Multi-Agent Trust Region Learning

1 INTRODUCTION . Multi-agent systems ( MAS ) ( Shoham & Leyton-Brown , 2008 ) have received much attention from the reinforcement learning community . In real-world , automated driving ( Cao et al. , 2012 ) , StarCraft II ( Vinyals et al. , 2019 ) and Dota 2 ( Berner et al. , 2019 ) are a few examples of the myriad of applications that can be modeled by MAS . Due to the complexity of multi-agent problems ( Chatterjee et al. , 2004 ) , investigating if agents can learn to behave effectively during interactions with environments and other agents is essential ( Fudenberg et al. , 1998 ) . This can be achieved naively through the independent learner ( IL ) ( Tan , 1993 ) , which ignores the other agents and optimizes the policy assuming a stable environment ( Buşoniu et al. , 2010 ; Hernandez-Leal et al. , 2017 ) . Due to their theoretical guarantee and good empirical performance in real-world applications , trust region methods ( e.g. , PPO ( Schulman et al. , 2015 ; 2017 ) ) based ILs are popular ( Vinyals et al. , 2019 ; Berner et al. , 2019 ) . In single-agent learning , trust region methods can produce a monotonic payoff improvement guarantee ( Kakade & Langford , 2002 ) via line search ( Schulman et al. , 2015 ) . In multi-agent scenarios , however , an agent ’ s improvement is affected by other agent ’ s adaptive behaviors ( i.e. , the multi-agent environment is non-stationary ( Hernandez-Leal et al. , 2017 ) ) . As a result , trust region learners can measure the policy improvements of the agents ’ current policies , but the improvements of the updated opponents ’ policies are unknown ( shown in Fig . 1 ) . Therefore , trust region based ILs act less well in MAS as in single-agent tasks . Moreover , the convergence to a fixed point , such as a Nash equilibrium ( Nash et al. , 1950 ; Bowling & Veloso , 2004 ; Mazumdar et al. , 2020 ) , is a common and widely accepted solution concept for multi-agent learning . Thus , although independent learners can best respond to other agents ’ current policies , they lose their convergence guarantee ( Laurent et al. , 2011 ) . One solution to address the convergence problem for independent learners is Empirical GameTheoretic Analysis ( EGTA ) ( Wellman , 2006 ) , which approximates the best response to the policies generated by the independent learners ( Lanctot et al. , 2017 ; Muller et al. , 2019 ) . 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Other multi-agent learning approaches collect or approximate additional information such as communication ( Foerster et al. , 2016 ) and centralized joint critics ( Lowe et al. , 2017 ; Foerster et al. , 2017 ; Sunehag et al. , 2018 ; Rashid et al. , 2018 ) . Nevertheless , these methods usually require centralized parameters or centralized communication assumptions . Thus , there is considerable interest in multi-agent learning to find an algorithm that , while having minimal requirements and computational cost as independent learners , also improves convergence performance at the same time . This paper presents the Multi-Agent Trust Region Learning ( MATRL ) algorithm that augments the trust-region ILs with a meta-game analysis to improve the stability and efficiency of learning . In MATRL , a trust region trial step for an agents ’ payoff improvement is implemented by independent learners , which gives a predicted policy based on the current policy . Then , an empirical policy-space meta-game is constructed to compare the expected advantage of predicted policies with the current policies . By solving the meta-game , MATRL finds a restricted step by aggregating the current and predicted policies using meta-game Nash Equilibrium . Finally , MATRL takes the best responses based on the aggregated policies from last step for each agent to explore because the found TSR is not always strict stable . MATRL is , therefore , able to provide a weak stable solution compared with the naive independent learners . Based on trust region independent learners , MATRL does not need extra parameters , simulations , or modifications to the independent learner itself . We provide insights into the empirical meta-game in Section 3.2 , showing that an approximated Nash equilibrium of the meta-game is a weak stable fixed point of the underlying game . Our experiments demonstrate that MATRL significantly outperforms deep independent learners ( Schulman et al. , 2017 ) with the same hyper-parameters , centralized VDN ( Sunehag et al. , 2018 ) , QMIX ( Rashid et al. , 2018 ) methods in discrete action grid-worlds , centralized MADDPG ( Lowe et al. , 2017 ) in a continuous action multi-agent MuJoCo task ( de Witt et al. , 2020 ) and zero-sum multi-agent Atari ( Terry & Black , 2020 ) . 2 PRELIMINARY . A Stochastic Game ( Shapley , 1953 ; Littman , 1994 ) can be defined as : G = 〈N , S , { Ai } , { Ri } , P , p0 , γ〉 , where N is a set of agents , n = |N | is the number of agents and S denotes the state space . Ai is the action space for agent i . A = A1 × · · · × An = Ai ×A−i is the joint action space , and for the simplicity we use −i denotes the other agents except agent i. Ri = Ri ( s , ai , a−i ) is the reward function for agent i ∈ N . P : S ×A×S → [ 0 , 1 ] is the transition function . p0 is the initial state distribution , γ ∈ [ 0 , 1 ] is a discount factor . Each agent i ∈ N has a stochastic policy πi ( ai|s ) : S ×Ai → [ 0 , 1 ] , and aims to maximize its long term discounted return : ηi ( πi , π−i ) = Es0 , a0i , a0−i··· [ ∞∑ t=0 γtRi ( s t , ati , a t −i ) ] , ( 1 ) where s0 ∼ p0 , st+1 ∼ P ( st+1|st , ati , at−i ) , ati ∼ πi ( ati|τ ti ) . We then can have the standard definitions of the state-action value function Qπi , π−ii ( s t , ati , a t −i ) = Est+1 , at+1i , at+1−i ··· [ ∑∞ l=0 γ lRi ( s t+l , at+li , a t+l −i ) ] , the value function V πi , π−i i ( s t ) = Eati , at−i , st+1··· [ ∑∞ l=0 γ lRi ( s t+l , at+li , a t+l −i ) ] , and the advantage function A πi , π−i i ( s t , ati , a t −i ) = Q πi , π−i i ( s t , ati , a t −i ) − V πi , π−i i ( s t ) given the state and joint action . 3 MULTI-AGENT TRUST REGION POLICY OPTIMIZATION . A trust region algorithm aims to answer two questions : how to compute a trust region trial step and whether a trial step should be accepted . In multi-agent learning , a trust region trial step towards agents ’ payoff improvement can be easily implemented with independent learners , and we call the independent payoff improvement area as Trust Payoff Region ( TPR ) . 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In other words , multi-agent trust region learning ( MATRL ) decomposes the trust region learning into two parts : firstly , find a trust payoff region between current policy πi and predicted policy π̂i ; then , with the help of the predicted policy , a precise method can , to some extent , approximate a weak stable fixed point . Instead of line searching in a single-agent payoff improvement , MATRL searches for the joint policy space to achieve a weak stable fixed point ( see Fig . 2 ) . Essentially , MATRL is a simple extension of the single-agent TRPO to MAS where independent learners learn to find a stable point between current policy and predicted policy . To solve the TSR , we assume the knowledge about other agents ’ policies during the training to find weak stable points via empirical meta-game analysis , while the execution can still be fully decentralized . We explain every step of MATRL in detail in the following sections . 3.1 INDEPENDENT TRUST PAYOFF IMPROVEMENT . Single-agent reinforcement learning algorithms can be straightforwardly applied to multi-agent learning , where we assume that all agents behave independently ( Tan , 1993 ) . In this section , we have chosen the policy-based reinforcement learning method as independent learners . In multi-agent games , the environment becomes a Markov decision process for agent i when each of the other agents plays according to a fixed policy . We set agent i ’ s to make a monotonic improvement against the fixed opponent policies . Thus , at each iteration , the policy is updated by maximizing the utility function ηi over a local neighborhood of the current joint policy πi , π−i : π̂i = arg maxπi∈Πi ηi ( πi , π−i ) based on the trajectories sampled by πi , π−i . We can adopt trust region policy optimization ( e.g. , PPO ( Schulman et al. , 2017 ) ) , which constrains step size in the policy update : π̂i = arg max πi∈Πθi ηi ( πi , π−i ) s.t . D ( πi , π̂i ) ≤ δi , ( 2 ) where D is a distance measurement and δi is a constraint . Independent trust region learners produce the monotonically improved policy π̂i which guarantees ηi ( π̂i , π−i ) ≥ ηi ( πi , π−i ) and give a trust payoff bound by π̂i . Due to the simultaneous policy improvement without awareness of other agents , however , the lower bound of payoff improvement from single-agent Schulman et al . ( 2015 ) no longer holds for multi-agent payoff improvement . Following the similar proof procedures , we can obtain a precise lower bound for a multi-agent simultaneous trust payoff region in Theorem 1 : Theorem 1 ( Independent Trust Payoff Region ) . Denote the expected advantage gain when πi , π−i → π̂i , π̂−i as : g πi , π−i i ( π̂i , π̂−i ) : = ∑ s pπi , π−i ( s ) ∑ ai π̂i ( ai|s ) ∑ a−i π̂−i ( a−i|s ) Aπi , π−ii ( s , ai , a−i ) . ( 3 ) Let αi = DmaxTV ( πi , π̂i ) = maxsDTV ( πi ( ·|s ) ‖π̂i ( ·|s ) ) for agent i , where DTV is total variation divergence ( Schulman et al. , 2015 ) . Then , the following lower bound can be derived for multi-agent independent trust region optimization : ηi ( π̂i , π̂−i ) − ηi ( πi , π−i ) ≥ gπi , π−ii ( π̂i , π̂−i ) − 4γ i ( 1− γ ) 2 ( αi + α−i − αiα−i ) 2 , ( 4 ) where i = maxs , a−i , a−i ∣∣Aπi , π−ii ( s , ai , a−i ) ∣∣ . Under review as a conference paper at ICLR 2021 Proof . See Appendix B . Based on the independent trust payoff improvement , although the predicted policy π̂i will guide us in determining the step size of the TPR , but the stability of ( π̂i , π̂−i ) is still unknown . As shown in Theorem 1 , an agent ’ s lower bound is roughly O ( 4α2 ) , which is four times larger than the single-agent lower bound trust region of O ( α2 ) ( Kakade & Langford , 2002 ) . Furthermore , i = maxs , a−i , a−i ∣∣Aπi , π−ii ( s , ai , a−i ) ∣∣ depends on the other agents ’ s action a−i that would be very large when agents have conflicting interests . Therefore , the most critical issue underlying the multi-agent trust region learning is to find a TSR after the TPR . In next section , we will illustrate how to search for a weak stable fixed point within the TPR , based on the policy-space meta-game analysis .

Summary: This paper proposes a modification of the Independent Learners trust region policy optimization method in general sum games. The modification consists of first forming a “meta game”—ie. the matrix game in which each agent’s options are his previous policy and his independent trust region optimized policy—-and then interpolating between the two for each agent according to a Nash equilibrium of the meta game. The paper shows that this algorithm results in each step generating a “weak stable fixed point”. The paper concludes by showing a number of experimental results indicating the convergence and overall performance of the method as compared with relevant baselines in a number of different games.