. , and Yang , Z . ( 2020 ) . A theoretical analysis of deep q-learning . In Proc . Learning for Dynamics and Control ( L4DC ) , pages 486–489 . Farahmand , A.-M. and Ghavamzadeh , M. ( 2021 ) . Pid accelerated value iteration algorithm . In Proc . International Conference on Machine Learning ( ICML ) , pages 3143–3153 . Geist , M. , Scherrer , B. , and Pietquin , O . ( 2019 ) . A theory of regularized markov decision processes . In Proc . International Conference on Machine Learning ( ICML ) , pages 2160–2169 . Ghosh , M. K. , Golui , S. , Pal , C. , and Pradhan , S. ( 2021 ) . Zero-sum games for continuous-time markov decision processes with risk-sensitive average cost criterion . ArXiv:2109.08837 . Gu , H. , Guo , X. , Wei , X. , and Xu , R. ( 2021 ) . Mean-field multi-agent reinforcement learning : A decentralized network approach . ArXiv:2108.02731 . Guo , H. , Fu , Z. , Yang , Z. , and Wang , Z . ( 2021 ) . Decentralized single-timescale actor-critic on zero-sum two-player stochastic games . In Proc . International Conference on Machine Learning ( ICML ) , pages 3899–3909 . Guo , X. , Hu , A. , Xu , R. , and Zhang , J . ( 2019 ) . Learning mean-field games . Proc . Advances in Neural Information Processing Systems ( Neurips ) , 32:4966–4976 . Hu , J. and Wellman , M. P. ( 2003 ) . Nash q-learning for general-sum stochastic games . Journal of machine learning research , 4 ( Nov ) :1039–1069 . Huang , B. , Lee , J. D. , Wang , Z. , and Yang , Z . ( 2021 ) . Towards general function approximation in zero-sum markov games . ArXiv:2107.14702 . Jafarnia-Jahromi , M. , Jain , R. , and Nayyar , A . ( 2021 ) . Learning zero-sum stochastic games with posterior sampling . ArXiv:2109.03396 . Jin , Y. and Sidford , A . ( 2021 ) . Towards tight bounds on the sample complexity of average-reward mdps . In Proc . International Conference on Machine Learning ( ICML ) , pages 5055–5064 . Kononen , V. and Oja , E. ( 2004 ) . Asymmetric multiagent reinforcement learning in pricing applications . In Proc . IEEE International Joint Conference on Neural Networks ( IEEE Cat . No . 04CH37541 ) , volume 2 , pages 1097–1102 . Leonardos , S. , Overman , W. , Panageas , I. , and Piliouras , G. ( 2021 ) . Global convergence of multiagent policy gradient in markov potential games . ArXiv:2106.01969 . Liu , B. , Yang , Z. , and Wang , Z . ( 2021 ) . Policy optimization in zero-sum markov games : Fictitious self-play provably attains nash equilibria . Mei , J. , Xiao , C. , Szepesvari , C. , and Schuurmans , D. ( 2020 ) . On the global convergence rates of softmax policy gradient methods . In Proc . International Conference on Machine Learning ( ICML ) , pages 6820–6829 . Mertikopoulos , P. and Sandholm , W. H. ( 2016 ) . Learning in games via reinforcement and regularization . Mathematics of Operations Research , 41 ( 4 ) :1297–1324 . Moerland , T. M. , Broekens , J. , Plaat , A. , and Jonker , C. M. ( 2018 ) . A0c : Alpha zero in continuous action space . ArXiv:1805.09613 . Neu , G. , Jonsson , A. , and Gómez , V. ( 2017 ) . A unified view of entropy-regularized markov decision processes . ArXiv:1705.07798 . Ortner , P. and Auer , R. ( 2007 ) . Logarithmic online regret bounds for undiscounted reinforcement learning . In Proc . Advances in Neural Information Processing Systems ( Neurips ) , volume 19 , page 49 . Ortner , R. ( 2020 ) . Regret bounds for reinforcement learning via markov chain concentration . Journal of Artificial Intelligence Research , 67:115–128 . Paulin , D. ( 2015 ) . Concentration inequalities for markov chains by marton couplings and spectral methods . Electronic Journal of Probability , 20:1–32 . Savas , Y. , Ahmadi , M. , Tanaka , T. , and Topcu , U . ( 2019 ) . Entropy-regularized stochastic games . In Proc . IEEE 58th Conference on Decision and Control ( CDC ) , pages 5955–5962 . Shapley , L. S. ( 1953 ) . Stochastic games . Proceedings of the national academy of sciences , 39 ( 10 ) :1095–1100 . Sidford , A. , Wang , M. , Yang , L. , and Ye , Y . ( 2020 ) . Solving discounted stochastic two-player games with near-optimal time and sample complexity . In Proc . International Conference on Artificial Intelligence and Statistics ( AISTATS ) , pages 2992–3002 . Silver , D. , Schrittwieser , J. , Simonyan , K. , Antonoglou , I. , Huang , A. , Guez , A. , Hubert , T. , Baker , L. , Lai , M. , Bolton , A. , et al . ( 2017 ) . Mastering the game of go without human knowledge . nature , 550 ( 7676 ) :354–359 . Sutton , R. S. and Barto , A. G. ( 2018 ) . Reinforcement learning : An introduction . MIT press . Tamar , A. , Wu , Y. , Thomas , G. , Levine , S. , and Abbeel , P. ( 2016 ) . Value iteration networks . In Proc . Advances in neural information processing systems ( Neurips ) , pages 2154–2162 . Vinyals , O. , Babuschkin , I. , Czarnecki , W. M. , Mathieu , M. , Dudzik , A. , Chung , J. , Choi , D. H. , Powell , R. , Ewalds , T. , Georgiev , P. , et al . ( 2019 ) . Grandmaster level in starcraft ii using multi-agent reinforcement learning . Nature , 575 ( 7782 ) :350–354 . Wang , X. and Sandholm , T. ( 2002 ) . Reinforcement learning to play an optimal nash equilibrium in team markov games . volume 15 , pages 1603–1610 . Wei , C.-Y. , Hong , Y.-T. , and Lu , C.-J . ( 2017 ) . Online reinforcement learning in stochastic games . In Proc . Advances in neural information processing systems ( Neurips ) , pages 4994–5004 . Wei , C.-Y. , Lee , C.-W. , Zhang , M. , and Luo , H. ( 2021 ) . Last-iterate convergence of decentralized optimistic gradient descent/ascent in infinite-horizon competitive markov games . In Proc . Conference on Learning Theory ( COLT ) . Xia , Y. , Ding , W. , Zhang , X.-D. , Yu , N. , and Qin , T. ( 2016 ) . Budgeted bandit problems with continuous random costs . In Proc . Asian conference on machine learning ( ACML ) , pages 317–332 . Xu , Y. , Deng , Z. , Wang , M. , Xu , W. , So , A. M.-C. , and Cui , S. ( 2020 ) . Voting-based multiagent reinforcement learning for intelligent iot . IEEE Internet of Things Journal , 8 ( 4 ) :2681–2693 . Zhang , K. , Kakade , S. , Basar , T. , and Yang , L. ( 2020 ) . Model-based multi-agent rl in zero-sum markov games with near-optimal sample complexity . In Proc . Advances in Neural Information Processing Systems ( Neurips ) , volume 33 . Zhang , K. , Yang , Z. , Liu , H. , Zhang , T. , and Basar , T. ( 2021 ) . Finite-sample analysis for decentralized batch multi-agent reinforcement learning with networked agents . IEEE Transactions on Automatic Control . Zhang , X. , Zhang , K. , Miehling , E. , and Basar , T. ( 2019 ) . Non-cooperative inverse reinforcement learning . In Proc . Advances in Neural Information Processing Systems ( Neurips ) , volume 32 , pages 9487–9497 . Zhao , Y. , Tian , Y. , Lee , J. D. , and Du , S. S. ( 2021 ) . Provably efficient policy gradient methods for two-player zero-sum markov games . ArXiv:2102.08903 . Zhu , Y. and Zhao , D. ( 2020 ) . Online minimax q network learning for two-player zero-sum markov games . IEEE Transactions on Neural Networks and Learning Systems . Zou , S. , Xu , T. , and Liang , Y . ( 2019 ) . Finite-sample analysis for sarsa and q-learning with linear function approximation . ArXiv:1902.02234 . Appendix Table of Contents A Analysis of stochastic PU ( SPU ) for entropy-regularized matrix game 13 B Estimation error bounds 18 C Properties of the Duality Gap 25 D Proof of Theorem 1 27 E Proof of Theorem 2 31 F Summary of Comparison of Sample Complexities 34 G Rationality 35 A ANALYSIS OF STOCHASTIC PU ( SPU ) FOR ENTROPY-REGULARIZED MATRIX GAME We first consider the following zero-sum entropy regularized matrix game , which can be considered as a simple special case of Markov game with only one state . max µ∈∆ ( d1 ) min ν∈∆ ( d2 ) hτ ( µ , ν ) : = µ > Qν + τH ( µ ) − τH ( ν ) . ( 22 ) where Q ∈ Rd1×d2 is fixed . The solution ( µ∗τ , ν∗τ ) of the above problem , also known as the quantal response equilibrium ( QRE ) , satisfies the following condition ( Cen et al. , 2021 ) µ∗τ ∝ exp ( Qν∗τ /τ ) , ν∗τ ∝ exp ( −Q > µ∗τ/τ ) . ( 23 ) Our stochastic PU ( SPU ) algorithm for the above matrix game is written as follows , µt+1 ( a ) ∝µt ( a ) 1−ητ exp ( η [ Qνt + δ ( 1 ) t ] a ) ( 24 ) νt+1 ( b ) ∝νt ( b ) 1−ητ exp ( − η [ Q > µt + δ ( 2 ) t ] b ) ( 25 ) µt+1 ( a ) ∝µt ( a ) 1−ητ exp ( η [ Qνt+1 + δ ( 1 ) t ] a ) ( 26 ) νt+1 ( b ) ∝νt ( b ) 1−ητ exp ( − η [ Q > µt+1 + δ ( 2 ) t ] b ) , ( 27 ) where δ ( 1 ) t is an additive noise to the underlying true quantity Qνt , and the other noises δ ( 2 ) t , δ ( 1 ) t , δ ( 2 ) t are similar . If all these noises are zero , the above stochastic PU algorithm reduces to the PU algorithm for matrix game defined in Section 2 of ( Cen et al. , 2021 ) . The convergence rate of stochastic PU for matrix game is shown below . The proof logic follows from that of ( Cen et al. , 2021 ) . Lemma 1 . Choose learning rate η ≤ [ 2 ( τ + ‖Q‖∞ ) ] −1 and use the uniform policy initialization µ0 = 1/d1 , ν0 = 1/d2 . Then , the SPU algorithm defined by eqs . ( 24 ) - ( 27 ) for entropy-regularized matrix game has the following convergence rates . KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) ≤ ( 1− ητ ) t ln ( d1d2 ) + t−1∑ j=0 ( 1− ητ ) t−1−j [ 2η2 ( ‖δ ( 1 ) j ‖ 2 ∞ + ‖δ ( 2 ) j ‖ 2 ∞ ) + 2η τ ( ‖δ ( 1 ) j ‖2∞ + ‖δ ( 2 ) j ‖2∞ ) ] . ( 28 ) ∣∣hτ ( µ ? τ , ν ? τ ) − hτ ( µt+1 , νt+1 ) ∣∣ ≤ 4 η ( 1− ητ ) t ln ( d1d2 ) + t∑ j=0 ( 1− ητ ) t−1−j [ 8η ( ‖δ ( 1 ) j ‖ 2 ∞ + ‖δ ( 2 ) j ‖ 2 ∞ ) + 33 4τ ( ‖δ ( 1 ) j ‖2∞ + ‖δ ( 2 ) j ‖2∞ ) ] . ( 29 ) max µ∈∆ ( d1 ) hτ ( µ , νt+1 ) − min ν∈∆ ( d2 ) hτ ( µt+1 , ν ) ≤ 2 η ( 1− ητ ) t ln ( d1d2 ) + t∑ j=0 ( 1− ητ ) t−1−j [ 16η ( ‖δ ( 1 ) j ‖ 2 ∞ + ‖δ ( 2 ) j ‖ 2 ∞ ) + 12 τ ( ‖δ ( 1 ) j ‖2∞ + ‖δ ( 2 ) j ‖2∞ ) ] . ( 30 ) Proof . Eq . ( 26 ) implies that for some constant c ∈ R , lnµt+1 = ( 1− ητ ) lnµt + η ( Qνt+1 + δ ( 1 ) t ) + c1 ⇒ 〈 lnµt+1 − ( 1− ητ ) lnµt − η ( Qνt+1 + δ ( 1 ) t ) , µt+1 − µ∗τ 〉 = 0 . ( 31 ) where we used 〈µt+1 − µ∗τ ,1〉 = 0 . Similarly , eqs . ( 27 ) & ( 23 ) respectively imply that〈 ln νt+1 − ( 1− ητ ) ln νt + η ( Q > µt+1 + δ ( 2 ) t ) , νt+1 − ν∗τ 〉 =0 ( 32 ) 〈 lnµ∗τ −Qν∗τ /τ , µt+1 − µ∗τ 〉 =0 ( 33 ) 〈 ln ν∗τ +Q > µ∗τ/τ , νt+1 − ν∗τ 〉 =0 . ( 34 ) Similarly , eqs . ( 24 ) & ( 26 ) imply that〈 ln ( µt+1/µt+1 ) , µt+1 − µt+1 〉 = 〈 η [ Q ( νt − νt+1 ) + δ ( 1 ) t − δ ( 1 ) t ] , µt+1 − µt+1 〉 ≤ η‖Q‖∞‖µt+1 − µt+1‖1‖νt − νt+1‖1 + η‖δ ( 1 ) t − δ ( 1 ) t ‖∞‖µt+1 − µt+1‖1 ≤ η 2 ‖Q‖∞ ( ‖µt+1 − µt+1‖21 + ‖νt − νt+1‖21 ) + η2 2 ( 1− η‖Q‖∞ ) ‖δ ( 1 ) t − δ ( 1 ) t ‖2∞ + 1 2 ( 1− η‖Q‖∞ ) ‖µt+1 − µt+1‖21 ( i ) ≤ KL ( µt+1‖µt+1 ) + η‖Q‖∞KL ( νt+1‖νt ) + 2η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 1 ) t ‖2∞ ) , ( 35 ) where ( i ) uses the Pinsker ’ s inequality and η ≤ [ 2 ( τ + ‖Q‖∞ ) ] −1 . Similarly to the above derivation , we can infer from eqs . ( 25 ) & ( 27 ) that〈 ln ( νt+1/νt+1 ) , νt+1 − νt+1 〉 ≤KL ( νt+1‖νt+1 ) + η‖Q‖∞KL ( µt+1‖µt ) + 2η2 ( ‖δ ( 2 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) . ( 36 ) Computing eq . ( 31 ) +eq . ( 32 ) -ητ [ eq . ( 33 ) +eq . ( 34 ) ] yields that η〈δ ( 1 ) t , µt+1 − µ∗τ 〉 − η〈δ ( 2 ) t , νt+1 − ν∗τ 〉 = 〈 lnµt+1 − ( 1− ητ ) lnµt − ητ lnµ∗τ − ηQ ( νt+1 − ν∗τ ) , µt+1 − µ∗τ 〉 + 〈 ln νt+1 − ( 1− ητ ) ln νt − ητ ln ν∗τ + ηQ > ( µt+1 − µ∗τ ) , νt+1 − ν∗τ 〉 = 〈 ln ( µt+1/µ ∗ τ ) − ( 1− ητ ) ln ( µt/µ∗τ ) , µt+1 − µ∗τ 〉 + 〈 ln ( νt+1/ν ∗ τ ) − ( 1− ητ ) ln ( νt/ν∗τ ) , νt+1 − ν∗τ 〉 =KL ( µ∗τ‖µt+1 ) − ( 1− ητ ) KL ( µ∗τ‖µt ) + 〈 ln ( µt+1/µ ∗ τ ) − ln ( µt+1/µt+1 ) − ( 1− ητ ) [ ln ( µt+1/µ ∗ τ ) − ln ( µt+1/µt ) ] , µt+1 〉 +KL ( ν∗τ ‖νt+1 ) − ( 1− ητ ) KL ( ν∗τ ‖νt ) + 〈 ln ( νt+1/ν ∗ τ ) − ln ( νt+1/νt+1 ) − ( 1− ητ ) [ ln ( νt+1/ν ∗ τ ) − ln ( νt+1/νt ) ] , νt+1 〉 =KL ( µ∗τ‖µt+1 ) − ( 1− ητ ) KL ( µ∗τ‖µt ) + ητKL ( µt+1‖µ∗τ ) + ( 1− ητ ) KL ( µt+1‖µt ) +KL ( µt+1‖µt+1 ) +KL ( ν∗τ ‖νt+1 ) − ( 1− ητ ) KL ( ν∗τ ‖νt ) + ητKL ( νt+1‖ν∗τ ) + ( 1− ητ ) KL ( νt+1‖νt ) +KL ( νt+1‖νt+1 ) − 〈 ln ( µt+1/µt+1 ) , µt+1 − µt+1 〉 − 〈 ln ( νt+1/νt+1 ) , νt+1 − νt+1 〉 ( i ) ≥KL ( µ∗τ‖µt+1 ) − ( 1− ητ ) KL ( µ∗τ‖µt ) + ητKL ( µt+1‖µ∗τ ) + ( 1− ητ − η‖Q‖∞ ) KL ( µt+1‖µt ) +KL ( ν∗τ ‖νt+1 ) − ( 1− ητ ) KL ( ν∗τ ‖νt ) + ητKL ( νt+1‖ν∗τ ) + ( 1− ητ − η‖Q‖∞ ) KL ( νt+1‖νt ) − 2η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) , ( 37 ) where ( i ) uses eqs . ( 35 ) & ( 36 ) . The left side of the above inequality has the following upper bound . η〈δ ( 1 ) t , µt+1 − µ∗τ 〉 − η〈δ ( 2 ) t , νt+1 − ν∗τ 〉 ≤η‖δ ( 1 ) t ‖∞‖µt+1 − µ∗τ‖1 + η‖δ ( 2 ) t ‖∞‖νt+1 − ν∗τ ‖1 ≤η 2 ( 2‖δ ( 1 ) t ‖2∞ τ + τ 2 ‖µt+1 − µ∗τ‖21 + 2‖δ ( 2 ) t ‖2∞ τ + τ 2 ‖νt+1 − ν∗τ ‖21 ) ≤η τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + ητ 2 ( KL ( µt+1‖µ∗τ ) +KL ( νt+1‖ν∗τ ) ) . ( 38 ) Substituting eq . ( 38 ) into eq . ( 37 ) and rearranging it yields that KL ( µ∗τ‖µt+1 ) +KL ( ν∗τ ‖νt+1 ) + ητ 2 [ KL ( µt+1‖µ∗τ ) +KL ( νt+1‖ν∗τ ) ] + 1 2 [ KL ( µt+1‖µt ) +KL ( νt+1‖νt ) ] ≤ ( 1− ητ ) [ KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) ] + 2η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 2η τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) , ( 39 ) where we use η ≤ [ 2 ( τ + ‖Q‖∞ ) ] −1 ≤ ( 2τ ) −1 . This further implies that KL ( µ∗τ‖µt+1 ) +KL ( ν∗τ ‖νt+1 ) ≤ ( 1− ητ ) [ KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) ] + 2η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 2η τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) . ( 40 ) Consequently , eq . ( 28 ) can be proved via iterating the above inequality and using the facts that KL ( µ∗τ‖µ0 ) ≤ ln d1 , KL ( ν∗τ ‖ν0 ) ≤ ln d2 ( Since µ0 and ν0 are uniform probability vectors ) . Next , we will prove eq . ( 29 ) . Computing eq . ( 31 ) −ητ ·eq . ( 33 ) yields that 〈Q ( νt+1 − ν∗τ ) , µt+1 − µ∗τ 〉 = η−1 〈 lnµt+1 − ( 1− ητ ) lnµt − ητ lnµ∗τ − ηδ ( 1 ) t , µt+1 − µ∗τ 〉 . ( 41 ) Hence , we conclude that hτ ( µt+1 , νt+1 ) − hτ ( µ ? τ , ν ? τ ) ≤ hτ ( µt+1 , νt+1 ) − hτ ( µt+1 , ν ? τ ) = µ > t+1Q ( νt+1 − ν ? τ ) + τν > t+1 ln νt+1 − τν ? > τ ln ν ? τ = ( µt+1 − µ ? τ ) > Q ( νt+1 − ν ? τ ) + µ ? > τ Q ( νt+1 − ν ? τ ) + τν > t+1 ln νt+1 − τν ? > τ ln ν ? τ ( i ) = η−1 〈 lnµt+1 − ( 1− ητ ) lnµt − ητ lnµ∗τ − ηδ ( 1 ) t , µt+1 − µ∗τ 〉 − τ 〈 ln ν∗τ , νt+1 − ν∗τ 〉 + τν > t+1 ln νt+1 − τν ? > τ ln ν ? τ ( ii ) = η−1 [ KL ( µ∗τ‖µt+1 ) − ( 1− ητ ) KL ( µ∗τ‖µt ) + ητKL ( µt+1‖µ∗τ ) + ( 1− ητ ) KL ( µt+1‖µt ) +KL ( µt+1‖µt+1 ) − 〈 ln ( µt+1/µt+1 ) , µt+1 − µt+1 〉 ] + τKL ( νt+1‖ν ? τ ) − 〈δ ( 1 ) t , µt+1 − µ∗τ 〉 ≤ η−1 [ KL ( µ∗τ‖µt+1 ) − ( 1− ητ ) KL ( µ∗τ‖µt ) + ητ ( KL ( µt+1‖µ∗τ ) +KL ( νt+1‖ν ? τ ) ) + ( 1− ητ ) KL ( µt+1‖µt ) −KL ( µt+1‖µt+1 ) ] + 1 2τ ‖δ ( 1 ) t ‖2∞ + τ 2 ‖µt+1 − µ∗τ‖21 ( iii ) ≤ η−1 [ KL ( µ∗τ‖µt+1 ) + 2ητ ( KL ( µt+1‖µ∗τ ) +KL ( νt+1‖ν ? τ ) ) +KL ( µt+1‖µt ) ] + 1 4τ ‖δ ( 1 ) t ‖2∞ ( iv ) ≤ 4η−1 [ KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) + 2η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 2η τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) ] + 1 4τ ‖δ ( 1 ) t ‖2∞ ( v ) ≤ 4η−1 [ KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) ] + 8η ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 33 4τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) , ( 42 ) where ( i ) uses eq . ( 41 ) & ( 34 ) , ( ii ) follows the derivation of eq . ( 37 ) , ( iii ) uses the Pinsker ’ s inequality , ( iv ) uses eq . ( 39 ) , and ( v ) uses η ≤ [ 2 ( τ + ‖Q‖∞ ) ] −1 . In a similar way , we can also prove that hτ ( µ ? τ , ν ? τ ) − hτ ( µt+1 , νt+1 ) ≤ hτ ( µ ? τ , νt+1 ) − hτ ( µt+1 , νt+1 ) ≤ 4η−1 [ KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) ] + 8η ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 33 4τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) . ( 43 ) Combining eqs . ( 42 ) & ( 43 ) yields that∣∣hτ ( µ ? τ , ν ? τ ) − hτ ( µt+1 , νt+1 ) ∣∣ ≤ 4η−1 [ KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) ] + 8η ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 33 4τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) ( i ) ≤ 4η−1 ( 1− ητ ) t ln ( d1d2 ) + t−1∑ j=0 ( 1− ητ ) t−1−j [ 8η ( ‖δ ( 1 ) j ‖ 2 ∞ + ‖δ ( 2 ) j ‖ 2 ∞ ) + 8 τ ( ‖δ ( 1 ) j ‖2∞ + ‖δ ( 2 ) j ‖2∞ ) ] + 8η ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 33 4τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) ≤ 4η−1 ( 1− ητ ) t ln ( d1d2 ) + t∑ j=0 ( 1− ητ ) t−1−j [ 8η ( ‖δ ( 1 ) j ‖ 2 ∞ + ‖δ ( 2 ) j ‖ 2 ∞ ) + 33 4τ ( ‖δ ( 1 ) j ‖2∞ + ‖δ ( 2 ) j ‖2∞ ) ] , where ( i ) uses eq . ( 28 ) . This proves eq . ( 29 ) . Next , to prove the duality gap ( 30 ) , we first derive an upper bound ofKL ( µ ? τ‖µt+1 ) +KL ( ν ? τ ‖νt+1 ) . KL ( ν ? τ ‖νt+1 ) =KL ( ν ? τ ‖νt+1 ) −KL ( νt+1‖νt+1 ) + 〈 νt+1 − ν ? τ , ln νt+1 − ln νt+1 〉 ≤KL ( ν ? τ ‖νt+1 ) + 〈 νt+1 − ν ? τ , ηQ > ( µt+1 − µt ) + η ( δ ( 2 ) t − δ ( 2 ) t ) 〉 ≤KL ( ν ? τ ‖νt+1 ) + ‖νt+1 − ν ? τ ‖1 ( η‖Q‖∞‖µt+1 − µt‖1 + η‖δ ( 2 ) t − δ ( 2 ) t ‖∞ ) ≤KL ( ν ? τ ‖νt+1 ) + 1 2 [ ( η‖Q‖∞ + 1 4 ) ‖νt+1 − ν ? τ ‖21 + η‖Q‖∞‖µt+1 − µt‖21 + 4η2‖δ ( 2 ) t − δ ( 2 ) t ‖2∞ ] ( i ) ≤KL ( ν ? τ ‖νt+1 ) + 3 4 KL ( ν ? τ ‖νt+1 ) + 1 2 KL ( µt+1‖µt ) + 4η2 ( ‖δ ( 2 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) , ( 44 ) where ( i ) uses the Pinker ’ s inequality and η ≤ [ 2 ( τ + ‖Q‖∞ ) ] −1 . Rearranging the above inequality yields that KL ( ν ? τ ‖νt+1 ) ≤4KL ( ν ? τ ‖νt+1 ) + 2KL ( µt+1‖µt ) + 16η2 ( ‖δ ( 2 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) . ( 45 ) Similarly , we can prove that KL ( µ ? τ‖µt+1 ) ≤4KL ( µ ? τ‖µt+1 ) + 2KL ( νt+1‖νt ) + 16η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 1 ) t ‖2∞ ) . ( 46 ) Summing up eqs . ( 45 ) & ( 46 ) yields that KL ( µ ? τ‖µt+1 ) +KL ( ν ? τ ‖νt+1 ) ≤ 4 [ KL ( µ∗τ‖µt+1 ) +KL ( ν∗τ ‖νt+1 ) ] + 2 [ KL ( µt+1‖µt ) +KL ( νt+1‖νt ) ] + 16η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) ( i ) ≤ 4 [ KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) ] + 24η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 16η τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) , ( 47 ) where ( i ) uses eq . ( 39 ) and η ≤ [ 2 ( τ + ‖Q‖∞ ) ] −1 ≤ ( 2τ ) −1 . Finally , we prove the duality gap bound ( 30 ) . hτ ( µ ? τ , ν ? τ ) − hτ ( µ , ν ? τ ) = ( µ ? τ − µ ) > Qν ? τ + τµ > lnµ− τµ ? > τ lnµ ? τ ( i ) = τ ( 〈µ ? τ − µ , lnµ ? τ 〉+ µ > lnµ− µ ? > τ lnµ ? τ ) = τKL ( µ‖µ ? τ ) , ( 48 ) where ( i ) uses eq . ( 33 ) . hτ ( µ , νt+1 ) − hτ ( µ , ν ? τ ) = µ > Q ( νt+1 − ν ? τ ) + τν > t+1 ln νt+1 − τν ? > τ ln ν ? τ = ( µ− µ ? τ ) > Q ( νt+1 − ν ? τ ) + µ ? > τ Q ( νt+1 − ν ? τ ) + τν > t+1 ln νt+1 − τν ? > τ ln ν ? τ ( i ) ≤ ‖µ− µ ? τ‖1‖Q‖∞‖νt+1 − ν ? τ ‖1 − τ 〈 ln ν∗τ , νt+1 − ν∗τ 〉 + τν > t+1 ln νt+1 − τν ? > τ ln ν ? τ ≤ 1 2 [ τ‖µ− µ ? τ‖21 + ‖Q‖2∞‖νt+1 − ν ? τ ‖21/τ ] + τKL ( νt+1‖ν ? τ ) ( ii ) ≤ τKL ( µ‖µ ? τ ) + τKL ( νt+1‖ν ? τ ) + τ−1‖Q‖2∞KL ( ν ? τ ‖νt+1 ) , ( 49 ) where ( i ) uses eq . ( 34 ) and ( ii ) uses the Pinsker ’ s inequality . Then , eq . ( 49 ) & eq . ( 48 ) imply that hτ ( µ , νt+1 ) − hτ ( µ ? τ , ν ? τ ) ≤ τKL ( νt+1‖ν ? τ ) + τ−1‖Q‖2∞KL ( ν ? τ ‖νt+1 ) , ( 50 ) where ( i ) uses eq . ( 45 ) , ( ii ) uses the Pinker ’ s inequality and η ≤ [ 2 ( τ + ‖Q‖∞ ) ] −1 . Similarly , it can be proved that hτ ( µ ? τ , ν ? τ ) − hτ ( µt+1 , ν ) ≤ τKL ( µt+1‖µ ? τ ) + τ−1‖Q‖2∞KL ( µ ? τ‖µt+1 ) . ( 51 ) Therefore , the duality gap ( 30 ) can be proved as follows . max µ∈∆ ( d1 ) hτ ( µ , νt+1 ) − min ν∈∆ ( d2 ) hτ ( µt+1 , ν ) = max µ∈∆ ( d1 ) , ν∈∆ ( d2 ) [ hτ ( µ , νt+1 ) − hτ ( µt+1 , ν ) ] ( i ) ≤ τ [ KL ( µt+1‖µ ? τ ) +KL ( νt+1‖ν ? τ ) ] + τ−1‖Q‖2∞ [ KL ( µ ? τ‖µt+1 ) +KL ( ν ? τ ‖νt+1 ) ] ( ii ) ≤ τ [ KL ( µt+1‖µ ? τ ) +KL ( νt+1‖ν ? τ ) ] + τ−1‖Q‖2∞ [ 4 [ KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) ] + 24η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 16η τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) ] ( iii ) ≤ max ( 2 η , 4‖Q‖2∞ τ ) [ KL ( µ∗τ‖µt ) +KL ( ν∗τ ‖νt ) + 2η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 2η τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) ] + τ−1‖Q‖2∞ [ 24η2 ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 16η τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) ] ( iv ) ≤ 2 η ( ( 1− ητ ) t ln ( d1d2 ) + t−1∑ j=0 ( 1− ητ ) t−1−j [ 2η2 ( ‖δ ( 1 ) j ‖ 2 ∞ + ‖δ ( 2 ) j ‖ 2 ∞ ) + 2η τ ( ‖δ ( 1 ) j ‖2∞ + ‖δ ( 2 ) j ‖2∞ ) ] ) + 16η ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) + 12 τ ( ‖δ ( 1 ) t ‖2∞ + ‖δ ( 2 ) t ‖2∞ ) ≤ 2 η ( 1− ητ ) t ln ( d1d2 ) + t∑ j=0 ( 1− ητ ) t−1−j [ 16η ( ‖δ ( 1 ) j ‖ 2 ∞ + ‖δ ( 2 ) j ‖ 2 ∞ ) + 12 τ ( ‖δ ( 1 ) j ‖2∞ + ‖δ ( 2 ) j ‖2∞ ) ] , ( 52 ) where ( i ) adds up eqs . ( 50 ) & ( 51 ) , ( ii ) uses eq . ( 47 ) , ( iii ) uses eq . ( 39 ) , and ( iv ) uses eq . ( 28 ) and 4τ−1‖Q‖2∞ ≤ 2/η ( since η ≤ [ 2 ( τ + ‖Q‖∞ ) ] −1 , τ ≤ 1 ) . This proves eq . ( 30 ) B ESTIMATION ERROR BOUNDS In this section , we derive error bounds for the estimators V̂k+1 ≈ V ′k+1 , Q̂ ( m ) k , t ≈ Q ( m ) k , t , Q̂ ( m ) k , t ≈ Q ( m ) k , t used in Algorithm 1 . For convenience , we define the following additional notations . µ̂k ( s ) : = 1 Nk+1 ∑ i∈Nk+1 1 { si = s } ≈ µk ( s ) ( 53 ) vk+1 ( s ) : = Eµk , π ( 1 ) k , π ( 2 ) k [ [ R ( s̃ , ã ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ] 1 { s̃ = s } ] ( 54 ) v′k+1 ( s ) : = Eµk , π ( 1 ) k , π ( 2 ) k [ [ R ( s̃ , ã ( 1 ) , ã ( 2 ) ) + γV̂k ( s ′ ) ] 1 { s̃ = s } ] ( 55 ) v̂k+1 ( s ) : = 1 Nk+1 ∑ i∈Nk+1 [ [ Ri + γV̂k ( si+1 ) ] 1 { si = s } ] ≈ v′k+1 ( s ) ≈ vk+1 ( s ) ( 56 ) Qk ( s , a ( 1 ) , a ( 2 ) ) : = R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) [ V ′k ( s ′ ) ] ( 57 ) V ′k+1 ( s ) = fτ ( Qk ( s ) ; π ( 1 ) k ( s ) , π ( 2 ) k ( s ) ) = vk+1 ( s ) µk ( s ) + τH ( π ( 1 ) k ( s ) ) − τH ( π ( 2 ) k ( s ) ) ( 58 ) V̂k+1 ( s ) : = v̂k+1 ( s ) µ̂k ( s ) + τH ( π ( 1 ) k ( s ) ) − τH ( π ( 2 ) k ( s ) ) ≈ V ′k+1 ( s ) ( 59 ) µ̂k , t ( s ) : = 1 Nk , t ∑ i∈Nk , t 1 { si = s } ≈ µ′k , t ( s ) ≈ µk , t ( s ) , ( 60 ) q ( m ) k , t ( s , a ( m ) ) : = E µ′k , t , π ′ ( 1 ) k , t , π ′ ( 2 ) k , t [ [ R ( s̃ , ã ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ] 1 { s̃ = s , ã ( m ) = a ( m ) } ] ( 61 ) q ′ ( m ) k , t ( s , a ( m ) ) : = E µ′k , t , π ′ ( 1 ) k , t , π ′ ( 2 ) k , t [ [ R ( s̃ , ã ( 1 ) , ã ( 2 ) ) + γV̂k ( s ′ ) ] 1 { s̃ = s , ã ( m ) = a ( m ) } ] ( 62 ) q̂ ( m ) k , t ( s , a ( m ) ) : = 1 Nk , t ∑ i∈Nk , t [ [ Ri + γV̂k ( si+1 ) ] 1 { si = s , a ( m ) i = a ( m ) } ] ≈ q′ ( m ) k , t ( s , a ( m ) ) ≈ q ( m ) k , t ( s , a ( m ) ) ( 63 ) Q ′ ( m ) k , t ( s , a ( m ) ) : = E a ( \m ) ∼π′ ( \m ) k , t ( s ) [ Qk ( s , a ( 1 ) , a ( 2 ) ) ] = q ( m ) k , t ( s , a ( m ) ) µ′k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) ( 64 ) Q̂ ( m ) k , t ( s , a ( m ) ) : = q̂ ( m ) k , t ( s , a ( m ) ) µ̂k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) ≈ Q′ ( m ) k , t ( s , a ( m ) ) ≈ Q ( m ) k , t ( s , a ( m ) ) , ( 65 ) where eqs . ( 53 ) - ( 59 ) and ( 60 ) - ( 65 ) are introduced for the description and error bound proof of the estimations V̂k+1 ≈ V ′k+1 and Q̂ ( m ) k , t ≈ Q ( m ) k , t , respectively . Specifically , eq . ( 53 ) estimates the stationary state distribution µk associated with the policy pair π ( 1 ) k , π ( 2 ) k using state frequency µ̂k ( s ) . In eqs . ( 54 ) & ( 55 ) , the expectation is taken over s̃ ∼ µk , ã ( 1 ) ∼ π ( 1 ) k ( s ) , ã ( 2 ) ∼ π ( 2 ) k ( s ) , s ′ ∼ P ( ·|s̃ , ã ( 1 ) , ã ( 2 ) ) . The definition of E µ′k , t , π ′ ( 1 ) k , t , π ′ ( 2 ) k , t in eqs . ( 61 ) & ( 62 ) is similar . Equations ( 58 ) & ( 64 ) give two equivalent definitions of V ′k+1 and Q ′ ( m ) k , t ( s , a ( m ) ) , respectively 1 ( We will prove the equivalence in Lemma 2 below ) . Equation ( 60 ) estimates the stationary state distribution µ′k , t associated with the smoothed policy pair π′ ( 1 ) k , t , π ′ ( 2 ) k , t , and thus approximates the stationary state distribution µk , t associated with the current policy pair π ( 1 ) k , t , π ( 2 ) k , t . To prove the estimation error bounds , we first prove the following two lemmas . Lemma 2 . Eq . ( 13 ) and the second “ = ” of eqs . ( 58 ) & ( 64 ) hold for all s , a ( 1 ) , a ( 2 ) . Proof . The second “ = ” of eq . ( 64 ) under m = 1 can be proved as follows . q ( 1 ) k , t ( s , a ( 1 ) ) ( i ) = E µ′k , t , π ′ ( 1 ) k , t , π ′ ( 2 ) k , t [ [ R ( s̃ , ã ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ] 1 { s̃ = s , ã ( m ) = a ( m ) } ] = E s̃∼µ′k , t , ã ( 1 ) ∼π ′ ( 1 ) k , t ( s ) , ã ( 2 ) ∼π′ ( 2 ) k , t ( s ) , s′∼P ( ·|s , a ( 1 ) , ã ( 2 ) ) [ [ R ( s , a ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ] 1 { s̃ = s , ã ( 1 ) = a ( 1 ) } ] ( ii ) = E s̃∼µ′k , t , ã ( 1 ) ∼π ′ ( 1 ) k , t ( s ) [ 1 { s̃ = s , ã ( 1 ) = a ( 1 ) } ] E ã ( 2 ) ∼π′ ( 2 ) k , t ( s ) , s′∼P ( ·|s , a ( 1 ) , ã ( 2 ) ) [ R ( s , a ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ] ( iii ) = µ′k , t ( s ) π ′ ( 1 ) k , t ( a ( 1 ) |s ) E ã ( 2 ) ∼π′ ( 2 ) k , t ( s ) [ Qk ( s , a ( 1 ) , ã ( 2 ) ) ] ( iv ) = µ′k , t ( s ) π ′ ( 1 ) k , t ( a ( 1 ) |s ) Q′k , t ( s , a ( 1 ) ) , where ( i ) uses eq . ( 61 ) which denotes E µ′k , t , π ′ ( 1 ) k , t , π ′ ( 2 ) k , t as expectation over s̃ ∼ µ′k , t , ã ( 1 ) ∼ π ′ ( 1 ) k , t ( s̃ ) , ã ( 2 ) ∼ π′ ( 2 ) k , t ( s̃ ) , s′ ∼ P ( ·|s̃ , ã ( 1 ) , ã ( 2 ) ) , ( ii ) uses independence between s̃ ∼ µ′k , t , ã ( 1 ) ∼ π ′ ( 1 ) k , t ( s ) and ã ( 2 ) ∼ π′ ( 2 ) k , t ( s ) , s′ ∼ P ( ·|s , a ( 1 ) , ã ( 2 ) ) , ( iii ) uses eq . ( 57 ) , and ( iv ) uses the first definition of eq . ( 64 ) . The second “ = ” of eq . ( 64 ) under m = 2 and eq . ( 13 ) can be proved in a similar way . The second “ = ” of eq . ( 58 ) can be proved as follows . vk+1 ( s ) ( i ) = E µk , π ( 1 ) k , π ( 2 ) k [ [ R ( s̃ , ã ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ] 1 { s̃ = s } ] = E s̃∼µk , ã ( 1 ) ∼π ( 1 ) k ( s ) , ã ( 2 ) ∼π ( 2 ) k ( s ) , s ′∼P ( ·|s , ã ( 1 ) , ã ( 2 ) ) [ [ R ( s , ã ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ] 1 { s̃ = s } ] 1The definition after the second “ = ” of eq . ( 58 ) is the same as eq . ( 11 ) ( ii ) = Es̃∼µk [ 1 { s̃ = s } ] E ã ( 1 ) ∼π ( 1 ) k ( s ) , ã ( 2 ) ∼π ( 2 ) k ( s ) , s ′∼P ( ·|s , ã ( 1 ) , ã ( 2 ) ) [ R ( s , ã ( 1 ) , ã ( 2 ) ) + γV ′k ( s ′ ) ] ( iii ) = µk ( s ) Eã ( 1 ) ∼π ( 1 ) k ( s ) , ã ( 2 ) ∼π ( 2 ) k ( s ) [ Qk ( s , ã ( 1 ) , ã ( 2 ) ) ] = µk , t ( s ) [ fτ ( Qk ( s ) ; π ( 1 ) k ( s ) , π ( 2 ) k ( s ) ) − τH ( π ( 1 ) k ( s ) ) + τH ( π ( 2 ) k ( s ) ) ] ( iv ) = µk ( s ) [ V ′k+1 ( s ) − τH ( π ( 1 ) k ( s ) ) + τH ( π ( 2 ) k ( s ) ) ] where ( i ) uses eq . ( 54 ) , ( ii ) uses independence between s′ ∼ µk and ã ( 1 ) ∼ π ( 1 ) k ( s ) , ã ( 2 ) ∼ π ( 2 ) k ( s ) , s ′ ∼ P ( ·|s , ã ( 1 ) , ã ( 2 ) ) , ( iii ) uses eq . ( 57 ) , and ( iv ) uses eq . ( 58 ) . Lemma 3 . If ‖V̂0‖∞ ≤ Vmax : = 1+τ lnAmax1−γ , then for all k ≥ 0 , 0 ≤ t ≤ Tk − 1 , we have ‖V̂k‖∞ , ‖V ′k‖∞ , ‖V ∗‖∞ ≤ Vmax , ( 66 ) |v̂k+1 ( s ) | ≤ 2Vmaxµ̂k ( s ) , ∀s ∈ S , ( 67 ) max ( ‖Qk‖∞ , ‖Q′ ( m ) k , t ‖∞ , ‖Q ∗‖∞ ) ≤ Qmax , ( 68 ) where Qmax : = 1 + γVmax = 1+γτ lnAmax1−γ with Amax : = max ( |A ( 1 ) | , |A ( 2 ) | ) . Proof . We will first prove ‖V̂k‖∞ ≤ Vmax in eq . ( 66 ) by induction . Suppose ‖V̂k′‖∞ ≤ Vmax holds for a certain k′ ∈ N. Then , eq . ( 59 ) implies that for all s ∈ S , |V̂k′+1 ( s ) | ≤ ∣∣∣ v̂k′+1 ( s ) µ̂′k′ ( s ) ∣∣∣+ τ [ H ( π ( 1 ) k′ ( s ) ) −H ( π ( 2 ) k′ ( s ) ) ] ≤ ∑ i∈Nk′+1 [ [ |Ri|+ γ|V̂k′ ( si+1 ) | ] 1 { si = s } ] ∑ i∈Nk′+1 1 { si = s } + τ lnAmax ( i ) ≤ ∑ i∈Nk′+1 [ ( 1 + γVmax ) 1 { si = s } ] ∑ i∈Nk′+1 1 { si = s } + τ lnAmax =1 + γ ( 1 + τ lnAmax ) 1− γ + τ lnAmax = Vmax , ( 69 ) where ( i ) uses the inequality that 0 ≤ H ( π ( m ) ) ≤ ln |A ( m ) | ≤ lnAmax , ∀π ( m ) ∈ ∆ ( |A ( m ) | ) . Since ‖V̂0‖∞ ≤ Vmax , ‖V̂k‖∞ ≤ Vmax for all k ∈ N. A similar induction yields that ‖V ′k‖∞ ≤ Vmax . Next , we prove ‖V ∗‖∞ ≤ Vmax , ‖Q∗‖∞ ≤ Qmax in eqs . ( 66 ) & ( 68 ) respectively . Notice that V ∗ and Q∗ have the following relation . Q∗ ( s , a ( 1 ) , a ( 2 ) ) = R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) V ∗ ( s′ ) , V ∗ ( s ) = π∗ ( 1 ) ( s ) > Q∗ ( s ) π∗ ( 2 ) ( s ) + τ [ H ( π∗ ( 1 ) ( s ) ) −H ( π∗ ( 2 ) ( s ) ) ] . Hence , ‖Q∗‖∞ ≤ 1 + γ‖V ∗‖∞ ( 70 ) ‖V ∗‖∞ ≤ ‖Q∗‖∞ + τ lnAmax ( 71 ) Substituting eq . ( 70 ) into eq . ( 71 ) yields that ‖V ∗‖∞ ≤ 1 + γ‖V ∗‖∞ + τ lnAmax ⇒ ‖V ∗‖∞ ≤ Vmax . The proof of eq . ( 66 ) is finished . Then , substituting ‖V ∗‖∞ ≤ Vmax into eq . ( 70 ) proves that ‖Q∗‖∞ ≤ Qmax . Equation ( 67 ) can be proved as follows.∣∣∣ v̂k+1 ( s ) µ̂k ( s ) ∣∣∣ ( i ) ≤|V̂k+1 ( s ) |+ τ ∣∣H ( π ( 2 ) k ( s ) ) −H ( π ( 1 ) k ( s ) ) ∣∣ ≤ Vmax + τ lnAmax ( ii ) ≤ 2Vmax , where ( i ) uses eq . ( 58 ) and ( ii ) uses Vmax : = 1+τ lnAmax1−γ ≥ τ lnAmax . Next , we prove eq . ( 68 ) . It can be easily seen from eqs . ( 57 ) & ( 66 ) that∣∣Qk ( s , a ( 1 ) , a ( 2 ) ) ∣∣ ≤∣∣R ( s , a ( 1 ) , a ( 2 ) ) ∣∣+ γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) ∣∣V ′k ( s′ ) ∣∣ ≤ 1 + γVmax = Qmax , which implies that ‖Qk‖∞ ≤ Qmax . Hence , ∣∣Q′ ( m ) k , t ( s , a ( m ) ) ∣∣ ( i ) ≤ Ea ( \m ) ∼π′ ( \m ) k , t ( s ) ∣∣Qk ( s , a ( 1 ) , a ( 2 ) ) ∣∣ ≤ Qmax where ( i ) uses eq . ( 64 ) . This proves eq . ( 68 ) . With the above two Lemmas , we can derive the estimation error bounds as follows . Lemma 4 . Use hyperparameter choices Nk , t , Nk , t ≥ 650tmixAmaxµmin ln ( 20Tsum|S|Amax δ √ µmin ) and Nk+1 ≥ 650tmix µmin ( 1−γ ) 2 ln ( 4 δ √ µmin ) in Algorithm 1 . Then with probability at least 1 − δ , all the following bounds hold for all 0 ≤ k ≤ K − 1 , 0 ≤ t ≤ Tk − 1 , m = 1 , 2 , s ∈ S , a ( m ) ∈ A ( m ) , where Tsum : = ∑K−1 k=0 Tk.∥∥V̂k − V ′k∥∥∞ ≤ 2Vmaxγk + 171Vmax1− γ √ tmix Nk+1µmin ln ( 4K|S| δ √ µmin ) ( 72 ) ∥∥Q̂ ( m ) k , t −Q ( m ) k , t ∥∥∞ ≤ Qmax [ 640tmixAmax Nk , tµmin ′ ln ( 20Tsum|S|Amax δ √ µmin ) + 25 √ tmixAmax Nk , tµmin ′ ln ( 20Tsum|S|Amax δ √ µmin ) + ′ ] + 2 ∥∥V̂k − V ′k∥∥∞ ( 73 ) ∥∥Q̂ ( m ) k , t −Q ( m ) k , t ∥∥∞ ≤ Qmax [ 640tmixAmax Nk , tµmin ′ ln ( 20Tsum|S|Amax δ √ µmin ) + 25 √ tmixAmax Nk , tµmin ′ ln ( 20Tsum|S|Amax δ √ µmin ) + ′ ] + 2 ∥∥V̂k − V ′k∥∥∞ . ( 74 ) More specifically , the upper bounds ( 73 ) & ( 74 ) can be simplified respectively as follows when ′ = [ 650tmixAmax Nk , tµmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 1/3 and ′ = [ 650tmixAmax Nk , tµmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 1/3 . ∥∥Q̂ ( m ) k , t −Q ( m ) k , t ∥∥∞ ≤18Qmax [ tmixAmaxNk , tµmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 1/3 + 2 ∥∥V̂k − V ′k∥∥∞ , ( 75 ) ∥∥Q̂ ( m ) k , t −Q ( m ) k , t ∥∥∞ ≤18Qmax [ tmixAmaxNk , tµmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 1/3 + 2 ∥∥V̂k − V ′k∥∥∞ . ( 76 ) Proof . If the initial state distribution of the minibatch Nk , t equals µ′k , t , i.e. , sk , t,0 ∼ µ′k , t , then Nk , tq̂ ( m ) k , t ( s , a ( m ) ) : = ∑ i∈Nk , t g ( si , a ( 1 ) i , a ( 2 ) i , si+1 ) ( g ( si , a ( 1 ) i , a ( 2 ) i , si+1 ) : = [ Ri + γV̂k ( si+1 ) ] 1 { si = s , a ( m ) i = a ( m ) } ) and its expectated value Nk , tq ′ ( m ) k , t ( s , a ( m ) ) satisfy the following concentration bound . Pµ′k , t { ∣∣Nk , tq̂ ( m ) k , t ( s , a ( m ) ) −Nk , tq′ ( m ) k , t ( s , a ( m ) ) ∣∣ ≥ u } ( i ) ≤ 2 exp [ − u 2γps 8 ( Nk , t + 1/γps ) Q2maxµ ′ k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) + 40uQmax ] ( ii ) ≤ 2 exp [ − u 2/ ( 2tmix ) 8 ( Nk , t + 2tmix ) Q2maxµ ′ k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) + 40uQmax ] , ( 77 ) where Pµ′k , t is under the initial state distribution µ ′ k , t , ( i ) uses Theorem 3.4 of ( Paulin , 2015 ) and the inequalities that ∣∣g ( si , a ( 1 ) i , a ( 2 ) i , si+1 ) − Esi∼µ′k , tg ( si , a ( 1 ) i , a ( 2 ) i , si+1 ) ∣∣ ≤ 2 ( 1 + γVmax ) = 2Qmax ( Based on Lemma 3 ) and that varsi∼µ′k , t [ g ( si , a ( 1 ) i , a ( 2 ) i , si+1 ) ] ≤ Esi∼µ′k , t [ [ Ri + γV ′ k ( si+1 ) ] 2 1 { si = s , a ( m ) i = a ( m ) } ] ≤ ( 1 + γVmax ) 2µ′k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) = Q2maxµ ′ k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) , ( ii ) uses Proposition 3.4 of ( Paulin , 2015 ) which states that the pseudo spectral gap γps has a lower bound 1/ ( 2tmix ) for any uniformly ergodic Markov chain ( This condition holds for our MDP with finitely many states and actions ) . Then , for any initial state distribution st , k,0 ∼ ξ , Proposition 3.10 of ( Paulin , 2015 ) implies that Pξ { ∣∣Nk , tq̂ ( m ) k , t ( s , a ( m ) ) −Nk , tq′ ( m ) k , t ( s , a ( m ) ) ∣∣ ≥ u } ≤ √ Es∼ξ [ ξ ( s ) µ′k , t ( s ) ] Pµ′k , t { ∣∣Nk , tq̂ ( m ) k , t ( s , a ( m ) ) −Nk , tq′ ( m ) k , t ( s , a ( m ) ) ∣∣ ≥ u } ( i ) ≤ √ 1 µmin exp [ − u 2/ ( 4tmix ) 8 ( Nk , t + 2tmix ) Q2maxµ ′ k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) + 40uQmax ] , ( 78 ) where ( i ) uses eq . ( 77 ) and the inequality that Es∼ξ [ ξ ( s ) µ′k , t ( s ) ] ≤ 1µmin . In a similar way , the following concentration bound can be proved for Nk , tµ̂k , t ( s ) : =∑ i∈Nk , t 1 { si = s } and Nk , tµ ′ k , t ( s ) = Eµ′k , t [ Nk , tµ̂k , t ( s ) ] . Pξ { ∣∣Nk , tµ̂k , t ( s ) −Nk , tµ′k , t ( s ) ∣∣ ≥ u } ≤√ 1µmin exp [ − u 2/ ( 4tmix ) 8 ( Nk , t + 2tmix ) µ′k , t ( s ) + 40u ] , ( 79 ) where we use the inequalities that varµ′k , t1 { si = s } ≤ µ ′ k , t ( s ) and that ∣∣1 { si = s } − Eµ′k , t1 { si = s } ∣∣ ≤ 1 . Letting the right hand sides of eqs . ( 78 ) & ( 79 ) be upper bounded by δ/4 and applying the union bound yields that , with probability at least 1−δ/2 , the following two inequalities simultaneously hold . ∣∣q̂ ( m ) k , t ( s , a ( m ) ) − q′ ( m ) k , t ( s , a ( m ) ) ∣∣ ≤ 160Qmax Nk , t tmix ln ( 4 δ √ µmin ) + 6Qmax√ Nk , t √ tmix ( 1 + 2tmix/Nk , t ) µ′k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) ln ( 4 δ √ µmin ) , ( 80 ) ∣∣µ̂k , t ( s ) − µ′k , t ( s ) ∣∣ ≤ 160 Nk , t tmix ln ( 4 δ √ µmin ) + 6√ Nk , t √ tmix ( 1 + 2tmix/Nk , t ) µ′k , t ( s ) ln ( 4 δ √ µmin ) ( i ) ≤ 1 2 µ′k , t ( s ) , ( 81 ) where ( i ) holds since Nk , t ≥ 650tmixAmaxµmin ln ( 20Tsum|S|Amax δ √ µmin ) ≥ 650tmix and µ′k , t ( s ) ≥ µmin . Similarly , it can be proved that the following two inequalities holds with probability at least 1− δ/2.∣∣v̂k+1 ( s ) − v′k+1 ( s ) ∣∣ ≤ 160Qmax Nk+1 tmix ln ( 4 δ √ µmin ) + 6Qmax√ Nk+1 √ tmix ( 1 + 2tmix/Nk+1 ) µk ( s ) ln ( 4 δ √ µmin ) , ( 82 ) ∣∣µ̂k ( s ) − µk ( s ) ∣∣ ≤ 160 Nk+1 tmix ln ( 4 δ √ µmin ) + 6√ Nk+1 √ tmix ( 1 + 2tmix/Nk+1 ) µk ( s ) ln ( 4 δ √ µmin ) . ( 83 ) Hence , eqs . ( 80 ) - ( 83 ) hold with probability at least 1− δ . In this case , we have that∣∣v̂k+1 ( s ) − vk+1 ( s ) ∣∣ ≤ ∣∣v̂k+1 ( s ) − v′k+1 ( s ) ∣∣+ ∣∣v′k+1 ( s ) − vk+1 ( s ) ∣∣ ( i ) ≤ 160Qmax Nk+1 tmix ln ( 4 δ √ µmin ) + 6Qmax√ Nk+1 √ tmix ( 1 + 2tmix/Nk+1 ) µk ( s ) ln ( 4 δ √ µmin ) + γE µk , π ( 1 ) k , π ( 2 ) k [ [ V̂k ( s ′ ) − V ′k ( s′ ) ] 1 { s̃ = s } ] ≤ 160Qmax Nk+1 tmix ln ( 4 δ √ µmin ) + 6Qmax√ Nk+1 √ tmix ( 1 + 2tmix/Nk+1 ) µk ( s ) ln ( 4 δ √ µmin ) + γmax s′ ∣∣V̂k ( s′ ) − V ′k ( s′ ) ∣∣Eµk1 { s̃ = s } ≤ 160Qmax Nk+1 tmix ln ( 4 δ √ µmin ) + 6Qmax√ Nk+1 √ tmix ( 1 + 2tmix/Nk+1 ) µk ( s ) ln ( 4 δ √ µmin ) + γµk ( s ) ∥∥V̂k − V ′k∥∥∞ , ( 84 ) where ( i ) uses eqs . ( 54 ) , ( 55 ) & ( 82 ) . |V̂k+1 ( s ) − V ′k+1 ( s ) | ( i ) = ∣∣∣ v̂k+1 ( s ) µ̂k ( s ) − vk+1 ( s ) µk ( s ) ∣∣∣ ≤ µ−1k ( s ) ∣∣v̂k+1 ( s ) − vk+1 ( s ) ∣∣+ ∣∣v̂k+1 ( s ) ∣∣∣∣∣µk ( s ) − µ̂k ( s ) µk ( s ) µ̂k ( s ) ∣∣∣ ( ii ) ≤ µ−1k ( s ) [ 160Qmax Nk+1 tmix ln ( 4 δ √ µmin ) + 6Qmax√ Nk+1 √ tmix ( 1 + 2tmix/Nk+1 ) µk ( s ) ln ( 4 δ √ µmin ) + γµk ( s ) ∥∥V̂k − V ′k∥∥∞ ] + 2Vmaxµ−1k ( s ) [ 160 Nk+1 tmix ln ( 4 δ √ µmin ) + 6√ Nk+1 √ tmix ( 1 + 2tmix/Nk+1 ) µk ( s ) ln ( 4 δ √ µmin ) ] ( iii ) ≤ γ‖V̂k − V ′k‖∞ + Vmax µk ( s ) [ 480tmix Nk+1 ln ( 4 δ √ µmin ) + 18 √ tmixµk ( s ) Nk+1 ( 1 + 2tmix/Nk+1 ) ln ( 4 δ √ µmin ) ] ( iv ) ≤ γ‖V̂k − V ′k‖∞ + Vmax [ 24 40tmix Nk+1µmin ln ( 4 δ √ µmin ) + 18 √ 1.05tmix Nk+1µmin ln ( 4 δ √ µmin ) ] ( v ) ≤ γ‖V̂k − V ′k‖∞ + Vmax [ 24 √ 40tmix Nk+1µmin ln ( 4 δ √ µmin ) + 18 √ 1.05tmix Nk+1µmin ln ( 4 δ √ µmin ) ] ≤ γ‖V̂k − V ′k‖∞ + 171Vmax √ tmix Nk+1µmin ln ( 4 δ √ µmin ) , ( 85 ) where ( i ) uses eqs . ( 58 ) & ( 59 ) , ( ii ) uses eqs . ( 67 ) , ( 83 ) & ( 84 ) , ( iii ) uses Qmax ≤ Vmax , ( iv ) uses µk ( s ) ≥ µmin and Nk+1 ≥ 40tmixµmin ln ( 4 δ √ µmin ) ≥ 40tmix , and ( v ) uses Nk+1 ≥ 40tmixµmin ln ( 4 δ √ µmin ) . Applying the union bound to the above inequality over all 0 ≤ k ≤ K − 1 , s ∈ S and taking maximum over s ∈ S , we obtain that with probability at least 1− δ , ‖V̂k+1 − V ′k+1‖∞ ≤γ‖V̂k − V ′k‖∞ + 171Vmax √ tmix Nk+1µmin ln ( 4 δ √ µmin ) . Iterating the above inequality yields that with probability at least 1− δ , ‖V̂k − V ′k‖∞ ≤γk‖V̂0 − V ′0‖∞ + 171Vmax 1− γ √ tmix Nk+1µmin ln ( 4K|S| δ √ µmin ) ( i ) ≤2Vmaxγk + 171Vmax 1− γ √ tmix Nk+1µmin ln ( 4K|S| δ √ µmin ) , ( 86 ) where ( i ) uses the fact that ‖V ′0‖∞ , ‖V̂0‖∞ ≤ Vmax . Hence , with probability at least 1− 2δ , eqs . ( 80 ) , ( 81 ) & ( 86 ) hold simultaneously . In this case , ∣∣Q̂ ( m ) k , t ( s , a ( m ) ) −Q′ ( m ) k , t ( s , a ( m ) ) ∣∣ ( i ) = ∣∣∣ q̂ ( m ) k , t ( s , a ( m ) ) µ̂k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) − q ( m ) k , t ( s , a ( m ) ) µ′k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) ∣∣∣ ( ii ) ≤ ∣∣q̂ ( m ) k , t ( s , a ( m ) ) − q ( m ) k , t ( s , a ( m ) ) ∣∣ µ̂k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) + |µ′k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) Q′ ( m ) k , t ( s , a ( m ) ) | ∣∣µ̂k , t ( s ) − µ′k , t ( s ) ∣∣ µ̂k , t ( s ) µ′k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) ( iii ) ≤ 2 ∣∣q̂ ( m ) k , t ( s , a ( m ) ) − q′ ( m ) k , t ( s , a ( m ) ) ∣∣+ 2∣∣q′ ( m ) k , t ( s , a ( m ) ) − q ( m ) k , t ( s , a ( m ) ) ∣∣ µ′k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) + 2Qmax ∣∣µ̂k , t ( s ) − µ′k , t ( s ) ∣∣ µ′k , t ( s ) ( iv ) ≤ 2Qmax µ′k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) [ 320 Nk , t tmix ln ( 4 δ √ µmin ) + 12√ Nk , t √ tmix ( 1 + 2tmix/Nk , t ) µ′k , t ( s ) π ′ ( m ) k , t ( a ( m ) |s ) ln ( 4 δ √ µmin ) ] + 2 ∥∥V̂k − V ′k∥∥∞ ( v ) ≤ Qmax [ 640tmixAmax Nk , tµmin ′ ln ( 4 δ √ µmin ) + 25 √ tmixAmax Nk , tµmin ′ ln ( 4 δ √ µmin ) ] + 2 ∥∥V̂k − V ′k∥∥∞ , ( 87 ) where ( i ) uses eqs . ( 64 ) & ( 65 ) , ( ii ) uses eq . ( 64 ) , ( iii ) uses eq . ( 68 ) and the inequality that µ̂k , t ( s ) ≥ µ′k , t ( s ) /2 implied by ( i ) of eq . ( 81 ) , ( iv ) uses eqs . ( 80 ) & ( 81 ) and the following inequality based on eqs . ( 61 ) & ( 62 ) , and ( v ) uses µ′k , t ( s ) ≥ µmin ( based on Assumption 1 ) , π ′ ( m ) k , t ( a ( m ) |s ) = ( 1 − ′ ) π ( m ) k , t ( a ( m ) |s ) + ′/|A ( m ) | ≥ ′/Amax and Nk , t ≥ 650tmix.∣∣q′ ( m ) k , t ( s , a ( m ) ) − q ( m ) k , t ( s , a ( m ) ) ∣∣ ≤ γE µ′k , t , π ′ ( 1 ) k , t , π ′ ( 2 ) k , t [ ∣∣V̂k ( s′ ) − V ′k ( s′ ) ∣∣1 { s̃ = s , ã ( m ) = a ( m ) } ] ≤ µ′k , t ( s ) π ′ ( m ) k , t ( s , a ( m ) ) ∥∥V̂k − V ′k∥∥∞ . Notice that the following inequality always holds.∣∣Q′ ( m ) k , t ( s , a ( m ) ) −Q ( m ) k , t ( s , a ( m ) ) ∣∣ ≤ ∑ a ( \m ) ∣∣Qk ( s , a ( 1 ) , a ( 2 ) ) ∣∣∣∣π′ ( \m ) k , t ( a ( \m ) |s ) − π ( \m ) k , t ( a ( \m ) |s ) ∣∣ ( i ) ≤ Qmax ′ ∣∣∣π ( \m ) k , t ( a ( \m ) |s ) − |A ( m ) |−1∣∣∣ ≤ Qmax ′ , ( 88 ) where ( i ) uses eq . ( 68 ) . Hence , combining eqs . ( 87 ) & ( 88 ) , it can be seen that with probability at least 1− 2δ , the following inequality holds∣∣Q̂ ( m ) k , t ( s , a ( m ) ) −Q ( m ) k , t ( s , a ( m ) ) ∣∣ ≤ Qmax [ 640tmixAmax Nk , tµmin ′ ln ( 4 δ √ µmin ) + 25 √ tmixAmax Nk , tµmin ′ ln ( 4 δ √ µmin ) + ′ ] + 2 ∥∥V̂k − V ′k∥∥∞ . ( 89 ) Similarly , it can be proved that with probability at least 1− 2δ , ∣∣Q̂ ( m ) k , t ( s , a ( m ) ) −Q ( m ) k , t ( s , a ( m ) ) ∣∣ ≤ Qmax [ 640tmixAmax Nk , tµmin ′ ln ( 4 δ √ µmin ) + 25 √ tmixAmax Nk , tµmin ′ ln ( 4 δ √ µmin ) + ′ ] + 2 ∥∥V̂k − V ′k∥∥∞ . ( 90 ) Finally , eqs . ( 72 ) - ( 74 ) are proved by applying a union bound to eq . ( 86 ) and to eqs . ( 89 ) & ( 90 ) over all 0 ≤ k ≤ T − 1 , 0 ≤ t ≤ Tk − 1 , m = 1 , 2 , s ∈ S , a ( m ) ∈ A ( m ) . Now we consider the hyperparameter choice ′ = [ 650tmixAmax Nk , tµmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 1/3 . First , this is a valid choice , since Nk , t ≥ 650tmixAmaxµmin ln ( 20Tsum|S|Amax δ √ µmin ) implies that ′ ∈ [ 0 , 1 ] . Then , for this ′ , the upper bound ( 73 ) is simplified as follows.∥∥Q̂ ( m ) k , t −Q ( m ) k , t ∥∥∞ ≤ Qmax [ 640tmixAmax Nk , tµmin ′ ln ( 20Tsum|S|Amax δ √ µmin ) + 25 √ tmixAmax Nk , tµmin ′ ln ( 20Tsum|S|Amax δ √ µmin ) + ′ ] + 2 ∥∥V̂k − V ′k∥∥∞ ( i ) ≤ Qmax ( 640 650 ′2 + 25√ 650 ′ ) + 2 ∥∥V̂k − V ′k∥∥∞ ( ii ) ≤ 2Qmax ′ + 2 ∥∥V̂k − V ′k∥∥∞ = 18Qmax [ tmixAmaxNk , tµmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 1/3 + 2 ∥∥V̂k − V ′k∥∥∞ , where ( i ) uses tmixAmaxNk , tµmin ln ( 20Tsum|S|Amax δ √ µmin ) = ′3/650 , and ( ii ) uses ′ ∈ [ 0 , 1 ] . This proves eq . ( 75 ) . The proof of eq . ( 76 ) is similar . C PROPERTIES OF THE DUALITY GAP In this section , we prove some useful properties of the duality gap D ( τ ) ( π ( 1 ) , π ( 2 ) ) : = max s , π′ ( 1 ) , π′ ( 2 ) [ V ( τ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ) π ( 1 ) , π′ ( 2 ) ( s ) ] . Lemma 5 . For any policy pair π ( 1 ) , π ( 2 ) , it holds that D ( τ ) ( π ( 1 ) , π ( 2 ) ) ≤ 1 1− γ max s , π′ ( 1 ) ( s ) , π′ ( 2 ) ( s ) [ fτ [ Q ( τ ) ∗ ( s ) , π ′ ( 1 ) ( s ) , π ( 2 ) ( s ) ] − fτ [ Q ( τ ) ∗ ( s ) , π ( 1 ) ( s ) , π′ ( 2 ) ( s ) ] ] . This Lemma generalized the Lemma 32 of ( Wei et al. , 2021 ) to entropy-regularized Markov game . Proof . Throughout this proof , we denote the policy pair ( π ( 1 ) ∗τ , π ( 2 ) ∗τ ) as the Nash equilibrium and their associated V-function and Q-function are respectively denoted as V ( τ ) ∗ and Q ( τ ) ∗ . Note that V ( τ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ) ∗ ( s ) = ∑ a ( 1 ) , a ( 2 ) [ Q ( τ ) π′ ( 1 ) , π ( 2 ) ( s , a ( 1 ) , a ( 2 ) ) π′ ( 1 ) ( a ( 1 ) |s ) π ( 2 ) ( a ( 2 ) |s ) −Q ( τ ) ∗ ( s , a ( 1 ) , a ( 2 ) ) π ( 1 ) ∗τ ( a ( 1 ) |s ) π ( 2 ) ∗τ ( a ( 2 ) |s ) ] + τ [ H ( π′ ( 1 ) ( s ) ) −H ( π ( 2 ) ( s ) ) −H ( π ( 1 ) ∗τ ( s ) ) +H ( π ( 2 ) ∗τ ( s ) ) ] = ∑ a ( 1 ) , a ( 2 ) [ Q ( τ ) π′ ( 1 ) , π ( 2 ) ( s , a ( 1 ) , a ( 2 ) ) −Q ( τ ) ∗ ( s , a ( 1 ) , a ( 2 ) ) ] π′ ( 1 ) ( a ( 1 ) |s ) π ( 2 ) ( a ( 2 ) |s ) + ∑ a ( 1 ) , a ( 2 ) Q ( τ ) ∗ ( s , a ( 1 ) , a ( 2 ) ) [ π′ ( 1 ) ( a ( 1 ) |s ) π ( 2 ) ( a ( 2 ) |s ) − π ( 1 ) ∗τ ( a ( 1 ) |s ) π ( 2 ) ∗τ ( a ( 2 ) |s ) ] + τ [ H ( π′ ( 1 ) ( s ) ) −H ( π ( 2 ) ( s ) ) −H ( π ( 1 ) ∗τ ( s ) ) +H ( π ( 2 ) ∗τ ( s ) ) ] = γ ∑ a ( 1 ) , a ( 2 ) [ π′ ( 1 ) ( a ( 1 ) |s ) π ( 2 ) ( a ( 2 ) |s ) Es′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) [ V ( τ ) π′ ( 1 ) , π ( 2 ) ( s′ ) − V ( τ ) ∗ ( s′ ) ] ] + π′ ( 1 ) ( s ) > Q ( τ ) ∗ ( s ) π ( 2 ) ( s ) − π ( 1 ) ∗τ ( s ) > Q ( τ ) ∗ ( s ) π ( 2 ) ∗τ ( s ) + τ [ H ( π′ ( 1 ) ( s ) ) −H ( π ( 2 ) ( s ) ) −H ( π ( 1 ) ∗τ ( s ) ) +H ( π ( 2 ) ∗τ ( s ) ) ] ≤ γmax s′ [ V ( τ ) π′ ( 1 ) , π ( 2 ) ( s′ ) − V ( τ ) ∗ ( s′ ) ] + τfτ ( Q ( τ ) ∗ ( s ) ; π ′ ( 1 ) ( s ) , π ( 2 ) ( s ) ) − τfτ ( Q ( τ ) ∗ ( s ) ; π ( 1 ) ∗τ ( s ) , π ( 2 ) ∗τ ( s ) ) ( i ) ≤ γmax s′ [ V ( τ ) π′ ( 1 ) , π ( 2 ) ( s′ ) − V ( τ ) ∗ ( s′ ) ] + τfτ ( Q ( τ ) ∗ ( s ) ; π ′ ( 1 ) ( s ) , π ( 2 ) ( s ) ) − τ min π′ ( 2 ) fτ ( Q ( τ ) ∗ ( s ) ; π ( 1 ) ( s ) , π′ ( 2 ) ( s ) ) , where ( i ) uses fτ ( Q ( τ ) ∗ ( s ) ; π ( 1 ) ∗τ ( s ) , π ( 2 ) ∗τ ( s ) ) = maxπ′′ ( 1 ) minπ′ ( 2 ) fτ ( Q ( τ ) ∗ ( s ) ; π ′′ ( 1 ) ( s ) , π′ ( 2 ) ( s ) ) ≥ minπ′ ( 2 ) fτ ( Q ( τ ) ∗ ( s ) ; π ( 1 ) ( s ) , π′ ( 2 ) ( s ) ) . Applying maxs , π′ ( 1 ) to both sides of the above inequality and rearranging it yields that max s , π′ ( 1 ) [ V ( τ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ) ∗ ( s ) ] ≤ 1 1− γ max s , π′ ( 1 ) ( s ) , π′ ( 2 ) ( s ) [ fτ [ Q ( τ ) ∗ ( s ) , π ′ ( 1 ) ( s ) , π ( 2 ) ( s ) ] − fτ [ Q ( τ ) ∗ ( s ) , π ( 1 ) ( s ) , π′ ( 2 ) ( s ) ] ] . ( 91 ) Similarly , we can obtain that max s , π′ ( 2 ) [ V ( τ ) ∗ ( s ) − V ( τ ) π ( 1 ) , π′ ( 2 ) ( s ) ] ≤ 1 1− γ max s , π′ ( 1 ) ( s ) , π′ ( 2 ) ( s ) [ fτ [ Q ( τ ) ∗ ( s ) , π ′ ( 1 ) ( s ) , π ( 2 ) ( s ) ] − fτ [ Q ( τ ) ∗ ( s ) , π ( 1 ) ( s ) , π′ ( 2 ) ( s ) ] ] . ( 92 ) Therefore , D ( τ ) ( π ( 1 ) , π ( 2 ) ) = max s , π′ ( 1 ) , π′ ( 2 ) [ V ( τ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ) π ( 1 ) , π′ ( 2 ) ( s ) ] ≤ max s , π′ ( 1 ) [ V ( τ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ) ∗ ( s ) ] + max s , π′ ( 2 ) [ V ( τ ) ∗ ( s ) − V ( τ ) π ( 1 ) , π′ ( 2 ) ( s ) ] ( i ) ≤ 2 1− γ max s , π′ ( 1 ) ( s ) , π′ ( 2 ) ( s ) [ fτ [ Q ( τ ) ∗ ( s ) , π ′ ( 1 ) ( s ) , π ( 2 ) ( s ) ] − fτ [ Q ( τ ) ∗ ( s ) , π ( 1 ) ( s ) , π′ ( 2 ) ( s ) ] ] where ( i ) uses eqs . ( 91 ) & ( 92 ) . Lemma 6 . D ( τ ) ( π ( 1 ) , π ( 2 ) ) is 2 lnAmax1−γ -Lipschitz continuous with regard to τ . Proof . The definition of the state value function V ( τ ) π ( 1 ) , π ( 2 ) in ( 3 ) can be rewritten as follows . V ( τ ) π ( 1 ) , π ( 2 ) ( s ) =E [ ∞∑ t=0 γtRt ∣∣∣s0 = s ] + ∞∑ t=0 γtτ [ H ( π ( 1 ) ( st ) ) −H ( π ( 2 ) ( st ) ) ] . ( 93 ) Hence , for any τ , τ ′ ≥ 0 , ∣∣V ( τ ′ ) π ( 1 ) , π ( 2 ) ( s ) − V ( τ ) π ( 1 ) , π ( 2 ) ( s ) ∣∣ ≤ ∞∑ t=0 γt|τ ′ − τ ||H ( π ( 1 ) ( st ) ) −H ( π ( 2 ) ( st ) ) | ( i ) ≤ ∞∑ t=0 γt|τ ′ − τ | lnAmax ≤ lnAmax 1− γ |τ ′ − τ | , ( 94 ) where ( i ) uses 0 ≤ H ( π ( m ) ( st ) ) ≤ ln |A ( m ) | ≤ lnAmax . Hence , this Lemma can be proved as follows . |D ( τ ′ ) ( π ( 1 ) , π ( 2 ) ) −D ( τ ) ( π ( 1 ) , π ( 2 ) ) | ≤ ∣∣∣ max s , π′ ( 1 ) , π′ ( 2 ) [ V ( τ ′ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ′ ) π ( 1 ) , π′ ( 2 ) ( s ) ] − max s , π′ ( 1 ) , π′ ( 2 ) [ V ( τ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ) π ( 1 ) , π′ ( 2 ) ( s ) ] ∣∣∣ ≤ max s , π′ ( 1 ) , π′ ( 2 ) ∣∣∣ [ V ( τ ′ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ′ ) π ( 1 ) , π′ ( 2 ) ( s ) ] − [ V ( τ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ) π ( 1 ) , π′ ( 2 ) ( s ) ] ∣∣∣ ≤ max s , π′ ( 1 ) , π′ ( 2 ) [ ∣∣V ( τ ′ ) π′ ( 1 ) , π ( 2 ) ( s ) − V ( τ ) π′ ( 1 ) , π ( 2 ) ( s ) ∣∣+ ∣∣V ( τ ) π ( 1 ) , π′ ( 2 ) ( s ) − V ( τ ′ ) π ( 1 ) , π′ ( 2 ) ( s ) ∣∣ ] ( i ) ≤ 2 lnAmax 1− γ |τ ′ − τ | , where ( i ) uses eq . ( 94 ) . D PROOF OF THEOREM 1 Theorem 1 ( Finite-time convergence rate ) . Apply Algorithm 1 to solve the entropy-regularized Markov game with τ ∈ ( 0 , 1 ] . Choose learning rate η = [ 2 ( τ +Qmax ) ] −1 , initialization ‖V̂0‖∞ ≤ Vmax and batch sizes Nk , t , Nk , t , Nk+1 that satisfy ( 18 ) & ( 19 ) . Then , the Nash equilibrium duality gap converges at the following rate with probability at least 1− δ . D ( τ ) ( π ( 1 ) K−1 , π ( 2 ) K−1 ) ≤ O ( Vmax lnAmax 1− γ K−1∑ k=0 γK−k ( 1− ητ ) Tk−1 + Vmax 1− γ [ tmixAmax µmin ln ( Tsum|S|Amax δµmin ) ] 2/3 K−1∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−2−t ( 1 N 2/3 k , t + Vmax τN 2/3 k , t+1 ) + V 3maxγ K τ2 ( 1− γ ) 2 + tmixV 3 max τ2µmin ( 1− γ ) 3 ln ( K|S| δµmin ) K−1∑ k=0 γK−k−1 Nk+1 ) . ( 20 ) Proof . First , consider the SPU iterations that are defined by replacingQ ( m ) k , t , Q ( m ) k , t+1 in ( 8 ) with Q̂ ( m ) k , t , Q̂ ( m ) k , t+1 , respectively , for m = 1 , 2 . We can apply Lemma 1 with the quantities being specified as follows . • Q : = Qk ( s ) , hτ ( µ , ν ) : = fτ ( Qk ( s ) ; µ , ν ) , • µt : = π ( 1 ) k , t ( s ) , νt : = π ( 2 ) k , t ( s ) , µt+1 : = π ( 1 ) k , t+1 ( s ) , νt+1 : = π ( 2 ) k , t+1 ( s ) , µ∗τ : = π ∗ ( 1 ) k ( s ) , ν ∗ τ : = π ∗ ( 2 ) k ( s ) , • dm : = |A ( m ) | , • δ ( m ) k , t : = Q̂ ( m ) k , t ( s ) −Q ( m ) k , t ( s ) , δ ( m ) k , t : = Q̂ ( m ) k , t+1 ( s ) −Q ( m ) k , t+1 ( s ) , where π ( 1 ) ∗k ( s ) , π ( 2 ) ∗ k ( s ) is the solution policy pair to the minimax optimization problem minπ ( 2 ) ( s ) maxπ ( 1 ) ( s ) fτ ( Qk ( s ) ; π ( 1 ) ( s ) , π ( 2 ) ( s ) ) . Consequently , eqs . ( 29 ) & ( 30 ) with t = Tk − 1 imply the following two inequalities , respectively.∣∣fτ ( Qk ( s ) ; π ( 1 ) k , Tk ( s ) , π ( 1 ) k , Tk ( s ) ) − fτ ( Qk ( s ) ; π∗ ( 1 ) k ( s ) , π∗ ( 2 ) k ( s ) ) ∣∣ = ∣∣V ′k+1 ( s ) − fτ ( Qk ( s ) ; π∗ ( 1 ) k ( s ) , π∗ ( 2 ) k ( s ) ) ∣∣ ≤ 8 η ( 1− ητ ) Tk−1 lnAmax + Tk−1∑ t=0 ( 1− ητ ) Tk−2−t 2∑ m=1 [ 8η ∥∥Q̂ ( m ) k , t ( s ) −Q ( m ) k , t ( s ) ∥∥2∞ + 334τ ∥∥Q̂ ( m ) k , t+1 ( s ) −Q ( m ) k , t+1 ( s ) ∥∥2∞ ] , ( 95 ) max π ( 1 ) ( s ) , π ( 2 ) ( s ) [ fτ ( Qk ( s ) ; π ( 1 ) ( s ) , π ( 2 ) k , Tk ( s ) ) − fτ ( Qk ( s ) ; π ( 1 ) k , Tk ( s ) , π ( 2 ) ( s ) ) ] = max π ( 1 ) ( s ) , π ( 2 ) ( s ) [ fτ ( Qk ( s ) ; π ( 1 ) ( s ) , π ( 2 ) k ( s ) ) − fτ ( Qk ( s ) ; π ( 1 ) k ( s ) , π ( 2 ) ( s ) ) ] ≤ 4 η ( 1− ητ ) Tk−1 lnAmax + Tk−1∑ t=0 ( 1− ητ ) Tk−2−t 2∑ m=1 [ 16η ∥∥Q̂ ( m ) k , t ( s ) −Q ( m ) k , t ( s ) ∥∥2∞ + 12τ ∥∥Q̂ ( m ) k , t+1 ( s ) −Q ( m ) k , t+1 ( s ) ∥∥2∞ ] . ( 96 ) Then , define the soft Bellman operator Bτ as follows . Bτ ( Q ) ( s , a ( 1 ) , a ( 2 ) ) : = R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) [ max π ( 1 ) ( s′ ) min π ( 2 ) ( s′ ) fτ ( Q ( s′ ) ; π ( 1 ) ( s′ ) , π ( 2 ) ( s′ ) ) ] , ( 97 ) where Q : S × A ( 1 ) × A ( 2 ) → R and Q ( s′ ) ∈ R|A ( 1 ) |×|A ( 2 ) | with each entry given by [ Q ( s′ ) ] a ( 1 ) , a ( 2 ) : = Q ( s , a ( 1 ) , a ( 2 ) ) . It can be proved that Bτ is a contraction operator and has a unique fixed point Q ( τ ) ∗ ( Cen et al. , 2021 ) , that is , ‖Bτ ( Q′ ) − Bτ ( Q ) ‖∞ ≤γ‖Q′ −Q‖∞ , ( 98 ) Bτ ( Q ( τ ) ∗ ) =Q ( τ ) ∗ . ( 99 ) As a result , we obtain that ‖Qk+1 −Q ( τ ) ∗ ‖∞ ( i ) ≤ ‖Qk+1 − Bτ ( Qk ) ‖∞ + ‖Bτ ( Qk ) − Bτ ( Q ( τ ) ∗ ) ‖∞ ( ii ) ≤ max s , a ( 1 ) , a ( 2 ) |Qk+1 ( s , a ( 1 ) , a ( 2 ) ) − Bτ ( Qk ) ( s , a ( 1 ) , a ( 2 ) ) |+ γ‖Qk −Q ( τ ) ∗ ‖∞ = max s , a ( 1 ) , a ( 2 ) ∣∣∣R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) V ′k+1 ( s′ ) − [ R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) fτ ( Qk ( s ′ ) ; π ∗ ( 1 ) k ( s ′ ) , π ∗ ( 2 ) k ( s ′ ) ) ] ∣∣∣+ γ‖Qk −Q ( τ ) ∗ ‖∞ ≤ γmax s′∈S ∣∣V ′k+1 ( s′ ) − fτ ( Qk ( s′ ) ; π∗ ( 1 ) k ( s′ ) , π∗ ( 2 ) k ( s′ ) ) ∣∣+ γ‖Qk −Q ( τ ) ∗ ‖∞ , ( 100 ) where ( i ) uses eq . ( 99 ) and ( ii ) uses eq . ( 98 ) . Throughout the proof , suppose that eqs . ( 72 ) , ( 75 ) & ( 76 ) hold simultaneously which occurs with probability at least 1− δ . In this case , we have∥∥Q̂ ( m ) k , t ( s ) −Q ( m ) k , t ( s ) ∥∥2∞ ( i ) ≤ 648Q2max [ tmixAmax Nk , tµmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 2/3 + 8 ∥∥V̂k − V ′k∥∥2∞ ( ii ) ≤ 648Q2max [ tmixAmax Nk , tµmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 2/3 + 32V 2maxγ 2k + 29241tmixV 2 max Nk+1µmin ( 1− γ ) 2 ln ( 4K|S| δ √ µmin ) , ( 101 ) where ( i ) uses eq . ( 75 ) and the inequality that ( a + b ) 2 ≤ 2a2 + 2b2 for any a , b ≥ 0 , ( ii ) uses eq . ( 72 ) and also ( a+ b ) 2 ≤ 2a2 + 2b2 for any a , b ≥ 0 . Similarly , we obtain that∥∥Q̂ ( m ) k , t+1 ( s ) −Q ( m ) k , t+1 ( s ) ∥∥2∞ ≤ 648Q2max [ tmixAmax Nk , t+1µmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 2/3 + 32V 2maxγ 2k + 29241tmixV 2 max Nk+1µmin ( 1− γ ) 2 ln ( 4K|S| δ √ µmin ) . ( 102 ) Then , iterating eq . ( 100 ) yields that ‖QK−1 −Q ( τ ) ∗ ‖∞ ≤ γK−1‖Q0 −Q ( τ ) ∗ ‖∞ + K−2∑ k=0 γK−k−1 max s′∈S ∣∣Vk+1 ( s′ ) − fτ ( Qk ( s′ ) ; π∗ ( 1 ) k ( s′ ) , π∗ ( 2 ) k ( s′ ) ) ∣∣ ( i ) ≤ 2γK−1Qmax + K−2∑ k=0 γK−k−1 [ 8 η ( 1− ητ ) Tk−1 lnAmax + Tk−1∑ t=0 ( 1− ητ ) Tk−2−t 2∑ m=1 [ 8η ∥∥Q̂ ( m ) k , t ( s ) −Q ( m ) k , t ( s ) ∥∥2∞ + 334τ ∥∥Q̂ ( m ) k , t+1 ( s ) −Q ( m ) k , t+1 ( s ) ∥∥2∞ ] ] ( ii ) ≤ 2γK−1Qmax + K−2∑ k=0 γK−k−1 [ 8 η ( 1− ητ ) Tk−1 lnAmax + Tk−1∑ t=0 ( 1− ητ ) Tk−2−t 2∑ m=1 ( 648Q2max [ tmixAmax µmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 2/3 ( 8η N 2/3 k , t + 33 4τN 2/3 k , t+1 ) + ( 8η + 33 4τ ) [ 32V 2maxγ 2k + 29241tmixV 2 max Nk+1µmin ( 1− γ ) 2 ln ( 4K|S| δ √ µmin ) ] ) ] ( iii ) ≤ 2γK−1Qmax + 8 η lnAmax K−2∑ k=0 γK−k−1 ( 1− ητ ) Tk−1 + 324Q2max [ tmixAmax µmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 2/3 K−2∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−2−t ( 32η N 2/3 k , t + 33 τN 2/3 k , t+1 ) + 1568γK−1V 2max ητ2 ( 1− γ ) + 1432809tmixV 2 max ητ2µmin ( 1− γ ) 2 ln ( 4K|S| δ √ µmin ) K−2∑ k=0 γK−k−1N−1k+1 ( 103 ) ( iv ) = O [ Vmax lnAmax K−2∑ k=0 γK−k ( 1− ητ ) Tk−1 + Vmax [ tmixAmax µmin ln ( Tsum|S|Amax δµmin ) ] 2/3 K−2∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−2−t ( 1 N 2/3 k , t + Vmax τN 2/3 k , t+1 ) + V 3maxγ K τ2 ( 1− γ ) + tmixV 3 max τ2µmin ( 1− γ ) 2 ln ( K|S| δµmin ) K−2∑ k=0 γK−k−1N−1k+1 ] ( 104 ) where ( i ) uses eqs . ( 68 ) & ( 95 ) , ( ii ) uses eqs . ( 101 ) & ( 102 ) , ( iii ) uses η = [ 2 ( τ +Qmax ) ] −1 ≤ 1/ ( 2τ ) , ∑Tk−1 t=0 ( 1 − ητ ) Tk−2−t ≤ 1 ητ ( 1−ητ ) ≤ 2 ητ and ∑K−2 k=0 γ K+k−1 ≤ γ K−1 1−γ , and ( iv ) uses Qmax = O ( Vmax ) for γ ≈ 1 and η = [ 2 ( τ + Qmax ) ] −1 = O ( Q−1max ) = O ( V −1max ) ( Since Qmax ≥ γτ = O ( τ ) ) . Using Lemma 5 , the convergence rate of the duality gap in eq . ( 20 ) can be proved as follows . D ( τ ) ( π ( 1 ) K−1 , π ( 2 ) K−1 ) ( i ) ≤ 2 1− γ max s , π ( 1 ) , π ( 2 ) [ fτ [ Q ( τ ) ∗ ( s ) , π ( 1 ) ( s ) , π ( 2 ) K−1 ( s ) ] − fτ [ Q ( τ ) ∗ ( s ) , π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ] ] ≤ 2 1− γ max s , π ( 1 ) , π ( 2 ) [ fτ ( QK−1 ( s ) ; π ( 1 ) ( s ) , π ( 2 ) K−1 ( s ) ) − fτ ( QK−1 ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) ] + 2 1− γ max s , π ( 1 ) , π ( 2 ) ∣∣fτ ( Q ( τ ) ∗ ( s ) ; π ( 1 ) ( s ) , π ( 2 ) K−1 ( s ) ) − fτ ( QK−1 ( s ) ; π ( 1 ) ( s ) , π ( 2 ) K−1 ( s ) ) ∣∣ + 2 1− γ max s , π ( 1 ) , π ( 2 ) ∣∣fτ ( QK−1 ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) − fτ ( Q ( τ ) ∗ ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) ∣∣ ( ii ) ≤ 8 η ( 1− γ ) ( 1− ητ ) TK−1−1 lnAmax + 2 1− γ TK−1−1∑ t=0 ( 1− ητ ) TK−1−2−t 2∑ m=1 [ 16η ∥∥Q̂ ( m ) K−1 , t ( s ) −Q ( m ) K−1 , t ( s ) ∥∥2∞ + 12τ ∥∥Q̂ ( m ) K−1 , t+1 ( s ) −Q ( m ) K−1 , t+1 ( s ) ∥∥2∞ ] + 2 1− γ max s , π ( 1 ) ∣∣π ( 1 ) ( s ) > [ Q ( τ ) ∗ ( s ) −QK−1 ( s ) ] π ( 2 ) K−1 ( s ) ∣∣ + 2 1− γ max s , π ( 2 ) ∣∣π ( 1 ) K−1 ( s ) > [ QK−1 ( s ) −Q ( τ ) ∗ ( s ) ] π ( 2 ) ( s ) ∣∣ ( iii ) ≤ 8 lnAmax η ( 1− γ ) ( 1− ητ ) TK−1−1 + 4 1− γ TK−1−1∑ t=0 ( 1− ητ ) TK−1−2−t ( 648Q2max [ tmixAmax µmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 2/3 ( 16η N 2/3 K−1 , t + 12 τN 2/3 K−1 , t+1 ) + ( 32η + 24 τ ) [ 32V 2maxγ 2 ( K−1 ) + 29241tmixV 2 max NKµmin ( 1− γ ) 2 ln ( 4K|S| δ √ µmin ) ] ) + 4 1− γ ‖QK−1 −Q ( τ ) ∗ ‖∞ ( iv ) ≤ 8 lnAmax η ( 1− γ ) ( 1− ητ ) TK−1−1 + 2592Q 2 max 1− γ [ tmixAmax µmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 2/3 TK−1−1∑ t=0 ( 1− ητ ) TK−1−2−t ( 16η N 2/3 K−1 , t + 12 τN 2/3 K−1 , t+1 ) + 160 ητ2 [ 32V 2max 1− γ γ2 ( K−1 ) + 29241tmixV 2 max NKµmin ( 1− γ ) 3 ln ( 4K|S| δ √ µmin ) ] + 8γK−1Qmax 1− γ + 32 lnAmax η ( 1− γ ) K−2∑ k=0 γK−k−1 ( 1− ητ ) Tk−1 + 1296Q2max 1− γ [ tmixAmax µmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 2/3 K−2∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−2−t ( 32η N 2/3 k , t + 33 τN 2/3 k , t+1 ) + 6272γK−1V 2max ητ2 ( 1− γ ) 2 + 5731236tmixV 2 max ητ2µmin ( 1− γ ) 3 ln ( 4K|S| δ √ µmin ) K−2∑ k=0 γK−k−1N−1k+1 ≤ 8γ K−1Qmax 1− γ + 32 lnAmax η ( 1− γ ) K−1∑ k=0 γK−k−1 ( 1− ητ ) Tk−1 + 1296Q2max 1− γ [ tmixAmax µmin ln ( 20Tsum|S|Amax δ √ µmin ) ] 2/3 K−1∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−2−t ( 32η N 2/3 k , t + 33 τN 2/3 k , t+1 ) + 11392γK−1V 2max ητ2 ( 1− γ ) 2 + 5731236tmixV 2 max ητ2µmin ( 1− γ ) 3 ln ( 4K|S| δ √ µmin ) K−1∑ k=0 γK−k−1N−1k+1 ( v ) = O [ Vmax lnAmax 1− γ K−1∑ k=0 γK−k ( 1− ητ ) Tk−1 + Vmax 1− γ [ tmixAmax µmin ln ( Tsum|S|Amax δµmin ) ] 2/3 K−1∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−2−t ( 1 N 2/3 k , t + Vmax τN 2/3 k , t+1 ) + V 3maxγ K τ2 ( 1− γ ) 2 + tmixV 3 max τ2µmin ( 1− γ ) 3 ln ( K|S| δµmin ) K−1∑ k=0 γK−k−1N−1k+1 ] , where ( i ) uses Lemma 5 , ( ii ) uses eq . ( 96 ) and the definition of the function fτ , ( iii ) uses eqs . ( 101 ) & ( 102 ) , ( iv ) uses eq . ( 103 ) and η = [ 2 ( τ + Qmax ) ] −1 ≤ 1/ ( 2τ ) , ∑Tk−1 t=0 ( 1 − ητ ) Tk−2−t ≤ 1 ητ ( 1−ητ ) ≤ 2 ητ , and ( v ) uses Qmax = O ( Vmax ) for γ ≈ 1 and η = [ 2 ( τ +Qmax ) ] −1 = O ( Q−1max ) = O ( V −1max ) ( Since Qmax ≥ γτ = O ( τ ) ) . E PROOF OF THEOREM 2 Theorem 2 ( Sample complexity ) . Implement Algorithm 1 with η = O ( 1 − γ ) , τ = O ( ( 1−γ ) lnAmax ) , K = O [ 1 1−γ ln ( lnAmax ( 1−γ ) ) ] and Tk = 1 + k ln γ −1 ln ( 1−ητ ) −1 . Choose the following adaptive batch sizes . Nk+1 = Õ ( tmix ( ln2Amax ) γ− k2 2µmin ( 1− γ ) 8 ) , Nk , t = Nk , t 3 2 ( 1− γ ) 3 = Õ ( tmixAmax ( 1− ητ ) −3 ( t+1 ) 5 µmin ( 1− γ ) 3 ) . Then , for any ≤ lnAmax1−γ , the overall sample complexity to achieve D ( 0 ) ( π ( 1 ) K−1 , π ( 2 ) K−1 ) ≤ is Õ ( tmixAmax µmin 5.5 ( 1−γ ) 13.5 ) . Please refer to ( 118 ) in Appendix E for a complete expression . Proof . Since τ = O ( ( 1−γ ) lnAmax ) , we have Vmax = 1 + τ lnAmax 1− γ = 1 1− γ +O ( ) = O ( ( 1− γ ) −1 ) . ( 105 ) Qmax = 1 + γτ lnAmax 1− γ = 1 1− γ +O ( ) = O ( ( 1− γ ) −1 ) . ( 106 ) Since Tk = 1 + k ln γ −1 ln ( 1−ητ ) −1 , we have Tsum = K−1∑ k=0 ( 1 + k ln γ−1 ln ( 1− ητ ) −1 ) =K + K ( K − 1 ) 2 O ( 1− γ ητ ) =K + K ( K − 1 ) 2 O ( lnAmax ( 1− γ ) ) =O ( K2 lnAmax ( 1− γ ) ) =O [ lnAmax ( 1− γ ) 3 ln2 ( lnAmax ( 1− γ ) ) ] . ( 107 ) Hence , ln ( Tsum|S|Amax δµmin ) = O [ ln ( |S|Amax lnAmax δµmin ( 1− γ ) 3 ) + ln ( 2 ln ( lnAmax ( 1− γ ) ) ) ] = O [ ln ( |S|Amax δµmin ( 1− γ ) ) ] . ( 108 ) Similarly , ln ( K|S| δµmin ) =O [ ln ( ( |S| δµmin ( 1− γ ) ) ln ( lnAmax ( 1− γ ) ) ) ] =O [ ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] +O [ ln ( ( |S| δµmin ( 1− γ ) ) ln ( 1 1− γ ) ) ] =O [ ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] +O [ ln ( |S| δµmin ( 1− γ ) ) ] =O [ ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] ( 109 ) where ( i ) uses ≤ lnAmax1−γ . With the above equalities , we will prove below that the hyperparameter choices in Theorem 2 are valid and satisfy the conditions of Theorem 1 with proper constants hidden in O ( · ) . η = [ 2 ( τ +Qmax ) ] −1 = O ( 1− γ ) ( 110 ) τ =O ( ( 1− γ ) lnAmax ) ∈ ( 0 , 1 ] ( 111 ) K = 1 ln γ−1 ln ( ln5/2Amax 5 ( 1− γ ) 7.5 ) = O [ 1 1− γ ln ( lnAmax ( 1− γ ) ) ] ≥ 1 ( 112 ) Tk =1 + k ln γ−1 ln ( 1− ητ ) −1 ≥ 1 ( 113 ) Nk+1 =O [ tmix ( ln2Amax ) γ−k/2 2µmin ( 1− γ ) 8 ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] ≥650tmix µmin ln ( 4 δ √ µmin ) = O [ tmix µmin ln ( 1 δµmin ) ] ( 114 ) Nk , t =O [ tmixAmax ( 1− ητ ) −3 ( t+1 ) /5 µmin ( 1− γ ) 3 ln ( |S|Amax δµmin ( 1− γ ) ) ] ≥650tmixAmax µmin ln ( 20Tsum|S|Amax δ √ µmin ) = O [ tmixAmax µmin ln ( |S|Amax δµmin ( 1− γ ) ) ] ( 115 ) Nk , t =O [ tmixAmax ( ln3/2Amax ) ( 1− ητ ) −3 ( t+1 ) /5 µmin 3/2 ( 1− γ ) 6 ln ( |S|Amax δµmin ( 1− γ ) ) ] ≥650tmixAmax µmin ln ( 20Tsum|S|Amax δ √ µmin ) = O [ tmixAmax µmin ln ( |S|Amax δµmin ( 1− γ ) ) ] , ( 116 ) With these hyperparameters and eqs . ( 108 ) & ( 109 ) , the duality gap bound ( 20 ) can be simplified as follows , D ( τ ) ( π ( 1 ) K−1 , π ( 2 ) K−1 ) ≤ O [ Vmax lnAmax 1− γ K−1∑ k=0 γK−k ( 1− ητ ) Tk−1 + Vmax 1− γ [ tmixAmax µmin ln ( Tsum|S|Amax δµmin ) ] 2/3 K−1∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−2−t ( 1 N 2/3 k , t + Vmax τN 2/3 k , t+1 ) + V 3maxγ K τ2 ( 1− γ ) 2 + tmixV 3 max τ2µmin ( 1− γ ) 3 ln ( K|S| δµmin ) K−1∑ k=0 γK−k−1 Nk+1 ] ( i ) ≤ O [ lnAmax ( 1− γ ) 2 K−1∑ k=0 γK−kγk + 1 ( 1− γ ) 2 [ tmixAmax µmin ln ( |S|Amax δµmin ( 1− γ ) ) ] 2/3 K−1∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−2−t ( O [ tmixAmax µmin ( 1− γ ) 3 ln ( |S|Amax δµmin ( 1− γ ) ) ] −2/3 ( 1− ητ ) 2 ( t+1 ) /5 +O ( lnAmax ( 1− γ ) 2 ) O [ tmixAmax ( ln3/2Amax ) µmin 3/2 ( 1− γ ) 6 ln ( |S|Amax δµmin ( 1− γ ) ) ] −2/3 ( 1− ητ ) 2 ( t+1 ) /5 ) + γK ( ln2Amax ) 2 ( 1− γ ) 7 + tmix ( ln 2Amax ) 2µmin ( 1− γ ) 8 ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) K−1∑ k=0 γK−k−1O [ tmix ( ln2Amax ) γ−k/2 2µmin ( 1− γ ) 8 ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] −1 ] ≤ O [ lnAmax ( 1− γ ) 2 KγK + K−1∑ k=0 γK−k−1 Tk−1∑ t=0 ( 1− ητ ) Tk−1−3 ( t+1 ) /5 + γK ( ln2Amax ) 2 ( 1− γ ) 7 + K−1∑ k=0 γK−k/2−1 ] ( ii ) ≤ O [ lnAmax ( 1− γ ) 2 KγK + K−1∑ k=0 γK−1 ( 1− ητ ) −3/5 [ ( 1− ητ ) −3Tk/5 − 1 ] ( 1− ητ ) −3/5 − 1 + γK ( ln2Amax ) 2 ( 1− γ ) 7 + γK−1 γ−K/2 − 1 γ−1/2 − 1 ] ( iii ) ≤ O [ lnAmax ( 1− γ ) 2 KγK + K−1∑ k=0 γK−1−3k/5 1− ( 1− ητ ) 3/5 + γK ( ln2Amax ) 2 ( 1− γ ) 7 + γ ( K−1 ) /2 1− γ1/2 ] ( iv ) ≤ O [ lnAmax ( 1− γ ) 2 KγK + γK−1 ( γ−3K/5 − 1 ) ητ ( γ−3/5 − 1 ) + γK ( ln2Amax ) 2 ( 1− γ ) 7 + γK/2 1− γ ] ( v ) ≤ O [ γK lnAmax ( 1− γ ) 3 ln ( lnAmax ( 1− γ ) ) + γ2K/5 lnAmax ( 1− γ ) 3 + γK ( ln2Amax ) 2 ( 1− γ ) 7 + γK/2 1− γ ] ( vi ) ≤ O [ + 3 √ 1− γ lnAmax + 5/2 ( 1− γ ) 2.75 ln5/4Amax ] = O ( ) ( vii ) ≤ c ( 117 ) where ( i ) , ( ii ) and ( iii ) use ( 1−ητ ) Tk−1 = γk based on eq . ( 113 ) , ( i ) also uses eqs . ( 105 ) , ( 108 ) - ( 111 ) & ( 114 ) - ( 116 ) , ( iv ) uses 1 − ( 1 − ητ ) 3/5 = O ( ητ ) , 1 − γ1/2 = O ( 1 − γ ) , ( v ) uses eqs . ( 110 ) - ( 112 ) , ( vi ) uses O [ γK lnAmax ( 1−γ ) 3 ln ( lnAmax ( 1−γ ) ) ] ≤ O [ γK ( ln2 Amax ) 2 ( 1−γ ) 7 ] and γK = 5 ( 1−γ ) 7.5 ln5/2 Amax implied by eq . ( 112 ) , and the numeric constant c > 0 in ( vii ) exists . By replacing with /c , we obtain that Dτ ( π ( 1 ) K−1 , π ( 2 ) K−1 ) ≤ which means ( π ( 1 ) K−1 , π ( 2 ) K−1 ) is an -Nash equilibrium policy pair , and the orders of all the hyperparameter choices remain the same . In this case , the overall sample complexity is given by K−1∑ k=0 [ Nk+1 + 2 Tk−1∑ t=0 ( Nk , t +Nk , t+1 ) ] ( i ) = O [ tmix ( ln2Amax ) 2µmin ( 1− γ ) 8 ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] K−1∑ k=0 γ−k/2 +O [ tmixAmax ( ln3/2Amax ) µmin 3/2 ( 1− γ ) 6 ln ( |S|Amax δµmin ( 1− γ ) ) ] K−1∑ k=0 Tk−1∑ t=0 ( 1− ητ ) −3 ( t+1 ) /5 = O [ tmix ( ln2Amax ) 2µmin ( 1− γ ) 8 ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] γ−K/2 − 1 γ−1/2 − 1 +O [ tmixAmax ( ln3/2Amax ) µmin 3/2 ( 1− γ ) 6 ln ( |S|Amax δµmin ( 1− γ ) ) ] K−1∑ k=0 [ ( 1− ητ ) −3Tk/5 − 1 ] 1− ( 1− ητ ) 3/5 ( ii ) ≤ O [ tmix ( ln2Amax ) 2µmin ( 1− γ ) 8 ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] γ−K/2 +O [ tmixAmax ( ln3/2Amax ) µmin 3/2 ( 1− γ ) 6 ln ( |S|Amax δµmin ( 1− γ ) ) lnAmax ( 1− γ ) 2 ] K−1∑ k=0 γ−3k/5 ≤ O [ tmix ( ln2Amax ) 2µmin ( 1− γ ) 8 ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] γ−K/2 +O [ tmixAmax ( ln5/2Amax ) µmin 5/2 ( 1− γ ) 8 ln ( |S|Amax δµmin ( 1− γ ) ) ] γ−3K/5 γ−3/5 − 1 ( iii ) ≤ O [ tmix ( ln2Amax ) 2µmin ( 1− γ ) 8 ln ( |S| ln ( −1 lnAmax ) δµmin ( 1− γ ) ) ] ln5/4Amax 5/2 ( 1− γ ) 3.75 +O [ tmixAmax ( ln5/2Amax ) µmin 5/2 ( 1− γ ) 8 ln ( |S|Amax δµmin ( 1− γ ) ) ] ln3/2Amax 3 ( 1− γ ) 5.5 ( iv ) = O [ tmixAmax ( ln4Amax ) µmin 5.5 ( 1− γ ) 13.5 ln ( |S|Amax δµmin ( 1− γ ) ) ] ( 118 ) = Õ ( tmixAmax µmin 5.5 ( 1− γ ) 13.5 ) where ( i ) usesNk , t ≤ Nk , t and eqs . ( 114 ) & ( 116 ) , ( ii ) uses γ−1/2−1 = O ( 1−γ ) , 1− ( 1−ητ ) 3/5 = O ( ητ ) = O ( ( 1−γ ) 2 lnAmax ) and ( 1−ητ ) Tk−1 = γk , ( iii ) uses γ−3/5−1 = O ( 1−γ ) and γK = 5 ( 1−γ ) 7.5 ln5/2 Amax implied by the hyperparameter choice ( 112 ) , and ( iv ) uses the fact that the second term of ( iii ) is larger than the first term of ( iii ) . F SUMMARY OF COMPARISON OF SAMPLE COMPLEXITIES In the column “ Model-free ” , a Xmeans that an algorithm does not need any prior knowledge of the environment , including reward mapping and transition kernel . In the column “ Private update ” , a Xmeans that the algorithm updates do not involve the opponent ’ s sensitive information , including action and policy . The sample complexity is defined as the total number of samples required to achieve an -duality gap for the Markov game . A “ – ” sign in this column means that such type of sample complexity is unavailable in that paper or its setting is different from ours . We use Õ to hide all the logarithm factors . We next explain how the sample complexity of ( Wei et al. , 2021 ) is calculated . Corollary 4 of ( Wei et al. , 2021 ) shows that the iterate-average duality gap is smaller than with the hyper-parameter choices : number of iterations T = Õ [ |S|2 η2 ( 1−γ ) 4 2 ] and number of samples queried per iteration L = Õ [ A3max|S| 6 ( 1−γ ) 13µminη3 6 ] ( we replace ξ with ) . Hence , the overall sample complexity is LT = Õ [ A3max|S|8 ( 1− γ ) 17µminη5 8 ] ( i ) ≥ Õ [ A3max|S|10.5 ( 1− γ ) 29.5µmin 8 ] , where ( i ) uses the choice of learning rate η ≤ O [ √ ( 1−γ ) 5 |S| ] required by their Algorithm 1 . G RATIONALITY In this section , we will prove that the policy extragradient ( PE ) algorithm Cen et al . ( 2021 ) is rational . The following analysis can be directly extended to our stochastic policy extragradient ( SPE ) algorithm by applying the estimation error bounds in Appendix B . An algorithm is called rational if , whenever the player 2 adopts an arbitrary stationary policy π ( 2 ) , the policy of the player 1 converges to the best response to π ( 2 ) , i.e. , π̃ ( 1 ) ∗τ ( s ) = arg maxπ ( 1 ) ( s ) V ( τ ) π ( 1 ) , π ( 2 ) ( s ) for all s ∈ S. We add tilde in π̃ ( 1 ) ∗τ ( s ) to differentiate it from the Nash policy pair ( π ( 1 ) ∗τ ( s ) , π ( 2 ) ∗τ ( s ) ) and some consequent notations are similar . Denote Q̃ ( τ ) ∗ : = Q ( τ ) π̃ ( 1 ) ∗τ , π ( 2 ) , Ṽ ( τ ) ∗ : = V ( τ ) π̃ ( 1 ) ∗τ , π ( 2 ) as the optimal value functions and define the soft Bellman operator as follows B̃τ ( Q ) ( s , a ( 1 ) , a ( 2 ) ) : = R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) [ max π ( 1 ) ( s′ ) fτ ( Q ( s′ ) ; π ( 1 ) ( s′ ) , π ( 2 ) ( s′ ) ) ] . ( 119 ) Similar to eqs . ( 98 ) & ( 99 ) , it can be proved that the soft Bellman operator B̃τ is also a contraction operator and has a unique fixed point Q̃ ( τ ) ∗ , i.e. , ‖B̃τ ( Q′ ) − B̃τ ( Q ) ‖∞ ≤γ‖Q′ −Q‖∞ , ( 120 ) B̃τ ( Q̃ ( τ ) ∗ ) =Q̃ ( τ ) ∗ . ( 121 ) To solve modified Markov game with fixed policy π ( 2 ) , we develop the single-player-perspective version of the PE algorithm in Algorithm 2 , which basically approximates the value iteration Qk+1 = B̃τ ( Qk ) . The main modification from the original PE algorithm is that the new policy update ( 126 ) only keeps the third expression of the PU steps ( 8 ) since only that affects the final policy of player 1 with fixed π ( 2 ) . This new policy update ( 126 ) aims to find π ( 1 ) that maximizes the following single-player Markov game problem with fixed Qk ( s ) and π ( 2 ) ( s ) for all s ∈ S , while the original PU algorithm seeks the Nash equilibrium ( π ( 1 ) , π ( 2 ) ) of the following function . max π ( 1 ) ( s ) fτ ( Qk ( s ) ; π ( 1 ) ( s ) , π ( 2 ) ( s ) ) : = [ π ( 1 ) ( s ) ] > Qk ( s ) π ( 2 ) ( s ) + τH ( π ( 1 ) ( s ) ) − τH ( π ( 2 ) ( s ) ) . ( 122 ) Similar to eq . ( 23 ) , it can be derived that the solution π̃∗ ( 1 ) k ( s ) to the above optimization problem satisfies the following condition Cen et al . ( 2021 ) . π̃ ∗ ( 1 ) k ( s ) ∝ exp ( Qk ( s ) π2 ( s ) /τ ) . ( 123 ) Algorithm 2 Single-player perspective PE algorithm for entropy-regularized Markov game . Initialize : V0 ( s ) for all s ∈ S. for value iterations k = 0 , 1 , . . . , K − 1 do Compute the following Q functions for all ( s , a ( 1 ) , a ( 2 ) ) . Qk ( s , a ( 1 ) , a ( 2 ) ) =R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( s , a ( 1 ) , a ( 2 ) ) Vk ( s′ ) ( 124 ) Q ( 1 ) k ( s , a ( 1 ) ) =Ea ( 2 ) ∼π ( 2 ) ( s ) Qk ( s , a ( 1 ) , a ( 2 ) ) ( 125 ) Initialize π ( 1 ) k,0 with uniform distribution . for PU iterations t = 0 , 1 , . . . , T − 1 do Player 1 implements the t-th policy update for all s , a ( 1 ) as follows . π ( 1 ) k , t+1 ( a ( 1 ) |s ) ∝ π ( 1 ) k , t ( a ( 1 ) |s ) 1−ητ exp ( ηQ ( 1 ) k ( s , a ( 1 ) ) ) , ( 126 ) end Let π ( 1 ) k = π ( 1 ) k , T , and perform the following value iteration for all s Vk+1 ( s ) : = fτ ( Qk ( s ) ; π ( 1 ) k ( s ) , π ( 2 ) ( s ) ) ( 127 ) end Output : π ( 1 ) K−1 . We first prove that the value functions Vk and Qk in Algorithm 2 are bounded as follows . Lemma 7 . If ‖V0‖∞ ≤ Vmax : = 1+τ lnAmax1−γ in Algorithm 2 , then for all k ≥ 0 , 0 ≤ t ≤ Tk − 1 , we have max ( ‖Vk‖∞ , ‖Ṽ ( τ ) ∗ ‖∞ ) ≤ Vmax , ( 128 ) max ( ‖Qk‖∞ , Q̃ ( τ ) ∗ ‖∞‖∞ ) ≤ Qmax , ( 129 ) where Qmax : = 1 + γVmax = 1+γτ lnAmax1−γ with Amax : = max ( |A ( 1 ) | , |A ( 2 ) | ) . Proof . The proof is similar to that of Lemma 3 . Similar to Lemma 1 , we can prove the following lemma about the convergence rates of the modified PU steps defined by eq . ( 126 ) . Lemma 8 . Apply Algorithm 2 to solve the entropy regularized game with τ > 0 . Choose learning rate η ≤ τ−1 and initialization ‖V0‖∞ ≤ Vmax . Then , the policy update defined by eq . ( 126 ) for maximizing the function ( 122 ) has the following convergence rates . KL ( π̃ ∗ ( 1 ) k ( s ) ‖π ( 1 ) k , T ( s ) ) +KL ( π ( 1 ) k , T ( s ) ‖π̃ ∗ ( 1 ) k ( s ) ) ≤ 2Qmax τ ( 1− ητ ) T ( 130 ) fτ ( Qk ( s ) ; π̃ ∗ ( 1 ) k ( s ) , π ( 2 ) ( s ) ) − Vk+1 ( s ) = τKL ( π ( 1 ) k , T ( s ) ‖π̃ ∗ ( 1 ) k ( s ) ) ≤ 2Qmax ( 1− ητ ) T ( 131 ) Proof . Taking logarithm of eq . ( 126 ) yields that there exists ck , t ∈ R such that lnπ ( 1 ) k , t+1 ( s ) = ( 1− ητ ) lnπ ( 1 ) k , t ( s ) + ηQk ( s ) π2 ( s ) + ck , t1 . Iterating the above equality over t = 0 , 1 , . . . , t yields that there exists c ∈ R such that lnπ ( 1 ) k , T ( s ) = 1− ( 1− ητ ) T τ Qk ( s ) π2 ( s ) + c1 ( 132 ) where we use the uniform policy initialization π ( 1 ) k,0 ( s ) = 1/|A ( 1 ) | . Taking logarithm of eq . ( 123 ) yields that there exists c∗ ∈ R such that ln π̃ ∗ ( 1 ) k ( s ) = Qk ( s ) π2 ( s ) /τ + c ∗1 ( 133 ) Therefore , eq . ( 130 ) can be proved as follows . KL ( π̃ ∗ ( 1 ) k ( s ) ‖π ( 1 ) k , T ( s ) ) +KL ( π ( 1 ) k , T ( s ) ‖π̃ ∗ ( 1 ) k ( s ) ) = 〈 π̃ ∗ ( 1 ) k ( s ) , ln π̃ ∗ ( 1 ) k ( s ) − lnπ ( 1 ) k , T ( s ) 〉 + 〈 π ( 1 ) k , T ( s ) , lnπ ( 1 ) k , T ( s ) − ln π̃ ∗ ( 1 ) k ( s ) 〉 ( i ) = 〈 π̃ ∗ ( 1 ) k ( s ) − π ( 1 ) k , T ( s ) , ( 1− ητ ) T τ Qk ( s ) π2 ( s ) + ( c ∗ − c ) 1 〉 ( ii ) = ( 1− ητ ) T τ [ π̃ ∗ ( 1 ) k ( s ) − π ( 1 ) k , T ( s ) ] > Qk ( s ) π2 ( s ) ( iii ) ≤ 2Qmax τ ( 1− ητ ) T where ( i ) uses eqs . ( 132 ) & ( 133 ) , ( ii ) uses 〈π̃∗ ( 1 ) k ( s ) − π ( 1 ) k , T ( s ) ,1〉 = 0 and ( iii ) uses ‖Qk ( s ) ‖∞ ≤ Qmax , ‖π̃∗ ( 1 ) k − π ( 1 ) k , T ‖1 ≤ 2 , ‖π2 ( s ) ‖1 = 1 and 1− ητ > 0. fτ ( Qk ( s ) ; π̃ ∗ ( 1 ) k ( s ) , π ( 2 ) ( s ) ) − Vk+1 ( s ) = fτ ( Qk ( s ) ; π̃ ∗ ( 1 ) k ( s ) , π ( 2 ) ( s ) ) − fτ ( Qk ( s ) ; π ( 1 ) k , T ( s ) , π ( 2 ) ( s ) ) = 〈 π̃ ∗ ( 1 ) k ( s ) − π ( 1 ) k , T ( s ) , Qk ( s ) π ( 2 ) ( s ) 〉 + τ 〈 π ( 1 ) k , T ( s ) , lnπ ( 1 ) k , T ( s ) 〉 − τ 〈 π̃ ∗ ( 1 ) k ( s ) , ln π̃ ∗ ( 1 ) k ( s ) 〉 ( i ) = 〈 π̃ ∗ ( 1 ) k ( s ) − π ( 1 ) k , T ( s ) , τ ln π̃ ∗ ( 1 ) k ( s ) 〉 + τ 〈 π ( 1 ) k , T ( s ) , lnπ ( 1 ) k , T ( s ) 〉 − τ 〈 π̃ ∗ ( 1 ) k ( s ) , ln π̃ ∗ ( 1 ) k ( s ) 〉 = τKL ( π ( 1 ) k , T ( s ) ‖π̃ ∗ ( 1 ) k ( s ) ) ( ii ) ≤ 2Qmax ( 1− ητ ) T where ( i ) uses eq . ( 133 ) and 〈 π̃ ∗ ( 1 ) k ( s ) − π ( 1 ) k , T ( s ) ,1 〉 = 0 and ( ii ) uses eq . ( 130 ) . Finally , we prove the convergence rate of Algorithm 2 as follows . Theorem 3 . Apply Algorithm 2 to solve the entropy regularized game with τ > 0 . Choose learning rate η ≤ τ−1 and initialization ‖V0‖∞ ≤ Vmax . Then , the Q function estimation error ‖QK−1 − Q̃ ( τ ) ∗ ‖∞ and the function value gap maxs [ Ṽ ( τ ) ∗ ( s ) − V ( τ ) π ( 1 ) K−1 , π ( 2 ) ( s ) ] converge at the following rates ‖QK−1 − Q̃ ( τ ) ∗ ‖∞ ≤ 2QmaxγK + 2γQmax 1− γ ( 1− ητ ) T , ( 134 ) max s [ Ṽ ( τ ) ∗ ( s ) − V ( τ ) π ( 1 ) K−1 , π ( 2 ) ( s ) ] ≤ 2Qmax 1− γ ( 2γK−1 + 1 + γ 1− γ ( 1− ητ ) T ) . ( 135 ) Proof . ‖Qk+1 − Q̃ ( τ ) ∗ ‖∞ ( i ) ≤ ‖Qk+1 − B̃τ ( Qk ) ‖∞ + ‖B̃τ ( Qk ) − B̃τ ( Q̃ ( τ ) ∗ ) ‖∞ ( ii ) ≤ max s , a ( 1 ) , a ( 2 ) |Qk+1 ( s , a ( 1 ) , a ( 2 ) ) − B̃τ ( Qk ) ( s , a ( 1 ) , a ( 2 ) ) |+ γ‖Qk − Q̃ ( τ ) ∗ ‖∞ = max s , a ( 1 ) , a ( 2 ) ∣∣∣R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) Vk+1 ( s′ ) − [ R ( s , a ( 1 ) , a ( 2 ) ) + γEs′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) fτ ( Qk ( s ′ ) ; π̃ ∗ ( 1 ) k ( s ′ ) , π ( 2 ) ( s′ ) ) ] ∣∣∣+ γ‖Qk − Q̃ ( τ ) ∗ ‖∞ ≤ γmax s′∈S ∣∣Vk+1 ( s′ ) − fτ ( Qk ( s′ ) ; π̃∗ ( 1 ) k ( s′ ) , π ( 2 ) ( s′ ) ) ∣∣+ γ‖Qk − Q̃ ( τ ) ∗ ‖∞ ( iii ) ≤ 2γQmax ( 1− ητ ) T + γ‖Qk − Q̃ ( τ ) ∗ ‖∞ , where ( i ) uses eq . ( 121 ) , ( ii ) uses eq . ( 120 ) , ( iii ) uses eq . ( 131 ) . Iterating the above inequality yields that ‖QK−1 − Q̃ ( τ ) ∗ ‖∞ ≤ γK‖Q0 − Q̃ ( τ ) ∗ ‖∞ + 2γQmax ( 1− ητ ) T 1− γT 1− γ ≤ 2QmaxγK + 2γQmax 1− γ ( 1− ητ ) T , which proves eq . ( 134 ) . Finally , eq . ( 135 ) can be proved as follows Ṽ ( τ ) ∗ ( s ) − V ( τ ) π ( 1 ) K−1 , π ( 2 ) ( s ) = ∑ a ( 1 ) , a ( 2 ) [ Q̃ ( τ ) ∗ ( s , a ( 1 ) , a ( 2 ) ) π̃ ( 1 ) ∗τ ( a ( 1 ) |s ) −Q ( τ ) π ( 1 ) K−1 , π ( 2 ) ( s , a ( 1 ) , a ( 2 ) ) π ( 1 ) K−1 ( a ( 1 ) |s ) ] π ( 2 ) ( a ( 2 ) |s ) + τ [ H ( π̃ ( 1 ) ∗τ ( s ) ) −H ( π ( 1 ) K−1 ( s ) ) ] = ∑ a ( 1 ) , a ( 2 ) [ Q̃ ( τ ) ∗ ( s , a ( 1 ) , a ( 2 ) ) −Q ( τ ) π ( 1 ) K−1 , π ( 2 ) ( s , a ( 1 ) , a ( 2 ) ) ] π ( 1 ) K−1 ( a ( 1 ) |s ) π ( 2 ) ( a ( 2 ) |s ) + ∑ a ( 1 ) , a ( 2 ) Q̃ ( τ ) ∗ ( s , a ( 1 ) , a ( 2 ) ) [ π̃ ( 1 ) ∗τ ( a ( 1 ) |s ) − π ( 1 ) K−1 ( a ( 1 ) |s ) ] π ( 2 ) ( a ( 2 ) |s ) + τ [ H ( π̃ ( 1 ) ∗τ ( s ) ) −H ( π ( 1 ) K−1 ( s ) ) ] = γ ∑ a ( 1 ) , a ( 2 ) [ π ( 1 ) K−1 ( a ( 1 ) |s ) π ( 2 ) ( a ( 2 ) |s ) Es′∼P ( ·|s , a ( 1 ) , a ( 2 ) ) [ Ṽ ( τ ) ∗ ( s ′ ) − V ( τ ) π ( 1 ) K−1 , π ( 2 ) ( s′ ) ] ] + ( π̃ ( 1 ) ∗τ ( s ) − π ( 1 ) K−1 ( s ) ) > Q̃ ( τ ) ∗ ( s ) π ( 2 ) ( s ) + τ [ H ( π̃ ( 1 ) ∗τ ( s ) ) −H ( π ( 1 ) K−1 ( s ) ) ] ≤ γmax s′ [ Ṽ ( τ ) ∗ ( s ′ ) − V ( τ ) π ( 1 ) K−1 , π ( 2 ) ( s′ ) ] + ( π̃ ( 1 ) ∗τ ( s ) − π ( 1 ) K−1 ( s ) ) > Q̃ ( τ ) ∗ ( s ) π ( 2 ) ( s ) + τ [ H ( π̃ ( 1 ) ∗τ ( s ) ) −H ( π ( 1 ) K−1 ( s ) ) ] . Applying maxs to both sides of the above inequality and rearranging it yields that max s [ Ṽ ( τ ) ∗ ( s ) − V ( τ ) π ( 1 ) K−1 , π ( 2 ) ( s ) ] ≤ 1 1− γ max s ( ( π̃ ( 1 ) ∗τ ( s ) − π ( 1 ) K−1 ( s ) ) > Q̃ ( τ ) ∗ ( s ) π ( 2 ) ( s ) + τ [ H ( π̃ ( 1 ) ∗τ ( s ) ) −H ( π ( 1 ) K−1 ( s ) ) ] ) ≤ 1 1− γ max s ( fτ ( Q̃ ( τ ) ∗ ( s ) ; π̃ ( 1 ) ∗τ ( s ) , π ( 2 ) ( s ) ) − fτ ( Q̃ ( τ ) ∗ ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) ) ≤ 1 1− γ max s ( fτ ( Q̃ ( τ ) ∗ ( s ) ; π̃ ( 1 ) ∗τ ( s ) , π ( 2 ) ( s ) ) − fτ ( QK−1 ( s ) ; π̃ ∗ ( 1 ) K−1 , π ( 2 ) ( s ) ) ) + 1 1− γ max s ( fτ ( QK−1 ( s ) ; π̃ ∗ ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) − fτ ( QK−1 ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) ) + 1 1− γ max s ( fτ ( QK−1 ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) − fτ ( Q̃ ( τ ) ∗ ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) ) ≤ 1 1− γ max s ∣∣∣ ( max π ( 1 ) ( s ) fτ ( Q̃ ( τ ) ∗ ( s ) ; π ( 1 ) ( s ) , π ( 2 ) ( s ) ) ) − ( max π ( 1 ) ( s ) fτ ( QK−1 ( s ) ; π ( 1 ) ( s ) , π ( 2 ) ( s ) ) ) ∣∣∣ + 1 1− γ max s ( fτ ( QK−1 ( s ) ; π̃ ∗ ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) − VK ( s ) ) + 1 1− γ max s ∣∣∣fτ ( QK−1 ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) − fτ ( Q̃ ( τ ) ∗ ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) ∣∣∣ ( i ) ≤ 2 1− γ ( 2Qmaxγ K + 2γQmax 1− γ ( 1− ητ ) T ) + 2Qmax 1− γ ( 1− ητ ) T ≤ 2Qmax 1− γ ( 2γK−1 + 1 + γ 1− γ ( 1− ητ ) T ) , where ( i ) uses eq . ( 131 ) and the following inequality that holds for all s , π ( 1 ) ( s ) , π ( 2 ) ( s ) , ( ii ) uses eq . ( 134 ) and VK−1 ( s ) = fτ ( Q ( τ ) K−1 ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) .∣∣∣fτ ( QK−1 ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) − fτ ( Q̃ ( τ ) ∗ ( s ) ; π ( 1 ) K−1 ( s ) , π ( 2 ) ( s ) ) ∣∣∣ ≤ ‖QK−1 ( s ) − Q̃ ( τ ) ∗ ( s ) ‖∞ eq . ( 134 ) ≤ 2QmaxγK + 2γQmax 1− γ ( 1− ητ ) T . In Theorem 3 , the convergence rates of both the Q function estimation error ‖QK−1−Q̃ ( τ ) ∗ ‖∞ and the function value gap maxs [ Ṽ ( τ ) ∗ ( s ) − V ( τ ) π ( 1 ) K−1 , π ( 2 ) ( s ) ] consist of two terms O ( γK ) and O ( 1− ητ ) T , which characterize the exponential convergence rates of the K outer value iterations and the T inner policy updates respectively . Such a convergence result indicates that Algorithm 2 converges to the optimal solution to the regularized Markov game , and thus can be arbitrarily close to the optimal solution of the unregularized Markov game with sufficiently small τ . As a result , the PU algorithm Cen et al . ( 2020 ) is rational . Similar result can be directly extended to our stochastic policy extragradient ( SPE ) algorithm by applying the estimation error bounds in Appendix B .

Summary: ## Summary   This paper studies a decentralized stochastic policy extra-gradient algorithm for solving two-player zero-sum Markov game. In comparison to the standard policy extra-gradient algorithm, this algorithm uses a set of stochastic estimators to estimate the value functions involved in the stochastic updates. Improved sample complexity guarantee is also derived.