| 1: Initialize: for any (s,a,b,h),Qn(s,a,b) ← H,Q,(s,a,b) ← 0,△ ← H,Nn(s,a,b) ←0. | |
| 2: | for episodek=1,...,Kdo for steph=H,H-1,...,1do |
| 3: | |
| 4: | for(s,a,b) ∈S× A× Bdo |
| 5: | t ←Nn(s,a,b). |
| 6: | if t>O then |
| 7: | β ← BoNUs(t,Vn[(Vh+1 +Vh+1)/2](s,a,b)). |
| 8: | γ ← (c/H)Ph(Vh+1 -Vh+1)(s,a,b). |
| 9: | Qn(s,a,b)←min{(rh +PnVh+1)(s,a,b)+γ+β,H}. |
| 10: | Q(s,a,b)←max{(rh+PhVh+1)(s,a,b)-γ-β,0}. |
| 11: | fors∈Sdo |
| 12: | Th(·,·|s) ← CCE(Qn(s,·,),Q(s,·, )). |
| 13: | Vn(s)← (DπnQn)(s)；Vn(s) ←(DπnQn)(s). |
| 14: 15: | if(V1-V1)(s1)<△ then △←(V1-V1)(s1) and πout ←π. |
| 16: | for step h =1,...,H do |
| 17: | take action (an, br) ~ πh(*,|sh),observe reward rh and next state Sh+1· |
| |
| 18: | add1 to Nn(sh,ah,bh) and Nh(sh,ah,bh,Sh+1). |
| 19: | Ph(-Ish,ah,bh) ← Nn(Sh,ah,bh,:)/Nn(sh,ah,bh). |
| 20: | Output (μout,vout) that are the marginal policies of πout. |
+
+to compute the Nash equilibrium for a two-player general-sum matrix game, which is in general PPAD-complete (Daskalakis, 2013).
+
+To overcome this computational challenge, we compute a relaxation of the Nash equilibrium— Coarse Correlated Equalibirum $( C C E )$ —instead, a technique first introduced by Xie et al. (2020) to address reinforcement learning problems in Markov Games. Formally, for any pair of matrices $\overline { { Q } } , \underline { { Q } } \in [ 0 , H ] ^ { A \times B }$ , ${ \mathrm { C C E } } ( \overline { { Q } } , \underline { { Q } } )$ returns a distribution $\pi \in \Delta _ { \mathcal { A } \times \mathcal { B } }$ such that
+
+$$
+\mathbb { E } _ { ( a , b ) \sim \pi } \overline { { Q } } ( a , b ) \geq \operatorname* { m a x } _ { a ^ { \star } } \mathbb { E } _ { ( a , b ) \sim \pi } \overline { { Q } } ( a ^ { \star } , b ) , \qquad \mathbb { E } _ { ( a , b ) \sim \pi } \underline { { Q } } ( a , b ) \leq \operatorname* { m i n } _ { b ^ { \star } } \mathbb { E } _ { ( a , b ) \sim \pi } \underline { { Q } } ( a , b ^ { \star } ) .
+$$
+
+Intuitively, in a CCE the players choose their actions in a potentially correlated way such that no one can benefit from unilateral unconditional deviation. A CCE always exists, since Nash equilibrium is also a CCE and a Nash equilibrium always exists. Furthermore, a CCE can be computed by linear programming in polynomial time. We remark that different from Nash equilibrium where the policies of each player are independent, the policies given by CCE are in general correlated for each player. Therefore, executing such a policy (line 17) requires the cooperation of two players.
+
+# 3.2 THEORETICAL GUARANTEES
+
+Now we are ready to present the theoretical guarantees for Algorithm 1. We let $\pi ^ { k }$ denote the policy computed in line 12 in the $k ^ { \mathrm { { t h } } }$ episode, and $\bar { \mu } ^ { k } , \nu ^ { k }$ denote the marginal policy of $\pi ^ { k }$ for each player.
+
+Theorem 3 (Nash-VI with Hoeffding bonus). For any $p \in ( 0 , 1 ]$ , letting $\iota = \log ( S A B T / p )$ , then with probability at least $1 - p$ , Algorithm $^ { l }$ with Hoeffding type bonus (3) (with some absolute $c > 0$ ) achieves:
+
+• $( V _ { 1 } ^ { \dagger , \nu ^ { o u t } } - V _ { 1 } ^ { \mu ^ { o u t } , \dagger } ) ( s _ { 1 } ) \leq \epsilon ,$ , if the number of episodes $K \geq \Omega ( H ^ { 4 } S A B \iota / \epsilon ^ { 2 } + H ^ { 3 } S ^ { 2 } A B \iota ^ { 2 } / \epsilon ) .$ $\begin{array} { r } { \bullet \operatorname { R e g r e t } ( K ) = \sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \dagger , \nu ^ { k } } - V _ { 1 } ^ { \mu ^ { k } , \dagger } ) ( s _ { 1 } ) \le \mathcal { O } ( \sqrt { H ^ { 3 } S A B T \iota } + H ^ { 3 } S ^ { 2 } A B \iota ^ { 2 } ) . } \end{array}$
+
+Theorem 3 provides both a sample complexity bound and a regret bound for Nash-VI to find an $\epsilon$ -approximate Nash equilibrium. For small $\epsilon \leq H / ( S \iota )$ , the sample complexity scales as ${ \tilde { \mathcal { O } } } ( H ^ { 4 } S A B / { \epsilon } ^ { 2 } )$ . Similarly, for large $T \geq H ^ { 3 } S ^ { 3 } A B \iota ^ { 3 }$ , the regret scales as $\tilde { \mathcal { O } } ( \sqrt { H ^ { 3 } S A B T } )$ , where $T = K H$ is the total number of steps played within $K$ episodes. Theorem 3 is significant in that it improves the sample complexity of the model-based algorithm in Markov games from √ $S ^ { 2 }$ to $S$ (and the regret from $S$ to $\sqrt { S }$ ). This is achieved by adding the new auxiliary bonus $\gamma$ in value iteration steps as explained in Section 3.1. The proof of Theorem 3 can be found in Appendix F.1.
+
+Our next theorem states that when using Bernstein bonus instead of Hoeffding bonus as in (3), the sample complexity of Nash-VI algorithm can be further improved by a $H$ factor in the leading order term (and the regret improved by a $\sqrt { H }$ factor).
+
+Theorem 4 (Nash-VI with the Bernstein bonus). For any $p \in ( 0 , 1 ]$ , letting $\iota = \log ( S A B T / p )$ , then with probability at least $1 - p ,$ , Algorithm $^ { l }$ with Bernstein type bonus (3) (with some absolute $c > 0$ ) achieves:
+
+• $( V _ { 1 } ^ { \dagger , \nu ^ { o u t } } - V _ { 1 } ^ { \mu ^ { o u t } , \dagger } ) ( s _ { 1 } ) \leq \epsilon ,$ , if the number of episodes $K \geq \Omega ( H ^ { 3 } S A B \iota / \epsilon ^ { 2 } + H ^ { 3 } S ^ { 2 } A B \iota ^ { 2 } / \epsilon ) .$ $\begin{array} { r } { \bullet \operatorname { R e g r e t } ( K ) = \sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \dagger , \nu ^ { k } } - V _ { 1 } ^ { \mu ^ { k } , \dagger } ) ( s _ { 1 } ) \le \mathcal { O } ( \sqrt { H ^ { 2 } S A B T \iota } + H ^ { 3 } S ^ { 2 } A B \iota ^ { 2 } ) . } \end{array}$
+
+Compared with the information-theoretic sample complexity lower bound $\Omega ( H ^ { 3 } S ( A + B ) \iota / \epsilon ^ { 2 } )$ and regret lower bound $\Omega ( \sqrt { H ^ { 2 } S ( A + B ) T } )$ (Bai $\&$ Jin, 2020), when $\epsilon$ is small, Nash-VI with Bernstein bonus achieves the optimal dependency on all of $H , S , \epsilon$ up to logarithmic factors in both the sample complexity and the regret, and the only gap that remains open is a $A B / ( A + B ) \leq$ $\operatorname* { m i n } \left\{ A , B \right\}$ factor. The proof of Theorem 4 can be found in Appendix F.2.
+
+Comparison with model-free approaches. Different from our model-based approach, a recently proposed model-free algorithm Nash V-Learning (Bai et al., 2020) achieves sample complexity $\bar { \mathcal { O } } ( \bar { H } ^ { 6 } S ( A + B ) \iota / \epsilon ^ { 2 } )$ , which has a tight $( A + B )$ dependency on $A , B$ . However, our Nash-VI has the following important advantages over Nash V-Learning: 1. Our sample complexity has a better dependency on horizon $H$ ; 2. Our algorithm outputs a single pair of Markov policies $( \mu ^ { \mathrm { o u t } } , \nu ^ { \mathrm { o u t } } )$ while their algorithm outputs a generic history-dependent policy that can be only written as a nested mixture of Markov policies; 3. The model-free algorithms in Bai et al. (2020) cannot be directly√ modified to obtain a $\sqrt { T }$ -regret (so that the exploration policies can be arbitrarily poor), while our√ model-based algorithm has the $\sqrt { T }$ -regret guarantee. We comment that although both Nash-VI and Nash V-Learning have polynomial running time, the later enjoys a better computational complexity because Nash-VI requires to solve LPs for computing CCEs in each episode.
+
+# 4 REWARD-FREE LEARNING
+
+In this section, we modify our model-based algorithm Nash-VI for the reward-free exploration setting (Jin et al., 2020b), which is also known as the task-agnostic (Zhang et al., 2020b) or reward-agnostic setting. Reward-free learning has two phases: In the exploration phase, the agent first collects a dataset of transitions $\mathcal { D } = \{ ( \bar { s } _ { k , h } , a _ { k , h } , \bar { b } _ { k , h } , s _ { k , h + 1 } ) \} _ { ( k , h ) \in [ K ] \times [ H ] }$ from a Markov game $\mathcal { M }$ without the guidance of reward information. After the exploration, in the planning phase, for each task $i \in [ N ]$ , $\mathcal { D }$ is augmented with stochastic reward information to become $\mathcal { D } ^ { i } = \{ \left( s _ { k , h } , a _ { k , h } , b _ { k , h } , s _ { k , h + 1 } , r _ { k , h } \right) \} _ { ( k , h ) \in [ K ] \times [ H ] }$ , where $r _ { k , h }$ is sampled from an unknown reward distribution with expectation equal to $r _ { h } ^ { i } ( s _ { k , h } , a _ { k , h } , b _ { k , h } )$ . Here, we use $r ^ { i }$ to refer to the unknown reward function of the $i ^ { \mathrm { t h } }$ task. The goal is to compute nearly-optimal policies for $N$ tasks under $\mathcal { M }$ , given the augmented datasets.
+
+There are strong practical motivations for considering the reward-free setting. First, in applications such as robotics, we face multiple tasks in sequential systems with shared transition dynamics (i.e. the world) but very different rewards. There, we prefer to learn the underlying transition independent of reward information. Second, from the algorithm design perspective, decoupling exploration and planning (i.e. performing exploration without reward information) can be valuable for designing new algorithms in more challenging settings (e.g. with function approximation).
+
+Due to space limits, we defer the description of our algorithm Optimistic Value Iteration with Zero Reward (VI-Zero, Algorithm 2) to Appendix B and only state its theoretical guarantees here. The following theorem claims that the empirical transition $\widehat { \mathbb { P } } ^ { \mathrm { o u t } }$ outputted by VI-Zero is close to the true transition $\mathbb { P }$ , in the sense that any Nash equilibrium of the ${ \mathcal { M } } ( { \widehat { \mathbb { P } } } , { \widehat { r } } ^ { i } )$ $\because [ i \in [ N ] )$ is also an approximate
+
+Nash equilibrium of the true underlying Markov game ${ \mathcal { M } } ( { \mathbb { P } } , r ^ { i } )$ , where $\widehat { r } ^ { i }$ is the empirical estimate of $r ^ { i }$ computed using $\mathcal { D } ^ { i }$ .
+
+Theorem 5 (Sample complexity of VI-Zero). There exists an absolute constant c, for any $p \in ( 0 , 1 ]$ , $\epsilon \in ( 0 , H ]$ , $N \in \mathbb N$ , if we choose bonus $\beta _ { t } = c ( \sqrt { H ^ { 2 } \iota / t } + H ^ { 2 } S \iota / t )$ with $\iota = \log ( N S A B T / p )$ and $K \ge c ( H ^ { 4 } S A B \iota / \epsilon ^ { 2 } + H ^ { 3 } S ^ { 2 } A B \iota ^ { 2 } / \epsilon )$ , then with probability at least $1 - p ,$ , the output $\widehat { \mathbb { P } } ^ { \mathrm { o u t } }$ of Algorithm 2 has the following property: for any $N$ fixed reward functions $\boldsymbol { r } ^ { \mathrm { 1 } } , \ldots , \boldsymbol { r } ^ { N }$ , a Nash equilibrium of Markov game $\mathcal { M } ( \widehat { \mathbb { P } } ^ { \mathrm { o u t } } , \widehat { r } ^ { i } )$ is also an $\epsilon$ -approximate Nash equilibrium of the true Markov game ${ \mathcal { M } } ( \mathbb { P } , r ^ { i } )$ for all $i \in [ N ]$ .
+
+Theorem 5 shows that, when $\epsilon$ is small, VI-Zero only needs ${ \tilde { \mathcal { O } } } ( H ^ { 4 } S A B / { \epsilon } ^ { 2 } )$ samples to learn an estimate of the transition $\widehat { \mathbb { P } } ^ { \mathrm { o u t } }$ , which is accurate enough to learn the approximate Nash equilibrium for any $N$ fixed rewards. The most important advantage of reward-free learning comes from the sample complexity only scaling polylogarithmically with respect to the number of tasks or reward functions $N$ . This is in sharp contrast to the reward-aware algorithms (e.g. Nash-VI), where the algorithm has to be rerun for each different task, and the total sample complexity must scale linearly in $N$ . In exchange for this benefit, compared to Nash-VI, VI-Zero loses a factor of $H$ in the leading term of sample complexity since we cannot use Bernstein bonus anymore due to the lack of reward information. VI-Zero also does not have a regret guarantee, since again without reward information, the exploration policies are naturally sub-optimal. The proof of Theorem 5 can be found in Appendix G.1.
+
+Connections with reward-free learning in MDPs. Since MDPs are special cases of Markov games, our algorithm VI-Zero directly applies to the single-agent setting, and yields a sample complexity similar to existing results (Zhang et al., 2020b; Wang et al., 2020). However, distinct from existing results which require both the exploration algorithm and the planning algorithm to be specially designed to work together, our algorithm allows an arbitrary planning algorithm as long as it computes the Nash equilibrium of a Markov game with known transition and reward. Therefore, our results completely decouple the exploration and the planning.
+
+Lower bound for reward-free learning. Finally, we comment that despite the sample complexity in Theorem 5 scaling as $A B$ instead of $A + B$ , our next theorem states that unlike the general rewardaware setting, this $A B$ scaling is unavoidable in the reward-free setting. This reveals an intrinsic gap between the reward-free and reward-aware learning: An $A + B$ dependency is only achievable via sampling schemes that are reward-aware. A similar lower bound is also presented in recent work (Zhang et al., 2020a) for the discounted setting with a different hard instance construction.
+
+Theorem 6 (Lower bound for reward-free learning of Markov games). There exists an absolute constant $c > 0$ such that for any $\epsilon \in ( 0 , c ]$ , there exists a family of Markov games $\mathfrak { M } ( \epsilon )$ satisfying that: for any reward-free algorithm $\mathfrak { A }$ using $K \le c H ^ { 2 } \bar { S } A B / \epsilon ^ { 2 }$ episodes, there exists a Markov game $\mathcal { M } \in \mathfrak { M } ( \epsilon )$ such that if we run $\mathfrak { A }$ on $\mathcal { M }$ and output policies $( \hat { \mu } , \hat { \nu } )$ , then with probability at least $1 / 4$ , we have $( V _ { 1 } ^ { \dagger , \hat { \nu } } - V _ { 1 } ^ { \hat { \mu } , \dagger } ) ( s _ { 1 } ) \geq \epsilon$ .
+
+This lower bound shows that the sample complexity in Theorem 5 is optimal in $S , A , B$ , and $\epsilon$ . The proof of Theorem 6 can be found in Appendix G.3.
+
+# 5 MULTI-PLAYER GENERAL-SUM GAMES
+
+We adapt our analysis to multi-player general-sum games and present the first lines of provably efficient algorithms. Concretely, we design two model-based algorithms Multi-Nash-VI and MultiVI-Zero (Algorithm 3 and Algorithm 4) that can find an $\epsilon$ -approximate) $\{ \mathrm { N A S H } , \mathrm { C E } , \mathrm { C C E } \}$ equilibrium for any multi-player general-sum Markov game in $\begin{array} { r } { \tilde { \mathcal { O } } ( H ^ { 4 } S ^ { 2 } \prod _ { i = 1 } ^ { m } A _ { i } / \epsilon ^ { 2 } ) } \end{array}$ episodes of game playing, where $A _ { i }$ is the number of actions for player $i \in \{ 1 , \ldots , m \}$ (Theorem 15 and Theorem 16). Due to space limit, we defer the detailed setups, algorithms and results to Appendix C.
+
+# 6 CONCLUSION
+
+In this paper, we provided a sharp analysis of model-based algorithms for Markov games. Our new algorithm Nash-VI can find an $\epsilon$ -approximate Nash equilibrium of a zero-sum Markov game in $\tilde { \mathcal { O } } ( H ^ { 3 } S A B / \epsilon ^ { 2 } )$ episodes of game playing, which almost matches the sample complexity lower bound except for the $A B$ vs. $A + B$ dependency. We also applied our analysis to derive new efficient algorithms for task-agnostic game playing, as well as the first line of multi-player generalsum Markov games. There are a number of compelling future directions to this work. For example, can we achieve $A + B$ instead of $A B$ sample complexity for zero-sum games using model-based approaches (thus closing the gap between lower and upper bounds)? How can we design more efficient algorithms for general-sum games with better sample complexity (e.g., $\mathcal O ( S )$ instead of $\mathcal { O } ( S ^ { 2 } ) \colon$ We leave these problems as future work.
+
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+
+# A RELATED WORK
+
+Markov games. Markov games (or stochastic games) are proposed in the early 1950s (Shapley, 1953). They are widely used to model multi-agent RL. Learning the Nash equilibria of Markov games has been studied in Littman (1994; 2001); Hu & Wellman (2003); Hansen et al. (2013); Lee et al. (2020), where the transition matrix and reward are assumed to be known, or in the asymptotic setting where the number of data goes to infinity. These results do not directly apply to the nonasymptotic setting where the transition and reward are unknown and only a limited amount of data are available for estimating them.
+
+Another line of work assumes certain strong reachability assumptions under which sophisticated exploration strategies are not required. A prevalent approach is to assume access to simulators (generative models) that enable the agent to directly sample transition and reward information for any state-action pair. In this setting, Jia et al. (2019); Sidford et al. (2019); Zhang et al. (2020a) provide non-asymptotic bounds on the number of calls to the simulator for finding an $\epsilon$ approximate Nash equilibrium. Wei et al. (2017) studies Markov games under an alternative assumption that no matter what strategy one agent sticks to, the other agent can always reach all states by playing a certain policy.
+
+Non-asymptotic guarantees without reachability assumptions. The recent work of Bai & Jin (2020); Xie et al. (2020) provide the first line of non-asymptotic sample complexity guarantees on learning Markov games without these reachability assumptions, in which exploration is essential. However, both results suffer from highly suboptimal sample complexity. The results of Xie et al. (2020) also apply to the linear function approximation setting. More recently, two model-free algorithms—Nash Q-Learning and Nash V-Learning—are shown to achieve better sample complexity guarantees (Bai et al., 2020). In particular, the Nash V-learning algorithm achieves the nearoptimal dependence on $S$ , $A$ and $B$ . However, the dependence on $H$ is worse than our results and the output policy is a nested mixture, which is hard to implement. We compare our results with existing non-asymptotic guarantees in Table 1.
+
+We remark that the classical R-max algorithm (Brafman & Tennenholtz, 2002) also provides provable guarantees for learning Markov games. However, Brafman & Tennenholtz (2002) uses a weaker definition of regret (similar to the online setting in Xie et al. (2020)), and consequently their result does not imply any sample complexity result for finding Nash equilibrium policies.
+
+Adversarial MDPs. Another way to model the multi-player bahavior is to use adversarial MDPs. Most work in this line considers the setting with adversarial rewards (Zimin & Neu, 2013; Rosenberg & Mansour, 2019; Jin et al., 2019), where the reward can be manipulated by an adversary arbitrarily and the goal is to compete with the optimal (stationary) policy in hindsight. Adversarial MDP with changing dynamics is computationally hard even under full-information feedback (Yadkori et al., 2013). Notice these results do not directly imply provable self-play algorithms in our setting, because the opponent in Markov games can affect both the reward and the transition.
+
+Single-agent RL. There is a rich literature on reinforcement learning in MDPs (see e.g., Jaksch et al., 2010; Osband et al., 2014; Azar et al., 2017; Dann et al., 2017; Strehl et al., 2006; Jin et al., 2018). MDP is a special case of Markov games, where only a single agent interacts with a stochastic environment. For the tabular episodic setting with nonstationary dynamics and no simulators, the best sample complexity is $\tilde { \mathcal { O } } ( \bar { H ^ { 3 } } S A / \epsilon ^ { 2 } )$ , achieved by model-based algorithm in Azar et al. (2017) and model-free algorithms in Zhang et al. (2020c), respectively, where $S$ is the number of states, $A$ is the number of actions, $H$ is the length of each episode. Both of them match the lower bound $\Omega ( H ^ { 3 } S A / \epsilon ^ { 2 } )$ (Jaksch et al., 2010; Osband & Van Roy, 2016; Jin et al., 2018).
+
+Reward-free and task-agnostic exploration. Jin et al. (2020a) proposes a new paradigm of learning an MDP, which they called reward-free exploration. In this setting, the agent goes through a two-stage process. In the exploration phase the agent can interacts with the environment without knowing the reward function and in the planning phase the reward function is given and the agent needs output a policy. The goal is to make the output policy near optimal for any given reward function. A closely related setting is task-agnostic learning, where the reward function is determined at the very beginning but not revealed until the planning phase. Notice algorithms for task-agnostic
+
+# Algorithm 2 Optimistic Value Iteration with Zero Reward (VI-Zero)
+
+Require: Bonus $\beta _ { t }$ .
+1: Initialize: for any $( s , a , b , h )$ $h ) , \widetilde { V } _ { h } ( s , a , b ) \gets H , \Delta \gets H , N _ { h } ( s , a , b ) \gets 0 .$ .
+2: for episode $k = 1 , \ldots , K$ do
+3: for step $h = H , H - 1 , \ldots , 1$ do
+4: for $( s , a , b ) \in \mathcal { S } \times \mathcal { A } \times \mathcal { B }$ do
+5: $t \gets N _ { h } ( s , a , b )$ .
+6: if $t > 0$ then
+7: $\begin{array} { r l } & { \quad \widetilde { Q } _ { h } ( s , a , b ) \gets \operatorname* { m i n } \{ ( \widehat { \mathbb { P } } _ { h } \widetilde { V } _ { h + 1 } ) ( s , a , b ) + \beta _ { t } , H \} . } \\ & { \quad \mathbf { r } s \in S \mathbf { d o } } \\ & { \quad \pi _ { h } ( s ) \gets \arg \operatorname* { m a x } _ { ( a , b ) \in A \times B } \widetilde { Q } _ { h } ( s , a , b ) . } \\ & { \quad \widetilde { V } _ { h } ( s ) \gets ( \mathbb { D } _ { \pi _ { h } } \widetilde { Q } _ { h } ) ( s ) . } \end{array}$
+8: fo
+9:
+10:
+11: if $\widetilde { V } _ { 1 } ( s _ { 1 } ) < \Delta$ then
+12: $\Delta \widetilde { V } _ { 1 } ( s _ { 1 } )$ and $\widehat { \mathbb { P } } ^ { \mathrm { o u t } } \widehat { \mathbb { P } }$ .
+13: for step $h = 1 , \ldots , H$ do
+14: take action $( a _ { h } , b _ { h } ) \sim \pi _ { h } ( \cdot , \cdot | s _ { h } )$ , observe next state $s _ { h + 1 }$ .
+15: add 1 to $N _ { h } ( s _ { h } , a _ { h } , b _ { h } )$ and $N _ { h } ( s _ { h } , a _ { h } , b _ { h } , s _ { h + 1 } )$ .
+16: $\widehat { \mathbb { P } } _ { h } ( \cdot | s _ { h } , a _ { h } , b _ { h } ) \gets N _ { h } ( s _ { h } , a _ { h } , b _ { h } , \cdot ) / N _ { h } ( s _ { h } , a _ { h } , b _ { h } ) .$
+17: Output $\widehat { \mathbb { P } } ^ { \mathrm { o u t } }$ .
+
+learning can also be transferred to reward-free exploration by taking union bound w.r.t. different possible reward function. In Table 1, VI-explore (Bai & Jin, 2020) and Algorithm 2 can also be applied to this setting.
+
+Jin et al. (2020a) also proposes an algorithm, which first finds a covering policy to maximize the probability to reach each state separately and then collects data following this policy. Zhang et al. (2020b) takes a different approach by first runs the optimistic Q-learning algorithm (Jin et al., 2018) with zero reward to explore the environment, and then they utilizes the trajectories collected to compute a policy in an incremental manner. Wang et al. (2020) follows a simialr scheme, but studies reward-free exploration in linear-parametrized MDPs.
+
+# B OPTIMISTIC VALUE ITERATION WITH ZERO REWARD – VI-ZERO
+
+We now describe our algorithm for reward-free learning in zero-sum Markov games.
+
+Exploration phase. In the first phase of reward-free learning, we deploy algorithm Optimistic Value Iteration with Zero Reward (VI-Zero, Algorithm 2). This algorithm differs from the rewardaware Nash-VI (Algorithm 1) in two important aspects. First, we use zero reward in the exploration phase (Line 7), and only maintains an upper bound of the (reward-free) value function instead of both upper and lower bounds. Second, our exploration policy is the maximizing (instead of CCE) policy of the value function (Line 9). We remark that the $\widetilde { Q } _ { h } ( s , a , b )$ maintained in the algorithm 2 is no longer an upper bound for any actual value function (as it has no reward), but rather a measure of uncertainty or suboptimality that the agent may suffer—if she takes action $( a , b )$ at state $s$ and step $h$ , and makes decisions by utilizing the empirical estimate $\widehat { \mathbb { P } }$ in the remaining steps (see a rigorous version of this statement in Lemma 27). Finally, the empirical transition $\widehat { \mathbb { P } }$ of the episode that minimizes $\widetilde { V } _ { 1 } ( s _ { 1 } )$ is outputted and passed to the planning phase.
+
+Planning phase. After obtaining the estimate of tranisiton $\widehat { \mathbb { P } }$ , our planning algorithm is rather simple. For the $i ^ { \mathrm { t h } }$ task, let ${ \widehat { r } } ^ { i }$ be the empirical estimate of $r ^ { i }$ computed using the $i ^ { \mathrm { t h } }$ augmented dataset $\mathcal { D } ^ { i }$ . Then we compute the Nash equilibrium of the Markov game ${ \mathcal { M } } ( { \widehat { \mathbb { P } } } , { \widehat { r } } ^ { i } )$ with estimated transition $\widehat { \mathbb { P } }$ and reward $\widehat { r } ^ { i }$ . Since both $\widehat { \mathbb { P } }$ and ${ \widehat { r } } ^ { i }$ are known exactly, this is a pure computation problem b bwithout any sampling error and can be efficiently solved by simple planning algorithms such as the vanilla Nash value iteration without optimism (see Appendix G.2 for more details).
+
+# C MULTIPLAYER GENERAL-SUM MARKOV GAMES
+
+In this section, we extend both our model-based algorithms (Algorithm 1 and Algorithm 2) to the setting of multiplayer general-sum Markov games, and present corresponding theoretical guarantees.
+
+# C.1 PROBLEM FORMULATION
+
+A general-sum Markov game (general-sum MG) with $m$ players is a tuple $\mathrm { M G } ( \ b { H } , \ b { S } , \{ \ b { A } _ { i } \} _ { i = 1 } ^ { m } , \mathbb { P } , \{ r _ { i } \} _ { i = 1 } ^ { m } )$ , where $H$ , $s$ denote the length of each episode and the state space. Different from the two-player zero-sum setting, we now have $m$ different action spaces, where $A _ { i }$ is the action space for the $i ^ { \mathrm { { t h } } }$ player and $| { \bar { \mathcal { A } } } _ { i } | = A _ { i }$ . We let $\pmb { a } : = ( a _ { 1 } , \cdots , a _ { m } )$ denote the (tuple of) joint actions by all $m$ players. $\mathbb { P } = \{ \mathbb { P } _ { h } \} _ { h \in [ H ] }$ is a collection of transition matrices, so that $\mathbb { P } _ { h } ( \cdot | s , \pmb { a } )$ gives the distribution of the next state if actions $^ { a }$ are taken at state $s$ at step $h$ , and $r _ { i } = \{ r _ { h , i } \} _ { h \in [ H ] }$ is a collection of reward functions for $i ^ { \mathrm { { t h } } }$ player, so that $r _ { h , i } ( s , \pmb { a } )$ gives the reward received by the $i ^ { \mathrm { { t h } } }$ player if actions $^ { a }$ are taken at state $s$ at step $h$ .
+
+In this section, we consider three versions of equlibrium for general-sum MGs: Nash equilibrium (NE), correlated equilibrium (CE), and coarse correlated equilibrium (CCE), all being standard solution notions in games (Nisan et al., 2007). These three notions coincide on two-player zero-sum games, but are not equivalent to each other on multi-player general-sum games; any one of them could be desired depending on the application at hand. Below we introduce their definitions.
+
+(Approximate) Nash equilibrium in general-sum MG. The policy of the $i ^ { \mathrm { { t h } } }$ player is denoted as πi := πh,i : S → ∆Ai h∈[H]. We denote the product policy of all the players as $\pi : =$ $\pi _ { 1 } \times \cdots \times \pi _ { M }$ , and denote the policy of the all the players except the $i ^ { \mathrm { { t h } } }$ player as $\pi _ { - i }$ . We define $V _ { h , i } ^ { \pi } ( s )$ as the expected cumulative reward that will be received by the $i ^ { \mathrm { { t h } } }$ player if starting at state $s$ at step $h$ and all players follow policy $\pi$ . For any strategy $\pi _ { - i }$ , there also exists a best response of the ith player, which is a policy µ†(π−i) satisfying V µ†(h,i $V _ { h , i } ^ { { \mu ^ { \dagger } } ( \pi _ { - i } ) , \pi _ { - i } } ( s ) = \operatorname* { s u p } _ { \pi _ { i } } V _ { h , i } ^ { \pi _ { i } , \pi _ { - i } } ( s )$ V πi,π−ih,i (s) for any $( s , h ) \in \mathcal { S } \times [ H ]$ . We denote $V _ { h , i } ^ { \dagger , \pi _ { - i } } : = V _ { h , i } ^ { \mu ^ { \dagger } ( \pi _ { - i } ) , \pi _ { - i } }$ i . The Q-functions of the best response can be defined similarly.
+
+Our first objective is to find an approximate Nash equilibrium of Markov games.
+
+approximate Nash equilibrium if Definition 7 ( -approximate Nash equilibrium in general-sum MG). A product policy $\begin{array} { r } { \operatorname* { m a x } _ { i \in [ m ] } ( V _ { 1 , i } ^ { \dagger , \pi _ { - i } } - V _ { 1 , i } ^ { \pi } ) ( s _ { 1 } ) \le \epsilon . } \end{array}$ . $\pi$ is an $\epsilon$ -
+
+The above definition requires the suboptimality gap $( V _ { 1 , i } ^ { \dag , \pi _ { - i } } \ - \ : V _ { 1 , i } ^ { \pi } ) ( s _ { 1 } )$ to be less than $\epsilon$ for all player . This is consistent with the two-player case (Definition 1) up to a constant of 2, since in the two-player zero-sum setting, we have $\bar { V } _ { 1 , 1 } ^ { \pi } ( s _ { 1 } ) = - V _ { 1 , 2 } ^ { \pi } ( s _ { 1 } )$ for any product policy $\pi = ( \mu , \nu )$ , and therefore $\begin{array} { r } { ( V _ { 1 , 1 } ^ { \dag , \nu } - V _ { 1 , 1 } ^ { \mu , \dag } ) ( s _ { 1 } ) \le 2 \operatorname* { m a x } _ { i \in [ 2 ] } { ( V _ { 1 , i } ^ { \dag , \pi _ { - i } } - V _ { 1 , i } ^ { \pi } ) ( s _ { 1 } ) } \le 2 ( V _ { 1 , 1 } ^ { \dag , \nu } - V _ { 1 , 1 } ^ { \mu , \dag } ) ( s _ { 1 } ) . } \end{array}$ We can similarly define the regret.
+
+Definition 8 (Nash-regret in general-sum MG). Let $\pi ^ { k }$ denote the (product) policy deployed by the algorithm in the $k ^ { \mathrm { { t h } } }$ episode. After a total of $K$ episodes, the regret is defined as
+
+$$
+\mathrm { R e g r e t } _ { \mathsf { N a s h } } ( K ) = \sum _ { k = 1 } ^ { K } \operatorname* { m a x } _ { i \in [ m ] } ( V _ { 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } - V _ { 1 , i } ^ { \pi ^ { k } } ) ( s _ { 1 } ) .
+$$
+
+(Approximate) CCE in general-sum MG. The coarse correlated equilibrium (CCE) is a relaxed version of Nash equilibrium in which we consider general correlated policies instead of product policies. Let $\mathcal { A } = \mathcal { A } _ { 1 } \times \cdot \cdot \cdot \times \mathcal { A } _ { m }$ denote the joint action space.
+
+n 9 (CCE inis a CCE if lated)for all $\pi : = \{ \pi _ { h } ( s ) \in \Delta _ { \mathcal { A } } : ( h , s ) \in$ $[ H ] \times S \}$ $\begin{array} { r } { \operatorname* { m a x } _ { i \in [ m ] } V _ { h , i } ^ { \dag , \pi - i } ( s ) \leq V _ { h , i } ^ { \pi } ( s ) } \end{array}$ $( s , h ) \in \mathcal { S } \times [ H ]$
+
+Compared with a Nash equilibrium, a CEE is not necessarily a product policy, that is, we may not have $\pi _ { h } ( s ) \in \Delta _ { { \cal A } _ { 1 } } \times \cdot \cdot \cdot \times \Delta _ { { \cal A } _ { m } }$ . Similarly, we also define $\epsilon$ -approximate CCE and CCE-regret below.
+
+n 10 is an $\cdot$ -approximate CCE in -approximate CCE if $\pi : = \{ \pi _ { h } ( s ) \in \Delta _ { \mathcal { A } } : ( h , s ) \in$ $[ H ] \times S \}$ $\epsilon$ $\begin{array} { r } { \operatorname* { m a x } _ { i \in [ m ] } { ( V _ { 1 , i } ^ { \dag , \pi _ { - i } } - V _ { 1 , i } ^ { \pi } ) ( s _ { 1 } ) } \le \epsilon . } \end{array}$
+
+Definition 11 (CCE-regret in general-sum MG). Let policy $\pi ^ { k }$ denote the (correlated) policy deployed by the algorithm in the $\bar { k } ^ { \mathrm { t h } }$ episode. After a total of $K$ episodes, the regret is defined as
+
+$$
+\mathrm { R e g r e t } _ { \mathsf { C C E } } ( K ) = \sum _ { k = 1 } ^ { K } \operatorname* { m a x } _ { i \in [ m ] } ( V _ { 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } - V _ { 1 , i } ^ { \pi ^ { k } } ) ( s _ { 1 } ) .
+$$
+
+(Approximate) CE in general-sum MG. The correlated equilibrium (CE) is another relaxation of the Nash equilibrium. To define CE, we first introduce the concept of strategy modification: A strategy modification $\phi : = \{ \phi _ { h , s } ( a ) \in \mathcal { A } _ { i } : ( h , s , a ) \in [ H ] \times \mathcal { S } \times \mathcal { A } _ { i } \}$ for player $i$ is a set of $S \times H$ injective functions from $\mathbf { \mathcal { A } } _ { i }$ to itself. Let $\Phi _ { i }$ denote the set of all possible strategy modifications for player $i$ .
+
+One can compose a strategy modification $\phi$ with any Markov policy $\pi$ and obtain a new policy $\phi \diamond \pi$ such that when policy $\pi$ chooses to play $\pmb { a } : = ( a _ { 1 } , \dots , a _ { m } )$ at state $s$ and step $h$ , policy $\phi \diamond \pi \mathbf { w } \mathrm { i l l }$ play $( a _ { 1 } , \dots , a _ { i - 1 } , \phi _ { h , s } ( a _ { i } ) , a _ { i + 1 } , \dots , a _ { m } )$ instead.
+
+Definition 12 (CE in general-sum MG). A policy $\pi : = \{ \pi _ { h } ( s ) \in \Delta _ { \mathcal { A } } : ( h , s ) \in [ H ] \times \mathcal { S } \}$ is a CE if $\begin{array} { r } { \operatorname* { m a x } _ { i \in [ m ] } \operatorname* { m a x } _ { \phi \in \Phi _ { i } } V _ { h , i } ^ { \phi \diamond \pi } ( s ) \leq V _ { h , i } ^ { \pi } ( s ) } \end{array}$ holds for all $( s , h ) \in \mathcal { S } \times [ H ]$ .
+
+Similarly, we have an approximate version of CE and CE-regret.
+
+Definition 13 ( $[ H ] \times S \}$ is an $\dot { \epsilon }$ $\epsilon$ -approximate CE in Markov games). A policy -approximate CE if $\begin{array} { r } { \operatorname* { m a x } _ { i \in [ m ] } \operatorname* { m a x } _ { \phi \in \Phi _ { i } } ( V _ { 1 , i } ^ { \phi \diamond \pi } - V _ { 1 , i } ^ { \pi } ) ( s _ { 1 } ) \leq \epsilon . } \end{array}$ $\pi : = \{ \pi _ { h } ( s ) \in \Delta _ { \mathcal { A } } : ( h , s ) \in$ .
+
+Definition 14 (CE-regret in multiplayer Markov games). Let product policy $\pi ^ { k }$ denote the policy deployed by the algorithm in the $\bar { k ^ { \mathrm { { t h } } } }$ episode. After a total of $K$ episodes, the regret is defined as
+
+$$
+{ \mathrm { R e g r e t } } _ { \mathsf { C E } } ( K ) = \sum _ { k = 1 } ^ { K } \operatorname* { m a x } _ { i \in [ m ] } \operatorname* { m a x } _ { \phi } { \big ( } V _ { 1 , i } ^ { \phi \diamond \pi ^ { k } } - V _ { 1 , i } ^ { \pi ^ { k } } { \big ) } ( s _ { 1 } ) .
+$$
+
+Relationship between Nash, CE, and CCE For general-sum MGs, we have $\{ \mathrm { N a s h } \} \subseteq \{ \mathrm { C E } \} \subseteq$ $\{ \mathrm { C C E } \}$ , so that they form a nested set of notions of equilibria (Nisan et al., 2007). Indeed, one can easily verify that if we restrict the choice of strategy modification $\phi$ to those consisting of only constant functions, i.e., $\phi _ { h , s } ( a )$ being independent of $a$ , Definition 12 will reduce to the definition of CCE policy. In addition, any Nash equilibrium is a CE by definition. Finally, since a Nash equilibrium always exists, so does CE and CCE.
+
+# C.2 MULTIPLAYER OPTIMISTIC NASH VALUE ITERATION
+
+Here we present the Multi-Nash-VI algorithm, which is an extension of Algorithm 1 for multi-player general-sum Markov games.
+
+The EQUILIBRIUM Subroutine. Our EQUILIBRIUM subroutine in Line 11 could be taken from either one of the $\{ \mathrm { N A S H , C E , C C E } \}$ subroutines for one-step games. When using NASH, we compute the Nash equilibrium of a one-step multi-player game (see, e.g., Berg & Sandholm (2016) for an overview of the available algorithms); the worst-case computational complexity of such a subroutine will be PPAD-hard (Daskalakis, 2013). When using CE or CCE, we find CEs or CCEs of the one-step games respectively, which can be solved in polynomial time using linear programming. However, the policies found are not guaranteed to be a product policy. We remark that in Algorithm 1 we used the CCE subroutine for finding Nash in two-player zero-sum games, which seemingly contrasts the principle of using the right subroutine for finding the right equilibrium, but nevertheless works as the Nash equilibrium and CCE are equivalent in zero-sum games.
+
+Now we are ready to present the theoretical guarantees for Algorithm 3. We let $\pi ^ { k }$ denote the policy computed in line 11 of Algorithm 3 in the $k ^ { \mathrm { { \bar { t h } } } }$ episode.
+
+Theorem 15 (Multi-Nash-VI). There exists an absolute constant $c ,$ , for any $p \in \mathsf { \Gamma } ( 0 , 1 ]$ , let $\iota =$ $\log ( S A B T / p )$ , then with probability at least $1 - p$ , Algorithm $^ 3$ with bonus $\beta _ { t } = c \sqrt { S H ^ { 2 } \iota / t }$ and EQUILIBRIUM being one of {NASH, CE, CCE} satisfies (repsectively):
+
+# Algorithm 3 Multiplayer Optimistic Nash Value Iteration (Multi-Nash-VI)
+
+1: Initialize: for any $( s , \pmb { a } , h , i )$ $\begin{array} { r } { \overline { { Q } } _ { h , i } ( s , { \pmb a } ) H , \underline { { Q } } _ { h , i } ( s , { \pmb a } ) 0 , \Delta H , N _ { h } ( s , { \pmb a } ) 0 . } \end{array}$ .
+2: for episode $k = 1 , \ldots , K$ do
+3: for step $h = H , H - 1 , \ldots , 1 \mathbf { d } \mathbf { 4 }$ o
+4: for $( s , \pmb { a } ) \in \pmb { S } \times \mathcal { A } _ { 1 } \times \cdot \cdot \cdot \times \mathcal { A } _ { m }$ do
+5: $t \gets N _ { h } ( s , \pmb { a } )$ ;
+6: if $t > 0$ then
+7: for player $i = 1 , 2 , \dots , m$ do
+8: $\overline { { Q } } _ { h , i } ( s , \pmb { a } ) \operatorname* { m i n } \{ ( r _ { h , i } + \widehat { \mathbb { P } } _ { h } \overline { { V } } _ { h + 1 , i } ) ( s , \pmb { a } ) + \beta _ { t } , H \} .$
+9: $\underline { { Q } } _ { h , i } ( s , \mathbf { a } ) \operatorname* { m a x } \{ ( r _ { h , i } + \widehat { \mathbb { P } } _ { h } \underline { { V } } _ { h + 1 , i } ) ( s , \mathbf { a } ) - \beta _ { t } , 0 \} .$
+10: for $s \in S$ do
+11: $\pi _ { h } ( \cdot | s ) \gets \mathrm { E Q U I L I B R I U M } ( \overline { { \boldsymbol { Q } } } _ { h , 1 } ( s , \cdot ) , \overline { { \boldsymbol { Q } } } _ { h , 2 } ( s , \cdot ) , \cdot \cdot , \overline { { \boldsymbol { Q } } } _ { h , M } ( s , \cdot ) ) .$
+12: for player $i = 1 , 2 , \dots , m$ do
+13: $\begin{array} { r } { \Vec { V } _ { h , i } ( s ) ( \mathbb { D } _ { \pi _ { h } } \overline { { Q } } _ { h , i } ) ( s ) ; \quad \underline { { V } } _ { h , i } ( s ) ( \mathbb { D } _ { \pi _ { h } } \underline { { Q } } _ { h , i } ) ( s ) . } \end{array}$
+14: if $\begin{array} { r } { \operatorname* { m a x } _ { i \in [ m ] } ( \overline { { V } } _ { 1 , i } - \underline { { V } } _ { 1 , i } ) ( s _ { 1 } ) < \Delta } \end{array}$ then
+15: $\Delta \gets \mathrm { m a x } _ { i \in [ m ] } ( \overline { { V } } _ { 1 , i } - \underline { { V } } _ { 1 , i } ) ( s _ { 1 } )$ and $\pi ^ { \mathrm { o u t } } \pi$ .
+16: for step $h = 1 , \ldots , H$ do
+17: take action $\mathbf { a } _ { h } \sim \pi _ { h } ( \cdot | s _ { h } )$ , observe reward $r _ { h }$ and next state $s _ { h + 1 }$ .
+18: add 1 to $N _ { h } ( s _ { h } , \pmb { a } _ { h } )$ and $\dot { N } _ { h } ( s _ { h } , \pmb { a } _ { h } , s _ { h + 1 } )$ .
+19: $\widehat { \mathbb { P } } _ { h } ( \cdot | s _ { h } , \pmb { a } _ { h } ) \gets N _ { h } ( s _ { h } , \pmb { a } _ { h } , \cdot ) / N _ { h } ( s _ { h } , \pmb { a } _ { h } )$ .
+20: Output $\pi ^ { \mathrm { { o u t } } }$ .
+
+• $\pi ^ { o u t }$ is an $\epsilon$ -approximate $\{ \mathrm { N A S H , C E , C C E } \}$ , if the number of episodes $K \_ { \mathbf { \alpha } } \geq$ $\begin{array} { r } { \Omega ( H ^ { 4 } S ^ { 2 } ( \prod _ { i = 1 } ^ { m } \tilde { A _ { i } } ) \iota / \epsilon ^ { 2 } ) } \end{array}$ .
+
+$$
+\bullet \ \operatorname { R e g r e t } _ { \{ \substack { \mathrm { N a s h } , \mathsf { C E } , \mathsf { C C E } \} } } ( K ) \leq \mathcal { O } ( \sqrt { H ^ { 3 } S ^ { 2 } ( \prod _ { i = 1 } ^ { m } A _ { i } ) T \iota } ) .
+$$
+
+In the situation where the EQUILIBRIUM subroutine is taken as NASH, Theorem 15 provides the sample complexity bound of Multi-Nash-VI algorithm to find a $\epsilon$ -approximate Nash equilibrium and its regret bound. Compared with our earlier result in two-player zero-sum games (Theorem 3), here the sample complexity scales as $S ^ { 2 } H ^ { 4 }$ instead of $S H ^ { 3 }$ . This is because the auxiliary bonus and Bernstein concentration technique do not apply here. Furthermore, the sample complexity is proportional to $\textstyle \prod _ { i = 1 } ^ { m } A _ { i }$ , which increases exponentially as the number of players increases.
+
+Runtime of Algorithm 3 We remark that while the Nash guarantee is the strongest among the three guarantees presented in Theorem 15, the runtime of Algorithm 3 in the Nash case is not guaranteed to be polynomial and in the worst case PPAD-hard (due to the hardness of the NASH subroutine). In contrast, the CE and CCE guarantees are weaker, but the corresponding algorithms are guaranteed to finish in polynomial time.
+
+# C.3 MULTIPLAYER REWARD-FREE LEARNING
+
+We can also generalize VI-Zero to the multiplayer setting and obtain Algorithm 4, Multi-VI-Zero, which is almost the same as VI-Zero except that its exploration bonus √ $\beta _ { t }$ is larger than that of VIZero by a $\sqrt { S }$ factor.
+
+Similar to Theorem 5, we have the following theoretical guarantee claiming that any $\{ \mathrm { N A S H , C C E , C E } \}$ of the ${ \mathcal { M } } ( { \widehat { \mathbb { P } } } , { \widehat { r } } ^ { i } )$ $( i \in [ N ] )$ is also an approximate $\{ \mathrm { N A S H , C C E , C E } \}$ of the true Markov game ${ \mathcal { M } } ( \mathbb { P } , r ^ { i } )$ , where $\widehat { \mathbb { P } } ^ { \mathrm { o u t } }$ is the empirical transition outputted by Algorithm 4 and $\widehat { r } ^ { i }$ is the empirical estimate of $r ^ { i }$ .
+
+Theorem 16 (Multi-VI-Zero). There exists an absolute constant $c _ { * }$ , for any $p ~ \in ~ ( 0 , 1 ] , ~ \epsilon ~ \in$ $( 0 , H ]$ , $N \in \mathbb { N }$ , if we choose bonus $\beta _ { t } ~ = ~ c \sqrt { H ^ { 2 } S \iota / t }$ with $\iota \ : = \ : \log ( N S A B T / p )$ and $K \geq$ $c ( H ^ { 4 } S ^ { 2 } ( \prod _ { i = 1 } ^ { m } A _ { i } ) \iota / \epsilon ^ { 2 } )$ , then with probability at least $1 - p ,$ , the output $\widehat { \mathbb { P } } ^ { o u t }$ of Algorithm $^ { 4 }$ has the following property: for any $N$ fixed reward functions $r ^ { 1 } , \ldots , r ^ { N }$ , any $\{ \mathrm { N A S H , C C E , C E } \}$ of
+
+# Algorithm 4 Multiplayer Optimistic Value Iteration with Zero Reward (Multi-VI-Zero)
+
+1: Initialize: for any $( s , \pmb { a } , h )$ , $\widetilde { V } _ { h } ( s , a ) \gets H$ , $\Delta H$ , Nh(s, a) ← 0.
+2: for episode $k = 1 , \ldots , K$ do
+3: for step $h = H , H - 1 , \ldots , 1$ do
+4: for $( s , \pmb { a } ) \in \mathcal { S } \times \mathcal { A } _ { 1 } \times \cdot \cdot \cdot \times \mathcal { A } _ { m } \ : \mathfrak { c }$ do
+5: $t \gets N _ { h } ( s , \pmb { a } )$ .
+6: if $t > 0$ then
+7: $\widetilde { Q } _ { h } ( s , \pmb { a } ) \gets \operatorname* { m i n } \{ ( \widehat { \mathbb { P } } _ { h } \widetilde { V } _ { h + 1 } ) ( s , \pmb { a } ) + \beta _ { t } , H \} .$
+8: for $s \in S$ do
+9: $\begin{array} { r } { \pi _ { h } ( s ) \gets \arg \operatorname* { m a x } _ { \pmb { a } \in \mathcal { A } _ { 1 } \times \dots \times \mathcal { A } _ { m } } \widetilde { Q } _ { h } ( s , \pmb { a } ) . } \end{array}$
+10: $\widetilde { V } _ { h } ( s ) ( \mathbb { D } _ { \pi _ { h } } \widetilde { Q } _ { h } ) ( s )$ .
+11: if $\widetilde { V } _ { 1 } ( s _ { 1 } ) < \Delta$ then
+12: $\Delta \widetilde { V } _ { 1 } ( s _ { 1 } )$ and $\widehat { \mathbb { P } } ^ { \mathrm { o u t } } \widehat { \mathbb { P } }$ .
+13: for step $h = 1 , \ldots , H$ do
+14: take action $\mathbf { \sigma } _ { \mathbf { a } _ { h } } \sim \pi _ { h } ( \cdot , \cdot | s _ { h } )$ , observe next state $s _ { h + 1 }$ .
+15: add 1 to $N _ { h } ( s _ { h } , \pmb { a } _ { h } )$ and $\dot { N _ { h } } ( s _ { h } , \pmb { a } _ { h } , s _ { h + 1 } )$ .
+16: $\widehat { \mathbb { P } } _ { h } ( \cdot | s _ { h } , \pmb { a } _ { h } ) \gets N _ { h } ( s _ { h } , \pmb { a } _ { h } , \cdot ) / N _ { h } ( s _ { h } , \pmb { a } _ { h } )$ .
+17: Output $\widehat { \mathbb { P } } ^ { \mathrm { o u t } }$ .
+
+Markov game $\mathcal { M } ( \widehat { \mathbb { P } } ^ { o u t } , \widehat { r } ^ { i } )$ is also an $\epsilon$ -approximate $\{ \mathrm { N A S H , C C E , C E } \}$ of the true Markov game ${ \mathcal { M } } ( \mathbb { P } , r ^ { i } )$ for all $i \in [ N ]$ b.
+
+The proof of Theorem 16 can be found in Appendix H.2. It is worth mentioning that the empirical Markov game $\mathcal { M } ( \widehat { \mathbb { P } } ^ { \mathrm { o u t } } , \widehat { r } ^ { i } )$ may have multiple $\{ \mathrm { N a s h }$ equilibria, ${ \mathrm { C C E s , C E s } } \}$ and Theorem 16 ensures that all of them are $\epsilon$ b-approximate $\{ \mathrm { N a s h }$ equilibria, $_ \mathrm { C C E s , C E s } \}$ of the true Markov game. Also, note that the sample complexity here is quadratic in the number of states because we are using the exploration bonus $\beta _ { t } = \sqrt { H ^ { 2 } S \iota / t }$ that is larger than usual by a $\sqrt { S }$ factor.
+
+# D BELLMAN EQUATIONS FOR MARKOV GAMES
+
+In this section, we present the Bellman equations for different types of values in Markov games.
+
+Fixed policies. For any pair of Markov policy $( \mu , \nu )$ , by definition of their values in (1) (2), we have the following Bellman equations:
+
+$$
+Q _ { h } ^ { \mu , \nu } ( s , a , b ) = ( r _ { h } + \mathbb { P } _ { h } V _ { h + 1 } ^ { \mu , \nu } ) ( s , a , b ) , \qquad V _ { h } ^ { \mu , \nu } ( s ) = ( \mathbb { D } _ { \mu _ { h } \times \nu _ { h } } Q _ { h } ^ { \mu , \nu } ) ( s )
+$$
+
+for all $( s , a , b , h ) \in \mathcal { S } \times \mathcal { A } \times \mathcal { B } \times [ H ]$ , where $V _ { H + 1 } ^ { \mu , \nu } ( s ) = 0$ for all $s \in S$ .
+
+Best responses. For any Markov policy $\mu$ of the max-player, by definition, we have the following Bellman equations for values of its best response:
+
+$$
+Q _ { h } ^ { \mu , \dagger } ( s , a , b ) = ( r _ { h } + \mathbb { P } _ { h } V _ { h + 1 } ^ { \mu , \dagger } ) ( s , a , b ) , \qquad V _ { h } ^ { \mu , \dagger } ( s ) = \operatorname* { i n f } _ { \nu \in \Delta _ { B } } ( \mathbb { D } _ { \mu _ { h } \times \nu } Q _ { h } ^ { \mu , \dagger } ) ( s ) ,
+$$
+
+for all $( s , a , b , h ) \in \mathcal { S } \times \mathcal { A } \times \mathcal { B } \times [ H ]$ , where $V _ { H + 1 } ^ { \mu , \dagger } ( s ) = 0$ for all $s \in S$
+
+Similarly, for any Markov policy $\nu$ of the min-player, we also have the following symmetric version of Bellman equations for values of its best response:
+
+$$
+Q _ { h } ^ { \dagger , \nu } ( s , a , b ) = ( r _ { h } + \mathbb { P } _ { h } V _ { h + 1 } ^ { \dagger , \nu } ) ( s , a , b ) , \qquad V _ { h } ^ { \dagger , \nu } ( s ) = \operatorname* { s u p } _ { \mu \in \Delta _ { A } } ( \mathbb { D } _ { \mu \times \nu _ { h } } Q _ { h } ^ { \dagger , \nu } ) ( s ) .
+$$
+
+for all $( s , a , b , h ) \in \mathcal { S } \times \mathcal { A } \times \mathcal { B } \times [ H ]$ , where $V _ { H + 1 } ^ { \dagger , \nu } ( s ) = 0$ for all $s \in S$
+
+Nash equilibria. Finally, by definition of Nash equilibria in Markov games, we have the following Bellman optimality equations:
+
+$$
+\begin{array} { r l } & { \textstyle Q _ { h } ^ { \star } ( s , a , b ) = ( r _ { h } + { \mathbb P } _ { h } V _ { h + 1 } ^ { \star } ) ( s , a , b ) } \\ & { \textstyle V _ { h } ^ { \star } ( s ) = \operatorname* { s u p } _ { \mu \in \Delta _ { \cal A } } \operatorname* { i n f } _ { \nu \in \Delta _ { \cal B } } ( { \mathbb P } _ { \mu \times \nu } Q _ { h } ^ { \star } ) ( s ) = \operatorname* { i n f } _ { \nu \in \Delta _ { \mathcal B } } \operatorname* { s u p } _ { \mu \in \Delta _ { \cal A } } ( { \mathbb D } _ { \mu \times \nu } Q _ { h } ^ { \star } ) ( s ) . } \end{array}
+$$
+
+for all $( s , a , b , h ) \in \mathcal { S } \times \mathcal { A } \times \mathcal { B } \times [ H ]$ , where $V _ { H + 1 } ^ { \star } ( s ) = 0$ for all $s \in S$ .
+
+# E PROPERTIES OF COARSE CORRELATED EQUILIBRIUM
+
+Recall the definition for CCE in our main paper (4), we restate it here after rescaling. For any pair of matrices $P , Q \in [ 0 , 1 ] ^ { n \times m }$ , the subroutine $\mathrm { C C E } ( P , Q )$ returns a distribution $\pi \in \Delta _ { n \times m }$ that satisfies:
+
+$$
+\begin{array} { l l } { { \mathbb { E } } _ { ( a , b ) \sim \pi } P ( a , b ) \geq \displaystyle \operatorname* { m a x } _ { a ^ { \star } } { \mathbb { E } } _ { ( a , b ) \sim \pi } P ( a ^ { \star } , b ) } \\ { { \mathbb { E } } _ { ( a , b ) \sim \pi } Q ( a , b ) \leq \displaystyle \operatorname* { m i n } _ { b ^ { \star } } { \mathbb { E } } _ { ( a , b ) \sim \pi } Q ( a , b ^ { \star } ) } \end{array}
+$$
+
+We make three remarks on CCE. First, a CCE always exists since a Nash equilibrium for a generalsum game with payoff matrices $( P , Q )$ is also a CCE defined by $( P , Q )$ , and a Nash equilibrium always exists. Second, a CCE can be efficiently computed, since above constraints (5) for CCE can be rewritten as $n + m$ linear constraints on $\pi \in \Delta _ { n \times m }$ , which can be efficiently resolved by standard linear programming algorithm. Third, a CCE in general-sum games needs not to be a Nash equilibrium. However, a CCE in zero-sum games is guaranteed to be a Nash equalibrium.
+
+Proposition 17. Let $\pi = \operatorname { C C E } ( Q , Q )$ , and $( \mu , \nu )$ be the marginal distribution over both players’ actions induced by $\pi$ . Then $( \mu , \nu )$ is a Nash equilibrium for payoff matrix $Q$ .
+
+Proof of Proposition $^ { I 7 }$ . Let $N ^ { \star }$ be the value of Nash equilibrium for $Q$ . Since $\pi = \operatorname { C C E } ( Q , Q )$ , by definition, we have:
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { ( a , b ) \sim \pi } Q ( a , b ) \geq \underset { a ^ { \star } } { \operatorname* { m a x } } \mathbb { E } _ { ( a , b ) \sim \pi } Q ( a ^ { \star } , b ) = \underset { a ^ { \star } } { \operatorname* { m a x } } \mathbb { E } _ { b \sim \nu } Q ( a ^ { \star } , b ) \geq N ^ { \star } } \\ & { \mathbb { E } _ { ( a , b ) \sim \pi } Q ( a , b ) \leq \underset { b ^ { \star } } { \operatorname* { m i n } } \mathbb { E } _ { ( a , b ) \sim \pi } Q ( a , b ^ { \star } ) = \underset { b ^ { \star } } { \operatorname* { m i n } } \mathbb { E } _ { a \sim \mu } Q ( a , b ^ { \star } ) \leq N ^ { \star } } \end{array}
+$$
+
+This gives:
+
+$$
+\operatorname* { m a x } _ { a ^ { \star } } \mathbb { E } _ { b \sim \nu } Q ( a ^ { \star } , b ) = \operatorname* { m i n } _ { b ^ { \star } } \mathbb { E } _ { a \sim \mu } Q ( a , b ^ { \star } ) = N ^ { \star }
+$$
+
+which finishes the proof.
+
+Intuitively, a CCE procedure can be used in Nash Q-learning for finding an approximate Nash equilibrium, because the values of upper confidence and lower confidence $\overline { { Q } }$ and $\underline { { \boldsymbol { Q } } }$ ) will be eventually very close, so that the preconditions of Proposition 17 becomes approximately satisfied.
+
+# F PROOF FOR SECTION 3 – OPTIMISTIC NASH VALUE ITERATION
+
+# F.1 PROOF OF THEOREM 3
+
+We denote $V ^ { k }$ $^ { r k } , Q ^ { k } , \pi ^ { k } ,$ , $\mu ^ { k }$ and $\nu ^ { k ^ { \ 4 } }$ for values and policies at the beginning of the $k$ -th episode. In particular, $N _ { h } ^ { k } ( s , a , b )$ is the number we have visited the state-action tuple $( s , a , b )$ at the $h$ -th step before the $k$ -th episode. $N _ { h } ^ { k } ( s , a , b , s ^ { \prime } )$ is defined by the same token. Using this notation, we can further define the empirical transition by $\widehat { \mathbb { P } } _ { h } ^ { k } \big ( s ^ { \prime } | s , a , b \big ) : = N _ { h } ^ { k } \big ( s , a , b , s ^ { \prime } \big ) / N _ { h } ^ { k } \big ( s , a , b \big )$ . If $N _ { h } ^ { k } ( s , a , b ) = 0$ , we set $\widehat { \mathbb { P } } _ { h } ^ { k } ( s ^ { \prime } | s , a , b ) = 1 / S$ .
+
+As a result, the bonus terms can be written as
+
+$$
+\beta _ { h } ^ { k } ( s , a , b ) : = C \left( \sqrt { \frac { \iota H ^ { 2 } } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { H ^ { 2 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \right)
+$$
+
+$$
+\gamma _ { h } ^ { k } ( s , a , b ) : = \frac { C } { H } \widehat { \mathbb { P } } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { V } _ { h + 1 } ^ { k } ) ( s , a , b )
+$$
+
+for some large absolute constant $C > 0$ .
+
+Lemma 18. Let $c _ { 1 }$ be some large absolute constant. Define event $E _ { 0 }$ to be: for all $h , s , a , b , s ^ { \prime }$ and $k \in [ K ]$ ,
+
+$$
+\begin{array} { r l } & { { ( | [ ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) V _ { h + 1 } ^ { \star } ] ( s , a , b ) | \leq c _ { 1 } \sqrt { \frac { H ^ { 2 } \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } , } } \\ & | ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) ( s ^ { \prime } \mid s , a , b ) | \leq c _ { 1 } ( \sqrt { \frac { \operatorname* { m i n } \{ \mathbb { P } _ { h } ( s ^ { \prime } \mid s , a , b ) , \widehat { \mathbb { P } } _ { h } ^ { k } ( s ^ { \prime } \mid s , a , b ) \} \} { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } ) . } \end{array}
+$$
+
+We have $\mathbb { P } ( E _ { 1 } ) \geq 1 - p .$ .
+
+Proof. The proof is standard and folklore: apply standard concentration inequalities and then take a union bound. For completeness, we provide the proof of the second one here.
+
+Consider a fixed $( s , a , b , h )$ tuple.
+
+Let’s consider the following equivalent random process: (a) before the agent starts, the environment samples $\{ s ^ { ( 1 ) } , s ^ { ( 2 ) } , \ldots , s ^ { ( \bar { K } ) } \}$ independently from $\mathbb { P } _ { h } ( \cdot \mid s , a , b )$ ; (b) during the interaction between the agent and environment, the time the agent reaches $( s , a , b , h )$ , the environment will make the agent transit to $s ^ { ( i ) }$ . Note that the randomness induced by this interaction procedure is exactly the same as the original one, which means the probability of any event in this context is the same as in the original problem. Therefore, it suffices to prove the target concentration inequality in this ’easy’ context. Denote by $\widehat { \mathbb { P } } _ { h } ^ { ( t ) } ( \cdot \mid s , a , b )$ the empirical estimate of $\mathbb { P } _ { h } ( \cdot \mid s , a , b )$ calculated using $\{ s ^ { ( 1 ) } , s ^ { ( 2 ) } , \ldots , s ^ { ( t ) } \}$ . For a fixed $t$ and $s ^ { \prime }$ , by applying the Bernstein inequality and its empirical version, we have with probability at least $1 - { \dot { p } } / { \bar { S } } ^ { 2 } { \dot { A } } { \dot { B T } }$ ,
+
+$$
+\vert ( \mathbb { P } _ { h } - \widehat { \mathbb { P } } _ { h } ^ { ( t ) } ) ( s ^ { \prime } \mid s , a , b ) \vert \leq \mathscr { O } \left( \sqrt { \frac { \operatorname* { m i n } \{ \mathbb { P } _ { h } ( s ^ { \prime } \mid s , a , b ) , \widehat { \mathbb { P } } _ { h } ^ { ( t ) } ( s ^ { \prime } \mid s , a , b ) \} s } { t } } + \frac { \iota } { t } \right) .
+$$
+
+Now we can take a union bound over all $s , a , b , h , s ^ { \prime }$ and $t \in [ K ]$ , and obtain that with probability at least $1 - p$ , for all $s , a , b , h , s ^ { \prime }$ and $t \in [ K ]$ ,
+
+$$
+\vert ( \mathbb { P } _ { h } - \widehat { \mathbb { P } } _ { h } ^ { ( t ) } ) ( s ^ { \prime } \mid s , a , b ) \vert \leq \mathscr { O } \left( \sqrt { \frac { \operatorname* { m i n } \{ \mathbb { P } _ { h } ( s ^ { \prime } \mid s , a , b ) , \widehat { \mathbb { P } } _ { h } ^ { ( t ) } ( s ^ { \prime } \mid s , a , b ) \} s } { t } } + \frac { \iota } { t } \right) .
+$$
+
+Note that the agent can reach each $( s , a , b , h )$ for at most $K$ times, this directly implies that the third inequality also holds with probability at least $1 - p$ . □
+
+We begin with an auxiliary lemma bounding the lower-order term.
+
+Lemma 19. Suppose event $E _ { 0 }$ holds, then there exists absolute constant $c _ { 2 }$ such that: if function $g ( s )$ satisfies $| g | ( s ) \leq ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s )$ for all $s$ , then
+
+$$
+\begin{array} { r l } & { \displaystyle \left| ( \widehat { \mathbb { P } } _ { h } ^ { k } - { \mathbb { P } } _ { h } ) g ( s , a , b ) \right| } \\ & { \displaystyle \leq c _ { 2 } \bigg ( \frac { 1 } { H } \operatorname* { m i n } \{ \widehat { \mathbb { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) , { \mathbb { P } } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) \} + \frac { H ^ { 2 } S { \iota } } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \bigg ) . } \end{array}
+$$
+
+Proof. By triangle inequality,
+
+$$
+\begin{array} { r l } & { | ( \widehat { \mathbb { P } } _ { h } ^ { h } - \mathbb { P } _ { h } ) | \mathcal { Q } ( s , \theta , \theta ) | } \\ & { \le \sum _ { \ell } | ( \widehat { \mathbb { P } } _ { h } ^ { h } - \mathbb { P } _ { h } ) ( s ^ { \prime } | s , \alpha , \theta , \theta ) | | g | ( s ^ { \prime } ) } \\ & { \le \sum _ { \ell } | ( \widehat { \mathbb { P } } _ { h } ^ { h } - \mathbb { P } _ { h } ) ( s ^ { \prime } | s , \alpha , \theta ) | ( \overline { { V } } _ { h + 1 } ^ { h } - { V } _ { h + 1 } ^ { h } ) ( s ^ { \prime } ) } \\ & \overset { ( i ) } { \le } \mathcal { O } ( \sum _ { \ell } \sqrt \frac { \widehat { \mathbb { P } } _ { h } ^ { h } ( s ^ { \prime } | s ^ { \prime } , \alpha , \theta ) | } { \widehat { \mathbb { P } } _ { h } \operatorname* { m a x } \{ N _ { h } ^ { h } ( s , \alpha , \theta , \theta ) , 1 \} } + \frac { s } { \operatorname* { m a x } \{ N _ { h } ^ { h } ( s , \alpha , \theta , \theta ) , 1 \} } ) ( \widehat { V } _ { h + 1 } ^ { h } - \frac { V _ { h + 1 } ^ { h } } { \sum _ { h + 1 } ^ { h } \} ) ( s ^ { \prime } ) ) } \\ & { \overset { ( i i ) } { \le } \mathcal { O } ( \sum _ { \ell } \frac { \widehat { \mathbb { P } } _ { h } ^ { h } ( s ^ { \prime } | s , \alpha , \theta , b ) } { H } + \frac { H _ { \ell } } { \operatorname* { m a x } \{ N _ { h } ^ { h } ( s , \alpha , \theta , b ) , 1 \} } ) ( \widehat { V } _ { h + 1 } ^ { h } - { V } _ { h + 1 } ^ { h } ) ( s ^ { \prime } ) ) } \\ & \le \mathcal { O } ( \frac { \widehat { \mathbb { P } } _ { h } ^ { h } ( { V } _ { h + 1 } ^ { h } - \frac { { V } _ { h + 1 } ^ { h } } { H } ) ( s , \alpha , \theta ) } { \widehat { \mathbb { P } } _ { H } } + \frac { H _ { \ell } } { \operatorname* { m a x } \{ N _ { h } ^ { h } ( s , \alpha , b ) , 1 \} } + \frac { H ^ { 2 } S _ { \ell } } \operatorname* { m a x } \{ N _ \end{array}
+$$
+
+where $( i )$ is by the second inequality in event $E _ { 0 }$ and $( i i )$ is by AM-GM inequality. This proves the empirical version. Similarly, we can show
+
+$$
+\vert ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) g ( s , a , b ) \vert \leq \mathcal { O } \left( \frac { \mathbb { P } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) } { H } + \frac { H ^ { 2 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \right) ,
+$$
+
+Combining the two bounds completes the proof.
+
+Now we can prove the upper and lower bounds are indeed upper and lower bounds of the best reponses.
+
+Lemma 20. Suppose event $E _ { 0 }$ holds. Then for all $h , s , a , b$ and $k \in [ K ]$ , we have
+
+$$
+\left\{ \begin{array} { l l } { \overline { { Q } } _ { h } ^ { k } ( s , a , b ) \geq Q _ { h } ^ { \dagger , \nu ^ { k } } ( s , a , b ) \geq Q _ { h } ^ { \mu ^ { k } , \dagger } ( s , a , b ) \geq \underline { { Q } } _ { h } ^ { k } ( s , a , b ) , } \\ { \overline { { V } } _ { h } ^ { k } ( s ) \geq V _ { h } ^ { \dagger , \nu ^ { k } } ( s ) \geq V _ { h } ^ { \mu ^ { k } , \dagger } ( s ) \geq \underline { { V } } _ { h } ^ { k } ( s ) . } \end{array} \right.
+$$
+
+Proof. The proof is by backward induction. Suppose the bounds hold for the $Q$ -values in the $( h +$ $1 ) ^ { \mathrm { t h } }$ step, we now establish the bounds for the $V$ -values in the $( h + 1 ) ^ { \mathrm { t h } }$ step and $Q$ -values in the $h ^ { \mathrm { t h } }$ -step. For any state $s$ :
+
+$$
+\begin{array} { r l } & { \overline { { V } } _ { h + 1 } ^ { k } ( s ) = \mathbb { D } _ { \pi _ { h + 1 } ^ { k } } \overline { { Q } } _ { h + 1 } ^ { k } ( s ) } \\ & { \qquad \geq \underset { \mu } { \operatorname* { m a x } } \mathbb { D } _ { \mu \times \nu _ { h + 1 } ^ { k } } \overline { { Q } } _ { h + 1 } ^ { k } ( s ) } \\ & { \qquad \geq \underset { \mu } { \operatorname* { m a x } } \mathbb { D } _ { \mu \times \nu _ { h + 1 } ^ { k } } Q _ { h + 1 } ^ { \dagger , \nu ^ { k } } ( s ) = V _ { h + 1 } ^ { \dagger , \nu ^ { k } } ( s ) . } \end{array}
+$$
+
+Similarly, we can show $\underline { { { V } } } _ { h + 1 } ^ { k } ( s ) \leq V _ { h + 1 } ^ { \mu ^ { k } , \dagger } ( s )$ . Therefore, we have: for all $s$
+
+$$
+\overline { { V } } _ { h + 1 } ^ { k } ( s ) \geq V _ { h + 1 } ^ { \dag , \nu ^ { k } } ( s ) \geq V _ { h + 1 } ^ { \star } ( s ) \geq V _ { h + 1 } ^ { \mu ^ { k } , \dag } ( s ) \geq \underline { { V } } _ { h + 1 } ^ { k } ( s ) .
+$$
+
+Now consider an arbitrary triple $( s , a , b )$ in the $h ^ { \mathrm { t h } }$ step. We have
+
+$$
+\begin{array} { r l } & { \quad ( \widetilde { Q } _ { h } ^ { k } - Q _ { h } ^ { \dagger , \nu ^ { k } } ) ( s , a , b ) } \\ & { \geq \operatorname* { m i n } \Bigg \{ ( \widehat { P } _ { h } ^ { k } \overline { { V } } _ { h + 1 } ^ { k } - \mathbb { P } _ { h } V _ { h + 1 } ^ { \dagger , \nu ^ { k } } + \beta _ { h } ^ { k } + \gamma _ { h } ^ { k } ) ( s , a , b ) , 0 \Bigg \} } \\ & { \geq \operatorname* { m i n } \Bigg \{ ( \widehat { P } _ { h } ^ { k } V _ { h + 1 } ^ { \dagger , \nu ^ { k } } - \mathbb { P } _ { h } V _ { h + 1 } ^ { \dagger , \nu ^ { k } } + \beta _ { h } ^ { k } + \gamma _ { h } ^ { k } ) ( s , a , b ) , 0 \Bigg \} } \\ & { = \operatorname* { m i n } \Bigg \{ ( \underbrace { ( \widehat { P } _ { h } ^ { k } - \mathbb { P } _ { h } ) ( V _ { h + 1 } ^ { \dagger , \nu ^ { k } } - V _ { h + 1 } ^ { \ast } ) ( s , a , b ) } _ { ( A ) } + \underbrace { ( \widehat { P } _ { h } ^ { k } - \mathbb { P } _ { h } ) V _ { h + 1 } ^ { \ast } ( s , a , b ) } _ { ( B ) } } \\ & { \qquad + ( \beta _ { h } ^ { k } + \gamma _ { h } ^ { k } ) ( s , a , b ) , 0 \Bigg \} . } \end{array}
+$$
+
+Invoking Lemma 19 with $g = V _ { h + 1 } ^ { \dag , \nu ^ { k } } - V _ { h + 1 } ^ { \star }$ ,
+
+$$
+\vert ( A ) \vert \leq \mathcal { O } \left( \frac { \widehat { \mathbb { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) } { H } + \frac { H ^ { 2 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \right) .
+$$
+
+By the first inequality in event $E _ { 0 }$
+
+$$
+\left| ( B ) \right| \leq \mathcal { O } \left( \sqrt { \frac { H ^ { 2 } \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } \right) .
+$$
+
+Plugging the two inequalities above back into (10) and recalling the definition of $\beta _ { h } ^ { k }$ and $\gamma _ { h } ^ { k }$ , we obtain $\overline { { Q } } _ { h } ^ { k } ( s , a , b ) \geq Q _ { h } ^ { \dagger , \nu ^ { k } } ( s , a , b )$ . Similarly, we can show ${ \underline { { Q } } } _ { h } ^ { k } ( s , a , b ) \leq Q _ { h } ^ { \mu ^ { k } , \dagger } ( s , a , b )$ . □
+
+Finally we come to the proof of Theorem 3.
+
+Proof of Theorem 3. Suppose event $E _ { 0 }$ holds. We first upper bound the regret. By Lemma 20, the regret can be upper bounded by
+
+$$
+\sum _ { k } ( V _ { 1 } ^ { \dagger , \nu ^ { k } } ( s _ { 1 } ^ { k } ) - V _ { 1 } ^ { \mu ^ { k } , \dagger } ( s _ { 1 } ^ { k } ) ) \leq \sum _ { k } ( \overline { { V } } _ { 1 } ^ { k } ( s _ { 1 } ^ { k } ) - \underline { { V } } _ { 1 } ^ { k } ( s _ { 1 } ^ { k } ) ) .
+$$
+
+For brevity’s sake, we define the following notations:
+
+$$
+\begin{array} { r l } & { \left\{ \begin{array} { l l } { \Delta _ { h } ^ { k } : = ( \overline { { V } } _ { h } ^ { k } - \underline { { V } } _ { h } ^ { k } ) ( s _ { h } ^ { k } ) , } \\ { \zeta _ { h } ^ { k } : = \Delta _ { h } ^ { k } - ( \overline { { Q } } _ { h } ^ { k } - \underline { { Q } } _ { h } ^ { k } ) ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) , } \\ { \xi _ { h } ^ { k } : = \mathbb { P } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) - \Delta _ { h + 1 } ^ { k } . } \end{array} \right. } \end{array}
+$$
+
+Let $\mathcal { F } _ { h } ^ { k }$ be the $\sigma$ -field generated by the following random variables:
+
+$$
+\begin{array} { r } { \{ \big ( s _ { i } ^ { j } , a _ { i } ^ { j } , b _ { i } ^ { j } , r _ { i } ^ { j } \big ) \} _ { ( i , j ) \in [ H ] \times [ k - 1 ] } \bigcup \{ \big ( s _ { i } ^ { k } , a _ { i } ^ { k } , b _ { i } ^ { k } , r _ { i } ^ { k } \big ) \} _ { i \in [ h - 1 ] } \bigcup \{ s _ { h } ^ { k } \} . } \end{array}
+$$
+
+It’s easy to check $\zeta _ { h } ^ { k }$ and $\xi _ { h } ^ { k }$ are martingale differences with respect to $\mathcal { F } _ { h } ^ { k }$ . With a slight abuse of notation, we use $\beta _ { h } ^ { k }$ to refer to $\beta _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } )$ and $N _ { h } ^ { k }$ to refer to $N _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } )$ in the following proof.
+
+We have
+
+$$
+\begin{array} { r l } & { \mathrm { i } \frac { \partial } { \partial x _ { i } } - \zeta _ { n } ^ { k } + ( \widetilde { Q } _ { i } ^ { 0 } - \frac { Q ^ { k } } { 2 } ) \Big ( \delta _ { i , j } ^ { k } , \delta _ { i , j } ^ { k } , \widetilde { Q } _ { i } ^ { k } \Big ) } \\ & { \leq \zeta _ { n } ^ { k } + 2 \delta _ { i ^ { k } } ^ { k } - 2 \gamma _ { k } ^ { k } + \widetilde { P } _ { i } ^ { k } ( \widetilde { V } _ { i - 1 } ^ { k } - \gamma _ { k } ^ { k } , \delta _ { i } ^ { k } ) \Big ( \delta _ { i , j } ^ { k } , \delta _ { i ^ { k } } ^ { k } , \delta _ { i ^ { k } } ^ { k } \Big ) } \\ & { \overset { ( a ) } { \leq } \delta _ { i ^ { k } } ^ { k } + 2 \delta _ { i ^ { k } } ^ { k } - 2 \gamma _ { k } ^ { k } + \widetilde { P } _ { i } ^ { k } ( \widetilde { V } _ { i - 1 } ^ { k } - \gamma _ { k } ^ { k } , \delta _ { i ^ { k } } ^ { k } ) \Big ( \delta _ { i ^ { k } , i ^ { k } } ^ { k } , \delta _ { i ^ { k } } ^ { k } \Big ) } \\ & { \qquad + \sigma _ { i ^ { k } } ^ { k } \Big ( \delta _ { i ^ { k } } ^ { k } ( \widetilde { V } _ { i - 1 } ^ { k } - \gamma _ { k , i } ^ { k } ) + \widetilde { V } _ { i } ^ { k } \widetilde { V } _ { i } ^ { k } \Big ) + \frac { \widetilde { P } _ { i ^ { k } } ^ { k } } { \operatorname* { m a x } \{ 1 , \widetilde { V } _ { i } ^ { k } , \delta _ { i ^ { k } } ^ { k } \} } } \\ & \qquad + \sigma _ { i ^ { k } } ^ { k } \Big ( \widetilde { V } _ { i } ^ { k } ( \widetilde { V } _ { i - 1 } ^ { k } - \widetilde { V } _ { i } ^ { k } ) \Big ( \delta _ { i ^ { k } } ^ { k } , \delta _ { i ^ { k } } ^ { k } \Big ) + \frac { \widetilde { P } _ { i ^ { k } } ^ { k } } { \operatorname* { m a x } \{ 1 , \widetilde { V } _ { i } ^ { k } , \delta _ { i ^ { k } } ^ { k } \} \Big ) } \\ & \overset { ( b ) } { \leq } \delta _ { i ^ { k } } ^ { k } + 2 \sigma \end{array}
+$$
+
+where $( i )$ and $( i i )$ follow from Lemma 19.
+
+Define $c _ { 3 } : = 1 + 2 c _ { 2 } C$ and $\kappa : = 1 + c _ { 3 } / H$ . Recursing this argument for $h \in [ H ]$ and summing over $k$ ,
+
+$$
+\sum _ { k = 1 } ^ { K } \Delta _ { 1 } ^ { k } \leq \sum _ { k = 1 } ^ { K } \sum _ { h = 1 } ^ { H } \left[ \kappa ^ { h - 1 } \zeta _ { h } ^ { k } + \kappa ^ { h } \xi _ { h } ^ { k } + \mathcal { O } \left( \sqrt { \frac { \iota H ^ { 2 } } { \operatorname* { m a x } \{ N _ { h } ^ { k } , 1 \} } } + \frac { H ^ { 2 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } , 1 \} } \right) \right] .
+$$
+
+By Azuma-Hoeffding inequality, with probability at least $1 - p$
+
+$$
+\begin{array}{c} \begin{array} { r l } & { \{ \displaystyle \sum _ { k = 1 } ^ { K } \sum _ { h = 1 } ^ { H } \kappa ^ { h - 1 } \zeta _ { h } ^ { k } \le \mathcal { O } \Big ( H \sqrt { H K \iota } \Big ) = \mathcal { O } \Big ( \sqrt { H ^ { 2 } T \iota } \Big ) , } \\ & { \lfloor \displaystyle \sum _ { k = 1 } ^ { K } \sum _ { h = 1 } ^ { H } \kappa ^ { h } \xi _ { h } ^ { k } \le \mathcal { O } \Big ( H \sqrt { H K \iota } \Big ) = \mathcal { O } \Big ( \sqrt { H ^ { 2 } T \iota } \Big ) . } \end{array} \end{array}
+$$
+
+By pigeon-hole argument,
+
+$$
+\begin{array} { l } { { \displaystyle \sum _ { k = 1 } ^ { K } \sum _ { h = 1 } ^ { H } \frac { 1 } { \sqrt { \operatorname* { m a x } \{ N _ { h } ^ { k } , 1 \} } } \le \sum _ { s , a , b , h \colon N _ { h } ^ { K } ( s , a , b ) > 0 } \sum _ { n = 1 } ^ { K } \frac { 1 } { \sqrt { n } } + H S A B } } \\ { { \displaystyle \qquad \le { \mathcal O } \Bigl ( \sqrt { H S A B T } + H S A B \Bigr ) , } } \\ { { \displaystyle \sum _ { k = 1 } ^ { K } \sum _ { h = 1 } ^ { H } \frac { 1 } { \operatorname* { m a x } \{ N _ { h } ^ { k } , 1 \} } \le \sum _ { s , a , b , h \colon N _ { h } ^ { K } ( s , a , b ) > 0 } \sum _ { n = 1 } ^ { K _ { h } ^ { K } ( s , a , b ) } \frac { 1 } { n } + H S A B } } \\ { { \displaystyle \qquad \le { \mathcal O } ( H S A B ) } . } \end{array}
+$$
+
+Put everything together, with probability at least $1 - 2 p$ (one $p$ comes from $\mathbb { P } ( E _ { 0 } ) \ge 1 - p$ and the other is for equation (12)),
+
+$$
+\sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \dagger , \nu ^ { k } } ( s _ { 1 } ^ { k } ) - V _ { 1 } ^ { \mu ^ { k } , \dagger } ( s _ { 1 } ^ { k } ) ) \leq \mathcal { O } \Big ( \sqrt { H ^ { 3 } S A B T \iota } + H ^ { 3 } S ^ { 2 } A B \iota ^ { 2 } \Big )
+$$
+
+For the PAC guarantee, recall that we choose $\pi ^ { \mathrm { o u t } } = \pi ^ { k ^ { \star } }$ such that $k ^ { \star } = \mathrm { a r g m i n } _ { k } \left( \overline { { V } } _ { 1 } ^ { k } - \underline { { V } } _ { 1 } ^ { k } \right) ( s _ { 1 } )$ As a result,
+
+$$
+( V _ { 1 } ^ { \dagger , \nu ^ { k ^ { \prime } } } - V _ { 1 } ^ { \mu ^ { k ^ { \prime } } , \dagger } ) ( s _ { 1 } ) \leq ( \overline { { V } } _ { 1 } ^ { k ^ { \star } } - \underline { { V } } _ { 1 } ^ { k ^ { \star } } ) ( s _ { 1 } ) \leq \frac { 1 } { K } \mathcal { O } \Big ( \sqrt { H ^ { 3 } S A B T \iota } + H ^ { 3 } S ^ { 2 } A B \iota ^ { 2 } \Big ) ,
+$$
+
+which concludes the proof.
+
+# F.2 PROOF OF THEOREM 4
+
+We use the same notation as in Appendix F.1 except the form of bonus. Besides, we define the empirical variance operator
+
+$$
+\begin{array} { r } { \widehat { \mathbb { V } } _ { h } ^ { k } V ( s , a , b ) : = \operatorname { V a r } _ { s ^ { \prime } \sim \widehat { \mathbb { P } } _ { h } ^ { k } ( \cdot \vert s , a , b ) } V ( s ^ { \prime } ) } \end{array}
+$$
+
+and the true (population) variance operator
+
+$$
+\begin{array} { r } { \mathbb { V } _ { h } V ( s , a , b ) : = \operatorname { V a r } _ { s ^ { \prime } \sim \mathbb { P } _ { h } ( \cdot \vert s , a , b ) } V ( s ^ { \prime } ) } \end{array}
+$$
+
+for any function $V \in \Delta ^ { S }$ . If $N _ { h } ^ { k } ( s , a , b ) = 0$ , we simply set $\widehat { \mathbb { V } } _ { h } ^ { k } V ( s , a , b ) : = H ^ { 2 }$ regardless of the choice of $V$ .
+
+As a result, the bonus terms can be written as
+
+$$
+\beta _ { h } ^ { k } ( s , a , b ) : = C \left( \sqrt { \frac { \iota \widehat { \mathbb { V } } _ { h } ^ { k } [ ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] ( s , a , b ) } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { H ^ { 2 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \right)
+$$
+
+for some absolute constant $C > 0$ .
+
+Lemma 21. Let $c _ { 1 }$ be some large absolute constant. Define event $E _ { 1 }$ to be: for all $h , s , a , b , s ^ { \prime }$ and $k \in [ K ]$ ,
+
+$$
+\begin{array} { r l } & { \displaystyle ( | [ \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) V _ { h + 1 } ^ { \star } ] ( s , a , b ) | \leq c _ { 1 } ( \sqrt { \frac { \widehat { \mathbb { V } } _ { h } ^ { k } V _ { h + 1 } ^ { \star } ( s , a , b ) \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { H \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } ) , } \\ & \displaystyle | [ \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) ( s ^ { \prime } | s , a , b ) | \leq c _ { 1 } ( \sqrt { \frac { \operatorname* { m i n } \{ \mathbb { P } _ { h } ( s ^ { \prime } | s , a , b ) , \widehat { \mathbb { P } } _ { h } ^ { k } ( s ^ { \prime } | s , a , b ) \} \} { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } ) , } \\ & { \displaystyle \| ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) ( \cdot | s , a , b ) \| _ { 1 } \leq c _ { 1 } \sqrt { \frac { S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } . } \end{array}
+$$
+
+We have $\mathbb { P } ( E _ { 1 } ) \ge 1 - p$ .
+
+The proof of Lemma 21 is highly similar to that of Lemma 18. Specifically, the first two can be proved by following basically the same argument in Lemma 18; the third one is standard (e.g., equation (12) in Azar et al. (2017)). We omit the proof here.
+
+Since the proof of Lemma 19 does not depend on the form of the bonus, it can also be applied in this section. As in Appendix F.1, we will prove the upper and lower bounds are indeed upper and lower bounds of the best reponses.
+
+Lemma 22. Suppose event $E _ { 1 }$ holds. Then for all $h , s , a , b$ and $k \in [ K ]$ , we have
+
+$$
+\left\{ \begin{array} { l l } { \overline { { Q } } _ { h } ^ { k } ( s , a , b ) \geq Q _ { h } ^ { \dagger , \nu ^ { k } } ( s , a , b ) \geq Q _ { h } ^ { \mu ^ { k } , \dagger } ( s , a , b ) \geq \underline { { Q } } _ { h } ^ { k } ( s , a , b ) , } \\ { \overline { { V } } _ { h } ^ { k } ( s ) \geq V _ { h } ^ { \dagger , \nu ^ { k } } ( s ) \geq V _ { h } ^ { \mu ^ { k } , \dagger } ( s ) \geq \underline { { V } } _ { h } ^ { k } ( s ) . } \end{array} \right.
+$$
+
+Proof. The proof is by backward induction and very similar to that of Lemma 20. Suppose the bounds hold for the $Q$ -values in the $( h + 1 ) ^ { \mathrm { t h } }$ step, we now establish the bounds for the $V$ -values in the $( h + 1 ) ^ { \mathrm { t h } }$ step and $Q$ -values in the $h ^ { \mathrm { t h } }$ -step.
+
+The proof for the $V$ -values is the same as (9).
+
+For the $Q$ -values, the decomposition (10) still holds and $( A )$ is bounded using Lemma 19 as before.
+The only difference is that we need to bound $( B )$ more carefully.
+
+First, by the first inequality in event $E _ { 1 }$ ,
+
+$$
+\vert ( B ) \vert \leq \mathcal { O } \left( \sqrt { \frac { \widehat { \mathbb { V } } _ { h } ^ { k } V _ { h + 1 } ^ { \star } ( s , a , b ) \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { H \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \right) .
+$$
+
+By the relation of $V$ -values in the $( h + 1 ) ^ { \mathrm { t h } }$ step,
+
+$$
+\begin{array} { r l } & { \ | [ \widehat { \Psi } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] - \widehat { \Psi } _ { h } ^ { k } V _ { h + 1 } ^ { \star } | ( s , a , b ) } \\ & { \leq | [ \widehat { \mathbb { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] ^ { 2 } - ( \widehat { \mathbb { P } } _ { h } ^ { k } V _ { h + 1 } ^ { \star } ) ^ { 2 } | ( s , a , b ) } \\ & { \quad + | \widehat { \mathbb { P } } _ { h } ^ { k } [ ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] ^ { 2 } - \widehat { \mathbb { P } } _ { h } ^ { k } ( V _ { h + 1 } ^ { \star } ) ^ { 2 } | ( s , a , b ) } \\ & { \leq 4 H \widehat { \mathbb { P } } _ { h } ^ { k } | ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) / 2 - V _ { h + 1 } ^ { \star } | ( s , a , b ) } \\ & { \leq 4 H \widehat { \mathbb { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) , } \end{array}
+$$
+
+which implies
+
+$$
+\begin{array} { r l } & { \sqrt { \frac { \hat { \mathcal { V } } _ { h } ^ { k } V _ { h + 1 } ^ { k } ( s , a , b ) } { \operatorname* { m a x } \{ \mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \} } } } \\ & { \leq \sqrt { \frac { \vert \hat { \mathcal { V } } _ { h } ^ { k } [ ( \widehat { V } _ { h + 1 } ^ { k } + \frac { V _ { h + 1 } ^ { k } } { \vert \mathcal { V } _ { h + 1 } ^ { k } \vert } ) / 2 ] + 4 H \hat { \mathcal { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { k } ) \vert ( s , a , b ) } { \operatorname* { m a x } \{ \mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \} } } } \\ & \leq \sqrt { \frac { \vert \hat { \mathcal { V } } _ { h } ^ { k } [ ( \overline { { V } } _ { h + 1 } ^ { k } + \frac { V _ { h + 1 } ^ { k } } { \vert \mathcal { V } _ { h + 1 } ^ { k } \vert } ) / 2 ] ( s , a , b ) } { \operatorname* { m a x } \{ \mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \sqrt { \frac { 4 H \hat { \mathbb { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } - \frac { V _ { h + 1 } ^ { k } } { \vert \mathcal { V } _ { h } ^ { k } ( s , a , b ) , 1 \} ) } { \operatorname* { m a x } \{ \mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \} } } } \\ & \overset { ( i ) } { \leq } \sqrt { \frac { \vert \hat { \mathcal { V } } _ { h } ^ { k } [ ( \overline { { V } } _ { h + 1 } ^ { k } + \frac { V _ { h + 1 } ^ { k } } { \vert \mathcal { V } _ { h + 1 } ^ { k } \vert } ) / 2 \vert ( s , a , b ) } { \operatorname* { m a x } \{ \mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { \vert \hat { \mathcal { V } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { k } ) \vert } { H } + \frac { 4 H ^ { 2 } \varepsilon } \operatorname* { m a x } \{ \mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \} \end{array}
+$$
+
+where $( i )$ is by AM-GM inequality.
+
+Plugging the above inequalities back into (10) and recalling the definition of $\beta _ { h } ^ { k }$ and $\gamma _ { h } ^ { k }$ completes the proof. □
+
+We need one more lemma to control the error of the empirical variance estimator:
+
+Lemma 23. Suppose event $E _ { 1 }$ holds. Then for all $h , s , a , b$ and $k \in [ K ]$ , we have
+
+$$
+\begin{array} { r l } & { \quad \vert \widehat { \mathbb { V } } _ { h } ^ { k } [ ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] - \mathbb { V } _ { h } V _ { h + 1 } ^ { \pi ^ { k } } \vert ( s , a , b ) } \\ & { \le 4 H \mathbb { P } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) + \mathcal { O } \bigg ( 1 + \frac { H ^ { 4 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \bigg ) . } \end{array}
+$$
+
+Proof. By Lemma 22, we have $\overline { { V } } _ { h } ^ { k } ( s ) \geq V _ { h } ^ { \pi ^ { k } } ( s ) \geq \underline { { V } } _ { h } ^ { k } ( s )$ . As a result,
+
+$$
+\begin{array} { r l } & { | \widehat { \nabla } _ { h } ^ { k } [ ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] - \mathbb { V } _ { h } V _ { h + 1 } ^ { \pi ^ { k } } | ( s , a , b ) } \\ & { | [ \widehat { \mathbb { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } / 4 - \mathbb { P } _ { h } ( V _ { h + 1 } ^ { \pi ^ { k } } ) ^ { 2 } ] ( s , a , b ) - [ ( \widehat { \mathbb { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) ) ^ { 2 } / 4 - ( \mathbb { P } _ { h } V _ { h + 1 } ^ { \pi ^ { k } } ) ^ { 2 } ] ( s , a , b ) | } \\ & { \widehat { \mathfrak { E } } [ \widehat { \mathbb { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - \mathbb { P } _ { h } ( \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \widehat { \mathbb { P } } _ { h } ^ { k } \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } + ( \mathbb { P } _ { h } \overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } ] ( s , a , b ) } \\ & { \widehat { \times } [ | ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) ( \overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | + | \mathbb { P } _ { h } [ ( \overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } ] | } \\ & \ + | ( \widehat { \mathbb { P } } _ { h } ^ { k } \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \mathbb { P } _ { h } \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | + | ( \mathbb { P } _ { h } \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \mathbb { P } _ { h } \overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } \end{array}
+$$
+
+These terms can be bounded separately by using event $E _ { 1 }$ :
+
+$$
+\begin{array} { r l } & { ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) ( \overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | ( s , a , b ) \leq H ^ { 2 } \| ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) ( \cdot \mid s , a , b ) \| _ { 1 } \leq \mathcal { O } ( H ^ { 2 } \sqrt { \frac { S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } ) , } \\ & { \mathbb { P } _ { h } [ ( \overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } ] | ( s , a , b ) \leq 2 H [ \mathbb { P } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ] ( s , a , b ) , } \\ & { ( \widehat { \mathbb { P } } _ { h } ^ { k } \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \mathbb { P } _ { h } \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | ( s , a , b ) \leq 2 H [ ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) \underline { { V } } _ { h + 1 } ^ { k } ] ( s , a , b ) \leq \mathcal { O } ( H ^ { 2 } \sqrt { \frac { S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } } \\ & { ( \mathbb { P } _ { h } \underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \mathbb { P } _ { h } \overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | ( s , a , b ) \leq 2 H [ \mathbb { P } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ] ( s , a , b ) . } \end{array}
+$$
+
+Combining with $\begin{array} { r } { H ^ { 2 } \sqrt { \frac { S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } \leq 1 + \frac { H ^ { 4 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } \end{array}$ completes the proof.
+
+Finally we come to the proof of Theorem 4.
+
+Proof of Theorem 4. Suppose event $E _ { 1 }$ holds. We define $\Delta _ { h } ^ { k }$ , $\zeta _ { h } ^ { k }$ abd $\xi _ { h } ^ { k }$ as in the proof of Theorem 3. As before we have
+
+$$
+\begin{array} { r l } & { \Delta _ { h } ^ { k } \leq \zeta _ { h } ^ { k } + \left( 1 + \frac { c _ { 3 } } { H } \right) \mathbb { P } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) \left( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \right) } \\ & { \qquad + 4 c _ { 2 } C \left( \sqrt { \frac { \iota \widehat { \mathbb { V } } _ { h } ^ { k } [ ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] \left( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \right) } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) , 1 \} } } + \frac { H ^ { 2 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } \left( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \right) , 1 \} } \right) . } \end{array}
+$$
+
+By Lemma 23,
+
+$$
+\begin{array} { r l } & { \sqrt { \frac { \varepsilon \hat { \Psi } _ { h } ^ { k } [ ( \overline { { V } } _ { h + 1 } ^ { k } + \underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] ( s , a , b ) } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } } \\ & { \leq \mathcal { O } \left( \sqrt { \frac { \varepsilon \hat { \Psi } _ { h } \hat { V } _ { h + 1 } ^ { k } ( s , a , b ) + \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \sqrt { \frac { H : \mathbb { P } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { H ^ { 2 } \sqrt { S } k } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \right) } \\ & { \leq c _ { 4 } \left( \sqrt { \frac { \varepsilon \hat { \Psi } _ { h } \hat { V } _ { h + 1 } ^ { \varepsilon } ( s , a , b ) + \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { \mathbb { P } _ { h } ( \overline { { V } } _ { h + 1 } ^ { k } - \underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) } { H } + \frac { H ^ { 2 } \sqrt { S } \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \right) , } \end{array}
+$$
+
+where $c _ { 4 }$ is some absolute constant. Define $c _ { 5 } : = 4 c _ { 2 } c _ { 4 } C + c _ { 3 }$ and $\kappa : = 1 + c _ { 5 } / H$ . Plugging (18) back into (17), we have
+
+$$
+\begin{array} { r l } & { \Delta _ { h } ^ { k } \leq \kappa \Delta _ { h + 1 } ^ { k } + \kappa \xi _ { h } ^ { k } + \zeta _ { h } ^ { k } } \\ & { \qquad + \ O \biggl ( \sqrt { \frac { \iota \Psi _ { h } V _ { h + 1 } ^ { \pi ^ { k } } \bigl ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \bigr ) } { N _ { h } ^ { k } \bigl ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \bigr ) } } + \sqrt { \frac { \iota } { N _ { h } ^ { k } \bigl ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \bigr ) } } + \frac { H ^ { 2 } S \iota } { N _ { h } ^ { k } \bigl ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \bigr ) } \biggr ) \biggr \} . } \end{array}
+$$
+
+Recursing this argument for $h \in [ H ]$ and summing over $k$ ,
+
+$$
+\begin{array} { r l r } { { \sum _ { k = 1 } ^ { K } \Delta _ { 1 } ^ { k } \le \sum _ { k = 1 } ^ { K } \sum _ { h = 1 } ^ { H } [ \kappa ^ { h - 1 } \zeta _ { h } ^ { k } + \kappa ^ { h } \xi _ { h } ^ { k } } } \\ & { } & { \quad + \operatorname { \mathcal { O } } ( \sqrt { \frac { \iota \nabla _ { h } V _ { h + 1 } ^ { \pi ^ { k } } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) } { \operatorname* { m a x } \{ N _ { h } ^ { k } , 1 \} } } + \sqrt { \frac { \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } , 1 \} } } + \frac { H ^ { 2 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } , 1 \} } ) ] . } \end{array}
+$$
+
+The remaining steps are the same as that in the proof of Theorem 3 except that we need to bound the sum of variance term.
+
+By Cauchy-Schwarz,
+
+$$
+\sum _ { = 1 } ^ { K } \sum _ { h = 1 } ^ { H } \sqrt { \frac { \mathbb { V } _ { h } V _ { h + 1 } ^ { \pi ^ { k } } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) , 1 \} } } \leq \sqrt \sum _ { k = 1 } ^ { K } \sum _ { h = 1 } ^ { H } \mathbb { V } _ { h } V _ { h + 1 } ^ { \pi ^ { k } } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) \cdot \sum _ { k = 1 } ^ { K } \sum _ { h = 1 } ^ { H } \frac { 1 } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) , 1 \} }
+$$
+
+By the Law of total variation and standard martingale concentration (see Lemma C.5 in Jin et al. (2018) for a formal proof), with probability at least $1 - p$ , we have
+
+$$
+\sum _ { k = 1 } ^ { K } \sum _ { h = 1 } ^ { H } \mathbb { V } _ { h } V _ { h + 1 } ^ { \pi ^ { k } } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) { \le } \mathcal { O } \big ( H T + H ^ { 3 } \iota \big ) .
+$$
+
+Putting all relations together, we obtain that with probability at least $1 - 2 p$ (one $p$ comes from $\begin{array} { r } { \mathbb { P } ( E _ { 1 } ) { \stackrel { - } { = } } 1 - p } \end{array}$ and the other comes from the inequality for bounding the variance term),
+
+$$
+\operatorname { R e g r e t } ( K ) = \sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \dagger , \nu ^ { k } } - V _ { 1 } ^ { \mu ^ { k } , \dagger } ) ( s _ { 1 } ) \leq \mathcal { O } ( \sqrt { H ^ { 2 } S A B T \iota } + H ^ { 3 } S ^ { 2 } A B \iota ^ { 2 } ) .
+$$
+
+Rescaling $p$ completes the proof.
+
+# G PROOF FOR SECTION 4 – REWARD-FREE LEARNING
+
+# G.1 PROOF OF THEOREM 5
+
+In this section, we prove Theorem 5 for the single reward function case, i.e., $N = 1$ . The proof for multiple reward functions $N > 1$ ) simply follows from taking a union bound, that is, replacing the failure probability $p$ by $N p$ .
+
+Let $( \mu ^ { k } , \nu ^ { k } )$ be an arbitrary Nash-equilibrium policy of $\widehat { \mathcal { M } } ^ { k } : = ( \widehat { \mathbb { P } } ^ { k } , \widehat { r } ^ { k } )$ , where ${ \widehat { \mathbb { P } } } ^ { k }$ and $\widehat { r } ^ { k }$ are our bempirical estimate of the transition and the reward at the beginning of the $k$ b’th episode in Algorithm 2, respectively. We use $N _ { h } ^ { k } ( s , a , b )$ to denote the number we have visited the state-action tuple $( s , a , b )$ at the $h$ -th step before the $k$ ’th episode. And the bonus used in the $k$ ’th episode can be written as
+
+$$
+\beta _ { h } ^ { k } ( s , a , b ) : = C \left( \sqrt { \frac { H ^ { 2 } \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { H ^ { 2 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \right) ,
+$$
+
+where $\iota = \log ( S A B T / p )$ and $C$ is some large absolute constant.
+
+We use ${ \widehat { Q } } ^ { k }$ and $\widehat { V } ^ { k }$ to denote the empirical optimal value functions of ${ \widehat { \mathcal { M } } } ^ { k }$ as following.
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \widehat { Q } _ { h } ^ { k } ( s , a , b ) = ( \widehat { \mathbb { P } } _ { h } ^ { k } \widehat { V } _ { h + 1 } ) ( s , a , b ) + \widehat { r } _ { h } ^ { k } ( s , a , b ) , } \\ { \widehat { V } _ { h } ^ { k } ( s ) = \displaystyle \operatorname* { m a x } _ { \mu } \operatorname* { m i n } _ { \nu } \mathbb { D } _ { \mu \times \nu } \widehat { Q } _ { h } ^ { k } ( s ) . } \end{array} \right. } \end{array}
+$$
+
+Since $( \mu ^ { k } , \nu ^ { k } )$ is a Nash-equilibrium policy of ${ \widehat { \mathcal { M } } } ^ { k }$ , we also have $\widehat { V } _ { h } ^ { k } ( s ) = \mathbb { D } _ { \mu ^ { k } \times \nu ^ { k } } \widehat { Q } _ { h } ^ { k } ( s )$ .
+
+We begin with stating a useful property of matrix game that will be frequently used in our analysis.
+Since its proof is quite simple, we omit it here.
+
+Lemma 24. Let $\mathbf { X } , \mathbf { Y } , \mathbf { Z } \in \mathbb { R } ^ { A \times B }$ and $\Delta _ { d }$ be the $d$ -dimensional simplex. Suppose $| \mathbf { X } - \mathbf { Y } | \leq \mathbf { Z }$ where the inequality is entry-wise. Then
+
+$$
+\left| \operatorname* { m a x } _ { \mu \in \triangle _ { A } } \operatorname* { m i n } _ { \nu \in \triangle _ { B } } \mu ^ { \top } \mathbf { X } \nu - \operatorname* { m a x } _ { \mu \in \triangle _ { A } } \operatorname* { m i n } _ { \nu \in \triangle _ { B } } \mu ^ { \top } \mathbf { Y } \nu \right| \leq \operatorname* { m a x } _ { i , j } \mathbf { Z } _ { i j } .
+$$
+
+Lemma 25. Let $c _ { 1 }$ be some large absolute constant such that $c _ { 1 } ^ { 2 } + c _ { 1 } \leq C$ . Define event $E _ { 1 }$ to $b e$ : for all $h , s , a , b , s ^ { \prime }$ and $k \in [ K ]$ ,
+
+$$
+\begin{array}{c} \begin{array} { r } { \left\{ \vert \left[ ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) V _ { h + 1 } ^ { \star } \right] ( s , a , b ) \vert \le \frac { c _ { 1 } } { 1 0 } \sqrt { \frac { H ^ { 2 } \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } , \right.} \\ { \left. \left( \widehat { \boldsymbol { r } } _ { h } ^ { k } - \boldsymbol { r } _ { h } \right) ( s , a , b ) \right. \le \frac { c _ { 1 } } { 1 0 } \sqrt { \frac { H ^ { 2 } \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } , } \\ { \left. \left( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } \right) ( s ^ { \prime } \mid s , a , b ) \right. \le \frac { c _ { 1 } } { 1 0 } \left( \sqrt { \frac { \widehat { \mathbb { P } } _ { h } ^ { k } ( s ^ { \prime } \mid s , a , b ) \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } + \frac { \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } \right) . } \end{array} \end{array}
+$$
+
+We have $\mathbb { P } ( E _ { 1 } ) \geq 1 - p .$
+
+Proof. The proof is standard and folklore: apply standard concentration inequalities and then take a union bound. For completeness, we provide the proof of the third one here.
+
+Consider a fixed $( s , a , b , h )$ tuple.
+
+Let’s consider the following equivalent random process: (a) before the agent starts, the environment samples $\{ s ^ { ( 1 ) } , s ^ { ( 2 ) } , \ldots , s ^ { ( \bar { K } ) } \}$ independently from $\mathbb { P } _ { h } ( \cdot \mid s , a , b )$ ; (b) during the interaction between the agent and environment, the $i ^ { \mathrm { t h } }$ time the agent reaches $( s , a , b , h )$ , the environment will make the agent transit to $s ^ { ( i ) }$ . Note that the randomness induced by this interaction procedure is exactly the same as the original one, which means the probability of any event in this context is the same as in the original problem. Therefore, it suffices to prove the target concentration inequality in this ’easy’ context. Denote by $\widehat { \mathbb { P } } _ { h } ^ { ( t ) } ( \cdot \mid s , a , b )$ the empirical estimate of $\mathbb { P } _ { h } ( \cdot \mid s , a , b )$ calculated using $\{ s ^ { ( 1 ) } , s ^ { ( 2 ) } , \ldots , s ^ { ( t ) } \}$ . For a fixed $t$ and $s ^ { \prime }$ , by the empirical Bernstein inequality, we have with probability at least $1 - p \big / S ^ { 2 } A B T$ ,
+
+$$
+\vert ( \mathbb { P } _ { h } - \widehat { \mathbb { P } } _ { h } ^ { ( t ) } ) ( s ^ { \prime } \mid s , a , b ) \vert \leq \mathcal { O } \left( \sqrt { \frac { \widehat { \mathbb { P } } _ { h } ^ { ( t ) } ( s ^ { \prime } \mid s , a , b ) \iota } { t } } + \frac { \iota } { t } \right) .
+$$
+
+Now we can take a union bound over all $s , a , b , h , s ^ { \prime }$ and $t \in [ K ]$ , and obtain that with probability at least $1 - p$ , for all $s , a , b , h , s ^ { \prime }$ and $t \in [ K ]$ ,
+
+$$
+\vert ( \mathbb { P } _ { h } - \widehat { \mathbb { P } } _ { h } ^ { ( t ) } ) ( s ^ { \prime } \mid s , a , b ) \vert \leq \mathcal { O } \left( \sqrt { \frac { \widehat { \mathbb { P } } _ { h } ^ { ( t ) } ( s ^ { \prime } \mid s , a , b ) \iota } { t } } + \frac { \iota } { t } \right) .
+$$
+
+Note that the agent can reach each $( s , a , b , h )$ for at most $K$ times, this directly implies that the third inequality also holds with probability at least $1 - p$ . □
+
+The following lemma states that the empirical optimal value functions are close to the true optimal ones, and their difference is controlled by the exploration value functions calculated in Algorithm 2.
+
+Lemma 26. Suppose event $E _ { 1 }$ (defined in Lemma 25) holds. Then for all $h , s , a , b$ and $k \in [ K ]$ , we have,
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \Big | \widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \star } ( s , a , b ) \Big | \le \widetilde { Q } _ { h } ^ { k } ( s , a , b ) , } \\ { \Big | \widehat { V } _ { h } ^ { k } ( s ) - V _ { h } ^ { \star } ( s ) \Big | \le \widetilde { V } _ { h } ^ { k } ( s ) . } \end{array} \right. } \end{array}
+$$
+
+Proof. Let’s prove by doing backward induction on $h$ . The case of $h = H + 1$ holds trivially.
+
+Assume the conclusion hold for $( h + 1 )$ ’th step. For $h$ ’th step,
+
+$$
+\begin{array} { r l } & { \quad \Bigl | \widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \star } ( s , a , b ) \Bigr | } \\ & { \le \operatorname* { m i n } \Big \{ \bigl | [ ( \widehat { \mathbb { P } } _ { h } ^ { k } - { \mathbb { P } } _ { h } ) V _ { h + 1 } ^ { \star } ] ( s , a , b ) \bigr | + | ( \widehat { r } _ { h } ^ { k } - r _ { h } ) ( s , a , b ) | + \bigl | [ \widehat { \mathbb { P } } _ { h } ^ { k } ( \widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \star } ) ] ( s , a , b ) \bigr | , H \Bigr \} } \\ & { \overset { ( i ) } { \le } \operatorname* { m i n } \Big \{ \beta _ { h } ^ { k } ( s , a , b ) + ( \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ) ( s , a , b ) , H \Big \} \overset { ( i i ) } { = } \widetilde { Q } _ { h } ^ { k } ( s , a , b ) , } \end{array}
+$$
+
+where $( i )$ follows from the induction hypothesis and event $E _ { 1 }$ , and $( i i )$ follows from the definition of $\widetilde { Q } _ { h } ^ { k }$ . By Lemma 24, we immediately obtain $| \widehat { V } _ { h } ^ { k } ( s ) - V _ { h } ^ { \star } ( s ) | \le \widetilde { V } _ { h } ^ { k } ( s )$ .
+
+Now, we are ready to establish the key lemma in our analysis using Lemma 26.
+
+Lemma 27. Suppose event $E _ { 1 }$ (defined in Lemma 25) holds. Then for all $h , s , a , b$ and $k \in [ K ]$ , we have
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \vert \widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \dagger , \nu ^ { k } } ( s , a , b ) \vert \le \alpha _ { h } \widetilde { Q } _ { h } ^ { k } ( s , a , b ) , } \\ { \vert \widehat { V } _ { h } ^ { k } ( s ) - V _ { h } ^ { \dagger , \nu ^ { k } } ( s ) \vert \le \alpha _ { h } \widetilde { V } _ { h } ^ { k } ( s ) , } \end{array} \right. } \end{array}
+$$
+
+and
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \vert \widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \mu ^ { k } , \dagger } ( s , a , b ) \vert \le \alpha _ { h } \widetilde { Q } _ { h } ^ { k } ( s , a , b ) , } \\ { \vert \widehat { V } _ { h } ^ { k } ( s ) - V _ { h } ^ { \mu ^ { k } , \dagger } ( s ) \vert \le \alpha _ { h } \widetilde { V } _ { h } ^ { k } ( s ) , } \end{array} \right. } \end{array}
+$$
+
+where $\alpha _ { H + 1 } = 0$ and $\begin{array} { r } { \alpha _ { h } = [ ( 1 + \frac { 1 } { H } ) \alpha _ { h + 1 } + \frac { 1 } { H } ] \leq 4 } \end{array}$ .
+
+Proof. We only prove the first set of inequalities. The second one follows exactly the same. Again, the proof is by performing backward induction on $h$ . It is trivial to see the conclusion holds for
+
+$( H + 1 ) ^ { \prime }$ ’th step with $\alpha _ { H + 1 } = 0$ . Now, assume the conclusion holds for $( h + 1 )$ ’th step. For $h$ ’th step,
+
+$$
+\begin{array} { r l } & { | \widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \dagger , \nu } ( s , a , b ) | } \\ & { \leq \operatorname* { m i n } \bigg \{ | [ ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) ( V _ { h + 1 } ^ { \dagger , \nu ^ { k } } - V _ { h + 1 } ^ { \ast } ) ] ( s , a , b ) | + | ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) V _ { h + 1 } ^ { \star } ( s , a , b ) | } \\ & { \quad + | ( \widehat { r } _ { h } ^ { k } - r _ { h } ) ( s , a , b ) | + | [ \widehat { \mathbb { P } } _ { h } ( \widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \dagger , \nu ^ { k } } ) ] ( s , a , b ) | , H \bigg \} } \\ & { \leq \operatorname* { m i n } \bigg \{ \underbrace { | [ ( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } ) ( V _ { h + 1 } ^ { \dagger , \nu ^ { k } } - V _ { h + 1 } ^ { \ast } ) ] ( s , a , b ) | } _ { ( T _ { 1 } ^ { k } ) } + c _ { 1 } \sqrt { \frac { H ^ { 2 } \iota } { \operatorname* { m a x } \{ N _ { h + 1 } ^ { k } ( s , a , b ) , 1 \} } } } \\ & { \qquad + \underbrace { | [ \widehat { \mathbb { P } } _ { h } ( \widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \dagger , \nu ^ { k } } ) ] ( s , a , b ) | } _ { ( T _ { 2 } ^ { k } ) } , H \bigg \} , } \end{array}
+$$
+
+where the second inequality follows from the definition of event $E _ { 1 }$ .
+
+We can control the term $( T _ { 1 } )$ by combining Lemma 26 and the induction hypothesis to bound $| V _ { h + 1 } ^ { \dag , \nu ^ { k } } - V _ { h + 1 } ^ { \star } |$ , and then applying the third inequality in event $E _ { 1 }$ :
+
+$$
+\begin{array} { r l } & { ( T _ { 1 } ) \leq \displaystyle \sum _ { s ^ { \prime } } | \widehat { \mathbb { P } } _ { h } ^ { k } ( s ^ { \prime } \mid s , a , b ) - { \mathbb { P } } _ { h } ( s ^ { \prime } \mid s , a , b ) | | V _ { h + 1 } ^ { \dagger , \nu ^ { k } } - V _ { h + 1 } ^ { \star } ( s ^ { \prime } ) | } \\ & { \quad \leq \displaystyle \sum _ { s ^ { \prime } } | \widehat { \mathbb { P } } _ { h } ^ { k } ( s ^ { \prime } \mid s , a , b ) - { \mathbb { P } } _ { h } ( s ^ { \prime } \mid s , a , b ) | \Big ( | V _ { h + 1 } ^ { \dagger , \nu ^ { k } } - \widehat { V } _ { h + 1 } ^ { k } ( s ^ { \prime } ) | + | \widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \star } ( s ^ { \prime } ) | \Big ) } \\ & { \quad \leq \displaystyle \sum _ { s ^ { \prime } } | \widehat { \mathbb { P } } _ { h } ^ { k } ( s ^ { \prime } \mid s , a , b ) - { \mathbb { P } } _ { h } ( s ^ { \prime } \mid s , a , b ) | ( \alpha _ { h + 1 } + 1 ) \widetilde { V } _ { h + 1 } ^ { k } } \\ & { \quad \leq \displaystyle \frac { ( \alpha _ { h + 1 } + 1 ) } { H } ( \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ) ( s , a , b ) + \displaystyle \frac { c _ { 1 } ^ { 2 } ( \alpha _ { h + 1 } + 1 ) H ^ { 2 } S _ { L } } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } . } \end{array}
+$$
+
+The term $( T _ { 2 } )$ is bounded by directly applying the induction hypothesis
+
+$$
+| [ \widehat { \mathbb { P } } _ { h } ( \widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \dagger , \nu ^ { k } } ) ] ( s , a , b ) | \leq \alpha _ { h + 1 } [ \widehat { \mathbb { P } } _ { h } \widetilde { V } _ { h + 1 } ^ { k } ] ( s , a , b ) .
+$$
+
+Plugging (29) and (30) into (28), we obtain
+
+$$
+\begin{array} { r l } & { ~ \left. \widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \dagger , s ^ { \prime } } ( s , a , b ) \right. } \\ & { \leq \operatorname* { m i n } \bigg \{ ( 1 + \frac { 1 } { H } ) \alpha _ { h + 1 } + \frac { 1 } { H } [ \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ] ( s , a , b ) + c _ { 1 } \sqrt { \frac { H ^ { 2 } \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } } } \\ & { ~ + \frac { c _ { 1 } ^ { 2 } ( \alpha _ { h + 1 } + 1 ) H ^ { 2 } S \iota } { \operatorname* { m a x } \{ N _ { h } ^ { k } ( s , a , b ) , 1 \} } , H \bigg \} } \\ & { \overset { ( i ) } { \leq } \operatorname* { m i n } \bigg \{ \bigg ( ( 1 + \frac { 1 } { H } ) \alpha _ { h + 1 } + \frac { 1 } { H } \bigg ) [ \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ] ( s , a , b ) + \beta _ { h } ^ { k } ( s , a , b ) , H \bigg \} } \\ & { \overset { ( i i ) } { \leq } \bigg ( ( 1 + \frac { 1 } { H } ) \alpha _ { h + 1 } + \frac { 1 } { H } \bigg ) \widehat { Q } _ { h } ^ { k } ( s , a , b ) , } \end{array}
+$$
+
+where $( i )$ follows from the definition of $\beta _ { h } ^ { k }$ , and $( i i )$ follows from the definition of $\widetilde Q _ { h } ^ { k }$ . Therefore, by (31), choosing $\begin{array} { r } { \alpha _ { h } = [ ( 1 + \frac { 1 } { H } ) \alpha _ { h + 1 } + \frac { \top } { H } ] } \end{array}$ suffices for the purpose of induction.
+
+Now, let’s prove the inequality for $V$ functions.
+
+$$
+\begin{array} { r l } & { \lvert ( \widehat { V } _ { h } ^ { k } - V _ { h } ^ { \dagger , \nu ^ { k } } ) ( s ) \rvert \stackrel { ( i ) } { = } \lvert \displaystyle \operatorname* { m a x } _ { \mu \in \triangle _ { A } } ( { \mathbb D } _ { \mu , \nu ^ { k } } \widehat { Q } _ { h } ^ { k } ) ( s ) - \displaystyle \operatorname* { m a x } _ { \mu \in \triangle _ { A } } ( { \mathbb D } _ { \mu , \nu ^ { k } } Q _ { h } ^ { \dagger , \nu ^ { k } } ) ( s ) \rvert } \\ & { \qquad \stackrel { ( i i ) } { \leq } \displaystyle \operatorname* { m a x } _ { a , b } \Big [ \alpha _ { h } \widetilde { Q } _ { h } ^ { k } ( s , a , b ) \Big ] = \alpha _ { h } \widetilde { V } _ { h } ^ { k } ( s ) , } \end{array}
+$$
+
+where $( i )$ follows from the definition of $\widehat { V } _ { h } ^ { k }$ and $V _ { h } ^ { \dag , \nu ^ { k } }$ , and $( i i )$ uses (31) and Lemma 24.
+
+Theorem 28 (Guarantee for UCB-VI from Azar et al. (2017)). For any $p \in \mathsf { \Gamma } ( 0 , 1 ]$ , choose the exploration bonus $\beta _ { t }$ in Algrothm $2 \ : a s$ (20). Then, with probability at least $1 - p$ ,
+
+$$
+\sum _ { k = 1 } ^ { K } \widetilde { V } _ { 1 } ^ { k } ( s _ { 1 } ) \leq \mathcal { O } ( \sqrt { H ^ { 4 } S A K \iota } + H ^ { 3 } S ^ { 2 } A \iota ^ { 2 } ) .
+$$
+
+Proof of Theorem 5. Recall that o ${ \mathfrak { x } } = \arg \operatorname* { m i n } _ { k \in [ K ] } { \widetilde { V } } _ { h } ^ { k } ( s )$ . By Lemma 27 and Theorem 28, with probability at least $1 - 2 p$ ,
+
+$$
+\begin{array} { r l } { { V _ { h } ^ { \dagger , \nu ^ { \mathrm { o u t } } } ( s ) - V _ { h } ^ { \mu ^ { \mathrm { o u t } } , \dagger } ( s ) \le | V _ { h } ^ { \dagger , \nu ^ { \mathrm { o u t } } } ( s ) - \widehat { V } _ { h } ^ { \mathrm { o u t } } ( s ) | + | \widehat { V } _ { h } ^ { \mathrm { o u t } } ( s ) - V _ { h } ^ { \mu ^ { \mathrm { o u t } } , \dagger } ( s ) | } } \\ & { \le 8 \widetilde { V } _ { h } ^ { \mathrm { o u t } } ( s ) \le \mathcal { O } ( \sqrt { \frac { H ^ { 4 } S A \iota } { K } } + \frac { H ^ { 3 } S ^ { 2 } A \iota ^ { 2 } } { K } ) . } \end{array}
+$$
+
+Rescaling $p$ completes the proof.
+
+# G.2 VANILLA NASH VALUE ITERATION
+
+Here, we provide one optional algorithm, Vanilla Nash VI, for computing the Nash equilibrium policy for a known model. Its only difference from the value iteration algorithm for MDPs is that the maximum operator is replaced by the minimax operator in Line 7. We remark that the Nash equilibrium for a two-player zero-sum game can be computed in polynomial time.
+
+# Algorithm 5 Vanilla Nash Value Iteration
+
+1: Input: model $\widehat { \mathcal { M } } = ( \widehat { \mathbb { P } } , \widehat { r } )$ .
+2: Initialize: for all $( s , a , b )$ , $V _ { H + 1 } ( s , a , b ) 0$ .
+3: for step $h = H , H - 1 , \ldots , 1$ do
+4: for $( s , a , b ) \in \mathcal { S } \times \mathcal { A } \times \mathcal { B }$ do
+5: $\begin{array} { r } { Q _ { h } ( s , a , b ) \gets [ \widehat { \mathbb { P } } _ { h } V _ { h + 1 } ] ( s , a , b ) + \widehat { r } _ { h } ( s , a , b ) . } \end{array}$
+6: for $s \in S$ do
+7: $\big ( \hat { \mu } _ { h } ( \cdot \vert \ s ) , \hat { \nu } _ { h } ( \cdot \ \vert \ s ) \big ) \gets \mathrm { N A S H - Z E R O - S U M } ( Q _ { h } ( s , \cdot , \cdot ) )$ .
+8: $V _ { h } ( s ) \gets \hat { \mu } _ { h } ( \cdot \mid s ) ^ { \top } Q _ { h } ( s , \cdot , \cdot ) \hat { \nu } _ { h } ( \cdot \mid s )$ .
+9: Output $( \hat { \mu } , \hat { \nu } ) \gets \{ ( \hat { \mu } _ { h } ( \cdot { | } s ) , \hat { \nu } _ { h } ( \cdot { | } s ) ) \} _ { ( h , s ) \in [ H ] \times { \mathcal { S } } } .$ .
+
+By recalling the definition of best responses in Appendix D, one can directly see that the output policy $( \hat { \mu } , \hat { \nu } )$ is a Nash equilibrium for $\widehat { \mathcal { M } }$ .
+
+# G.3 PROOF OF THEOREM 6
+
+In this section, we first prove a $\Theta ( A B / \epsilon ^ { 2 } )$ lower bound for reward-free matrix games, i.e., $S =$ $H = 1$ , and then generalize it to $\Theta ( S A \dot { B } H ^ { 2 } / \epsilon ^ { 2 } )$ for the Markov games setting.
+
+# G.3.1 REWARD-FREE MATRIX GAMES
+
+In the matrix game, let the max-player pick row and the min-player pick column. We consider the following family of Bernoulli matrix games:
+
+$$
+\mathfrak { M } ( \epsilon ) = \left\{ \mathcal { M } \in \mathbb { R } ^ { A \times B } \mathrm { ~ w i t h ~ } \mathcal { M } _ { a b } = \frac { 1 } { 2 } + ( 1 - 2 \cdot \mathbf { 1 } \{ a \neq a ^ { \star } \mathfrak { E } b = b ^ { \star } \} ) \epsilon \colon ( a ^ { \star } , b ^ { \star } ) \in [ A ] \times [ B ] \right\} ,
+$$
+
+where in matrix game $\mathcal { M }$ , the reward is sampled from Bernoulli $( \mathcal { M } _ { a b } )$ if the max-player picks the $a$ ’th row and the min-player picks the $b ^ { \mathrm { : } }$ ’th column.
+
+# Min-player
+
+$$
+\begin{array} { r l } { \mathrm { ~ a c t i o n } \quad 1 } & { \dots \quad b ^ { \star } - 1 \quad b ^ { \star } \quad b ^ { \star } + 1 \quad \dots \quad B } \\ { \mathrm { ~ 1 } \quad + \quad \dots \quad + \quad - \quad - \quad + \quad \dots \quad + } & { \dots \quad + } \\ { \vdots \quad \vdots \quad \ddots \quad } & { \vdots \quad \vdots \quad \vdots \quad \ddots \quad \vdots } \\ { \mathrm { ~ 2 } \quad a ^ { \star } - 1 \quad + \quad \dots \quad + \quad - \quad - } & { + \quad \dots \quad + } \\ { \mathrm { ~ a ^ { \star } ~ - p l a y e r } \quad a ^ { \star } \quad + \quad \dots \quad + \quad + \quad + \quad + \quad \dots \quad + } \\ { \ a ^ { \star } + 1 \quad + \quad \dots \quad + \quad - \quad + \quad - \quad + \quad \dots \quad + } \\ { \vdots \quad \vdots \quad \vdots \quad \ddots \quad } & { \vdots \quad \vdots \quad \vdots \quad \ddots \quad \vdots } \\ { \quad \ A \quad + \quad \dots \quad + \quad - \quad - \quad + \quad \dots \quad + \quad \dots \quad + } \end{array}
+$$
+
+Above, we visualize the hard instance by using $^ +$ and − to represent $1 / 2 + \epsilon$ and $1 / 2 { - \epsilon }$ , respectively. It is direct to see that the optimal policy for the max-player is always picking the $a ^ { \star }$ ’th row and the optimal policy for the min-player is always picking the $b ^ { \star }$ ’th column. If the max-player picks the $a ^ { \star }$ ’th row with probability smaller than $2 / 3$ , it is at least $\epsilon / 1 0$ suboptimal.
+
+Lemma 29. For any fixed matrix game $\mathcal { M }$ from $\mathfrak { M } ( \epsilon )$ and $N \in \mathbb { N }$ , if an algorithm $\mathcal { A }$ can output a policy that is at most $\epsilon / 1 0$ suboptimal with probability at least $p$ using at most $N$ samples, then there exists an algorithm $\hat { A }$ that can identify the best row in $\mathcal { M }$ with probability at least $p$ using at most $N$ samples.
+
+Proof. We simply define $\hat { A }$ as running algorithm $\mathcal { A }$ and choosing the most played row by its outputted policy as the guess for the best row. By simple calculation, one can show $\hat { A }$ will output the best row in $\mathcal { M }$ with probability at least $p$ . □
+
+Lemma 29 directly implies that in order to prove the desired lower bound for matrix games:
+
+Claim 30. for any algorithm $\mathcal { A }$ using at most $N = A B / ( 1 0 ^ { 3 } \epsilon ^ { 2 } )$ samples, there exists a matrix game $\mathcal { M }$ in $\mathfrak { M } ( \epsilon )$ such that when running $\mathcal { A }$ on $\mathcal { M }$ , it will output a policy that is at least $\epsilon / 1 0$ suboptimal for the max-player with probability at least $1 / 4$ ,
+
+it suffices to prove the following claim:
+
+Claim 31. for any algorithm $\hat { A }$ using at most $N = A B / ( 1 0 ^ { 3 } \epsilon ^ { 2 } )$ samples, there exists a matrix game $\mathcal { M }$ in $\mathfrak { M } ( \epsilon )$ such that when running $\hat { A }$ on $\mathcal { M }$ , it will fail to identify the optimal row with probability at least $1 / 4$ .
+
+Proof of Claim 31. WLOG, we assume $\hat { A }$ is deterministic. Since this is the reward-free setting, being deterministic means that algorithm $\hat { A }$ will always pull each arm $( a , b )$ for some fixed $n ( a , b )$ times and then output a guess for $a ^ { \star }$ which is a function of the reward revealed.
+
+Denote by $L$ the reward revealed after algorithm $\hat { A }$ ’s pulling. Denote by $\mathbb { P } _ { \star }$ the probability induced by picking $\mathcal { M }$ uniformly at random from $\mathfrak { M } ( \epsilon )$ and running $\hat { A }$ on $\mathcal { M }$ . Denote by $\mathbb { P } _ { a , b }$ the probability induced by running $\hat { A }$ on $\mathcal { M }$ , whose indices of the special row and the special column are $a$ and $b$ , respectively. Denote by $\mathbb { P } _ { 0 , b }$ the probability induced by running $\mathcal { A }$ on $\mathcal { M }$ , whose $b '$ th column are all $1 / \bar { 2 } - \epsilon$ and other columns are all $1 / 2 + \epsilon$ . We want to mention that the $\mathcal { M }$ we use to define $\mathbb { P } _ { 0 , b }$ does not belong to $\mathfrak { M } ( \epsilon )$ .
+
+We have
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { P } _ { \star } ( \hat { A } ( L ) \neq a ^ { \star } ) \geq \frac { 1 } { A B } \sum _ { u , b } \mathbb { P } _ { 0 , b } ( \hat { A } ( L ) \neq a ) - \frac { 1 } { A B } \sum _ { a , b } \| \mathbb { P } _ { a , b } - \mathbb { P } _ { 0 , b } \| _ { 1 } } \\ { \displaystyle \geq 1 - \frac { 1 } { A } - \frac { 1 } { A B } \sum _ { u , b } \sqrt { 2 \mathrm { K I } ( \mathbb { P } _ { 0 , b } \| \mathbb { P } _ { a , b } ) } } \\ { \displaystyle = 1 - \frac { 1 } { A } - \frac { 1 } { A B } \sum _ { a , b } \sqrt { 2 n ( a , b ) [ ( \frac { 1 } { 2 } - \epsilon ) \log \frac { \frac { 1 } { 2 } - \epsilon } { 2 } + ( \frac { 1 } { 2 } + \epsilon ) \log \frac { \frac { 1 } { 2 } + \epsilon } { 2 } ] } } \\ { \displaystyle \geq 1 - \frac { 1 } { A } - \frac { 1 0 } { A B } \sum _ { a , b } \sqrt { n ( a , b ) \epsilon ^ { 2 } } } \\ { \displaystyle \geq 1 - \frac { 1 } { A } - \sqrt { \frac { 1 0 0 N \epsilon ^ { 2 } } { a B } } . } \end{array}
+$$
+
+Choosing $N = A B / ( 1 0 ^ { 3 } \epsilon ^ { 2 } )$ concludes the proof.
+
+# G.3.2 REWARD-FREE MARKOV GAMES
+
+Now let’s generalize the $\Theta ( A B / \epsilon ^ { 2 } )$ lower bound to $\Theta ( S A B H ^ { 2 } / \epsilon ^ { 2 } )$ for reward-free Markov games. We define the following family of MDPs:
+
+$$
+\Im ( \epsilon ) : = \left\{ \mathcal { I } ( \boldsymbol { a } ^ { \star } , \boldsymbol { b } ^ { \star } ) : ( \boldsymbol { a } ^ { \star } , \boldsymbol { b } ^ { \star } ) \in [ A ] ^ { H \times S } \times [ B ] ^ { H \times S } \right\} ,
+$$
+
+where MDP $\mathcal { I } ( a ^ { \star } , b ^ { \star } )$ is defined as below:
+
+• States and actions: $\mathcal { I } ( a ^ { \star } , b ^ { \star } )$ is a finite-horizon MDP with $S + 1$ states of length $H + 1$ . There is a fixed initial state $s _ { 0 }$ in the first step, $S$ states $\{ s _ { 1 } , \ldots , s _ { S } \}$ for the remaining steps. The two players have $A$ and $B$ actions, respectively.
+
+• Rewards: there is no reward in the first step. For the remaining steps $h \in \{ 2 , \dots , H + 1 \}$ , if the agent takes action $( a , b )$ at state $s _ { i }$ in the $h ^ { \mathrm { t h } }$ step, it will receive a binary reward sampled from
+
+$$
+\operatorname { B e r n o u l l i } { \left( { \frac { 1 } { 2 } } + ( 1 - 2 \cdot \mathbf { 1 } \{ a \neq a _ { h - 1 , i } ^ { \star } \& b = b _ { h - 1 , i } ^ { \star } \} ) { \frac { \epsilon } { H } } \right) }
+$$
+
+• Transitions: Regardless of the current state, actions and index of steps, the agent will always transit to one of $s _ { 1 } , \ldots , s _ { S }$ uniformly at random.
+
+It is direct to see that $\mathcal { I } ( a ^ { \star } , b ^ { \star } )$ is a collection of $S H$ independent matrix games from $\mathfrak { M } ( \epsilon / H )$ . Therefore, the optimal policy for the max player is to always pick action $\pmb { a } _ { h - 1 , i } ^ { \star }$ whenever it reaches state $s _ { i }$ in the $h ^ { \mathrm { t h } }$ step $\left( h \geq 2 \right)$ ). In other words, $\pmb { a } _ { h - 1 , i } ^ { \star }$ is the unique optimal action for the step-state pair $( h , i )$ .
+
+At a high level, in order to find an $\epsilon$ -optimal policy for the above Markov game, we need to identify at least half of the entries of $\mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { \delta } \mathbf { a } \left( \frac { \mathcal { \nu } } { \mathcal { \delta } } \right) \delta \mathbf { \delta } \left( \frac { \mathcal { \nu } } { \mathcal { \delta } } \right) .$ . Therefore, the number of episodes should be at least
+
+$$
+\Theta \bigg ( \frac { A B } { ( \epsilon / H ) ^ { 2 } } \bigg ) \times \frac { S } { 2 } = \Theta \bigg ( \frac { A B S H ^ { 2 } } { \epsilon ^ { 2 } } \bigg ) .
+$$
+
+Below we provide a formal proof of this argument, which is almost the same as that for the setting of reward-free matrix games.
+
+We start by proving an analogue of Lemma 29.
+
+Lemma 32. For any fixed matrix game $\mathcal { I } ( a ^ { \star } , b ^ { \star } )$ from $\Im ( \epsilon )$ and $N \in \mathbb { N }$ , if an algorithm $\mathcal { A }$ can output a policy that is at most $\epsilon / 1 0 ^ { 3 }$ suboptimal with probability at least $p$ using at most $N$ samples, then there exists an algorithm $\hat { A }$ that can correctly identify at least $S H - \lfloor S H / 5 0 0 \rfloor$ entries of $\mathbf { \pmb { a } } ^ { \star }$ with probability at least $p$ using at most $N$ samples.
+
+Proof. Denote by $\pi$ the output policy for the max player. Denote by $Z$ the collection of $( h , i )$ ’s in $[ H ] \times [ S ]$ such that $\pi _ { h + 1 } ( \boldsymbol { a } _ { h , i } ^ { \star } \mid s _ { i } ) \le 2 / 3$ .
+
+Observe that each time the max player picks a suboptimal action, it will incur an $2 \epsilon / H$ suboptimality in expectation. As a result, if $\pi$ is at most $\epsilon / 1 0 ^ { 3 }$ -suboptimal, we must have
+
+$$
+\frac { 1 } { S } \sum _ { ( h , i ) \in \cal Z } ( 1 - \pi _ { h + 1 } ( { \pmb a } _ { h , i } ^ { \star } \mid s _ { i } ) ) \times \frac { 2 \epsilon } { H } \leq \frac { \epsilon } { 1 0 ^ { 3 } } ,
+$$
+
+which implies $| Z | \le S H / 5 0 0$ , that is, for at most $\lfloor S H / 5 0 0 \rfloor$ different $i$ ’s, $\pi ( \mathbf { a } _ { i } ^ { \star } \mid s _ { i } ) \ \leq \ 2 / 3$ . Therefore, we can simply pick $\operatorname { a r g m a x } _ { a } \pi _ { h + 1 } ( a \mid s _ { i } )$ as the guess for $\pmb { a } _ { h , i } ^ { \star }$ . Since policy $\pi$ is at most $\epsilon / 1 0 ^ { 3 }$ suboptimal with probability at least $p$ , we can correctly identify the optimal actions for at least $S H - \lfloor S H / 5 0 0 \rfloor$ different $( s , h )$ pairs also with probability no smaller than $p$ . □
+
+Lemma 32 directly implies that in order to prove the desired lower bound for reward-free Markov games:
+
+Claim 33. for any algorithm $\mathcal { A }$ interacting with the environment for at most $K = A B S H ^ { 2 } / ( 1 0 ^ { 4 } \epsilon ^ { 2 } )$ episodes, there exists $\mathcal { I } \in \Im ( \epsilon )$ such that when running $\mathcal { A }$ on $\mathcal { I }$ , it will output a policy that is at least $\epsilon / 1 0 ^ { 3 }$ suboptimal for the max-player with probability at least $1 / 4$ ,
+
+it suffices to prove the following claim:
+
+Claim 34. for any algorithm $\hat { A }$ interacting with the environment for at most $K = A B S H ^ { 2 } / ( 1 0 ^ { 4 } \epsilon ^ { 2 } )$ episodes, there exists $\mathcal { I } \in \Im ( \epsilon )$ such that when running $\hat { A }$ on $\mathcal { I }$ , it will fail to identify the optimal actions for at least $\lfloor S H / 5 0 0 \rfloor + 1$ different $( s , h )$ pairs with probability at least $1 / 4$ .
+
+Proof of Claim 34. Denote by $\mathbb { P } _ { \star }$ $( \mathbb { E } _ { \star } )$ the probability (expectation) induced by picking $\mathcal { I } ( a ^ { \star } , b ^ { \star } )$ uniformly at random from $\Im ( \epsilon )$ and running $\hat { A }$ on $\mathcal { I }$ . Denote by $n _ { \mathrm { w r o n g } }$ the number of $( s , h )$ pairs for which $\hat { A }$ fails to identify the optimal actions. Denote by $\mathrm { e r r o r } _ { h , i }$ the indicator function of the event that $\hat { A }$ fails to identify the optimal action for $( h + 1 , i )$ .
+
+We prove by contradiction. Suppose for any $\mathcal { I } \in \Im ( \epsilon )$ , $\hat { A }$ can identify the optimal actions for at least $S H - \lfloor S H / 5 0 0 \rfloor$ different $( s , h )$ pairs with probability larger than $3 / 4$ . Then we have
+
+$$
+\mathbb { E } _ { \star } [ n _ { \mathrm { w r o n g } } ] \leq \frac { 1 } { 4 } \times S H + \frac { 3 } { 4 } \times \left\lfloor \frac { S H } { 5 0 0 } \right\rfloor \leq \frac { 1 0 1 S H } { 4 0 0 } .
+$$
+
+Since $\begin{array} { r } { \sum _ { ( h , i ) \in [ H ] \times [ S ] } \mathbb { E } _ { \star } [ \mathrm { e r r o r } _ { h , i } ] = \mathbb { E } _ { \star } [ n _ { \mathrm { w r o n g } } ] } \end{array}$ , there must exists $( h ^ { \prime } , i ^ { \prime } ) \in [ H ] \times [ S ]$ such that $\mathbb { E } _ { \star } [ \mathrm { e r r o r } _ { h ^ { \prime } , i ^ { \prime } } ] \le 1 0 1 / 4 0 0$ . However, in the following, we show that for every $( h , i ) \in [ H ] \times [ S ] , \hat { \mathcal { A } }$ fails to identify the optimal action for the step-state pair $( h + 1 , i )$ with probability at least $1 / 3$ , which directly implies $\mathbb { E } _ { \star } [ \mathrm { e r r o r } _ { h , i } ] \ge 1 / 3$ for all $( h , i ) \in [ H ] \times [ S ]$ . As a result, we obtain a contraction and Claim 34 holds.
+
+Now, let us prove that for every $( h , i ) \in [ H ] \times [ S ]$ , $\mathbb { E } _ { \star } [ \mathrm { e r r o r } _ { h , i } ] \ge 1 / 6$ . WLOG, we assume $\hat { A }$ is deterministic and it runs for exactly $K = A B S H ^ { 2 } / ( 1 0 ^ { 3 } \epsilon ^ { 2 } )$ episodes. In the following, we consider a fixed $( h ^ { \prime } , i ^ { \prime } )$ pair. For technical reason, we define MDP $\mathcal { T } _ { - ( h ^ { \prime } , i ^ { \prime } ) } ( \boldsymbol { a } ^ { \star } , \boldsymbol { b } ^ { \star } )$ as below:
+
+• States, actions and transitions: same as $\mathcal { I } ( a ^ { \star } , b ^ { \star } )$ .
+
+• Rewards: there is no reward in the first step. For the remaining steps $h \in \{ 2 , \dots , H + 1 \}$ , if the agent takes action $( a , b )$ at state $s _ { i }$ in the $h ^ { \mathrm { t h } }$ step such that $( h - 1 , i ) \neq ( h ^ { \prime } , i ^ { \prime } )$ , it will receive a binary reward sampled from
+
+$$
+\operatorname { B e r n o u l l i } \Big ( \frac { 1 } { 2 } + ( 1 - 2 \cdot \mathbf { 1 } \{ a \neq a _ { h - 1 , i } ^ { \star } \& b = b _ { h - 1 , i } ^ { \star } \} ) \frac { \epsilon } { H } \Big ) ,
+$$
+
+otherwise it will receive a binary reward sampled from
+
+$$
+\operatorname { B e r n o u l l i } \Big ( \frac { 1 } { 2 } + ( 1 - 2 \cdot { \bf 1 } \{ b = b _ { h - 1 , i } ^ { \star } \} ) \frac { \epsilon } { H } \Big ) .
+$$
+
+Intuitively, $\mathcal { T } _ { - ( h ^ { \prime } , i ^ { \prime } ) } ( \boldsymbol { a } ^ { \star } , \boldsymbol { b } ^ { \star } )$ is the same as $\mathcal { I } ( a ^ { \star } , b ^ { \star } )$ except that for the max player, all its actions at state $s _ { i ^ { \prime } }$ in the $h ^ { \prime } ^ { \mathrm { t h } }$ step are equivalent. In other words, $\mathcal { T } _ { - ( h ^ { \prime } , i ^ { \prime } ) } ( \boldsymbol { a } ^ { \star } , \boldsymbol { b } ^ { \star } )$ is independent of $\pmb { a } _ { h ^ { \prime } , i ^ { \prime } } ^ { \star }$ . To proceed, we need to define the following notations: denote by $n ( a , b )$ the number of times $\hat { A }$ picks action $( a , b )$ at state $s _ { i ^ { \prime } }$ in the $\left( h ^ { \prime } + 1 \right) ^ { \mathrm { t h } }$ step; denote by $\mathbb { P } ( \cdot \mid \mathcal { I } ( a ^ { \star } , b ^ { \star } ) ) ( \mathbb { E } [ \cdot \mid \mathcal { I } ( a ^ { \star } , b ^ { \star } ) ] )$ the probability (expectation) induced by running algorithm $\hat { A }$ on $\mathcal { I } ( a ^ { \star } , b ^ { \star } )$ ; similarly, we define $\mathbb { P } ( \cdot \mid$ $\mathcal { T } _ { - ( h ^ { \prime } , i ^ { \prime } ) } ( { \pmb a } ^ { \star } , { \pmb b } ^ { \star } ) ) \ ( \mathbb { E } [ \cdot \ \vert \ { \mathcal { T } } _ { - ( h ^ { \prime } , i ^ { \prime } ) } ( { \pmb a } ^ { \star } , { \pmb b } ^ { \star } ) ] )$ ; also recall that we denote by $\mathbb { P } _ { \star }$ $( \mathbb { E } _ { \star } )$ the probability (expectation) induced by picking $\mathcal { I } ( a ^ { \star } , b ^ { \star } )$ uniformly at random from $\Im ( \epsilon )$ and running $\hat { A }$ on $\mathcal { I }$ ; denote by $L$ the whole trajectory of states, actions and rewards produced by algorithm $\hat { A }$ in $N$ episodes; with slight abuse of notation, denote by $\hat { \mathcal { A } } ( L )$ the guess of $\hat { A }$ for $\pmb { a } _ { h ^ { \prime } , i ^ { \prime } } ^ { \star }$ based on $L$ .
+
+First, note that for any $( a , b ) ~ \in ~ [ A ] \times [ B ] , \mathbb { E } [ n ( s , a ) ~ | ~ \mathcal { J } _ { - ( h ^ { \prime } , i ^ { \prime } ) } ( { \pmb a } ^ { \star } , { \pmb b } ^ { \star } ) ]$ is independent of $( h ^ { \prime } , i ^ { \prime } , a ^ { \star } , b ^ { \star } )$ because the agent cannot observe any reward when interacting with the environment and the transition dynamics of different $\mathcal { T } _ { - ( h ^ { \prime } , i ^ { \prime } ) } ( \boldsymbol { a } ^ { \star } , \boldsymbol { b } ^ { \star } )$ ’s are the same. For simplicity of notation, we denote this expectation by $m ( a , b )$ . Note that $\begin{array} { r } { \sum _ { a , b } m ( a , b ) = K / S } \end{array}$ because the agent always reach state $s _ { i ^ { \prime } }$ in the $\left( h ^ { \prime } + 1 \right) ^ { \mathrm { t h } }$ step with probability $1 / S$ regardless of the actions taken.
+
+We have
+
+$$
+\begin{array} { r l } & { \qquad \quad - [ - \sum _ { j \in \mathbb { R } _ { + } } ( a _ { j } ^ { \top } - \sum _ { j \in \mathbb { R } _ { + } } ( a _ { j } ^ { \top } - \sum _ { j } - \lfloor ( j \cdot \theta ) ^ { \top } + k _ { j } ^ { \top } ) \lfloor \frac { 1 } { 2 } \rfloor ) } \\ { = } & { \frac { 1 } { 3 } ( \underbrace { \sin ^ { 2 } ( \theta ) } _ { \mathrm { { \mathbb { R } _ { + } } } } \cos ( \theta ) ) \cos ( \theta ) } \\ & { \qquad \quad \sin ^ { 2 } ( \theta ) } \\ & { \le } & { \frac { 1 } { 3 } ( \underbrace { \sin ^ { 2 } ( \theta ) } _ { \mathrm { { \mathbb { R } _ { + } } } } \cos ( \theta ) \cos ( \theta ) - \sin ^ { 2 } ( \theta ) ( \theta ) \cos ( \theta ) ) } \\ & { \qquad \quad - \frac { 1 } { 3 } ( \sin ^ { 2 } ( \theta ) \cos ( \theta ) ) } \\ & { = } & { \frac { 1 } { 3 } ( \frac { 1 } { 3 } ( \sin ^ { 2 } ( \theta ) - \sin ^ { 2 } ( \theta ) ) ) \cos ( \theta ) } \\ & { \qquad \cos ( \theta ) \cos ( \theta ) } \\ & { = } & { \frac { 1 } { 3 } ( \frac { 1 } { 3 } ( \sin ^ { 2 } ( \theta ) \cos ( \theta ) ) ) \cos ( \theta ) \sin ^ { 2 } ( \theta ) \sin ( \theta ) \sin ^ { 2 } ( \theta ) \sin ^ { 3 } ( \theta ) } \\ & { \qquad \quad \sin ^ { 2 } ( \theta ) } \\ & { = } & { \frac { 1 } { 3 } ( \frac { 1 } { 3 } ( \sin ^ { 2 } ( \theta ) \cos ( \theta ) ) \cos ( \theta ) \sin ^ { 2 } ( \theta ) + \frac { 1 } { 3 } ( \sin ^ { 2 } ( \theta ) ( \theta ) \sin ^ { 2 } ( \theta ) ) } \\ & { \qquad \quad \sin ^ { 2 } ( \theta ) \cos ( \theta ) } \\ & { \qquad \quad \sin ^ { 2 } ( \theta ) \cos ( \theta ) \sin ^ { 2 } ( \theta ) \sin ^ { 2 } ( \theta ) \sin ^ { 2 } \theta ) } \\ & \qquad \quad \sin ^ { 2 } ( \theta ) \cos \end{array}
+$$
+
+Plugging in $K = S A B H ^ { 2 } / ( 1 0 ^ { 4 } \epsilon ^ { 2 } )$ completes the proof.
+
+# H PROOF FOR APPENDIX C – MULTI-PLAYER GENERAL-SUM MARKOV GAMES
+
+# H.1 PROOF OF THEOREM 15
+
+# H.1.1 NE VERSION
+
+In this section, we prove Theorem 15 (NE version). As before, we begin with proving the optimistic estimations are indeed upper bounds of corresponding value and Q-value functions.
+
+Lemma 35. With probability $1 - p ,$ , for any $( s , a , h , k , i )$ :
+
+$$
+\begin{array} { r l } & { \overline { { Q } } _ { h , i } ^ { k } \left( \boldsymbol { s } , \boldsymbol { a } \right) \geq Q _ { h , i } ^ { \dag , \pi _ { - i } ^ { k } } \left( \boldsymbol { s } , \boldsymbol { a } \right) , \underline { { Q } } _ { h , i } ^ { k } \left( \boldsymbol { s } , \boldsymbol { a } \right) \leq Q _ { h , i } ^ { \pi ^ { k } } \left( \boldsymbol { s } , \boldsymbol { a } \right) , } \\ & { \qquad \overline { { V } } _ { h , i } ^ { k } \left( \boldsymbol { s } \right) \geq V _ { h , i } ^ { \dag , \pi _ { - i } ^ { k } } \left( \boldsymbol { s } \right) , \underline { { V } } _ { h , i } ^ { k } \left( \boldsymbol { s } \right) \leq V _ { h , i } ^ { \pi ^ { k } } \left( \boldsymbol { s } \right) . } \end{array}
+$$
+
+Proof. know a d - $k$ , we step, $h = H + 1$ to um $h = 1$ . For base case, weequality (40) holds $( H + 1 )$ $V _ { H + 1 , i } ^ { k } \left( s \right) = \overline { { { V } } } _ { H + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s \right) = 0$ $( h + 1 )$ -th step, for the $h$ -th step, by definition of the $Q$ functions,
+
+$$
+\begin{array} { r l } & { \overline { { Q } } _ { h , i } ^ { k } \left( s , \boldsymbol { a } \right) - Q _ { h , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s , \boldsymbol { a } \right) = \left[ \widehat { \mathbb { P } } _ { h } ^ { k } \overline { { V } } _ { h + 1 , i } ^ { k } \right] \left( s , \boldsymbol { a } \right) - \left[ \mathbb { P } _ { h } V _ { h + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \right] \left( s , \boldsymbol { a } \right) + \beta _ { t } } \\ & { \qquad = \underbrace { \widehat { \mathbb { P } } _ { h } ^ { k } \left( \overline { { V } } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \right) \left( s , \boldsymbol { a } \right) } _ { ( A ) } + \underbrace { \left( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } \right) V _ { h + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s , \boldsymbol { a } \right) } _ { ( B ) } + \beta _ { t } . } \end{array}
+$$
+
+By induction hypothesis, for any $s ^ { \prime }$ , $\left( \overline { { V } } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \right) ( s ^ { \prime } ) \geq 0$ , and thus $( A ) \geq 0$ . By uniform concentration (e.g., Lemma 12 in Bai & Jin (2020)), $( B ) \leq C \sqrt { S H ^ { 2 } \iota / N _ { h } ^ { k } ( s , \pmb { a } ) } = \beta _ { t }$ . Putting everything together we have $Q _ { h , i } ^ { k } \left( s , { \pmb a } \right) - Q _ { h , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s , { \pmb a } \right) \geq 0$ . The second inequality can be proved similarly.
+
+Now assume inequality (39) holds for the $h$ -th step, by definition of value functions and Nash equilibrium,
+
+$$
+\overline { { V } } _ { h , i } ^ { k } \left( s \right) = { \mathbb D } _ { \pi ^ { k } } \overline { { Q } } _ { h , i } ^ { k } \left( s \right) = \operatorname* { m a x } _ { \mu } { \mathbb D } _ { \mu \times \pi _ { - i } ^ { k } } \overline { { Q } } _ { h , i } ^ { k } \left( s \right) .
+$$
+
+By Bellman equation,
+
+$$
+V _ { h , i } ^ { \dag , \pi _ { - i } ^ { k } } \left( s \right) = \operatorname* { m a x } _ { \mu } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } Q _ { h , i } ^ { \dag , \pi _ { - i } ^ { k } } \left( s \right) .
+$$
+
+Since by induction hypothesis, for any $( s , \pmb { a } ) , \overline { { Q } } _ { h , i } ^ { k } \left( s , \pmb { a } \right) \geq Q _ { h , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s , \pmb { a } \right)$ . As a result, we also have $\overline { { V } } _ { h , i } ^ { k } \left( s \right) \geq V _ { h , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s \right)$ , which is exactly inequality (40) for the $h$ -th step. The second inequality can be proved similarly.
+
+Proof of Theorem 15. Let us focus on the $i$ -th player and ignore the subscript when there is no confusion. To bound
+
+$$
+\operatorname* { m a x } _ { i } \left( V _ { 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } - V _ { 1 , i } ^ { \pi ^ { k } } \right) \left( s _ { h } ^ { k } \right) \leq \operatorname* { m a x } _ { i } \left( \overline { { V } } _ { 1 , i } ^ { k } - \underline { { V } } _ { 1 , i } ^ { k } \right) \left( s _ { h } ^ { k } \right) ,
+$$
+
+we notice the following propogation:
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { ( \overline { { Q } } _ { h , i } ^ { k } - \underline { { Q } } _ { h , i } ^ { k } ) ( s , { \mathbf { a } } ) \leq \widehat { \mathbb { P } } _ { h } ^ { k } ( \overline { { V } } _ { h + 1 , i } ^ { k } - \underline { { V } } _ { h + 1 , i } ^ { k } ) ( s , { \mathbf { a } } ) + 2 \beta _ { h } ^ { k } ( s , { \mathbf { a } } ) , } \\ { ( \overline { { V } } _ { h , i } - \underline { { V } } _ { h , i } ) ( s ) = [ \mathbb { D } _ { \pi _ { h } } ( \overline { { Q } } _ { h , i } ^ { k } - \underline { { Q } } _ { h , i } ^ { k } ) ] ( s ) . } \end{array} \right. } \end{array}
+$$
+
+We can define $\widetilde { Q } _ { h } ^ { k }$ and $\widetilde { V } _ { h } ^ { k }$ recursively by $\widetilde V _ { H + 1 } ^ { k } = 0$ and
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { \widetilde { Q } _ { h } ^ { k } ( s , \pmb { a } ) = \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ( s , \pmb { a } ) + 2 \beta _ { h } ^ { k } ( s , \pmb { a } ) , } \\ { \widetilde { V } _ { h } ^ { k } ( s ) = [ \mathbb { D } _ { \pi _ { h } } \widetilde { Q } _ { h } ^ { k } ] ( s ) . } \end{array} \right. } \end{array}
+$$
+
+Then we can prove inductively that for any $k , h , s$ and $\textbf { \em a }$ we have
+
+$$
+\left\{ \begin{array} { l l } { \operatorname* { m a x } _ { i } ( \overline { { Q } } _ { h , i } ^ { k } - \underline { { Q } } _ { h , i } ^ { k } ) ( s , \pmb { a } ) \leq \widetilde { Q } _ { h } ^ { k } ( s , \pmb { a } ) , } \\ { \operatorname* { m a x } _ { i } ( \overline { { V } } _ { h , i } - \underline { { V } } _ { h , i } ) ( s ) \leq \widetilde { V } _ { h } ^ { k } ( s ) . } \end{array} \right.
+$$
+
+Thus we only need to bound $\textstyle \sum _ { k = 1 } ^ { K } { \widetilde { V } } _ { 1 } ^ { k } ( s )$ . Define the shorthand notation
+
+$$
+\left\{ \begin{array} { l l } { \beta _ { h } ^ { k } : = \beta _ { h } ^ { k } ( s _ { h } ^ { k } , \boldsymbol { a } _ { h } ^ { k } ) , } \\ { \Delta _ { h } ^ { k } : = \widetilde { V } _ { h } ^ { k } ( s _ { h } ^ { k } ) , } \\ { \zeta _ { h } ^ { k } : = \mathbb { D } _ { \pi ^ { k } } \widetilde { Q } _ { h } ^ { k } \left( s _ { h } ^ { k } \right) - \widetilde { Q } _ { h } ^ { k } ( s _ { h } ^ { k } , \boldsymbol { a } _ { h } ^ { k } ) , } \\ { \xi _ { h } ^ { k } : = \mathbb { P } _ { h } \widetilde { V } _ { h } ^ { k } ( s _ { h } ^ { k } , \boldsymbol { a } _ { h } ^ { k } ) - \Delta _ { h + 1 } ^ { k } . } \end{array} \right.
+$$
+
+We can check $\zeta _ { h } ^ { k }$ and $\xi _ { h } ^ { k }$ are martingale difference sequences. As a result,
+
+$$
+\begin{array} { r l } & { \Delta _ { h } ^ { k } = \mathbb { D } _ { \pi ^ { k } } \widetilde { Q } _ { h } ^ { k } \left( s _ { h } ^ { k } \right) } \\ & { \quad = \zeta _ { h } ^ { k } + \widetilde { Q } _ { h } ^ { k } \left( s _ { h } ^ { k } , \pmb { a } _ { h } ^ { k } \right) } \\ & { \quad = \zeta _ { h } ^ { k } + 2 \beta _ { h } ^ { k } + \mathbb { \widehat { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } \left( s _ { h } ^ { k } , \pmb { a } _ { h } ^ { k } \right) } \\ & { \quad \le \zeta _ { h } ^ { k } + 3 \beta _ { h } ^ { k } + \mathbb { P } _ { h } \widetilde { V } _ { h + 1 } ^ { k } \left( s _ { h } ^ { k } , \pmb { a } _ { h } ^ { k } \right) } \\ & { \quad = \zeta _ { h } ^ { k } + 3 \beta _ { h } ^ { k } + \xi _ { h } ^ { k } + \Delta _ { h + 1 } ^ { k } . } \end{array}
+$$
+
+Recursing this argument for $h \in [ H ]$ and taking the sum,
+
+$$
+\sum _ { k = 1 } ^ { K } \Delta _ { 1 } ^ { k } \leq \sum _ { k = 1 } ^ { K } \left( \zeta _ { h } ^ { k } + 3 \beta _ { h } ^ { k } + \xi _ { h } ^ { k } \right) \leq O \left( S \sqrt { H ^ { 3 } T \iota \prod _ { i = 1 } ^ { M } A _ { i } } \right) .
+$$
+
+# H.1.2 CCE VERSION
+
+The proof is very similar to the NE version. Specifically, the only part that uses the properties of NE there is Lemma 35. We prove a counterpart here.
+
+Lemma 36. With probability $1 - p ,$ , for any $( s , a , h , k , i )$ :
+
+$$
+\begin{array} { r l } & { \overline { { Q } } _ { h , i } ^ { k } \left( \boldsymbol { s } , \boldsymbol { a } \right) \geq Q _ { h , i } ^ { \dag , \pi _ { - i } ^ { k } } \left( \boldsymbol { s } , \boldsymbol { a } \right) , \underline { { Q } } _ { h , i } ^ { k } \left( \boldsymbol { s } , \boldsymbol { a } \right) \leq Q _ { h , i } ^ { \pi ^ { k } } \left( \boldsymbol { s } , \boldsymbol { a } \right) , } \\ & { \qquad \overline { { V } } _ { h , i } ^ { k } \left( \boldsymbol { s } \right) \geq V _ { h , i } ^ { \dag , \pi _ { - i } ^ { k } } \left( \boldsymbol { s } \right) , \underline { { V } } _ { h , i } ^ { k } \left( \boldsymbol { s } \right) \leq V _ { h , i } ^ { \pi ^ { k } } \left( \boldsymbol { s } \right) . } \end{array}
+$$
+
+Proof. know a d - $k$ , we step, $h = H + 1$ to um $h = 1$ . For base case, weequality (40) holds $( H + 1 )$ $V _ { H + 1 , i } ^ { k } \left( s \right) = \overline { { { V } } } _ { H + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s \right) = 0$ $( h + 1 )$ $h$ $Q$
+
+$$
+\begin{array} { r l } & { \overline { { Q } } _ { h , i } ^ { k } \left( s , \boldsymbol { a } \right) - Q _ { h , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s , \boldsymbol { a } \right) = \left[ \widehat { \mathbb { P } } _ { h } ^ { k } \overline { { V } } _ { h + 1 , i } ^ { k } \right] \left( s , \boldsymbol { a } \right) - \left[ \mathbb { P } _ { h } V _ { h + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \right] \left( s , \boldsymbol { a } \right) + \beta _ { t } } \\ & { \qquad = \underbrace { \widehat { \mathbb { P } } _ { h } ^ { k } \left( \overline { { V } } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \right) \left( s , \boldsymbol { a } \right) } _ { ( A ) } + \underbrace { \left( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } \right) V _ { h + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s , \boldsymbol { a } \right) } _ { ( B ) } + \beta _ { t } . } \end{array}
+$$
+
+By induction hypothesis, for any $s ^ { \prime }$ , $\left( \overline { { V } } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \dagger , \pi _ { - i } ^ { k } } \right) ( s ^ { \prime } ) \geq 0$ , and thus $( A ) \geq 0$ . By uniform concentration, $( B ) \leq C \sqrt { S H ^ { 2 } \iota / N _ { h } ^ { k } ( s , \pmb { a } ) } = \beta _ { t }$ . Putting everything together we have $Q _ { h , i } ^ { k } \left( s , \pmb { a } \right) -$ $Q _ { h , i } ^ { \dag , \pi _ { - i } ^ { k } } \left( s , { \pmb a } \right) \geq 0$ . The second inequality can be proved similarly.
+
+Now assume inequality (45) holds for the $h$ -th step, by definition of value functions and CCE,
+
+$$
+\overline { { V } } _ { h , i } ^ { k } \left( s \right) = { \mathbb D } _ { \pi ^ { k } } \overline { { Q } } _ { h , i } ^ { k } \left( s \right) = \operatorname* { m a x } _ { \mu } { \mathbb D } _ { \mu \times \pi _ { - i } ^ { k } } \overline { { Q } } _ { h , i } ^ { k } \left( s \right) .
+$$
+
+By Bellman equation,
+
+$$
+V _ { h , i } ^ { \dag , \pi _ { - i } ^ { k } } \left( s \right) = \operatorname* { m a x } _ { \mu } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } Q _ { h , i } ^ { \dag , \pi _ { - i } ^ { k } } \left( s \right) .
+$$
+
+Since by induction hypothesis, for any $( s , \pmb { a } ) , \overline { { Q } } _ { h , i } ^ { k } \left( s , \pmb { a } \right) \geq Q _ { h , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s , \pmb { a } \right)$ . As a result, we also have $\overline { { V } } _ { h , i } ^ { k } \left( s \right) \geq V _ { h , i } ^ { \dagger , \pi _ { - i } ^ { k } } \left( s \right)$ , which is exactly inequality (40) for the $h$ -th step. The second inequality can be proved similarly. □
+
+# H.1.3 CE VERSION
+
+The proof is very similar to the NE version. Specifically, the only part that uses the properties of NE there is Lemma 35. We prove a counterpart here.
+
+Lemma 37. With probability $1 - p ,$ for any $( s , a , h , k , i )$ :
+
+$$
+\begin{array} { r l } & { \overline { { Q } } _ { h , i } ^ { k } \left( s , \pmb { a } \right) \geq \operatorname* { m a x } _ { \phi } { Q } _ { h , i } ^ { \phi \diamond \pi ^ { k } } \left( s , \pmb { a } \right) , \underline { { Q } } _ { h , i } ^ { k } \left( s , \pmb { a } \right) \leq { Q } _ { h , i } ^ { \pi ^ { k } } \left( s , \pmb { a } \right) , } \\ & { } \\ & { \overline { { V } } _ { h , i } ^ { k } \left( s \right) \geq \operatorname* { m a x } _ { \phi } { V } _ { h , i } ^ { \phi \diamond \pi ^ { k } } \left( s \right) , \underline { { V } } _ { h , i } ^ { k } \left( s \right) \leq { V } _ { h , i } ^ { \pi ^ { k } } \left( s \right) . } \end{array}
+$$
+
+Proof. For each fixed $k$ , we prove this by induction from $h = H + 1$ to $h = 1$ . For base case, we know at the $( H + 1 )$ -th step, $\overline { { { V } } } _ { H + 1 , i } ^ { k } \left( s \right) = \underset { \phi } { \operatorname* { m a x } } \overline { { { V } } } _ { H + 1 , i } ^ { \phi \diamond \pi ^ { k } } \left( s \right) = 0$ . Now, assume the inequality (40) holds for the $( h + 1 )$ -th step, for the $h$ -th step, by definition of the $Q$ functions,
+
+$$
+\begin{array} { r l } & { \overline { { Q } } _ { h , i } ^ { k } \left( s , \mathfrak { a } \right) - \underset { \phi } { \operatorname* { m a x } } Q _ { h , i } ^ { \phi \phi \pi ^ { k } } \left( s , \mathfrak { a } \right) } \\ & { = \left[ \widehat { { \mathbb { P } } } _ { h } ^ { k } \overline { { V } } _ { h + 1 , i } ^ { k } \right] \left( s , \mathfrak { a } \right) - \left[ \mathbb { P } _ { h } \underset { \phi } { \operatorname* { m a x } } V _ { h + 1 , i } ^ { \phi \varphi \pi ^ { k } } \right] \left( s , \mathfrak { a } \right) + \beta _ { t } } \\ & { = \underbrace { \widehat { { \mathbb { P } } } _ { h } ^ { k } \left( \overline { { V } } _ { h + 1 , i } ^ { k } - \underset { \phi } { \operatorname* { m a x } } V _ { h + 1 , i } ^ { \phi \varphi \pi ^ { k } } \right) \left( s , \mathfrak { a } \right) } _ { ( A ) } + \underbrace { \left( \widehat { { \mathbb { P } } } _ { h } ^ { k } - \mathbb { P } _ { h } \right) \underset { \phi } { \operatorname* { m a x } } V _ { h + 1 , i } ^ { \phi \varphi \pi ^ { k } } \left( s , \mathfrak { a } \right) } _ { ( B ) } + \beta _ { t } . } \end{array}
+$$
+
+By induction hypothesis, for any $s ^ { \prime }$ , $\left( \overline { { V } } _ { h + 1 , i } ^ { k } - \underset { \phi } { \operatorname* { m a x } } V _ { h + 1 , i } ^ { \phi \diamond \pi ^ { k } } \right) ( s ^ { \prime } ) \geq 0$ , and thus $( A ) ~ \geq ~ 0$ . By uniform concentration, $( B ) \leq C \sqrt { S H ^ { 2 } \iota / N _ { h } ^ { k } ( s , \pmb { a } ) } = \beta _ { t }$ . Putting everything together we have $\overline { { Q } } _ { h , i } ^ { k } \left( s , { \pmb a } \right) - \operatorname* { m a x } _ { \phi } Q _ { h , i } ^ { \phi \diamond \pi ^ { k } } \left( s , { \pmb a } \right) \geq 0$ . The second inequality can be proved similarly.
+
+Now assume inequality (47) holds for the $h$ -th step, by definition of value functions and CE,
+
+$$
+\overline { { V } } _ { h , i } ^ { k } \left( s \right) = { \mathbb D } _ { \pi ^ { k } } \overline { { Q } } _ { h , i } ^ { k } \left( s \right) = \operatorname* { m a x } _ { \phi } { \mathbb D } _ { \phi \circ \pi ^ { k } } \overline { { Q } } _ { h , i } ^ { k } \left( s \right) .
+$$
+
+By Bellman equation,
+
+$$
+\displaystyle \operatorname* { m a x } _ { \phi } V _ { h , i } ^ { \phi \diamond \pi ^ { k } } \left( s \right) = \displaystyle \operatorname* { m a x } _ { \phi } \mathbb { D } _ { \phi \diamond \pi ^ { k } } \displaystyle \operatorname* { m a x } _ { \phi ^ { \prime } } Q _ { h , i } ^ { \phi ^ { \prime } \diamond \pi ^ { k } } \left( s \right) .
+$$
+
+Since by induction hypothesis, for any ${ \mathfrak { s } } , \pmb { a } ) , \overline { { Q } } _ { h , i } ^ { k } \left( { \mathfrak { s } } , \pmb { a } \right) \geq \operatorname* { m a x } _ { \phi } { Q _ { h , i } ^ { \phi \diamond \pi ^ { k } } \left( { \mathfrak { s } } , \pmb { a } \right) }$ . As a result, we also have V kh,i (s) ≥ maxV φπkh,i , which is exactly inequality (40) for the $h$ -th step. The second inequality can be proved similarly. □
+
+# H.2 PROOF OF THEOREM 16
+
+In this section, we prove each theorem for the single reward function case, i.e., $N = 1$ . The proof for the case of multiple reward functions $N > 1 \AA$ ) simply follows from taking a union bound, that is, replacing the failure probability $p$ by $N p$ .
+
+# H.2.1 NE VERSION
+
+Let $( \mu ^ { k } , \nu ^ { k } )$ be an arbitrary Nash-equilibrium policy of $\widehat { \mathcal { M } } ^ { k } : = ( \widehat { \mathbb { P } } ^ { k } , \widehat { r } ^ { k } )$ , where $\widehat { \mathbb { P } } ^ { k }$ and $\widehat { r } ^ { k }$ are our bempirical estimate of the transition and the reward at the beginning of the $k$ b’th episode in Algorithm 4. Given an arbitrary Nash equilibrium $\pi ^ { k }$ of ${ \widehat { \mathcal { M } } } ^ { k }$ , we use $\widehat { Q } _ { h , i } ^ { k }$ and $\widehat { V } _ { h , i } ^ { k }$ to denote its value functions of the $\because$ ’th player at the $h$ ’th step in ${ \widehat { \mathcal { M } } } ^ { k }$ .
+
+We prove the following two lemmas, which together imply the conclusion about Nash equilibriums in Theorem 16 as in the proof of Theorem 5.
+
+Lemma 38. With probability $1 - p ,$ , for any $( h , s , \pmb { a } , i , k )$ , we have
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { | \widehat { Q } _ { h , i } ^ { k } ( s , \pmb { a } ) - Q _ { h , i } ^ { \pi ^ { k } } ( s , \pmb { a } ) | \leq \widetilde { Q } _ { h } ^ { k } ( s , \pmb { a } ) , } \\ { | \widehat { V } _ { h , i } ^ { k } ( s ) - V _ { h , i } ^ { \pi ^ { k } } ( s ) | \leq \widetilde { V } _ { h } ^ { k } ( s ) . } \end{array} \right. } \end{array}
+$$
+
+Proof. For eachwe know at the $k$ $h = H + 1$ $h = 1$ . For base case,Now, assume the $( H + 1 )$ $\widehat { V } _ { H + 1 , i } ^ { k } = V _ { H + 1 , i } ^ { \pi ^ { k } } = \widehat { Q } _ { H + 1 , i } ^ { k } = Q _ { H + 1 , i } ^ { \pi ^ { k } } = 0$ $( h + 1 ) ^ { \ }$ $h$ $Q$
+
+$$
+\begin{array} { r l } & { \quad \left| \widehat { Q } _ { h , i } ^ { k } \left( s , a \right) - Q _ { h , i } ^ { \pi ^ { k } } \left( s , a \right) \right| } \\ & { \leq \left| \left[ \widehat { \mathbb { P } } _ { h } ^ { k } \widehat { V } _ { h + 1 , i } ^ { k } \right] \left( s , a \right) - \left[ \mathbb { P } _ { h } V _ { h + 1 , i } ^ { \pi ^ { k } } \right] \left( s , a \right) \right| + \left| r _ { h } ( s , a ) - \widehat { r } _ { h } ^ { k } ( s , a ) \right| } \\ & { \leq \left| \underbrace { \widehat { \mathbb { P } } _ { h } ^ { k } \left( \widehat { V } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \pi ^ { k } } \right) \left( s , a \right) } _ { ( A ) } \right| + \underbrace { \left| \left( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } \right) V _ { h + 1 , i } ^ { \pi ^ { k } } \left( s , a \right) \right| + \left| r _ { h } ( s , a ) - \widehat { r } _ { h } ^ { k } ( s , a ) \right| } _ { ( B ) } } \end{array}
+$$
+
+By the induction hypothesis,
+
+$$
+\begin{array} { r } { ( A ) \leq \widehat { \mathbb { P } } _ { h } ^ { k } \left| \widehat { V } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \pi ^ { k } } \right| ( s , \pmb { a } ) \leq ( \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ) ( s , \pmb { a } ) . } \end{array}
+$$
+
+By uniform concentration (e.g., Lemma 12 in Bai & Jin (2020)), $( B ) \leq \sqrt { S H ^ { 2 } \iota / N _ { h } ^ { k } ( s , \pmb { a } ) } = \beta _ { t } .$ Putting everything together we have
+
+$$
+\begin{array} { r } { \left| Q _ { h , i } ^ { \pi ^ { k } } \left( s , \pmb { a } \right) - \widehat { Q } _ { h , i } ^ { k } \left( s , \pmb { a } \right) \right| \leq \operatorname* { m i n } \left\{ ( \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ) ( s , \pmb { a } ) + \beta _ { t } , H \right\} = \widetilde { Q } _ { h } ^ { k } ( s , \pmb { a } ) , } \end{array}
+$$
+
+which proves the first inequality in (49). The inequality for $V$ functions follows directly by noting that the value functions are computed using the same policy $\pi ^ { k }$ . □
+
+Lemma 39. With probability $1 - p ,$ , for any $( h , s , \pmb { a } , i , k )$ , we have
+
+$$
+\begin{array} { r } { \left\{ \begin{array} { l l } { | \widehat { Q } _ { h , i } ^ { k } ( s , \pmb { a } ) - Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } ( s , \pmb { a } ) | \leq \widetilde { Q } _ { h } ^ { k } ( s , \pmb { a } ) , } \\ { | \widehat { V } _ { h , i } ^ { k } ( s ) - V _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } ( s ) | \leq \widetilde { V } _ { h } ^ { k } ( s ) . } \end{array} \right. } \end{array}
+$$
+
+Proof. For eachwe know at the $k$ we pro-th step, $h = H + 1$ $h = 1$ . For base case,Now, assume the $( H + 1 )$ $\widehat { V } _ { H + 1 , i } ^ { k } = V _ { H + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } = \widehat { Q } _ { H + 1 , i } ^ { k } = Q _ { H + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } = 0 .$ conclusion holds for the ’th step, for the $h$ ’th step, by definition of the $Q$ functions,
+
+$$
+\begin{array} { r l } & { \quad \left| \widehat { Q } _ { h , i } ^ { k } \left( s , a \right) - Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dagger } \left( s , a \right) \right| } \\ & { = \left| \left[ \widehat { \mathbb { P } } _ { h } ^ { k } \widehat { V } _ { h + 1 , i } ^ { k } \right] \left( s , a \right) - \left[ \mathbb { P } _ { h } V _ { h + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } \right] \left( s , a \right) \right| + \left| r _ { h } ( s , a ) - \widehat { r } _ { h } ^ { k } ( s , a ) \right| } \\ & { \leq \left| \widehat { \mathbb { P } } _ { h } ^ { k } \left( \widehat { V } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } \right) \left( s , a \right) \right| + \left| \left( \widehat { \mathbb { P } } _ { h } ^ { k } - \mathbb { P } _ { h } \right) V _ { h + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } \left( s , a \right) \right| + \left| r _ { h } ( s , a ) - \widehat { r } _ { h } ^ { k } ( s , a ) \right| } \\ & { \quad \qquad ( A ) } \end{array}
+$$
+
+By the induction hypothesis,
+
+$$
+\begin{array} { r } { ( A ) \leq \widehat { \mathbb { P } } _ { h } ^ { k } \left| \widehat { V } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } \right| ( s , \pmb { a } ) \leq ( \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ) ( s , \pmb { a } ) . } \end{array}
+$$
+
+By uniform concentration, $( B ) \leq \sqrt { S H ^ { 2 } \iota / N _ { h } ^ { k } ( s , \pmb { a } ) } = \beta _ { t }$ . Putting everything together we have
+
+$$
+\left| Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s , \pmb { a } \right) - \widehat { Q } _ { h , i } ^ { k } \left( s , \pmb { a } \right) \right| \leq \operatorname* { m i n } \left\{ ( \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ) ( s , \pmb { a } ) + \beta _ { t } , H \right\} = \widetilde { Q } _ { h } ^ { k } ( s , \pmb { a } ) ,
+$$
+
+which proves the first inequality in (50). It remains to show the inequality for $V$ functions also hold in $h$ ’th step.
+
+Since $\pi ^ { k }$ is a Nash-equilibrium policy, we have
+
+$$
+\widehat V _ { h , i } ^ { k } \left( s \right) = \operatorname* { m a x } _ { \mu } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } \widehat Q _ { h , i } ^ { k } \left( s \right) .
+$$
+
+By Bellman equation,
+
+$$
+V _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s \right) = \operatorname* { m a x } _ { \mu } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s \right) .
+$$
+
+Combining the two equations above, and utilizing the bound we just proved for $Q$ functions, we obtain
+
+$$
+\begin{array} { r l } & { \left| \widehat { V } _ { h , i } ^ { k } \left( s \right) - V _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s \right) \right| \leq \left| \underset { \mu } { \operatorname* { m a x } } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } \widehat { Q } _ { h , i } ^ { k } \left( s \right) - \underset { \mu } { \operatorname* { m a x } } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s \right) \right| } \\ & { \qquad \leq \underset { a } { \operatorname* { m a x } } \widetilde { Q } _ { h } ^ { k } ( s , a ) = \widetilde { V } _ { h } ^ { k } ( s ) , } \end{array}
+$$
+
+which completes the whole proof.
+
+# H.2.2 CCE VERSION
+
+The proof is almost the same as that for Nash equilibriums. We will reuse Lemma 38 and prove an analogue of Lemma 39. The conclusion for CCEs will follow directly by combining the two lemmas as in the proof of Theorem 5.
+
+Lemma 40. With probability $1 - p ,$ for any $( h , s , \pmb { a } , i , k )$ , we have
+
+$$
+\begin{array} { r } { \left\{ Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dagger } ( s , \pmb { a } ) - \widehat { Q } _ { h , i } ^ { k } ( s , \pmb { a } ) \leq \widetilde { Q } _ { h } ^ { k } ( s , \pmb { a } ) , \right. } \\ { \left. V _ { h , i } ^ { \pi _ { - i } ^ { k } , \dagger } ( s ) - \widehat { V } _ { h , i } ^ { k } ( s ) \leq \widetilde { V } _ { h } ^ { k } ( s ) . \right. } \end{array}
+$$
+
+Proof. For eachwe know at the $k$ we pro-th step, $h = H + 1$ $h = 1$ . For base case,Now, assume the $( H + 1 )$ $\widehat { V } _ { H + 1 , i } ^ { k } = V _ { H + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } = \widehat { Q } _ { H + 1 , i } ^ { k } = Q _ { H + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } = 0$
+
+conclusion holds for the $( h + 1 ) ^ { \dagger }$ th step, for the $h$ ’th step, by definition of the $Q$ functions,
+
+$$
+\begin{array} { r l } & { \quad Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dagger } \left( s , a \right) - \widehat { Q } _ { h , i } ^ { k } \left( s , a \right) } \\ & { \le \left[ \mathbb { P } _ { h } V _ { h + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } \right] \left( s , a \right) - \left[ \widehat { \mathbb { P } } _ { h } ^ { k } \widehat { V } _ { h + 1 , i } ^ { k } \right] \left( s , a \right) + \left| r _ { h } ( s , a ) - \widehat { r } _ { h } ^ { k } ( s , a ) \right| } \\ & { \le \underbrace { \widehat { \mathbb { P } } _ { h } ^ { k } \left( V _ { h + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } - \widehat { V } _ { h + 1 , i } ^ { k } \right) \left( s , a \right) } _ { ( A ) } + \underbrace { \left( \mathbb { P } _ { h } - \widehat { \mathbb { P } } _ { h } ^ { k } \right) V _ { h + 1 , i } ^ { \pi _ { - i } ^ { k } , \dagger } \left( s , a \right) + \left| r _ { h } ( s , a ) - \widehat { r } _ { h } ^ { k } ( s , a ) \right| } _ { ( B ) } . } \end{array}
+$$
+
+By the induction hypothesis, $( A ) \leq ( \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ) ( s , \pmb { a } )$ .
+
+By uniform concentration, $( B ) \leq \sqrt { S H ^ { 2 } \iota / N _ { h } ^ { k } ( s , \pmb { a } ) } = \beta _ { t }$ . Putting everything together we have
+
+$$
+\begin{array} { r } { Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s , \boldsymbol { a } \right) - \widehat { Q } _ { h , i } ^ { k } \left( s , \boldsymbol { a } \right) \leq \operatorname* { m i n } \left\{ ( \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ) ( s , \boldsymbol { a } ) + \beta _ { t } , H \right\} = \widetilde { Q } _ { h } ^ { k } ( s , \boldsymbol { a } ) , } \end{array}
+$$
+
+which proves the first inequality in (51). It remains to show the inequality for $V$ functions also hold in $h$ ’th step.
+
+Since $\pi ^ { k }$ is a CCE, we have
+
+$$
+\widehat V _ { h , i } ^ { k } \left( s \right) \geq \operatorname* { m a x } _ { \mu } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } \widehat Q _ { h , i } ^ { k } \left( s \right) .
+$$
+
+Observe that $V _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag }$ obeys the Bellman optimality equation, so we have
+
+$$
+V _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s \right) = \operatorname* { m a x } _ { \mu } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s \right) .
+$$
+
+Combining the two equations above, and utilizing the bound we just proved for $Q$ functions, we obtain
+
+$$
+\begin{array} { r l } & { V _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s \right) - \widehat { V } _ { h , i } ^ { k } \left( s \right) \leq \underset { \mu } { \operatorname* { m a x } } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } Q _ { h , i } ^ { \pi _ { - i } ^ { k } , \dag } \left( s \right) - \underset { \mu } { \operatorname* { m a x } } \mathbb { D } _ { \mu \times \pi _ { - i } ^ { k } } \widehat { Q } _ { h , i } ^ { k } \left( s \right) } \\ & { \qquad \leq \underset { a } { \operatorname* { m a x } } \widetilde { Q } _ { h } ^ { k } ( s , a ) = \widetilde { V } _ { h } ^ { k } ( s ) , } \end{array}
+$$
+
+which completes the whole proof.
+
+# H.2.3 CE VERSION
+
+The proof is almost the same as that for Nash equilibriums. We will reuse Lemma 38 and prove an analogue of Lemma 39. The conclusion for CEs will follow directly by combining the two lemmas as in the proof of Theorem 5.
+
+Lemma 41. With probability $1 - p ,$ for any $( h , s , \pmb { a } , i , k )$ and strategy modification $\phi$ for player $i$ , we have
+
+$$
+\begin{array} { r } { \left\{ Q _ { h , i } ^ { \phi \diamond \pi ^ { k } } ( s , \pmb { a } ) - \widehat { Q } _ { h , i } ^ { k } ( s , \pmb { a } ) \leq \widetilde { Q } _ { h } ^ { k } ( s , \pmb { a } ) , \right. } \\ { \left. V _ { h , i } ^ { \phi \diamond \pi ^ { k } } ( s ) - \widehat { V } _ { h , i } ^ { k } ( s ) \leq \widetilde { V } _ { h } ^ { k } ( s ) . \right. } \end{array}
+$$
+
+we know at the Proof. For each fixed $( H + 1 )$ $k$ , we prove this by induction from -th step, $\widehat { V } _ { H + 1 , i } ^ { k } \ : = V _ { H + 1 , i } ^ { \phi \diamond \pi ^ { k } } \ : = \widehat { Q } _ { H + 1 , i } ^ { k } \ : = Q _ { H + 1 , i } ^ { \phi \diamond \pi ^ { k } } \ : = \ : 0$ $h = H + 1$ to $h = 1$ . Now, assume the . For base case, conclusion holds for the $( h + 1 )$ ’th step, for the $h$ ’th step, following exactly the same argument as Lemma 40, we can show
+
+$$
+\begin{array} { r } { Q _ { h , i } ^ { \phi \circ \pi ^ { k } } ( s , a ) - \widehat { Q } _ { h , i } ^ { k } ( s , a ) \leq \operatorname* { m i n } \left\{ ( \widehat { \mathbb { P } } _ { h } ^ { k } \widetilde { V } _ { h + 1 } ^ { k } ) ( s , a ) + \beta _ { t } , H \right\} = \widetilde { Q } _ { h } ^ { k } ( s , a ) , } \end{array}
+$$
+
+which proves the first inequality in (52). It remains to show the inequality for $V$ functions also hold in $h$ ’th step.
+
+Since $\pi ^ { k }$ is a CE, we have
+
+$$
+\widehat { V } _ { h , i } ^ { k } \left( s \right) = \operatorname* { m a x } _ { \widetilde { \phi } _ { h , s } } \mathbb { D } _ { \widetilde { \phi } _ { h , s } \diamond \pi ^ { k } } \widehat { Q } _ { h , i } ^ { k } \left( s \right) ,
+$$
+
+where the maximum is take over all possible injective functions from $\mathbf { \mathcal { A } } _ { i }$ to itself.
+
+Observe that V φπk obeys the Bellman optimality equation, so we have
+
+$$
+V _ { h , i } ^ { \phi \diamond \pi ^ { k } } \left( s \right) = \operatorname* { m a x } _ { \widetilde { \phi } _ { h , s } } \mathbb { D } _ { \widetilde { \phi } _ { h , s } \diamond \pi ^ { k } } Q _ { h , i } ^ { \phi \diamond \pi ^ { k } } \left( s \right) .
+$$
+
+Combining the two equations above, and utilizing the bound we just proved for $Q$ functions, we obtain
+
+$$
+\begin{array} { r l } & { V _ { h , i } ^ { \phi \circ \pi ^ { k } } \left( s \right) - \widehat { V } _ { h , i } ^ { k } \left( s \right) = \underset { \bar { \phi } _ { h , s } } { \operatorname* { m a x } } \mathbb { D } _ { \widetilde { \phi } _ { h , s } \circ \pi ^ { k } } Q _ { h , i } ^ { \phi \circ \pi ^ { k } } \left( s \right) - \underset { \bar { \phi } _ { h , s } } { \operatorname* { m a x } } \mathbb { D } _ { \widetilde { \phi } _ { h , s } \circ \pi ^ { k } } \widehat { Q } _ { h , i } ^ { k } \left( s \right) } \\ & { \qquad \leq \underset { a } { \operatorname* { m a x } } \widetilde { Q } _ { h } ^ { k } ( s , a ) = \widetilde { V } _ { h } ^ { k } ( s ) , } \end{array}
+$$
+
+which completes the whole proof.
\ No newline at end of file
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new file mode 100644
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+ "type": "text",
+ "text": "A SHARP ANALYSIS OF MODEL-BASED REINFORCEMENT LEARNING WITH SELF-PLAY ",
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+ "text": "Anonymous authors Paper under double-blind review ",
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+ "type": "text",
+ "text": "ABSTRACT ",
+ "text_level": 1,
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+ "text": "Model-based algorithms—algorithms that explore the environment through building and utilizing an estimated model—are widely used in reinforcement learning practice and theoretically shown to achieve optimal sample efficiency for singleagent reinforcement learning in Markov Decision Processes (MDPs). However, for multi-agent reinforcement learning in Markov games, the current best known sample complexity for model-based algorithms is rather suboptimal and compares unfavorably against recent model-free approaches. In this paper, we present a sharp analysis of model-based self-play algorithms for multi-agent Markov games. We design an algorithm Optimistic Nash Value Iteration (Nash-VI) for two-player zero-sum Markov games that is able to output an $\\epsilon$ -approximate Nash policy in $\\tilde { \\mathcal { O } } ( H ^ { 3 } S A B / \\epsilon ^ { 2 } )$ episodes of game playing, where $S$ is the number of states, $A , B$ are the number of actions for the two players respectively, and $H$ is the horizon length. This significantly improves over the best known model-based guarantee of $\\tilde { \\mathcal { O } } ( H ^ { 4 } S ^ { 2 } \\bar { A } B / \\epsilon ^ { 2 } )$ , and is the first that matches the information-theoretic lower bound $\\Omega ( H ^ { 3 } \\dot { S } ( A + B ) / \\epsilon ^ { 2 } )$ except for a $\\operatorname* { m i n } \\left\\{ A , B \\right\\}$ factor. In addition, our guarantee compares favorably against the best known model-free algorithm if $\\operatorname* { m i n } \\bar { \\{ A , B \\} } = o ( \\bar { H } ^ { 3 } )$ , and outputs a single Markov policy while existing sampleefficient model-free algorithms output a nested mixture of Markov policies that is in general non-Markov and rather inconvenient to store and execute. We further adapt our analysis to designing a provably efficient task-agnostic algorithm for zero-sum Markov games, and designing the first line of provably sample-efficient algorithms for multi-player general-sum Markov games. ",
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+ "type": "text",
+ "text": "1 INTRODUCTION ",
+ "text_level": 1,
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+ "text": "This paper is concerned with the problem of multi-agent reinforcement learning (multi-agent RL), in which multiple agents learn to make decisions in an unknown environment in order to maximize their (own) cumulative rewards. Multi-agent RL has achieved significant recent success in traditionally hard AI challenges including large-scale strategy games (such as GO) (Silver et al., 2016; 2017), real-time video games involving team play such as Starcraft and Dota2 (OpenAI, 2018; Vinyals et al., 2019), as well as behavior learning in complex social scenarios (Baker et al., 2020). Achieving human-like (or super-human) performance in these games using multi-agent RL typically requires a large number of samples (steps of game playing) due to the necessity of exploration, and how to improve the sample complexity of multi-agent RL has been an important research question. ",
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+ "text": "One prevalent approach towards solving multi-agent RL is model-based methods, that is, to use the existing visitation data to build an estimate of the model (i.e. transition dynamics and rewards), run an offline planning algorithm on the estimated model to obtain the policy, and play the policy in the environment. Such a principle underlies some of the earliest single-agent online RL algorithms such as E3 (Kearns & Singh, 2002) and RMax (Brafman & Tennenholtz, 2002), and is conceptually appealing for multi-agent RL too since the multi-agent structure does not add complexity onto the model estimation part and only requires an appropriate multi-agent planning algorithm (such as value iteration for games (Shapley, 1953)) in a black-box fashion. On the other hand, modelfree methods do not directly build estimates of the model, but instead directly estimate the value functions or action-value (Q) functions of the problem at the optimal/equilibrium policies, and play the greedy policies with respect to the estimated value functions. Model-free algorithms have also ",
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+ "text": "Table 1: Sample complexity (the required number of episodes) for algorithms to find $\\epsilon$ -approximate Nash equlibrium policies in zero-sum Markov games: VI-explore and VI-UCLB by Bai $\\&$ Jin (2020), OMVI-SM by Xie et al. (2020), and Nash Q/V-learning by Bai et al. (2020). The lower bound was proved by Jin et al. (2018); Domingues et al. (2020). ",
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+ "type": "table",
+ "img_path": "images/29b3c0cebbabebba54f2c2bafea169bd6cc2eb2571c25896fd26102f4d75f302.jpg",
+ "table_caption": [],
+ "table_footnote": [],
+ "table_body": "| 1: Initialize: for any (s,a,b,h),Qn(s,a,b) ← H,Q,(s,a,b) ← 0,△ ← H,Nn(s,a,b) ←0. | |
| 2: | for episodek=1,...,Kdo for steph=H,H-1,...,1do |
| 3: | |
| 4: | for(s,a,b) ∈S× A× Bdo |
| 5: | t ←Nn(s,a,b). |
| 6: | if t>O then |
| 7: | β ← BoNUs(t,Vn[(Vh+1 +Vh+1)/2](s,a,b)). |
| 8: | γ ← (c/H)Ph(Vh+1 -Vh+1)(s,a,b). |
| 9: | Qn(s,a,b)←min{(rh +PnVh+1)(s,a,b)+γ+β,H}. |
| 10: | Q(s,a,b)←max{(rh+PhVh+1)(s,a,b)-γ-β,0}. |
| 11: | fors∈Sdo |
| 12: | Th(·,·|s) ← CCE(Qn(s,·,),Q(s,·, )). |
| 13: | Vn(s)← (DπnQn)(s)；Vn(s) ←(DπnQn)(s). |
| 14: 15: | if(V1-V1)(s1)<△ then △←(V1-V1)(s1) and πout ←π. |
| 16: | for step h =1,...,H do |
| 17: | take action (an, br) ~ πh(*,|sh),observe reward rh and next state Sh+1· |
| |
| 18: | add1 to Nn(sh,ah,bh) and Nh(sh,ah,bh,Sh+1). |
| 19: | Ph(-Ish,ah,bh) ← Nn(Sh,ah,bh,:)/Nn(sh,ah,bh). |
| 20: | Output (μout,vout) that are the marginal policies of πout. |
",
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+ {
+ "type": "text",
+ "text": "to compute the Nash equilibrium for a two-player general-sum matrix game, which is in general PPAD-complete (Daskalakis, 2013). ",
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+ "type": "text",
+ "text": "To overcome this computational challenge, we compute a relaxation of the Nash equilibrium— Coarse Correlated Equalibirum $( C C E )$ —instead, a technique first introduced by Xie et al. (2020) to address reinforcement learning problems in Markov Games. Formally, for any pair of matrices $\\overline { { Q } } , \\underline { { Q } } \\in [ 0 , H ] ^ { A \\times B }$ , ${ \\mathrm { C C E } } ( \\overline { { Q } } , \\underline { { Q } } )$ returns a distribution $\\pi \\in \\Delta _ { \\mathcal { A } \\times \\mathcal { B } }$ such that ",
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+ "type": "equation",
+ "img_path": "images/bd3bf15f7e6ca5c19e5a53fa3e9e54c57bb4b047b6537dcb20150a2257f87661.jpg",
+ "text": "$$\n\\mathbb { E } _ { ( a , b ) \\sim \\pi } \\overline { { Q } } ( a , b ) \\geq \\operatorname* { m a x } _ { a ^ { \\star } } \\mathbb { E } _ { ( a , b ) \\sim \\pi } \\overline { { Q } } ( a ^ { \\star } , b ) , \\qquad \\mathbb { E } _ { ( a , b ) \\sim \\pi } \\underline { { Q } } ( a , b ) \\leq \\operatorname* { m i n } _ { b ^ { \\star } } \\mathbb { E } _ { ( a , b ) \\sim \\pi } \\underline { { Q } } ( a , b ^ { \\star } ) .\n$$",
+ "text_format": "latex",
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+ "text": "Intuitively, in a CCE the players choose their actions in a potentially correlated way such that no one can benefit from unilateral unconditional deviation. A CCE always exists, since Nash equilibrium is also a CCE and a Nash equilibrium always exists. Furthermore, a CCE can be computed by linear programming in polynomial time. We remark that different from Nash equilibrium where the policies of each player are independent, the policies given by CCE are in general correlated for each player. Therefore, executing such a policy (line 17) requires the cooperation of two players. ",
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+ "type": "text",
+ "text": "3.2 THEORETICAL GUARANTEES ",
+ "text_level": 1,
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+ "text": "Now we are ready to present the theoretical guarantees for Algorithm 1. We let $\\pi ^ { k }$ denote the policy computed in line 12 in the $k ^ { \\mathrm { { t h } } }$ episode, and $\\bar { \\mu } ^ { k } , \\nu ^ { k }$ denote the marginal policy of $\\pi ^ { k }$ for each player. ",
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+ "text": "Theorem 3 (Nash-VI with Hoeffding bonus). For any $p \\in ( 0 , 1 ]$ , letting $\\iota = \\log ( S A B T / p )$ , then with probability at least $1 - p$ , Algorithm $^ { l }$ with Hoeffding type bonus (3) (with some absolute $c > 0$ ) achieves: ",
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+ "text": "• $( V _ { 1 } ^ { \\dagger , \\nu ^ { o u t } } - V _ { 1 } ^ { \\mu ^ { o u t } , \\dagger } ) ( s _ { 1 } ) \\leq \\epsilon ,$ , if the number of episodes $K \\geq \\Omega ( H ^ { 4 } S A B \\iota / \\epsilon ^ { 2 } + H ^ { 3 } S ^ { 2 } A B \\iota ^ { 2 } / \\epsilon ) .$ $\\begin{array} { r } { \\bullet \\operatorname { R e g r e t } ( K ) = \\sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \\dagger , \\nu ^ { k } } - V _ { 1 } ^ { \\mu ^ { k } , \\dagger } ) ( s _ { 1 } ) \\le \\mathcal { O } ( \\sqrt { H ^ { 3 } S A B T \\iota } + H ^ { 3 } S ^ { 2 } A B \\iota ^ { 2 } ) . } \\end{array}$ ",
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+ "type": "text",
+ "text": "Theorem 3 provides both a sample complexity bound and a regret bound for Nash-VI to find an $\\epsilon$ -approximate Nash equilibrium. For small $\\epsilon \\leq H / ( S \\iota )$ , the sample complexity scales as ${ \\tilde { \\mathcal { O } } } ( H ^ { 4 } S A B / { \\epsilon } ^ { 2 } )$ . Similarly, for large $T \\geq H ^ { 3 } S ^ { 3 } A B \\iota ^ { 3 }$ , the regret scales as $\\tilde { \\mathcal { O } } ( \\sqrt { H ^ { 3 } S A B T } )$ , where $T = K H$ is the total number of steps played within $K$ episodes. Theorem 3 is significant in that it improves the sample complexity of the model-based algorithm in Markov games from √ $S ^ { 2 }$ to $S$ (and the regret from $S$ to $\\sqrt { S }$ ). This is achieved by adding the new auxiliary bonus $\\gamma$ in value iteration steps as explained in Section 3.1. The proof of Theorem 3 can be found in Appendix F.1. ",
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+ "text": "Our next theorem states that when using Bernstein bonus instead of Hoeffding bonus as in (3), the sample complexity of Nash-VI algorithm can be further improved by a $H$ factor in the leading order term (and the regret improved by a $\\sqrt { H }$ factor). ",
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+ "text": "Theorem 4 (Nash-VI with the Bernstein bonus). For any $p \\in ( 0 , 1 ]$ , letting $\\iota = \\log ( S A B T / p )$ , then with probability at least $1 - p ,$ , Algorithm $^ { l }$ with Bernstein type bonus (3) (with some absolute $c > 0$ ) achieves: ",
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+ "type": "text",
+ "text": "• $( V _ { 1 } ^ { \\dagger , \\nu ^ { o u t } } - V _ { 1 } ^ { \\mu ^ { o u t } , \\dagger } ) ( s _ { 1 } ) \\leq \\epsilon ,$ , if the number of episodes $K \\geq \\Omega ( H ^ { 3 } S A B \\iota / \\epsilon ^ { 2 } + H ^ { 3 } S ^ { 2 } A B \\iota ^ { 2 } / \\epsilon ) .$ $\\begin{array} { r } { \\bullet \\operatorname { R e g r e t } ( K ) = \\sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \\dagger , \\nu ^ { k } } - V _ { 1 } ^ { \\mu ^ { k } , \\dagger } ) ( s _ { 1 } ) \\le \\mathcal { O } ( \\sqrt { H ^ { 2 } S A B T \\iota } + H ^ { 3 } S ^ { 2 } A B \\iota ^ { 2 } ) . } \\end{array}$ ",
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+ "type": "text",
+ "text": "Compared with the information-theoretic sample complexity lower bound $\\Omega ( H ^ { 3 } S ( A + B ) \\iota / \\epsilon ^ { 2 } )$ and regret lower bound $\\Omega ( \\sqrt { H ^ { 2 } S ( A + B ) T } )$ (Bai $\\&$ Jin, 2020), when $\\epsilon$ is small, Nash-VI with Bernstein bonus achieves the optimal dependency on all of $H , S , \\epsilon$ up to logarithmic factors in both the sample complexity and the regret, and the only gap that remains open is a $A B / ( A + B ) \\leq$ $\\operatorname* { m i n } \\left\\{ A , B \\right\\}$ factor. The proof of Theorem 4 can be found in Appendix F.2. ",
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+ "type": "text",
+ "text": "Comparison with model-free approaches. Different from our model-based approach, a recently proposed model-free algorithm Nash V-Learning (Bai et al., 2020) achieves sample complexity $\\bar { \\mathcal { O } } ( \\bar { H } ^ { 6 } S ( A + B ) \\iota / \\epsilon ^ { 2 } )$ , which has a tight $( A + B )$ dependency on $A , B$ . However, our Nash-VI has the following important advantages over Nash V-Learning: 1. Our sample complexity has a better dependency on horizon $H$ ; 2. Our algorithm outputs a single pair of Markov policies $( \\mu ^ { \\mathrm { o u t } } , \\nu ^ { \\mathrm { o u t } } )$ while their algorithm outputs a generic history-dependent policy that can be only written as a nested mixture of Markov policies; 3. The model-free algorithms in Bai et al. (2020) cannot be directly√ modified to obtain a $\\sqrt { T }$ -regret (so that the exploration policies can be arbitrarily poor), while our√ model-based algorithm has the $\\sqrt { T }$ -regret guarantee. We comment that although both Nash-VI and Nash V-Learning have polynomial running time, the later enjoys a better computational complexity because Nash-VI requires to solve LPs for computing CCEs in each episode. ",
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+ "type": "text",
+ "text": "4 REWARD-FREE LEARNING ",
+ "text_level": 1,
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+ "text": "In this section, we modify our model-based algorithm Nash-VI for the reward-free exploration setting (Jin et al., 2020b), which is also known as the task-agnostic (Zhang et al., 2020b) or reward-agnostic setting. Reward-free learning has two phases: In the exploration phase, the agent first collects a dataset of transitions $\\mathcal { D } = \\{ ( \\bar { s } _ { k , h } , a _ { k , h } , \\bar { b } _ { k , h } , s _ { k , h + 1 } ) \\} _ { ( k , h ) \\in [ K ] \\times [ H ] }$ from a Markov game $\\mathcal { M }$ without the guidance of reward information. After the exploration, in the planning phase, for each task $i \\in [ N ]$ , $\\mathcal { D }$ is augmented with stochastic reward information to become $\\mathcal { D } ^ { i } = \\{ \\left( s _ { k , h } , a _ { k , h } , b _ { k , h } , s _ { k , h + 1 } , r _ { k , h } \\right) \\} _ { ( k , h ) \\in [ K ] \\times [ H ] }$ , where $r _ { k , h }$ is sampled from an unknown reward distribution with expectation equal to $r _ { h } ^ { i } ( s _ { k , h } , a _ { k , h } , b _ { k , h } )$ . Here, we use $r ^ { i }$ to refer to the unknown reward function of the $i ^ { \\mathrm { t h } }$ task. The goal is to compute nearly-optimal policies for $N$ tasks under $\\mathcal { M }$ , given the augmented datasets. ",
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+ "text": "There are strong practical motivations for considering the reward-free setting. First, in applications such as robotics, we face multiple tasks in sequential systems with shared transition dynamics (i.e. the world) but very different rewards. There, we prefer to learn the underlying transition independent of reward information. Second, from the algorithm design perspective, decoupling exploration and planning (i.e. performing exploration without reward information) can be valuable for designing new algorithms in more challenging settings (e.g. with function approximation). ",
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+ "type": "text",
+ "text": "Due to space limits, we defer the description of our algorithm Optimistic Value Iteration with Zero Reward (VI-Zero, Algorithm 2) to Appendix B and only state its theoretical guarantees here. The following theorem claims that the empirical transition $\\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } }$ outputted by VI-Zero is close to the true transition $\\mathbb { P }$ , in the sense that any Nash equilibrium of the ${ \\mathcal { M } } ( { \\widehat { \\mathbb { P } } } , { \\widehat { r } } ^ { i } )$ $\\because [ i \\in [ N ] )$ is also an approximate ",
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+ "text": "Nash equilibrium of the true underlying Markov game ${ \\mathcal { M } } ( { \\mathbb { P } } , r ^ { i } )$ , where $\\widehat { r } ^ { i }$ is the empirical estimate of $r ^ { i }$ computed using $\\mathcal { D } ^ { i }$ . ",
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+ "text": "Theorem 5 (Sample complexity of VI-Zero). There exists an absolute constant c, for any $p \\in ( 0 , 1 ]$ , $\\epsilon \\in ( 0 , H ]$ , $N \\in \\mathbb N$ , if we choose bonus $\\beta _ { t } = c ( \\sqrt { H ^ { 2 } \\iota / t } + H ^ { 2 } S \\iota / t )$ with $\\iota = \\log ( N S A B T / p )$ and $K \\ge c ( H ^ { 4 } S A B \\iota / \\epsilon ^ { 2 } + H ^ { 3 } S ^ { 2 } A B \\iota ^ { 2 } / \\epsilon )$ , then with probability at least $1 - p ,$ , the output $\\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } }$ of Algorithm 2 has the following property: for any $N$ fixed reward functions $\\boldsymbol { r } ^ { \\mathrm { 1 } } , \\ldots , \\boldsymbol { r } ^ { N }$ , a Nash equilibrium of Markov game $\\mathcal { M } ( \\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } } , \\widehat { r } ^ { i } )$ is also an $\\epsilon$ -approximate Nash equilibrium of the true Markov game ${ \\mathcal { M } } ( \\mathbb { P } , r ^ { i } )$ for all $i \\in [ N ]$ . ",
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+ "text": "Theorem 5 shows that, when $\\epsilon$ is small, VI-Zero only needs ${ \\tilde { \\mathcal { O } } } ( H ^ { 4 } S A B / { \\epsilon } ^ { 2 } )$ samples to learn an estimate of the transition $\\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } }$ , which is accurate enough to learn the approximate Nash equilibrium for any $N$ fixed rewards. The most important advantage of reward-free learning comes from the sample complexity only scaling polylogarithmically with respect to the number of tasks or reward functions $N$ . This is in sharp contrast to the reward-aware algorithms (e.g. Nash-VI), where the algorithm has to be rerun for each different task, and the total sample complexity must scale linearly in $N$ . In exchange for this benefit, compared to Nash-VI, VI-Zero loses a factor of $H$ in the leading term of sample complexity since we cannot use Bernstein bonus anymore due to the lack of reward information. VI-Zero also does not have a regret guarantee, since again without reward information, the exploration policies are naturally sub-optimal. The proof of Theorem 5 can be found in Appendix G.1. ",
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+ "type": "text",
+ "text": "Connections with reward-free learning in MDPs. Since MDPs are special cases of Markov games, our algorithm VI-Zero directly applies to the single-agent setting, and yields a sample complexity similar to existing results (Zhang et al., 2020b; Wang et al., 2020). However, distinct from existing results which require both the exploration algorithm and the planning algorithm to be specially designed to work together, our algorithm allows an arbitrary planning algorithm as long as it computes the Nash equilibrium of a Markov game with known transition and reward. Therefore, our results completely decouple the exploration and the planning. ",
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+ "type": "text",
+ "text": "Lower bound for reward-free learning. Finally, we comment that despite the sample complexity in Theorem 5 scaling as $A B$ instead of $A + B$ , our next theorem states that unlike the general rewardaware setting, this $A B$ scaling is unavoidable in the reward-free setting. This reveals an intrinsic gap between the reward-free and reward-aware learning: An $A + B$ dependency is only achievable via sampling schemes that are reward-aware. A similar lower bound is also presented in recent work (Zhang et al., 2020a) for the discounted setting with a different hard instance construction. ",
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+ "text": "Theorem 6 (Lower bound for reward-free learning of Markov games). There exists an absolute constant $c > 0$ such that for any $\\epsilon \\in ( 0 , c ]$ , there exists a family of Markov games $\\mathfrak { M } ( \\epsilon )$ satisfying that: for any reward-free algorithm $\\mathfrak { A }$ using $K \\le c H ^ { 2 } \\bar { S } A B / \\epsilon ^ { 2 }$ episodes, there exists a Markov game $\\mathcal { M } \\in \\mathfrak { M } ( \\epsilon )$ such that if we run $\\mathfrak { A }$ on $\\mathcal { M }$ and output policies $( \\hat { \\mu } , \\hat { \\nu } )$ , then with probability at least $1 / 4$ , we have $( V _ { 1 } ^ { \\dagger , \\hat { \\nu } } - V _ { 1 } ^ { \\hat { \\mu } , \\dagger } ) ( s _ { 1 } ) \\geq \\epsilon$ . ",
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+ "type": "text",
+ "text": "This lower bound shows that the sample complexity in Theorem 5 is optimal in $S , A , B$ , and $\\epsilon$ . The proof of Theorem 6 can be found in Appendix G.3. ",
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+ "type": "text",
+ "text": "5 MULTI-PLAYER GENERAL-SUM GAMES ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "We adapt our analysis to multi-player general-sum games and present the first lines of provably efficient algorithms. Concretely, we design two model-based algorithms Multi-Nash-VI and MultiVI-Zero (Algorithm 3 and Algorithm 4) that can find an $\\epsilon$ -approximate) $\\{ \\mathrm { N A S H } , \\mathrm { C E } , \\mathrm { C C E } \\}$ equilibrium for any multi-player general-sum Markov game in $\\begin{array} { r } { \\tilde { \\mathcal { O } } ( H ^ { 4 } S ^ { 2 } \\prod _ { i = 1 } ^ { m } A _ { i } / \\epsilon ^ { 2 } ) } \\end{array}$ episodes of game playing, where $A _ { i }$ is the number of actions for player $i \\in \\{ 1 , \\ldots , m \\}$ (Theorem 15 and Theorem 16). Due to space limit, we defer the detailed setups, algorithms and results to Appendix C. ",
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+ "type": "text",
+ "text": "6 CONCLUSION ",
+ "text_level": 1,
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+ {
+ "type": "text",
+ "text": "In this paper, we provided a sharp analysis of model-based algorithms for Markov games. Our new algorithm Nash-VI can find an $\\epsilon$ -approximate Nash equilibrium of a zero-sum Markov game in $\\tilde { \\mathcal { O } } ( H ^ { 3 } S A B / \\epsilon ^ { 2 } )$ episodes of game playing, which almost matches the sample complexity lower bound except for the $A B$ vs. $A + B$ dependency. We also applied our analysis to derive new efficient algorithms for task-agnostic game playing, as well as the first line of multi-player generalsum Markov games. There are a number of compelling future directions to this work. For example, can we achieve $A + B$ instead of $A B$ sample complexity for zero-sum games using model-based approaches (thus closing the gap between lower and upper bounds)? How can we design more efficient algorithms for general-sum games with better sample complexity (e.g., $\\mathcal O ( S )$ instead of $\\mathcal { O } ( S ^ { 2 } ) \\colon$ We leave these problems as future work. ",
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+ "type": "text",
+ "text": "A RELATED WORK ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Markov games. Markov games (or stochastic games) are proposed in the early 1950s (Shapley, 1953). They are widely used to model multi-agent RL. Learning the Nash equilibria of Markov games has been studied in Littman (1994; 2001); Hu & Wellman (2003); Hansen et al. (2013); Lee et al. (2020), where the transition matrix and reward are assumed to be known, or in the asymptotic setting where the number of data goes to infinity. These results do not directly apply to the nonasymptotic setting where the transition and reward are unknown and only a limited amount of data are available for estimating them. ",
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Another line of work assumes certain strong reachability assumptions under which sophisticated exploration strategies are not required. A prevalent approach is to assume access to simulators (generative models) that enable the agent to directly sample transition and reward information for any state-action pair. In this setting, Jia et al. (2019); Sidford et al. (2019); Zhang et al. (2020a) provide non-asymptotic bounds on the number of calls to the simulator for finding an $\\epsilon$ approximate Nash equilibrium. Wei et al. (2017) studies Markov games under an alternative assumption that no matter what strategy one agent sticks to, the other agent can always reach all states by playing a certain policy. ",
+ "bbox": [
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+ "page_idx": 11
+ },
+ {
+ "type": "text",
+ "text": "Non-asymptotic guarantees without reachability assumptions. The recent work of Bai & Jin (2020); Xie et al. (2020) provide the first line of non-asymptotic sample complexity guarantees on learning Markov games without these reachability assumptions, in which exploration is essential. However, both results suffer from highly suboptimal sample complexity. The results of Xie et al. (2020) also apply to the linear function approximation setting. More recently, two model-free algorithms—Nash Q-Learning and Nash V-Learning—are shown to achieve better sample complexity guarantees (Bai et al., 2020). In particular, the Nash V-learning algorithm achieves the nearoptimal dependence on $S$ , $A$ and $B$ . However, the dependence on $H$ is worse than our results and the output policy is a nested mixture, which is hard to implement. We compare our results with existing non-asymptotic guarantees in Table 1. ",
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+ {
+ "type": "text",
+ "text": "We remark that the classical R-max algorithm (Brafman & Tennenholtz, 2002) also provides provable guarantees for learning Markov games. However, Brafman & Tennenholtz (2002) uses a weaker definition of regret (similar to the online setting in Xie et al. (2020)), and consequently their result does not imply any sample complexity result for finding Nash equilibrium policies. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Adversarial MDPs. Another way to model the multi-player bahavior is to use adversarial MDPs. Most work in this line considers the setting with adversarial rewards (Zimin & Neu, 2013; Rosenberg & Mansour, 2019; Jin et al., 2019), where the reward can be manipulated by an adversary arbitrarily and the goal is to compete with the optimal (stationary) policy in hindsight. Adversarial MDP with changing dynamics is computationally hard even under full-information feedback (Yadkori et al., 2013). Notice these results do not directly imply provable self-play algorithms in our setting, because the opponent in Markov games can affect both the reward and the transition. ",
+ "bbox": [
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+ "type": "text",
+ "text": "Single-agent RL. There is a rich literature on reinforcement learning in MDPs (see e.g., Jaksch et al., 2010; Osband et al., 2014; Azar et al., 2017; Dann et al., 2017; Strehl et al., 2006; Jin et al., 2018). MDP is a special case of Markov games, where only a single agent interacts with a stochastic environment. For the tabular episodic setting with nonstationary dynamics and no simulators, the best sample complexity is $\\tilde { \\mathcal { O } } ( \\bar { H ^ { 3 } } S A / \\epsilon ^ { 2 } )$ , achieved by model-based algorithm in Azar et al. (2017) and model-free algorithms in Zhang et al. (2020c), respectively, where $S$ is the number of states, $A$ is the number of actions, $H$ is the length of each episode. Both of them match the lower bound $\\Omega ( H ^ { 3 } S A / \\epsilon ^ { 2 } )$ (Jaksch et al., 2010; Osband & Van Roy, 2016; Jin et al., 2018). ",
+ "bbox": [
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+ "type": "text",
+ "text": "Reward-free and task-agnostic exploration. Jin et al. (2020a) proposes a new paradigm of learning an MDP, which they called reward-free exploration. In this setting, the agent goes through a two-stage process. In the exploration phase the agent can interacts with the environment without knowing the reward function and in the planning phase the reward function is given and the agent needs output a policy. The goal is to make the output policy near optimal for any given reward function. A closely related setting is task-agnostic learning, where the reward function is determined at the very beginning but not revealed until the planning phase. Notice algorithms for task-agnostic ",
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+ "type": "text",
+ "text": "Algorithm 2 Optimistic Value Iteration with Zero Reward (VI-Zero) ",
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+ "page_idx": 12
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+ "type": "text",
+ "text": "Require: Bonus $\\beta _ { t }$ . \n1: Initialize: for any $( s , a , b , h )$ $h ) , \\widetilde { V } _ { h } ( s , a , b ) \\gets H , \\Delta \\gets H , N _ { h } ( s , a , b ) \\gets 0 .$ . \n2: for episode $k = 1 , \\ldots , K$ do \n3: for step $h = H , H - 1 , \\ldots , 1$ do \n4: for $( s , a , b ) \\in \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { B }$ do \n5: $t \\gets N _ { h } ( s , a , b )$ . \n6: if $t > 0$ then \n7: $\\begin{array} { r l } & { \\quad \\widetilde { Q } _ { h } ( s , a , b ) \\gets \\operatorname* { m i n } \\{ ( \\widehat { \\mathbb { P } } _ { h } \\widetilde { V } _ { h + 1 } ) ( s , a , b ) + \\beta _ { t } , H \\} . } \\\\ & { \\quad \\mathbf { r } s \\in S \\mathbf { d o } } \\\\ & { \\quad \\pi _ { h } ( s ) \\gets \\arg \\operatorname* { m a x } _ { ( a , b ) \\in A \\times B } \\widetilde { Q } _ { h } ( s , a , b ) . } \\\\ & { \\quad \\widetilde { V } _ { h } ( s ) \\gets ( \\mathbb { D } _ { \\pi _ { h } } \\widetilde { Q } _ { h } ) ( s ) . } \\end{array}$ \n8: fo \n9: \n10: \n11: if $\\widetilde { V } _ { 1 } ( s _ { 1 } ) < \\Delta$ then \n12: $\\Delta \\widetilde { V } _ { 1 } ( s _ { 1 } )$ and $\\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } } \\widehat { \\mathbb { P } }$ . \n13: for step $h = 1 , \\ldots , H$ do \n14: take action $( a _ { h } , b _ { h } ) \\sim \\pi _ { h } ( \\cdot , \\cdot | s _ { h } )$ , observe next state $s _ { h + 1 }$ . \n15: add 1 to $N _ { h } ( s _ { h } , a _ { h } , b _ { h } )$ and $N _ { h } ( s _ { h } , a _ { h } , b _ { h } , s _ { h + 1 } )$ . \n16: $\\widehat { \\mathbb { P } } _ { h } ( \\cdot | s _ { h } , a _ { h } , b _ { h } ) \\gets N _ { h } ( s _ { h } , a _ { h } , b _ { h } , \\cdot ) / N _ { h } ( s _ { h } , a _ { h } , b _ { h } ) .$ \n17: Output $\\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } }$ . ",
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+ "type": "text",
+ "text": "learning can also be transferred to reward-free exploration by taking union bound w.r.t. different possible reward function. In Table 1, VI-explore (Bai & Jin, 2020) and Algorithm 2 can also be applied to this setting. ",
+ "bbox": [
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+ "type": "text",
+ "text": "Jin et al. (2020a) also proposes an algorithm, which first finds a covering policy to maximize the probability to reach each state separately and then collects data following this policy. Zhang et al. (2020b) takes a different approach by first runs the optimistic Q-learning algorithm (Jin et al., 2018) with zero reward to explore the environment, and then they utilizes the trajectories collected to compute a policy in an incremental manner. Wang et al. (2020) follows a simialr scheme, but studies reward-free exploration in linear-parametrized MDPs. ",
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "B OPTIMISTIC VALUE ITERATION WITH ZERO REWARD – VI-ZERO ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "We now describe our algorithm for reward-free learning in zero-sum Markov games. ",
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+ "text": "Exploration phase. In the first phase of reward-free learning, we deploy algorithm Optimistic Value Iteration with Zero Reward (VI-Zero, Algorithm 2). This algorithm differs from the rewardaware Nash-VI (Algorithm 1) in two important aspects. First, we use zero reward in the exploration phase (Line 7), and only maintains an upper bound of the (reward-free) value function instead of both upper and lower bounds. Second, our exploration policy is the maximizing (instead of CCE) policy of the value function (Line 9). We remark that the $\\widetilde { Q } _ { h } ( s , a , b )$ maintained in the algorithm 2 is no longer an upper bound for any actual value function (as it has no reward), but rather a measure of uncertainty or suboptimality that the agent may suffer—if she takes action $( a , b )$ at state $s$ and step $h$ , and makes decisions by utilizing the empirical estimate $\\widehat { \\mathbb { P } }$ in the remaining steps (see a rigorous version of this statement in Lemma 27). Finally, the empirical transition $\\widehat { \\mathbb { P } }$ of the episode that minimizes $\\widetilde { V } _ { 1 } ( s _ { 1 } )$ is outputted and passed to the planning phase. ",
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+ "text": "Planning phase. After obtaining the estimate of tranisiton $\\widehat { \\mathbb { P } }$ , our planning algorithm is rather simple. For the $i ^ { \\mathrm { t h } }$ task, let ${ \\widehat { r } } ^ { i }$ be the empirical estimate of $r ^ { i }$ computed using the $i ^ { \\mathrm { t h } }$ augmented dataset $\\mathcal { D } ^ { i }$ . Then we compute the Nash equilibrium of the Markov game ${ \\mathcal { M } } ( { \\widehat { \\mathbb { P } } } , { \\widehat { r } } ^ { i } )$ with estimated transition $\\widehat { \\mathbb { P } }$ and reward $\\widehat { r } ^ { i }$ . Since both $\\widehat { \\mathbb { P } }$ and ${ \\widehat { r } } ^ { i }$ are known exactly, this is a pure computation problem b bwithout any sampling error and can be efficiently solved by simple planning algorithms such as the vanilla Nash value iteration without optimism (see Appendix G.2 for more details). ",
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+ "type": "text",
+ "text": "C MULTIPLAYER GENERAL-SUM MARKOV GAMES ",
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+ "text": "In this section, we extend both our model-based algorithms (Algorithm 1 and Algorithm 2) to the setting of multiplayer general-sum Markov games, and present corresponding theoretical guarantees. ",
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+ "type": "text",
+ "text": "C.1 PROBLEM FORMULATION ",
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+ "text": "A general-sum Markov game (general-sum MG) with $m$ players is a tuple $\\mathrm { M G } ( \\ b { H } , \\ b { S } , \\{ \\ b { A } _ { i } \\} _ { i = 1 } ^ { m } , \\mathbb { P } , \\{ r _ { i } \\} _ { i = 1 } ^ { m } )$ , where $H$ , $s$ denote the length of each episode and the state space. Different from the two-player zero-sum setting, we now have $m$ different action spaces, where $A _ { i }$ is the action space for the $i ^ { \\mathrm { { t h } } }$ player and $| { \\bar { \\mathcal { A } } } _ { i } | = A _ { i }$ . We let $\\pmb { a } : = ( a _ { 1 } , \\cdots , a _ { m } )$ denote the (tuple of) joint actions by all $m$ players. $\\mathbb { P } = \\{ \\mathbb { P } _ { h } \\} _ { h \\in [ H ] }$ is a collection of transition matrices, so that $\\mathbb { P } _ { h } ( \\cdot | s , \\pmb { a } )$ gives the distribution of the next state if actions $^ { a }$ are taken at state $s$ at step $h$ , and $r _ { i } = \\{ r _ { h , i } \\} _ { h \\in [ H ] }$ is a collection of reward functions for $i ^ { \\mathrm { { t h } } }$ player, so that $r _ { h , i } ( s , \\pmb { a } )$ gives the reward received by the $i ^ { \\mathrm { { t h } } }$ player if actions $^ { a }$ are taken at state $s$ at step $h$ . ",
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+ "text": "In this section, we consider three versions of equlibrium for general-sum MGs: Nash equilibrium (NE), correlated equilibrium (CE), and coarse correlated equilibrium (CCE), all being standard solution notions in games (Nisan et al., 2007). These three notions coincide on two-player zero-sum games, but are not equivalent to each other on multi-player general-sum games; any one of them could be desired depending on the application at hand. Below we introduce their definitions. ",
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+ "text": "(Approximate) Nash equilibrium in general-sum MG. The policy of the $i ^ { \\mathrm { { t h } } }$ player is denoted as πi := \bπh,i : S → ∆Ai \th∈[H]. We denote the product policy of all the players as $\\pi : =$ $\\pi _ { 1 } \\times \\cdots \\times \\pi _ { M }$ , and denote the policy of the all the players except the $i ^ { \\mathrm { { t h } } }$ player as $\\pi _ { - i }$ . We define $V _ { h , i } ^ { \\pi } ( s )$ as the expected cumulative reward that will be received by the $i ^ { \\mathrm { { t h } } }$ player if starting at state $s$ at step $h$ and all players follow policy $\\pi$ . For any strategy $\\pi _ { - i }$ , there also exists a best response of the ith player, which is a policy µ†(π−i) satisfying V µ†(h,i $V _ { h , i } ^ { { \\mu ^ { \\dagger } } ( \\pi _ { - i } ) , \\pi _ { - i } } ( s ) = \\operatorname* { s u p } _ { \\pi _ { i } } V _ { h , i } ^ { \\pi _ { i } , \\pi _ { - i } } ( s )$ V πi,π−ih,i (s) for any $( s , h ) \\in \\mathcal { S } \\times [ H ]$ . We denote $V _ { h , i } ^ { \\dagger , \\pi _ { - i } } : = V _ { h , i } ^ { \\mu ^ { \\dagger } ( \\pi _ { - i } ) , \\pi _ { - i } }$ i . The Q-functions of the best response can be defined similarly. ",
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+ "type": "text",
+ "text": "Our first objective is to find an approximate Nash equilibrium of Markov games. ",
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+ "text": "approximate Nash equilibrium if Definition 7 ( -approximate Nash equilibrium in general-sum MG). A product policy $\\begin{array} { r } { \\operatorname* { m a x } _ { i \\in [ m ] } ( V _ { 1 , i } ^ { \\dagger , \\pi _ { - i } } - V _ { 1 , i } ^ { \\pi } ) ( s _ { 1 } ) \\le \\epsilon . } \\end{array}$ . $\\pi$ is an $\\epsilon$ - ",
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+ "text": "The above definition requires the suboptimality gap $( V _ { 1 , i } ^ { \\dag , \\pi _ { - i } } \\ - \\ : V _ { 1 , i } ^ { \\pi } ) ( s _ { 1 } )$ to be less than $\\epsilon$ for all player . This is consistent with the two-player case (Definition 1) up to a constant of 2, since in the two-player zero-sum setting, we have $\\bar { V } _ { 1 , 1 } ^ { \\pi } ( s _ { 1 } ) = - V _ { 1 , 2 } ^ { \\pi } ( s _ { 1 } )$ for any product policy $\\pi = ( \\mu , \\nu )$ , and therefore $\\begin{array} { r } { ( V _ { 1 , 1 } ^ { \\dag , \\nu } - V _ { 1 , 1 } ^ { \\mu , \\dag } ) ( s _ { 1 } ) \\le 2 \\operatorname* { m a x } _ { i \\in [ 2 ] } { ( V _ { 1 , i } ^ { \\dag , \\pi _ { - i } } - V _ { 1 , i } ^ { \\pi } ) ( s _ { 1 } ) } \\le 2 ( V _ { 1 , 1 } ^ { \\dag , \\nu } - V _ { 1 , 1 } ^ { \\mu , \\dag } ) ( s _ { 1 } ) . } \\end{array}$ We can similarly define the regret. ",
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+ "text": "Definition 8 (Nash-regret in general-sum MG). Let $\\pi ^ { k }$ denote the (product) policy deployed by the algorithm in the $k ^ { \\mathrm { { t h } } }$ episode. After a total of $K$ episodes, the regret is defined as ",
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+ "text": "$$\n\\mathrm { R e g r e t } _ { \\mathsf { N a s h } } ( K ) = \\sum _ { k = 1 } ^ { K } \\operatorname* { m a x } _ { i \\in [ m ] } ( V _ { 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } - V _ { 1 , i } ^ { \\pi ^ { k } } ) ( s _ { 1 } ) .\n$$",
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+ "text": "(Approximate) CCE in general-sum MG. The coarse correlated equilibrium (CCE) is a relaxed version of Nash equilibrium in which we consider general correlated policies instead of product policies. Let $\\mathcal { A } = \\mathcal { A } _ { 1 } \\times \\cdot \\cdot \\cdot \\times \\mathcal { A } _ { m }$ denote the joint action space. ",
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+ "text": "n 9 (CCE inis a CCE if lated)for all $\\pi : = \\{ \\pi _ { h } ( s ) \\in \\Delta _ { \\mathcal { A } } : ( h , s ) \\in$ $[ H ] \\times S \\}$ $\\begin{array} { r } { \\operatorname* { m a x } _ { i \\in [ m ] } V _ { h , i } ^ { \\dag , \\pi - i } ( s ) \\leq V _ { h , i } ^ { \\pi } ( s ) } \\end{array}$ $( s , h ) \\in \\mathcal { S } \\times [ H ]$ ",
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+ "text": "Compared with a Nash equilibrium, a CEE is not necessarily a product policy, that is, we may not have $\\pi _ { h } ( s ) \\in \\Delta _ { { \\cal A } _ { 1 } } \\times \\cdot \\cdot \\cdot \\times \\Delta _ { { \\cal A } _ { m } }$ . Similarly, we also define $\\epsilon$ -approximate CCE and CCE-regret below. ",
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+ "text": "n 10 is an $\\cdot$ -approximate CCE in -approximate CCE if $\\pi : = \\{ \\pi _ { h } ( s ) \\in \\Delta _ { \\mathcal { A } } : ( h , s ) \\in$ $[ H ] \\times S \\}$ $\\epsilon$ $\\begin{array} { r } { \\operatorname* { m a x } _ { i \\in [ m ] } { ( V _ { 1 , i } ^ { \\dag , \\pi _ { - i } } - V _ { 1 , i } ^ { \\pi } ) ( s _ { 1 } ) } \\le \\epsilon . } \\end{array}$ ",
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+ "text": "Definition 11 (CCE-regret in general-sum MG). Let policy $\\pi ^ { k }$ denote the (correlated) policy deployed by the algorithm in the $\\bar { k } ^ { \\mathrm { t h } }$ episode. After a total of $K$ episodes, the regret is defined as ",
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+ "text": "$$\n\\mathrm { R e g r e t } _ { \\mathsf { C C E } } ( K ) = \\sum _ { k = 1 } ^ { K } \\operatorname* { m a x } _ { i \\in [ m ] } ( V _ { 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } - V _ { 1 , i } ^ { \\pi ^ { k } } ) ( s _ { 1 } ) .\n$$",
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+ "text": "(Approximate) CE in general-sum MG. The correlated equilibrium (CE) is another relaxation of the Nash equilibrium. To define CE, we first introduce the concept of strategy modification: A strategy modification $\\phi : = \\{ \\phi _ { h , s } ( a ) \\in \\mathcal { A } _ { i } : ( h , s , a ) \\in [ H ] \\times \\mathcal { S } \\times \\mathcal { A } _ { i } \\}$ for player $i$ is a set of $S \\times H$ injective functions from $\\mathbf { \\mathcal { A } } _ { i }$ to itself. Let $\\Phi _ { i }$ denote the set of all possible strategy modifications for player $i$ . ",
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+ "text": "One can compose a strategy modification $\\phi$ with any Markov policy $\\pi$ and obtain a new policy $\\phi \\diamond \\pi$ such that when policy $\\pi$ chooses to play $\\pmb { a } : = ( a _ { 1 } , \\dots , a _ { m } )$ at state $s$ and step $h$ , policy $\\phi \\diamond \\pi \\mathbf { w } \\mathrm { i l l }$ play $( a _ { 1 } , \\dots , a _ { i - 1 } , \\phi _ { h , s } ( a _ { i } ) , a _ { i + 1 } , \\dots , a _ { m } )$ instead. ",
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+ "text": "Definition 12 (CE in general-sum MG). A policy $\\pi : = \\{ \\pi _ { h } ( s ) \\in \\Delta _ { \\mathcal { A } } : ( h , s ) \\in [ H ] \\times \\mathcal { S } \\}$ is a CE if $\\begin{array} { r } { \\operatorname* { m a x } _ { i \\in [ m ] } \\operatorname* { m a x } _ { \\phi \\in \\Phi _ { i } } V _ { h , i } ^ { \\phi \\diamond \\pi } ( s ) \\leq V _ { h , i } ^ { \\pi } ( s ) } \\end{array}$ holds for all $( s , h ) \\in \\mathcal { S } \\times [ H ]$ . ",
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+ "type": "text",
+ "text": "Similarly, we have an approximate version of CE and CE-regret. ",
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+ "text": "Definition 13 ( $[ H ] \\times S \\}$ is an $\\dot { \\epsilon }$ $\\epsilon$ -approximate CE in Markov games). A policy -approximate CE if $\\begin{array} { r } { \\operatorname* { m a x } _ { i \\in [ m ] } \\operatorname* { m a x } _ { \\phi \\in \\Phi _ { i } } ( V _ { 1 , i } ^ { \\phi \\diamond \\pi } - V _ { 1 , i } ^ { \\pi } ) ( s _ { 1 } ) \\leq \\epsilon . } \\end{array}$ $\\pi : = \\{ \\pi _ { h } ( s ) \\in \\Delta _ { \\mathcal { A } } : ( h , s ) \\in$ . ",
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+ "text": "Definition 14 (CE-regret in multiplayer Markov games). Let product policy $\\pi ^ { k }$ denote the policy deployed by the algorithm in the $\\bar { k ^ { \\mathrm { { t h } } } }$ episode. After a total of $K$ episodes, the regret is defined as ",
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+ "text": "$$\n{ \\mathrm { R e g r e t } } _ { \\mathsf { C E } } ( K ) = \\sum _ { k = 1 } ^ { K } \\operatorname* { m a x } _ { i \\in [ m ] } \\operatorname* { m a x } _ { \\phi } { \\big ( } V _ { 1 , i } ^ { \\phi \\diamond \\pi ^ { k } } - V _ { 1 , i } ^ { \\pi ^ { k } } { \\big ) } ( s _ { 1 } ) .\n$$",
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+ "text": "Relationship between Nash, CE, and CCE For general-sum MGs, we have $\\{ \\mathrm { N a s h } \\} \\subseteq \\{ \\mathrm { C E } \\} \\subseteq$ $\\{ \\mathrm { C C E } \\}$ , so that they form a nested set of notions of equilibria (Nisan et al., 2007). Indeed, one can easily verify that if we restrict the choice of strategy modification $\\phi$ to those consisting of only constant functions, i.e., $\\phi _ { h , s } ( a )$ being independent of $a$ , Definition 12 will reduce to the definition of CCE policy. In addition, any Nash equilibrium is a CE by definition. Finally, since a Nash equilibrium always exists, so does CE and CCE. ",
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+ "type": "text",
+ "text": "C.2 MULTIPLAYER OPTIMISTIC NASH VALUE ITERATION ",
+ "text_level": 1,
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+ "text": "Here we present the Multi-Nash-VI algorithm, which is an extension of Algorithm 1 for multi-player general-sum Markov games. ",
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+ "text": "The EQUILIBRIUM Subroutine. Our EQUILIBRIUM subroutine in Line 11 could be taken from either one of the $\\{ \\mathrm { N A S H , C E , C C E } \\}$ subroutines for one-step games. When using NASH, we compute the Nash equilibrium of a one-step multi-player game (see, e.g., Berg & Sandholm (2016) for an overview of the available algorithms); the worst-case computational complexity of such a subroutine will be PPAD-hard (Daskalakis, 2013). When using CE or CCE, we find CEs or CCEs of the one-step games respectively, which can be solved in polynomial time using linear programming. However, the policies found are not guaranteed to be a product policy. We remark that in Algorithm 1 we used the CCE subroutine for finding Nash in two-player zero-sum games, which seemingly contrasts the principle of using the right subroutine for finding the right equilibrium, but nevertheless works as the Nash equilibrium and CCE are equivalent in zero-sum games. ",
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+ "text": "Now we are ready to present the theoretical guarantees for Algorithm 3. We let $\\pi ^ { k }$ denote the policy computed in line 11 of Algorithm 3 in the $k ^ { \\mathrm { { \\bar { t h } } } }$ episode. ",
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+ "text": "Theorem 15 (Multi-Nash-VI). There exists an absolute constant $c ,$ , for any $p \\in \\mathsf { \\Gamma } ( 0 , 1 ]$ , let $\\iota =$ $\\log ( S A B T / p )$ , then with probability at least $1 - p$ , Algorithm $^ 3$ with bonus $\\beta _ { t } = c \\sqrt { S H ^ { 2 } \\iota / t }$ and EQUILIBRIUM being one of {NASH, CE, CCE} satisfies (repsectively): ",
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+ "type": "text",
+ "text": "Algorithm 3 Multiplayer Optimistic Nash Value Iteration (Multi-Nash-VI) ",
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+ "text": "1: Initialize: for any $( s , \\pmb { a } , h , i )$ $\\begin{array} { r } { \\overline { { Q } } _ { h , i } ( s , { \\pmb a } ) H , \\underline { { Q } } _ { h , i } ( s , { \\pmb a } ) 0 , \\Delta H , N _ { h } ( s , { \\pmb a } ) 0 . } \\end{array}$ . \n2: for episode $k = 1 , \\ldots , K$ do \n3: for step $h = H , H - 1 , \\ldots , 1 \\mathbf { d } \\mathbf { 4 }$ o \n4: for $( s , \\pmb { a } ) \\in \\pmb { S } \\times \\mathcal { A } _ { 1 } \\times \\cdot \\cdot \\cdot \\times \\mathcal { A } _ { m }$ do \n5: $t \\gets N _ { h } ( s , \\pmb { a } )$ ; \n6: if $t > 0$ then \n7: for player $i = 1 , 2 , \\dots , m$ do \n8: $\\overline { { Q } } _ { h , i } ( s , \\pmb { a } ) \\operatorname* { m i n } \\{ ( r _ { h , i } + \\widehat { \\mathbb { P } } _ { h } \\overline { { V } } _ { h + 1 , i } ) ( s , \\pmb { a } ) + \\beta _ { t } , H \\} .$ \n9: $\\underline { { Q } } _ { h , i } ( s , \\mathbf { a } ) \\operatorname* { m a x } \\{ ( r _ { h , i } + \\widehat { \\mathbb { P } } _ { h } \\underline { { V } } _ { h + 1 , i } ) ( s , \\mathbf { a } ) - \\beta _ { t } , 0 \\} .$ \n10: for $s \\in S$ do \n11: $\\pi _ { h } ( \\cdot | s ) \\gets \\mathrm { E Q U I L I B R I U M } ( \\overline { { \\boldsymbol { Q } } } _ { h , 1 } ( s , \\cdot ) , \\overline { { \\boldsymbol { Q } } } _ { h , 2 } ( s , \\cdot ) , \\cdot \\cdot , \\overline { { \\boldsymbol { Q } } } _ { h , M } ( s , \\cdot ) ) .$ \n12: for player $i = 1 , 2 , \\dots , m$ do \n13: $\\begin{array} { r } { \\Vec { V } _ { h , i } ( s ) ( \\mathbb { D } _ { \\pi _ { h } } \\overline { { Q } } _ { h , i } ) ( s ) ; \\quad \\underline { { V } } _ { h , i } ( s ) ( \\mathbb { D } _ { \\pi _ { h } } \\underline { { Q } } _ { h , i } ) ( s ) . } \\end{array}$ \n14: if $\\begin{array} { r } { \\operatorname* { m a x } _ { i \\in [ m ] } ( \\overline { { V } } _ { 1 , i } - \\underline { { V } } _ { 1 , i } ) ( s _ { 1 } ) < \\Delta } \\end{array}$ then \n15: $\\Delta \\gets \\mathrm { m a x } _ { i \\in [ m ] } ( \\overline { { V } } _ { 1 , i } - \\underline { { V } } _ { 1 , i } ) ( s _ { 1 } )$ and $\\pi ^ { \\mathrm { o u t } } \\pi$ . \n16: for step $h = 1 , \\ldots , H$ do \n17: take action $\\mathbf { a } _ { h } \\sim \\pi _ { h } ( \\cdot | s _ { h } )$ , observe reward $r _ { h }$ and next state $s _ { h + 1 }$ . \n18: add 1 to $N _ { h } ( s _ { h } , \\pmb { a } _ { h } )$ and $\\dot { N } _ { h } ( s _ { h } , \\pmb { a } _ { h } , s _ { h + 1 } )$ . \n19: $\\widehat { \\mathbb { P } } _ { h } ( \\cdot | s _ { h } , \\pmb { a } _ { h } ) \\gets N _ { h } ( s _ { h } , \\pmb { a } _ { h } , \\cdot ) / N _ { h } ( s _ { h } , \\pmb { a } _ { h } )$ . \n20: Output $\\pi ^ { \\mathrm { { o u t } } }$ . ",
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+ "text": "• $\\pi ^ { o u t }$ is an $\\epsilon$ -approximate $\\{ \\mathrm { N A S H , C E , C C E } \\}$ , if the number of episodes $K \\_ { \\mathbf { \\alpha } } \\geq$ $\\begin{array} { r } { \\Omega ( H ^ { 4 } S ^ { 2 } ( \\prod _ { i = 1 } ^ { m } \\tilde { A _ { i } } ) \\iota / \\epsilon ^ { 2 } ) } \\end{array}$ . ",
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+ "text": "$$\n\\bullet \\ \\operatorname { R e g r e t } _ { \\{ \\substack { \\mathrm { N a s h } , \\mathsf { C E } , \\mathsf { C C E } \\} } } ( K ) \\leq \\mathcal { O } ( \\sqrt { H ^ { 3 } S ^ { 2 } ( \\prod _ { i = 1 } ^ { m } A _ { i } ) T \\iota } ) .\n$$",
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+ "text": "In the situation where the EQUILIBRIUM subroutine is taken as NASH, Theorem 15 provides the sample complexity bound of Multi-Nash-VI algorithm to find a $\\epsilon$ -approximate Nash equilibrium and its regret bound. Compared with our earlier result in two-player zero-sum games (Theorem 3), here the sample complexity scales as $S ^ { 2 } H ^ { 4 }$ instead of $S H ^ { 3 }$ . This is because the auxiliary bonus and Bernstein concentration technique do not apply here. Furthermore, the sample complexity is proportional to $\\textstyle \\prod _ { i = 1 } ^ { m } A _ { i }$ , which increases exponentially as the number of players increases. ",
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+ "text": "Runtime of Algorithm 3 We remark that while the Nash guarantee is the strongest among the three guarantees presented in Theorem 15, the runtime of Algorithm 3 in the Nash case is not guaranteed to be polynomial and in the worst case PPAD-hard (due to the hardness of the NASH subroutine). In contrast, the CE and CCE guarantees are weaker, but the corresponding algorithms are guaranteed to finish in polynomial time. ",
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+ "type": "text",
+ "text": "C.3 MULTIPLAYER REWARD-FREE LEARNING ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "We can also generalize VI-Zero to the multiplayer setting and obtain Algorithm 4, Multi-VI-Zero, which is almost the same as VI-Zero except that its exploration bonus √ $\\beta _ { t }$ is larger than that of VIZero by a $\\sqrt { S }$ factor. ",
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+ "type": "text",
+ "text": "Similar to Theorem 5, we have the following theoretical guarantee claiming that any $\\{ \\mathrm { N A S H , C C E , C E } \\}$ of the ${ \\mathcal { M } } ( { \\widehat { \\mathbb { P } } } , { \\widehat { r } } ^ { i } )$ $( i \\in [ N ] )$ is also an approximate $\\{ \\mathrm { N A S H , C C E , C E } \\}$ of the true Markov game ${ \\mathcal { M } } ( \\mathbb { P } , r ^ { i } )$ , where $\\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } }$ is the empirical transition outputted by Algorithm 4 and $\\widehat { r } ^ { i }$ is the empirical estimate of $r ^ { i }$ . ",
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+ "text": "Theorem 16 (Multi-VI-Zero). There exists an absolute constant $c _ { * }$ , for any $p ~ \\in ~ ( 0 , 1 ] , ~ \\epsilon ~ \\in$ $( 0 , H ]$ , $N \\in \\mathbb { N }$ , if we choose bonus $\\beta _ { t } ~ = ~ c \\sqrt { H ^ { 2 } S \\iota / t }$ with $\\iota \\ : = \\ : \\log ( N S A B T / p )$ and $K \\geq$ $c ( H ^ { 4 } S ^ { 2 } ( \\prod _ { i = 1 } ^ { m } A _ { i } ) \\iota / \\epsilon ^ { 2 } )$ , then with probability at least $1 - p ,$ , the output $\\widehat { \\mathbb { P } } ^ { o u t }$ of Algorithm $^ { 4 }$ has the following property: for any $N$ fixed reward functions $r ^ { 1 } , \\ldots , r ^ { N }$ , any $\\{ \\mathrm { N A S H , C C E , C E } \\}$ of ",
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+ "type": "text",
+ "text": "Algorithm 4 Multiplayer Optimistic Value Iteration with Zero Reward (Multi-VI-Zero) ",
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+ "text": "1: Initialize: for any $( s , \\pmb { a } , h )$ , $\\widetilde { V } _ { h } ( s , a ) \\gets H$ , $\\Delta H$ , Nh(s, a) ← 0. \n2: for episode $k = 1 , \\ldots , K$ do \n3: for step $h = H , H - 1 , \\ldots , 1$ do \n4: for $( s , \\pmb { a } ) \\in \\mathcal { S } \\times \\mathcal { A } _ { 1 } \\times \\cdot \\cdot \\cdot \\times \\mathcal { A } _ { m } \\ : \\mathfrak { c }$ do \n5: $t \\gets N _ { h } ( s , \\pmb { a } )$ . \n6: if $t > 0$ then \n7: $\\widetilde { Q } _ { h } ( s , \\pmb { a } ) \\gets \\operatorname* { m i n } \\{ ( \\widehat { \\mathbb { P } } _ { h } \\widetilde { V } _ { h + 1 } ) ( s , \\pmb { a } ) + \\beta _ { t } , H \\} .$ \n8: for $s \\in S$ do \n9: $\\begin{array} { r } { \\pi _ { h } ( s ) \\gets \\arg \\operatorname* { m a x } _ { \\pmb { a } \\in \\mathcal { A } _ { 1 } \\times \\dots \\times \\mathcal { A } _ { m } } \\widetilde { Q } _ { h } ( s , \\pmb { a } ) . } \\end{array}$ \n10: $\\widetilde { V } _ { h } ( s ) ( \\mathbb { D } _ { \\pi _ { h } } \\widetilde { Q } _ { h } ) ( s )$ . \n11: if $\\widetilde { V } _ { 1 } ( s _ { 1 } ) < \\Delta$ then \n12: $\\Delta \\widetilde { V } _ { 1 } ( s _ { 1 } )$ and $\\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } } \\widehat { \\mathbb { P } }$ . \n13: for step $h = 1 , \\ldots , H$ do \n14: take action $\\mathbf { \\sigma } _ { \\mathbf { a } _ { h } } \\sim \\pi _ { h } ( \\cdot , \\cdot | s _ { h } )$ , observe next state $s _ { h + 1 }$ . \n15: add 1 to $N _ { h } ( s _ { h } , \\pmb { a } _ { h } )$ and $\\dot { N _ { h } } ( s _ { h } , \\pmb { a } _ { h } , s _ { h + 1 } )$ . \n16: $\\widehat { \\mathbb { P } } _ { h } ( \\cdot | s _ { h } , \\pmb { a } _ { h } ) \\gets N _ { h } ( s _ { h } , \\pmb { a } _ { h } , \\cdot ) / N _ { h } ( s _ { h } , \\pmb { a } _ { h } )$ . \n17: Output $\\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } }$ . ",
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+ "text": "Markov game $\\mathcal { M } ( \\widehat { \\mathbb { P } } ^ { o u t } , \\widehat { r } ^ { i } )$ is also an $\\epsilon$ -approximate $\\{ \\mathrm { N A S H , C C E , C E } \\}$ of the true Markov game ${ \\mathcal { M } } ( \\mathbb { P } , r ^ { i } )$ for all $i \\in [ N ]$ b. ",
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+ "type": "text",
+ "text": "The proof of Theorem 16 can be found in Appendix H.2. It is worth mentioning that the empirical Markov game $\\mathcal { M } ( \\widehat { \\mathbb { P } } ^ { \\mathrm { o u t } } , \\widehat { r } ^ { i } )$ may have multiple $\\{ \\mathrm { N a s h }$ equilibria, ${ \\mathrm { C C E s , C E s } } \\}$ and Theorem 16 ensures that all of them are $\\epsilon$ b-approximate $\\{ \\mathrm { N a s h }$ equilibria, $_ \\mathrm { C C E s , C E s } \\}$ of the true Markov game. Also, note that the sample complexity here is quadratic in the number of states because we are using the exploration bonus $\\beta _ { t } = \\sqrt { H ^ { 2 } S \\iota / t }$ that is larger than usual by a $\\sqrt { S }$ factor. ",
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+ "type": "text",
+ "text": "D BELLMAN EQUATIONS FOR MARKOV GAMES ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "In this section, we present the Bellman equations for different types of values in Markov games. ",
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+ "text": "Fixed policies. For any pair of Markov policy $( \\mu , \\nu )$ , by definition of their values in (1) (2), we have the following Bellman equations: ",
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+ "text": "$$\nQ _ { h } ^ { \\mu , \\nu } ( s , a , b ) = ( r _ { h } + \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\mu , \\nu } ) ( s , a , b ) , \\qquad V _ { h } ^ { \\mu , \\nu } ( s ) = ( \\mathbb { D } _ { \\mu _ { h } \\times \\nu _ { h } } Q _ { h } ^ { \\mu , \\nu } ) ( s )\n$$",
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+ "type": "text",
+ "text": "for all $( s , a , b , h ) \\in \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { B } \\times [ H ]$ , where $V _ { H + 1 } ^ { \\mu , \\nu } ( s ) = 0$ for all $s \\in S$ . ",
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+ "text": "Best responses. For any Markov policy $\\mu$ of the max-player, by definition, we have the following Bellman equations for values of its best response: ",
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+ "text": "$$\nQ _ { h } ^ { \\mu , \\dagger } ( s , a , b ) = ( r _ { h } + \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\mu , \\dagger } ) ( s , a , b ) , \\qquad V _ { h } ^ { \\mu , \\dagger } ( s ) = \\operatorname* { i n f } _ { \\nu \\in \\Delta _ { B } } ( \\mathbb { D } _ { \\mu _ { h } \\times \\nu } Q _ { h } ^ { \\mu , \\dagger } ) ( s ) ,\n$$",
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+ "type": "text",
+ "text": "for all $( s , a , b , h ) \\in \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { B } \\times [ H ]$ , where $V _ { H + 1 } ^ { \\mu , \\dagger } ( s ) = 0$ for all $s \\in S$ ",
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+ "type": "text",
+ "text": "Similarly, for any Markov policy $\\nu$ of the min-player, we also have the following symmetric version of Bellman equations for values of its best response: ",
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+ "text": "$$\nQ _ { h } ^ { \\dagger , \\nu } ( s , a , b ) = ( r _ { h } + \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\dagger , \\nu } ) ( s , a , b ) , \\qquad V _ { h } ^ { \\dagger , \\nu } ( s ) = \\operatorname* { s u p } _ { \\mu \\in \\Delta _ { A } } ( \\mathbb { D } _ { \\mu \\times \\nu _ { h } } Q _ { h } ^ { \\dagger , \\nu } ) ( s ) .\n$$",
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+ "type": "text",
+ "text": "for all $( s , a , b , h ) \\in \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { B } \\times [ H ]$ , where $V _ { H + 1 } ^ { \\dagger , \\nu } ( s ) = 0$ for all $s \\in S$ ",
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+ "text": "Nash equilibria. Finally, by definition of Nash equilibria in Markov games, we have the following Bellman optimality equations: ",
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+ "text": "$$\n\\begin{array} { r l } & { \\textstyle Q _ { h } ^ { \\star } ( s , a , b ) = ( r _ { h } + { \\mathbb P } _ { h } V _ { h + 1 } ^ { \\star } ) ( s , a , b ) } \\\\ & { \\textstyle V _ { h } ^ { \\star } ( s ) = \\operatorname* { s u p } _ { \\mu \\in \\Delta _ { \\cal A } } \\operatorname* { i n f } _ { \\nu \\in \\Delta _ { \\cal B } } ( { \\mathbb P } _ { \\mu \\times \\nu } Q _ { h } ^ { \\star } ) ( s ) = \\operatorname* { i n f } _ { \\nu \\in \\Delta _ { \\mathcal B } } \\operatorname* { s u p } _ { \\mu \\in \\Delta _ { \\cal A } } ( { \\mathbb D } _ { \\mu \\times \\nu } Q _ { h } ^ { \\star } ) ( s ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "for all $( s , a , b , h ) \\in \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { B } \\times [ H ]$ , where $V _ { H + 1 } ^ { \\star } ( s ) = 0$ for all $s \\in S$ . ",
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+ "type": "text",
+ "text": "E PROPERTIES OF COARSE CORRELATED EQUILIBRIUM",
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+ "text": "Recall the definition for CCE in our main paper (4), we restate it here after rescaling. For any pair of matrices $P , Q \\in [ 0 , 1 ] ^ { n \\times m }$ , the subroutine $\\mathrm { C C E } ( P , Q )$ returns a distribution $\\pi \\in \\Delta _ { n \\times m }$ that satisfies: ",
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+ "text": "$$\n\\begin{array} { l l } { { \\mathbb { E } } _ { ( a , b ) \\sim \\pi } P ( a , b ) \\geq \\displaystyle \\operatorname* { m a x } _ { a ^ { \\star } } { \\mathbb { E } } _ { ( a , b ) \\sim \\pi } P ( a ^ { \\star } , b ) } \\\\ { { \\mathbb { E } } _ { ( a , b ) \\sim \\pi } Q ( a , b ) \\leq \\displaystyle \\operatorname* { m i n } _ { b ^ { \\star } } { \\mathbb { E } } _ { ( a , b ) \\sim \\pi } Q ( a , b ^ { \\star } ) } \\end{array}\n$$",
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+ "text": "We make three remarks on CCE. First, a CCE always exists since a Nash equilibrium for a generalsum game with payoff matrices $( P , Q )$ is also a CCE defined by $( P , Q )$ , and a Nash equilibrium always exists. Second, a CCE can be efficiently computed, since above constraints (5) for CCE can be rewritten as $n + m$ linear constraints on $\\pi \\in \\Delta _ { n \\times m }$ , which can be efficiently resolved by standard linear programming algorithm. Third, a CCE in general-sum games needs not to be a Nash equilibrium. However, a CCE in zero-sum games is guaranteed to be a Nash equalibrium. ",
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+ "text": "Proposition 17. Let $\\pi = \\operatorname { C C E } ( Q , Q )$ , and $( \\mu , \\nu )$ be the marginal distribution over both players’ actions induced by $\\pi$ . Then $( \\mu , \\nu )$ is a Nash equilibrium for payoff matrix $Q$ . ",
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+ "text": "Proof of Proposition $^ { I 7 }$ . Let $N ^ { \\star }$ be the value of Nash equilibrium for $Q$ . Since $\\pi = \\operatorname { C C E } ( Q , Q )$ , by definition, we have: ",
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+ "text": "$$\n\\begin{array} { r l } & { \\mathbb { E } _ { ( a , b ) \\sim \\pi } Q ( a , b ) \\geq \\underset { a ^ { \\star } } { \\operatorname* { m a x } } \\mathbb { E } _ { ( a , b ) \\sim \\pi } Q ( a ^ { \\star } , b ) = \\underset { a ^ { \\star } } { \\operatorname* { m a x } } \\mathbb { E } _ { b \\sim \\nu } Q ( a ^ { \\star } , b ) \\geq N ^ { \\star } } \\\\ & { \\mathbb { E } _ { ( a , b ) \\sim \\pi } Q ( a , b ) \\leq \\underset { b ^ { \\star } } { \\operatorname* { m i n } } \\mathbb { E } _ { ( a , b ) \\sim \\pi } Q ( a , b ^ { \\star } ) = \\underset { b ^ { \\star } } { \\operatorname* { m i n } } \\mathbb { E } _ { a \\sim \\mu } Q ( a , b ^ { \\star } ) \\leq N ^ { \\star } } \\end{array}\n$$",
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+ "type": "text",
+ "text": "This gives: ",
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+ "img_path": "images/b92f1cbeef554dcf947404c5d1b43eb2b046cfe5e5f89914ca4e1c8a623276f3.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { a ^ { \\star } } \\mathbb { E } _ { b \\sim \\nu } Q ( a ^ { \\star } , b ) = \\operatorname* { m i n } _ { b ^ { \\star } } \\mathbb { E } _ { a \\sim \\mu } Q ( a , b ^ { \\star } ) = N ^ { \\star }\n$$",
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+ "type": "text",
+ "text": "which finishes the proof. ",
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+ "type": "text",
+ "text": "Intuitively, a CCE procedure can be used in Nash Q-learning for finding an approximate Nash equilibrium, because the values of upper confidence and lower confidence $\\overline { { Q } }$ and $\\underline { { \\boldsymbol { Q } } }$ ) will be eventually very close, so that the preconditions of Proposition 17 becomes approximately satisfied. ",
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+ "type": "text",
+ "text": "F PROOF FOR SECTION 3 – OPTIMISTIC NASH VALUE ITERATION ",
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+ "type": "text",
+ "text": "F.1 PROOF OF THEOREM 3 ",
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+ "type": "text",
+ "text": "We denote $V ^ { k }$ $^ { r k } , Q ^ { k } , \\pi ^ { k } ,$ , $\\mu ^ { k }$ and $\\nu ^ { k ^ { \\ 4 } }$ for values and policies at the beginning of the $k$ -th episode. In particular, $N _ { h } ^ { k } ( s , a , b )$ is the number we have visited the state-action tuple $( s , a , b )$ at the $h$ -th step before the $k$ -th episode. $N _ { h } ^ { k } ( s , a , b , s ^ { \\prime } )$ is defined by the same token. Using this notation, we can further define the empirical transition by $\\widehat { \\mathbb { P } } _ { h } ^ { k } \\big ( s ^ { \\prime } | s , a , b \\big ) : = N _ { h } ^ { k } \\big ( s , a , b , s ^ { \\prime } \\big ) / N _ { h } ^ { k } \\big ( s , a , b \\big )$ . If $N _ { h } ^ { k } ( s , a , b ) = 0$ , we set $\\widehat { \\mathbb { P } } _ { h } ^ { k } ( s ^ { \\prime } | s , a , b ) = 1 / S$ . ",
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+ "type": "text",
+ "text": "As a result, the bonus terms can be written as ",
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+ "img_path": "images/e9298315be7acd0303cf3089f837ed2e6fb4609e218c24551075956c25e3ef52.jpg",
+ "text": "$$\n\\beta _ { h } ^ { k } ( s , a , b ) : = C \\left( \\sqrt { \\frac { \\iota H ^ { 2 } } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { H ^ { 2 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\right)\n$$",
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+ "text": "$$\n\\gamma _ { h } ^ { k } ( s , a , b ) : = \\frac { C } { H } \\widehat { \\mathbb { P } } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { V } _ { h + 1 } ^ { k } ) ( s , a , b )\n$$",
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+ "type": "text",
+ "text": "for some large absolute constant $C > 0$ . ",
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+ "text": "Lemma 18. Let $c _ { 1 }$ be some large absolute constant. Define event $E _ { 0 }$ to be: for all $h , s , a , b , s ^ { \\prime }$ and $k \\in [ K ]$ , ",
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+ "type": "equation",
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+ "text": "$$\n\\begin{array} { r l } & { { ( | [ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) V _ { h + 1 } ^ { \\star } ] ( s , a , b ) | \\leq c _ { 1 } \\sqrt { \\frac { H ^ { 2 } \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } , } } \\\\ & | ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) ( s ^ { \\prime } \\mid s , a , b ) | \\leq c _ { 1 } ( \\sqrt { \\frac { \\operatorname* { m i n } \\{ \\mathbb { P } _ { h } ( s ^ { \\prime } \\mid s , a , b ) , \\widehat { \\mathbb { P } } _ { h } ^ { k } ( s ^ { \\prime } \\mid s , a , b ) \\} \\} { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "We have $\\mathbb { P } ( E _ { 1 } ) \\geq 1 - p .$ . ",
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+ "type": "text",
+ "text": "Proof. The proof is standard and folklore: apply standard concentration inequalities and then take a union bound. For completeness, we provide the proof of the second one here. ",
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+ "type": "text",
+ "text": "Consider a fixed $( s , a , b , h )$ tuple. ",
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+ "type": "text",
+ "text": "Let’s consider the following equivalent random process: (a) before the agent starts, the environment samples $\\{ s ^ { ( 1 ) } , s ^ { ( 2 ) } , \\ldots , s ^ { ( \\bar { K } ) } \\}$ independently from $\\mathbb { P } _ { h } ( \\cdot \\mid s , a , b )$ ; (b) during the interaction between the agent and environment, the time the agent reaches $( s , a , b , h )$ , the environment will make the agent transit to $s ^ { ( i ) }$ . Note that the randomness induced by this interaction procedure is exactly the same as the original one, which means the probability of any event in this context is the same as in the original problem. Therefore, it suffices to prove the target concentration inequality in this ’easy’ context. Denote by $\\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ( \\cdot \\mid s , a , b )$ the empirical estimate of $\\mathbb { P } _ { h } ( \\cdot \\mid s , a , b )$ calculated using $\\{ s ^ { ( 1 ) } , s ^ { ( 2 ) } , \\ldots , s ^ { ( t ) } \\}$ . For a fixed $t$ and $s ^ { \\prime }$ , by applying the Bernstein inequality and its empirical version, we have with probability at least $1 - { \\dot { p } } / { \\bar { S } } ^ { 2 } { \\dot { A } } { \\dot { B T } }$ , ",
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+ "text": "$$\n\\vert ( \\mathbb { P } _ { h } - \\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ) ( s ^ { \\prime } \\mid s , a , b ) \\vert \\leq \\mathscr { O } \\left( \\sqrt { \\frac { \\operatorname* { m i n } \\{ \\mathbb { P } _ { h } ( s ^ { \\prime } \\mid s , a , b ) , \\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ( s ^ { \\prime } \\mid s , a , b ) \\} s } { t } } + \\frac { \\iota } { t } \\right) .\n$$",
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+ "type": "text",
+ "text": "Now we can take a union bound over all $s , a , b , h , s ^ { \\prime }$ and $t \\in [ K ]$ , and obtain that with probability at least $1 - p$ , for all $s , a , b , h , s ^ { \\prime }$ and $t \\in [ K ]$ , ",
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+ "img_path": "images/a0e03f089a3ec8f7e59ac8e38d24e00c677789c80a954869b738e63beefaf7f5.jpg",
+ "text": "$$\n\\vert ( \\mathbb { P } _ { h } - \\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ) ( s ^ { \\prime } \\mid s , a , b ) \\vert \\leq \\mathscr { O } \\left( \\sqrt { \\frac { \\operatorname* { m i n } \\{ \\mathbb { P } _ { h } ( s ^ { \\prime } \\mid s , a , b ) , \\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ( s ^ { \\prime } \\mid s , a , b ) \\} s } { t } } + \\frac { \\iota } { t } \\right) .\n$$",
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+ "type": "text",
+ "text": "Note that the agent can reach each $( s , a , b , h )$ for at most $K$ times, this directly implies that the third inequality also holds with probability at least $1 - p$ . □ ",
+ "bbox": [
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+ "type": "text",
+ "text": "We begin with an auxiliary lemma bounding the lower-order term. ",
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+ "text": "Lemma 19. Suppose event $E _ { 0 }$ holds, then there exists absolute constant $c _ { 2 }$ such that: if function $g ( s )$ satisfies $| g | ( s ) \\leq ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s )$ for all $s$ , then ",
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+ "type": "equation",
+ "img_path": "images/4879d0618ab56907b0ea1f7b76ca839e8c53601142221ade62710e07aa488189.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\left| ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - { \\mathbb { P } } _ { h } ) g ( s , a , b ) \\right| } \\\\ & { \\displaystyle \\leq c _ { 2 } \\bigg ( \\frac { 1 } { H } \\operatorname* { m i n } \\{ \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) , { \\mathbb { P } } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) \\} + \\frac { H ^ { 2 } S { \\iota } } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\bigg ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Proof. By triangle inequality, ",
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+ "img_path": "images/214990b3fdeae023960c37bc3f6e3e40ce09d703e9fa70bbba7b1212017f73de.jpg",
+ "text": "$$\n\\begin{array} { r l } & { | ( \\widehat { \\mathbb { P } } _ { h } ^ { h } - \\mathbb { P } _ { h } ) | \\mathcal { Q } ( s , \\theta , \\theta ) | } \\\\ & { \\le \\sum _ { \\ell } | ( \\widehat { \\mathbb { P } } _ { h } ^ { h } - \\mathbb { P } _ { h } ) ( s ^ { \\prime } | s , \\alpha , \\theta , \\theta ) | | g | ( s ^ { \\prime } ) } \\\\ & { \\le \\sum _ { \\ell } | ( \\widehat { \\mathbb { P } } _ { h } ^ { h } - \\mathbb { P } _ { h } ) ( s ^ { \\prime } | s , \\alpha , \\theta ) | ( \\overline { { V } } _ { h + 1 } ^ { h } - { V } _ { h + 1 } ^ { h } ) ( s ^ { \\prime } ) } \\\\ & \\overset { ( i ) } { \\le } \\mathcal { O } ( \\sum _ { \\ell } \\sqrt \\frac { \\widehat { \\mathbb { P } } _ { h } ^ { h } ( s ^ { \\prime } | s ^ { \\prime } , \\alpha , \\theta ) | } { \\widehat { \\mathbb { P } } _ { h } \\operatorname* { m a x } \\{ N _ { h } ^ { h } ( s , \\alpha , \\theta , \\theta ) , 1 \\} } + \\frac { s } { \\operatorname* { m a x } \\{ N _ { h } ^ { h } ( s , \\alpha , \\theta , \\theta ) , 1 \\} } ) ( \\widehat { V } _ { h + 1 } ^ { h } - \\frac { V _ { h + 1 } ^ { h } } { \\sum _ { h + 1 } ^ { h } \\} ) ( s ^ { \\prime } ) ) } \\\\ & { \\overset { ( i i ) } { \\le } \\mathcal { O } ( \\sum _ { \\ell } \\frac { \\widehat { \\mathbb { P } } _ { h } ^ { h } ( s ^ { \\prime } | s , \\alpha , \\theta , b ) } { H } + \\frac { H _ { \\ell } } { \\operatorname* { m a x } \\{ N _ { h } ^ { h } ( s , \\alpha , \\theta , b ) , 1 \\} } ) ( \\widehat { V } _ { h + 1 } ^ { h } - { V } _ { h + 1 } ^ { h } ) ( s ^ { \\prime } ) ) } \\\\ & \\le \\mathcal { O } ( \\frac { \\widehat { \\mathbb { P } } _ { h } ^ { h } ( { V } _ { h + 1 } ^ { h } - \\frac { { V } _ { h + 1 } ^ { h } } { H } ) ( s , \\alpha , \\theta ) } { \\widehat { \\mathbb { P } } _ { H } } + \\frac { H _ { \\ell } } { \\operatorname* { m a x } \\{ N _ { h } ^ { h } ( s , \\alpha , b ) , 1 \\} } + \\frac { H ^ { 2 } S _ { \\ell } } \\operatorname* { m a x } \\{ N _ \\end{array}\n$$",
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+ "type": "text",
+ "text": "where $( i )$ is by the second inequality in event $E _ { 0 }$ and $( i i )$ is by AM-GM inequality. This proves the empirical version. Similarly, we can show ",
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+ "img_path": "images/c5ad715833e367ddb6a5b059020bfe2926ec6c5e95cfdcb9db8de9c2757ee5be.jpg",
+ "text": "$$\n\\vert ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) g ( s , a , b ) \\vert \\leq \\mathcal { O } \\left( \\frac { \\mathbb { P } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) } { H } + \\frac { H ^ { 2 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\right) ,\n$$",
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+ "type": "text",
+ "text": "Combining the two bounds completes the proof. ",
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+ },
+ {
+ "type": "text",
+ "text": "Now we can prove the upper and lower bounds are indeed upper and lower bounds of the best reponses. ",
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+ "type": "text",
+ "text": "Lemma 20. Suppose event $E _ { 0 }$ holds. Then for all $h , s , a , b$ and $k \\in [ K ]$ , we have ",
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+ "img_path": "images/e9c0b66c5fdf1e64ca3470957757439e97c87ec8fd378963b52d3d8174059f42.jpg",
+ "text": "$$\n\\left\\{ \\begin{array} { l l } { \\overline { { Q } } _ { h } ^ { k } ( s , a , b ) \\geq Q _ { h } ^ { \\dagger , \\nu ^ { k } } ( s , a , b ) \\geq Q _ { h } ^ { \\mu ^ { k } , \\dagger } ( s , a , b ) \\geq \\underline { { Q } } _ { h } ^ { k } ( s , a , b ) , } \\\\ { \\overline { { V } } _ { h } ^ { k } ( s ) \\geq V _ { h } ^ { \\dagger , \\nu ^ { k } } ( s ) \\geq V _ { h } ^ { \\mu ^ { k } , \\dagger } ( s ) \\geq \\underline { { V } } _ { h } ^ { k } ( s ) . } \\end{array} \\right.\n$$",
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+ "type": "text",
+ "text": "Proof. The proof is by backward induction. Suppose the bounds hold for the $Q$ -values in the $( h +$ $1 ) ^ { \\mathrm { t h } }$ step, we now establish the bounds for the $V$ -values in the $( h + 1 ) ^ { \\mathrm { t h } }$ step and $Q$ -values in the $h ^ { \\mathrm { t h } }$ -step. For any state $s$ : ",
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+ "img_path": "images/f50c562d688982b45c9ed7522fa98905d8a91dbbabbd09a04baa5fac77b208f8.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\overline { { V } } _ { h + 1 } ^ { k } ( s ) = \\mathbb { D } _ { \\pi _ { h + 1 } ^ { k } } \\overline { { Q } } _ { h + 1 } ^ { k } ( s ) } \\\\ & { \\qquad \\geq \\underset { \\mu } { \\operatorname* { m a x } } \\mathbb { D } _ { \\mu \\times \\nu _ { h + 1 } ^ { k } } \\overline { { Q } } _ { h + 1 } ^ { k } ( s ) } \\\\ & { \\qquad \\geq \\underset { \\mu } { \\operatorname* { m a x } } \\mathbb { D } _ { \\mu \\times \\nu _ { h + 1 } ^ { k } } Q _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } ( s ) = V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } ( s ) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Similarly, we can show $\\underline { { { V } } } _ { h + 1 } ^ { k } ( s ) \\leq V _ { h + 1 } ^ { \\mu ^ { k } , \\dagger } ( s )$ . Therefore, we have: for all $s$ ",
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+ "img_path": "images/4b074eb005fc6c2f6c1da3578e010b78afd9312821e3ca3228e63358666f68b2.jpg",
+ "text": "$$\n\\overline { { V } } _ { h + 1 } ^ { k } ( s ) \\geq V _ { h + 1 } ^ { \\dag , \\nu ^ { k } } ( s ) \\geq V _ { h + 1 } ^ { \\star } ( s ) \\geq V _ { h + 1 } ^ { \\mu ^ { k } , \\dag } ( s ) \\geq \\underline { { V } } _ { h + 1 } ^ { k } ( s ) .\n$$",
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+ "type": "text",
+ "text": "Now consider an arbitrary triple $( s , a , b )$ in the $h ^ { \\mathrm { t h } }$ step. We have ",
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+ "img_path": "images/2fdb38789723a532e7513dd268f548c11ce7935ee5cfeec89934e2fd461406a4.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\quad ( \\widetilde { Q } _ { h } ^ { k } - Q _ { h } ^ { \\dagger , \\nu ^ { k } } ) ( s , a , b ) } \\\\ & { \\geq \\operatorname* { m i n } \\Bigg \\{ ( \\widehat { P } _ { h } ^ { k } \\overline { { V } } _ { h + 1 } ^ { k } - \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } + \\beta _ { h } ^ { k } + \\gamma _ { h } ^ { k } ) ( s , a , b ) , 0 \\Bigg \\} } \\\\ & { \\geq \\operatorname* { m i n } \\Bigg \\{ ( \\widehat { P } _ { h } ^ { k } V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } - \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } + \\beta _ { h } ^ { k } + \\gamma _ { h } ^ { k } ) ( s , a , b ) , 0 \\Bigg \\} } \\\\ & { = \\operatorname* { m i n } \\Bigg \\{ ( \\underbrace { ( \\widehat { P } _ { h } ^ { k } - \\mathbb { P } _ { h } ) ( V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } - V _ { h + 1 } ^ { \\ast } ) ( s , a , b ) } _ { ( A ) } + \\underbrace { ( \\widehat { P } _ { h } ^ { k } - \\mathbb { P } _ { h } ) V _ { h + 1 } ^ { \\ast } ( s , a , b ) } _ { ( B ) } } \\\\ & { \\qquad + ( \\beta _ { h } ^ { k } + \\gamma _ { h } ^ { k } ) ( s , a , b ) , 0 \\Bigg \\} . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Invoking Lemma 19 with $g = V _ { h + 1 } ^ { \\dag , \\nu ^ { k } } - V _ { h + 1 } ^ { \\star }$ , ",
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+ "type": "equation",
+ "img_path": "images/2687648fe8a3f18fa1b664ecde7c0775650970b4192175f29417f9c207885714.jpg",
+ "text": "$$\n\\vert ( A ) \\vert \\leq \\mathcal { O } \\left( \\frac { \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) } { H } + \\frac { H ^ { 2 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\right) .\n$$",
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+ "type": "text",
+ "text": "By the first inequality in event $E _ { 0 }$ ",
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+ "type": "equation",
+ "img_path": "images/11aa388647059e1b92970c3420c97f7e2b099f0b7cc249f55026e3aacc175b85.jpg",
+ "text": "$$\n\\left| ( B ) \\right| \\leq \\mathcal { O } \\left( \\sqrt { \\frac { H ^ { 2 } \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } \\right) .\n$$",
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+ },
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+ "type": "text",
+ "text": "Plugging the two inequalities above back into (10) and recalling the definition of $\\beta _ { h } ^ { k }$ and $\\gamma _ { h } ^ { k }$ , we obtain $\\overline { { Q } } _ { h } ^ { k } ( s , a , b ) \\geq Q _ { h } ^ { \\dagger , \\nu ^ { k } } ( s , a , b )$ . Similarly, we can show ${ \\underline { { Q } } } _ { h } ^ { k } ( s , a , b ) \\leq Q _ { h } ^ { \\mu ^ { k } , \\dagger } ( s , a , b )$ . □ ",
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+ },
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+ "type": "text",
+ "text": "Finally we come to the proof of Theorem 3. ",
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+ },
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+ "type": "text",
+ "text": "Proof of Theorem 3. Suppose event $E _ { 0 }$ holds. We first upper bound the regret. By Lemma 20, the regret can be upper bounded by ",
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+ "img_path": "images/8b0843612a23a27a10b816c7cab3081040ba5b1f066b479e5fe618275c58542f.jpg",
+ "text": "$$\n\\sum _ { k } ( V _ { 1 } ^ { \\dagger , \\nu ^ { k } } ( s _ { 1 } ^ { k } ) - V _ { 1 } ^ { \\mu ^ { k } , \\dagger } ( s _ { 1 } ^ { k } ) ) \\leq \\sum _ { k } ( \\overline { { V } } _ { 1 } ^ { k } ( s _ { 1 } ^ { k } ) - \\underline { { V } } _ { 1 } ^ { k } ( s _ { 1 } ^ { k } ) ) .\n$$",
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+ "type": "text",
+ "text": "For brevity’s sake, we define the following notations: ",
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+ "img_path": "images/0f103a552abd62a7ea36148c154d07aeb7dbf916bac4053bf4e03defa5cf3f1f.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\left\\{ \\begin{array} { l l } { \\Delta _ { h } ^ { k } : = ( \\overline { { V } } _ { h } ^ { k } - \\underline { { V } } _ { h } ^ { k } ) ( s _ { h } ^ { k } ) , } \\\\ { \\zeta _ { h } ^ { k } : = \\Delta _ { h } ^ { k } - ( \\overline { { Q } } _ { h } ^ { k } - \\underline { { Q } } _ { h } ^ { k } ) ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) , } \\\\ { \\xi _ { h } ^ { k } : = \\mathbb { P } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) - \\Delta _ { h + 1 } ^ { k } . } \\end{array} \\right. } \\end{array}\n$$",
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+ {
+ "type": "text",
+ "text": "Let $\\mathcal { F } _ { h } ^ { k }$ be the $\\sigma$ -field generated by the following random variables: ",
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+ "type": "equation",
+ "img_path": "images/9818827a6b3a5eba7d0f8d8a4c200ee17f120a5969c191d4ce0c70d3c369482e.jpg",
+ "text": "$$\n\\begin{array} { r } { \\{ \\big ( s _ { i } ^ { j } , a _ { i } ^ { j } , b _ { i } ^ { j } , r _ { i } ^ { j } \\big ) \\} _ { ( i , j ) \\in [ H ] \\times [ k - 1 ] } \\bigcup \\{ \\big ( s _ { i } ^ { k } , a _ { i } ^ { k } , b _ { i } ^ { k } , r _ { i } ^ { k } \\big ) \\} _ { i \\in [ h - 1 ] } \\bigcup \\{ s _ { h } ^ { k } \\} . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "It’s easy to check $\\zeta _ { h } ^ { k }$ and $\\xi _ { h } ^ { k }$ are martingale differences with respect to $\\mathcal { F } _ { h } ^ { k }$ . With a slight abuse of notation, we use $\\beta _ { h } ^ { k }$ to refer to $\\beta _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } )$ and $N _ { h } ^ { k }$ to refer to $N _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } )$ in the following proof. ",
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+ {
+ "type": "text",
+ "text": "We have ",
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+ },
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+ "type": "equation",
+ "img_path": "images/14a1baeaf858739f0e89301684b2e198cee0d85f9f94b4c57bbc2c69da205dd8.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\mathrm { i } \\frac { \\partial } { \\partial x _ { i } } - \\zeta _ { n } ^ { k } + ( \\widetilde { Q } _ { i } ^ { 0 } - \\frac { Q ^ { k } } { 2 } ) \\Big ( \\delta _ { i , j } ^ { k } , \\delta _ { i , j } ^ { k } , \\widetilde { Q } _ { i } ^ { k } \\Big ) } \\\\ & { \\leq \\zeta _ { n } ^ { k } + 2 \\delta _ { i ^ { k } } ^ { k } - 2 \\gamma _ { k } ^ { k } + \\widetilde { P } _ { i } ^ { k } ( \\widetilde { V } _ { i - 1 } ^ { k } - \\gamma _ { k } ^ { k } , \\delta _ { i } ^ { k } ) \\Big ( \\delta _ { i , j } ^ { k } , \\delta _ { i ^ { k } } ^ { k } , \\delta _ { i ^ { k } } ^ { k } \\Big ) } \\\\ & { \\overset { ( a ) } { \\leq } \\delta _ { i ^ { k } } ^ { k } + 2 \\delta _ { i ^ { k } } ^ { k } - 2 \\gamma _ { k } ^ { k } + \\widetilde { P } _ { i } ^ { k } ( \\widetilde { V } _ { i - 1 } ^ { k } - \\gamma _ { k } ^ { k } , \\delta _ { i ^ { k } } ^ { k } ) \\Big ( \\delta _ { i ^ { k } , i ^ { k } } ^ { k } , \\delta _ { i ^ { k } } ^ { k } \\Big ) } \\\\ & { \\qquad + \\sigma _ { i ^ { k } } ^ { k } \\Big ( \\delta _ { i ^ { k } } ^ { k } ( \\widetilde { V } _ { i - 1 } ^ { k } - \\gamma _ { k , i } ^ { k } ) + \\widetilde { V } _ { i } ^ { k } \\widetilde { V } _ { i } ^ { k } \\Big ) + \\frac { \\widetilde { P } _ { i ^ { k } } ^ { k } } { \\operatorname* { m a x } \\{ 1 , \\widetilde { V } _ { i } ^ { k } , \\delta _ { i ^ { k } } ^ { k } \\} } } \\\\ & \\qquad + \\sigma _ { i ^ { k } } ^ { k } \\Big ( \\widetilde { V } _ { i } ^ { k } ( \\widetilde { V } _ { i - 1 } ^ { k } - \\widetilde { V } _ { i } ^ { k } ) \\Big ( \\delta _ { i ^ { k } } ^ { k } , \\delta _ { i ^ { k } } ^ { k } \\Big ) + \\frac { \\widetilde { P } _ { i ^ { k } } ^ { k } } { \\operatorname* { m a x } \\{ 1 , \\widetilde { V } _ { i } ^ { k } , \\delta _ { i ^ { k } } ^ { k } \\} \\Big ) } \\\\ & \\overset { ( b ) } { \\leq } \\delta _ { i ^ { k } } ^ { k } + 2 \\sigma \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "where $( i )$ and $( i i )$ follow from Lemma 19. ",
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+ },
+ {
+ "type": "text",
+ "text": "Define $c _ { 3 } : = 1 + 2 c _ { 2 } C$ and $\\kappa : = 1 + c _ { 3 } / H$ . Recursing this argument for $h \\in [ H ]$ and summing over $k$ , ",
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+ "type": "equation",
+ "img_path": "images/f606a9d394296a6f17b418e4eb6975c875de6bf406c5b80154e3a2df75542f8f.jpg",
+ "text": "$$\n\\sum _ { k = 1 } ^ { K } \\Delta _ { 1 } ^ { k } \\leq \\sum _ { k = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } \\left[ \\kappa ^ { h - 1 } \\zeta _ { h } ^ { k } + \\kappa ^ { h } \\xi _ { h } ^ { k } + \\mathcal { O } \\left( \\sqrt { \\frac { \\iota H ^ { 2 } } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } , 1 \\} } } + \\frac { H ^ { 2 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } , 1 \\} } \\right) \\right] .\n$$",
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+ "type": "text",
+ "text": "By Azuma-Hoeffding inequality, with probability at least $1 - p$ ",
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+ "img_path": "images/c7e90b268663dce59651d2bf3017f98c6d04c81c15a2462363639f0dade8bce8.jpg",
+ "text": "$$\n\\begin{array}{c} \\begin{array} { r l } & { \\{ \\displaystyle \\sum _ { k = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } \\kappa ^ { h - 1 } \\zeta _ { h } ^ { k } \\le \\mathcal { O } \\Big ( H \\sqrt { H K \\iota } \\Big ) = \\mathcal { O } \\Big ( \\sqrt { H ^ { 2 } T \\iota } \\Big ) , } \\\\ & { \\lfloor \\displaystyle \\sum _ { k = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } \\kappa ^ { h } \\xi _ { h } ^ { k } \\le \\mathcal { O } \\Big ( H \\sqrt { H K \\iota } \\Big ) = \\mathcal { O } \\Big ( \\sqrt { H ^ { 2 } T \\iota } \\Big ) . } \\end{array} \\end{array}\n$$",
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+ {
+ "type": "text",
+ "text": "By pigeon-hole argument, ",
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+ "type": "equation",
+ "img_path": "images/004449ef7f376502e2dc67ee792b5774a6f0abd49feaeb1317c38bdcece668fb.jpg",
+ "text": "$$\n\\begin{array} { l } { { \\displaystyle \\sum _ { k = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } \\frac { 1 } { \\sqrt { \\operatorname* { m a x } \\{ N _ { h } ^ { k } , 1 \\} } } \\le \\sum _ { s , a , b , h \\colon N _ { h } ^ { K } ( s , a , b ) > 0 } \\sum _ { n = 1 } ^ { K } \\frac { 1 } { \\sqrt { n } } + H S A B } } \\\\ { { \\displaystyle \\qquad \\le { \\mathcal O } \\Bigl ( \\sqrt { H S A B T } + H S A B \\Bigr ) , } } \\\\ { { \\displaystyle \\sum _ { k = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } \\frac { 1 } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } , 1 \\} } \\le \\sum _ { s , a , b , h \\colon N _ { h } ^ { K } ( s , a , b ) > 0 } \\sum _ { n = 1 } ^ { K _ { h } ^ { K } ( s , a , b ) } \\frac { 1 } { n } + H S A B } } \\\\ { { \\displaystyle \\qquad \\le { \\mathcal O } ( H S A B ) } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Put everything together, with probability at least $1 - 2 p$ (one $p$ comes from $\\mathbb { P } ( E _ { 0 } ) \\ge 1 - p$ and the other is for equation (12)), ",
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+ "type": "equation",
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+ "text": "$$\n\\sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \\dagger , \\nu ^ { k } } ( s _ { 1 } ^ { k } ) - V _ { 1 } ^ { \\mu ^ { k } , \\dagger } ( s _ { 1 } ^ { k } ) ) \\leq \\mathcal { O } \\Big ( \\sqrt { H ^ { 3 } S A B T \\iota } + H ^ { 3 } S ^ { 2 } A B \\iota ^ { 2 } \\Big )\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "For the PAC guarantee, recall that we choose $\\pi ^ { \\mathrm { o u t } } = \\pi ^ { k ^ { \\star } }$ such that $k ^ { \\star } = \\mathrm { a r g m i n } _ { k } \\left( \\overline { { V } } _ { 1 } ^ { k } - \\underline { { V } } _ { 1 } ^ { k } \\right) ( s _ { 1 } )$ As a result, ",
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+ "type": "equation",
+ "img_path": "images/539142e04a7f4766cb16ed449b56e452d70dbce0791a19250653e91d58bfc895.jpg",
+ "text": "$$\n( V _ { 1 } ^ { \\dagger , \\nu ^ { k ^ { \\prime } } } - V _ { 1 } ^ { \\mu ^ { k ^ { \\prime } } , \\dagger } ) ( s _ { 1 } ) \\leq ( \\overline { { V } } _ { 1 } ^ { k ^ { \\star } } - \\underline { { V } } _ { 1 } ^ { k ^ { \\star } } ) ( s _ { 1 } ) \\leq \\frac { 1 } { K } \\mathcal { O } \\Big ( \\sqrt { H ^ { 3 } S A B T \\iota } + H ^ { 3 } S ^ { 2 } A B \\iota ^ { 2 } \\Big ) ,\n$$",
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+ "type": "text",
+ "text": "which concludes the proof. ",
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+ {
+ "type": "text",
+ "text": "F.2 PROOF OF THEOREM 4 ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 21
+ },
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+ "type": "text",
+ "text": "We use the same notation as in Appendix F.1 except the form of bonus. Besides, we define the empirical variance operator ",
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+ "img_path": "images/4b4674a01870fd2d681b12ad821dbb50d305372569584f54857d9083f6bc28e5.jpg",
+ "text": "$$\n\\begin{array} { r } { \\widehat { \\mathbb { V } } _ { h } ^ { k } V ( s , a , b ) : = \\operatorname { V a r } _ { s ^ { \\prime } \\sim \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\cdot \\vert s , a , b ) } V ( s ^ { \\prime } ) } \\end{array}\n$$",
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+ "bbox": [
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+ "type": "text",
+ "text": "and the true (population) variance operator ",
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+ },
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+ "type": "equation",
+ "img_path": "images/5848e1d4a9ecc1eb24abdc25d35bd66ad21d1768451a01fe40e18cda3098b81a.jpg",
+ "text": "$$\n\\begin{array} { r } { \\mathbb { V } _ { h } V ( s , a , b ) : = \\operatorname { V a r } _ { s ^ { \\prime } \\sim \\mathbb { P } _ { h } ( \\cdot \\vert s , a , b ) } V ( s ^ { \\prime } ) } \\end{array}\n$$",
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+ },
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+ "type": "text",
+ "text": "for any function $V \\in \\Delta ^ { S }$ . If $N _ { h } ^ { k } ( s , a , b ) = 0$ , we simply set $\\widehat { \\mathbb { V } } _ { h } ^ { k } V ( s , a , b ) : = H ^ { 2 }$ regardless of the choice of $V$ . ",
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+ "type": "text",
+ "text": "As a result, the bonus terms can be written as ",
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+ {
+ "type": "equation",
+ "img_path": "images/47beec78ba31046899297daddf5c0e99d0b874ef4ef056716a42f8d7257e13a7.jpg",
+ "text": "$$\n\\beta _ { h } ^ { k } ( s , a , b ) : = C \\left( \\sqrt { \\frac { \\iota \\widehat { \\mathbb { V } } _ { h } ^ { k } [ ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] ( s , a , b ) } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { H ^ { 2 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\right)\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "for some absolute constant $C > 0$ . ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "Lemma 21. Let $c _ { 1 }$ be some large absolute constant. Define event $E _ { 1 }$ to be: for all $h , s , a , b , s ^ { \\prime }$ and $k \\in [ K ]$ , ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/73b635bb7f3701068bc510e8cbf3ed7b4612fb01ad67dc37acedf84b62786fd3.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\displaystyle ( | [ \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) V _ { h + 1 } ^ { \\star } ] ( s , a , b ) | \\leq c _ { 1 } ( \\sqrt { \\frac { \\widehat { \\mathbb { V } } _ { h } ^ { k } V _ { h + 1 } ^ { \\star } ( s , a , b ) \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { H \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } ) , } \\\\ & \\displaystyle | [ \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) ( s ^ { \\prime } | s , a , b ) | \\leq c _ { 1 } ( \\sqrt { \\frac { \\operatorname* { m i n } \\{ \\mathbb { P } _ { h } ( s ^ { \\prime } | s , a , b ) , \\widehat { \\mathbb { P } } _ { h } ^ { k } ( s ^ { \\prime } | s , a , b ) \\} \\} { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } ) , } \\\\ & { \\displaystyle \\| ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) ( \\cdot | s , a , b ) \\| _ { 1 } \\leq c _ { 1 } \\sqrt { \\frac { S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "We have $\\mathbb { P } ( E _ { 1 } ) \\ge 1 - p$ . ",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "The proof of Lemma 21 is highly similar to that of Lemma 18. Specifically, the first two can be proved by following basically the same argument in Lemma 18; the third one is standard (e.g., equation (12) in Azar et al. (2017)). We omit the proof here. ",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Since the proof of Lemma 19 does not depend on the form of the bonus, it can also be applied in this section. As in Appendix F.1, we will prove the upper and lower bounds are indeed upper and lower bounds of the best reponses. ",
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Lemma 22. Suppose event $E _ { 1 }$ holds. Then for all $h , s , a , b$ and $k \\in [ K ]$ , we have ",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "img_path": "images/affa7445a66c5e5afc44160670c1821d6610a2a0b466a1454f414faa0cc118cc.jpg",
+ "text": "$$\n\\left\\{ \\begin{array} { l l } { \\overline { { Q } } _ { h } ^ { k } ( s , a , b ) \\geq Q _ { h } ^ { \\dagger , \\nu ^ { k } } ( s , a , b ) \\geq Q _ { h } ^ { \\mu ^ { k } , \\dagger } ( s , a , b ) \\geq \\underline { { Q } } _ { h } ^ { k } ( s , a , b ) , } \\\\ { \\overline { { V } } _ { h } ^ { k } ( s ) \\geq V _ { h } ^ { \\dagger , \\nu ^ { k } } ( s ) \\geq V _ { h } ^ { \\mu ^ { k } , \\dagger } ( s ) \\geq \\underline { { V } } _ { h } ^ { k } ( s ) . } \\end{array} \\right.\n$$",
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+ },
+ {
+ "type": "text",
+ "text": "Proof. The proof is by backward induction and very similar to that of Lemma 20. Suppose the bounds hold for the $Q$ -values in the $( h + 1 ) ^ { \\mathrm { t h } }$ step, we now establish the bounds for the $V$ -values in the $( h + 1 ) ^ { \\mathrm { t h } }$ step and $Q$ -values in the $h ^ { \\mathrm { t h } }$ -step. ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "The proof for the $V$ -values is the same as (9). ",
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+ },
+ {
+ "type": "text",
+ "text": "For the $Q$ -values, the decomposition (10) still holds and $( A )$ is bounded using Lemma 19 as before. \nThe only difference is that we need to bound $( B )$ more carefully. ",
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+ "type": "text",
+ "text": "First, by the first inequality in event $E _ { 1 }$ , ",
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+ },
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+ "type": "equation",
+ "img_path": "images/0646474805cb8ab8bdd0d4231bb9a2f00c8d690dcf95435f0e16703168faa935.jpg",
+ "text": "$$\n\\vert ( B ) \\vert \\leq \\mathcal { O } \\left( \\sqrt { \\frac { \\widehat { \\mathbb { V } } _ { h } ^ { k } V _ { h + 1 } ^ { \\star } ( s , a , b ) \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { H \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\right) .\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "By the relation of $V$ -values in the $( h + 1 ) ^ { \\mathrm { t h } }$ step, ",
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+ "text": "$$\n\\begin{array} { r l } & { \\ | [ \\widehat { \\Psi } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] - \\widehat { \\Psi } _ { h } ^ { k } V _ { h + 1 } ^ { \\star } | ( s , a , b ) } \\\\ & { \\leq | [ \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] ^ { 2 } - ( \\widehat { \\mathbb { P } } _ { h } ^ { k } V _ { h + 1 } ^ { \\star } ) ^ { 2 } | ( s , a , b ) } \\\\ & { \\quad + | \\widehat { \\mathbb { P } } _ { h } ^ { k } [ ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] ^ { 2 } - \\widehat { \\mathbb { P } } _ { h } ^ { k } ( V _ { h + 1 } ^ { \\star } ) ^ { 2 } | ( s , a , b ) } \\\\ & { \\leq 4 H \\widehat { \\mathbb { P } } _ { h } ^ { k } | ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) / 2 - V _ { h + 1 } ^ { \\star } | ( s , a , b ) } \\\\ & { \\leq 4 H \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) , } \\end{array}\n$$",
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+ "bbox": [
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+ "type": "text",
+ "text": "which implies ",
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+ "type": "equation",
+ "img_path": "images/c2994ebdc1d2894d71a9b54efea93a715741e2430cd197f4110dccff1ba0fb68.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\sqrt { \\frac { \\hat { \\mathcal { V } } _ { h } ^ { k } V _ { h + 1 } ^ { k } ( s , a , b ) } { \\operatorname* { m a x } \\{ \\mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \\} } } } \\\\ & { \\leq \\sqrt { \\frac { \\vert \\hat { \\mathcal { V } } _ { h } ^ { k } [ ( \\widehat { V } _ { h + 1 } ^ { k } + \\frac { V _ { h + 1 } ^ { k } } { \\vert \\mathcal { V } _ { h + 1 } ^ { k } \\vert } ) / 2 ] + 4 H \\hat { \\mathcal { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { k } ) \\vert ( s , a , b ) } { \\operatorname* { m a x } \\{ \\mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \\} } } } \\\\ & \\leq \\sqrt { \\frac { \\vert \\hat { \\mathcal { V } } _ { h } ^ { k } [ ( \\overline { { V } } _ { h + 1 } ^ { k } + \\frac { V _ { h + 1 } ^ { k } } { \\vert \\mathcal { V } _ { h + 1 } ^ { k } \\vert } ) / 2 ] ( s , a , b ) } { \\operatorname* { m a x } \\{ \\mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\sqrt { \\frac { 4 H \\hat { \\mathbb { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\frac { V _ { h + 1 } ^ { k } } { \\vert \\mathcal { V } _ { h } ^ { k } ( s , a , b ) , 1 \\} ) } { \\operatorname* { m a x } \\{ \\mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \\} } } } \\\\ & \\overset { ( i ) } { \\leq } \\sqrt { \\frac { \\vert \\hat { \\mathcal { V } } _ { h } ^ { k } [ ( \\overline { { V } } _ { h + 1 } ^ { k } + \\frac { V _ { h + 1 } ^ { k } } { \\vert \\mathcal { V } _ { h + 1 } ^ { k } \\vert } ) / 2 \\vert ( s , a , b ) } { \\operatorname* { m a x } \\{ \\mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { \\vert \\hat { \\mathcal { V } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { k } ) \\vert } { H } + \\frac { 4 H ^ { 2 } \\varepsilon } \\operatorname* { m a x } \\{ \\mathcal { N } _ { h } ^ { k } ( s , a , b ) , 1 \\} \\end{array}\n$$",
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+ "type": "text",
+ "text": "where $( i )$ is by AM-GM inequality. ",
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+ "type": "text",
+ "text": "Plugging the above inequalities back into (10) and recalling the definition of $\\beta _ { h } ^ { k }$ and $\\gamma _ { h } ^ { k }$ completes the proof. □ ",
+ "bbox": [
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+ "type": "text",
+ "text": "We need one more lemma to control the error of the empirical variance estimator: ",
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+ "text": "Lemma 23. Suppose event $E _ { 1 }$ holds. Then for all $h , s , a , b$ and $k \\in [ K ]$ , we have ",
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+ "type": "equation",
+ "img_path": "images/ee13de770bef0de09a0461aae768dc32d714b1fc57fd88fd0bf5749c8c58095e.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\quad \\vert \\widehat { \\mathbb { V } } _ { h } ^ { k } [ ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] - \\mathbb { V } _ { h } V _ { h + 1 } ^ { \\pi ^ { k } } \\vert ( s , a , b ) } \\\\ & { \\le 4 H \\mathbb { P } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) + \\mathcal { O } \\bigg ( 1 + \\frac { H ^ { 4 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\bigg ) . } \\end{array}\n$$",
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+ "text": "Proof. By Lemma 22, we have $\\overline { { V } } _ { h } ^ { k } ( s ) \\geq V _ { h } ^ { \\pi ^ { k } } ( s ) \\geq \\underline { { V } } _ { h } ^ { k } ( s )$ . As a result, ",
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+ "img_path": "images/c79b973607eeb0785ff75ea7aaef7d5fa80d9a98af8843ed2b2329ba4838fa6d.jpg",
+ "text": "$$\n\\begin{array} { r l } & { | \\widehat { \\nabla } _ { h } ^ { k } [ ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] - \\mathbb { V } _ { h } V _ { h + 1 } ^ { \\pi ^ { k } } | ( s , a , b ) } \\\\ & { | [ \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } / 4 - \\mathbb { P } _ { h } ( V _ { h + 1 } ^ { \\pi ^ { k } } ) ^ { 2 } ] ( s , a , b ) - [ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) ) ^ { 2 } / 4 - ( \\mathbb { P } _ { h } V _ { h + 1 } ^ { \\pi ^ { k } } ) ^ { 2 } ] ( s , a , b ) | } \\\\ & { \\widehat { \\mathfrak { E } } [ \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - \\mathbb { P } _ { h } ( \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } + ( \\mathbb { P } _ { h } \\overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } ] ( s , a , b ) } \\\\ & { \\widehat { \\times } [ | ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) ( \\overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | + | \\mathbb { P } _ { h } [ ( \\overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } ] | } \\\\ & \\ + | ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \\mathbb { P } _ { h } \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | + | ( \\mathbb { P } _ { h } \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \\mathbb { P } _ { h } \\overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } \\end{array}\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "These terms can be bounded separately by using event $E _ { 1 }$ : ",
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+ "type": "equation",
+ "img_path": "images/975a0920ebf32b1d7be996d731cdab6df355045ddf0b75cff455dcf94b0750c3.jpg",
+ "text": "$$\n\\begin{array} { r l } & { ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) ( \\overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | ( s , a , b ) \\leq H ^ { 2 } \\| ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) ( \\cdot \\mid s , a , b ) \\| _ { 1 } \\leq \\mathcal { O } ( H ^ { 2 } \\sqrt { \\frac { S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } ) , } \\\\ & { \\mathbb { P } _ { h } [ ( \\overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } ] | ( s , a , b ) \\leq 2 H [ \\mathbb { P } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ] ( s , a , b ) , } \\\\ & { ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \\mathbb { P } _ { h } \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | ( s , a , b ) \\leq 2 H [ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) \\underline { { V } } _ { h + 1 } ^ { k } ] ( s , a , b ) \\leq \\mathcal { O } ( H ^ { 2 } \\sqrt { \\frac { S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } } \\\\ & { ( \\mathbb { P } _ { h } \\underline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } - ( \\mathbb { P } _ { h } \\overline { { V } } _ { h + 1 } ^ { k } ) ^ { 2 } | ( s , a , b ) \\leq 2 H [ \\mathbb { P } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ] ( s , a , b ) . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "Combining with $\\begin{array} { r } { H ^ { 2 } \\sqrt { \\frac { S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } \\leq 1 + \\frac { H ^ { 4 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } \\end{array}$ completes the proof. ",
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+ {
+ "type": "text",
+ "text": "Finally we come to the proof of Theorem 4. ",
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+ "type": "text",
+ "text": "Proof of Theorem 4. Suppose event $E _ { 1 }$ holds. We define $\\Delta _ { h } ^ { k }$ , $\\zeta _ { h } ^ { k }$ abd $\\xi _ { h } ^ { k }$ as in the proof of Theorem 3. As before we have ",
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+ "img_path": "images/cabb9e8437dc7f27dd9b44bd286dd7c59d4eb3665690ea2af7c03b998312d7f8.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\Delta _ { h } ^ { k } \\leq \\zeta _ { h } ^ { k } + \\left( 1 + \\frac { c _ { 3 } } { H } \\right) \\mathbb { P } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) \\left( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \\right) } \\\\ & { \\qquad + 4 c _ { 2 } C \\left( \\sqrt { \\frac { \\iota \\widehat { \\mathbb { V } } _ { h } ^ { k } [ ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] \\left( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \\right) } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) , 1 \\} } } + \\frac { H ^ { 2 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } \\left( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \\right) , 1 \\} } \\right) . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "By Lemma 23, ",
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+ "type": "equation",
+ "img_path": "images/e162b9c36a9b8f00fcc1a959f5bb2d9c28cf3575aa810d91073fbc2c7032b498.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\sqrt { \\frac { \\varepsilon \\hat { \\Psi } _ { h } ^ { k } [ ( \\overline { { V } } _ { h + 1 } ^ { k } + \\underline { { V } } _ { h + 1 } ^ { k } ) / 2 ] ( s , a , b ) } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } } \\\\ & { \\leq \\mathcal { O } \\left( \\sqrt { \\frac { \\varepsilon \\hat { \\Psi } _ { h } \\hat { V } _ { h + 1 } ^ { k } ( s , a , b ) + \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\sqrt { \\frac { H : \\mathbb { P } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { H ^ { 2 } \\sqrt { S } k } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\right) } \\\\ & { \\leq c _ { 4 } \\left( \\sqrt { \\frac { \\varepsilon \\hat { \\Psi } _ { h } \\hat { V } _ { h + 1 } ^ { \\varepsilon } ( s , a , b ) + \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { \\mathbb { P } _ { h } ( \\overline { { V } } _ { h + 1 } ^ { k } - \\underline { { V } } _ { h + 1 } ^ { k } ) ( s , a , b ) } { H } + \\frac { H ^ { 2 } \\sqrt { S } \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\right) , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where $c _ { 4 }$ is some absolute constant. Define $c _ { 5 } : = 4 c _ { 2 } c _ { 4 } C + c _ { 3 }$ and $\\kappa : = 1 + c _ { 5 } / H$ . Plugging (18) back into (17), we have ",
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+ "img_path": "images/51647fbab9bcb9a4b0c849b9f2d95070b52d14319516606c8d1415cf8432da9a.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\Delta _ { h } ^ { k } \\leq \\kappa \\Delta _ { h + 1 } ^ { k } + \\kappa \\xi _ { h } ^ { k } + \\zeta _ { h } ^ { k } } \\\\ & { \\qquad + \\ O \\biggl ( \\sqrt { \\frac { \\iota \\Psi _ { h } V _ { h + 1 } ^ { \\pi ^ { k } } \\bigl ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \\bigr ) } { N _ { h } ^ { k } \\bigl ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \\bigr ) } } + \\sqrt { \\frac { \\iota } { N _ { h } ^ { k } \\bigl ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \\bigr ) } } + \\frac { H ^ { 2 } S \\iota } { N _ { h } ^ { k } \\bigl ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } \\bigr ) } \\biggr ) \\biggr \\} . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Recursing this argument for $h \\in [ H ]$ and summing over $k$ , ",
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+ "img_path": "images/3c29bb8d884c2ede7231573f5cd53e9f195156cb34b67bae4a59c321022a7b5f.jpg",
+ "text": "$$\n\\begin{array} { r l r } { { \\sum _ { k = 1 } ^ { K } \\Delta _ { 1 } ^ { k } \\le \\sum _ { k = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } [ \\kappa ^ { h - 1 } \\zeta _ { h } ^ { k } + \\kappa ^ { h } \\xi _ { h } ^ { k } } } \\\\ & { } & { \\quad + \\operatorname { \\mathcal { O } } ( \\sqrt { \\frac { \\iota \\nabla _ { h } V _ { h + 1 } ^ { \\pi ^ { k } } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } , 1 \\} } } + \\sqrt { \\frac { \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } , 1 \\} } } + \\frac { H ^ { 2 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } , 1 \\} } ) ] . } \\end{array}\n$$",
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+ "type": "text",
+ "text": "The remaining steps are the same as that in the proof of Theorem 3 except that we need to bound the sum of variance term. ",
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+ "type": "text",
+ "text": "By Cauchy-Schwarz, ",
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+ "img_path": "images/0734b65eb4b33edf287c8a0acdd26f67b0929843d718216a724e02d097cf48fe.jpg",
+ "text": "$$\n\\sum _ { = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } \\sqrt { \\frac { \\mathbb { V } _ { h } V _ { h + 1 } ^ { \\pi ^ { k } } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) , 1 \\} } } \\leq \\sqrt \\sum _ { k = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } \\mathbb { V } _ { h } V _ { h + 1 } ^ { \\pi ^ { k } } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) \\cdot \\sum _ { k = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } \\frac { 1 } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) , 1 \\} }\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "By the Law of total variation and standard martingale concentration (see Lemma C.5 in Jin et al. (2018) for a formal proof), with probability at least $1 - p$ , we have ",
+ "bbox": [
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+ {
+ "type": "equation",
+ "img_path": "images/6f1c2537606d586029973853298e623f94a42a764a6eae97e4369a5d98cec121.jpg",
+ "text": "$$\n\\sum _ { k = 1 } ^ { K } \\sum _ { h = 1 } ^ { H } \\mathbb { V } _ { h } V _ { h + 1 } ^ { \\pi ^ { k } } ( s _ { h } ^ { k } , a _ { h } ^ { k } , b _ { h } ^ { k } ) { \\le } \\mathcal { O } \\big ( H T + H ^ { 3 } \\iota \\big ) .\n$$",
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+ "type": "text",
+ "text": "Putting all relations together, we obtain that with probability at least $1 - 2 p$ (one $p$ comes from $\\begin{array} { r } { \\mathbb { P } ( E _ { 1 } ) { \\stackrel { - } { = } } 1 - p } \\end{array}$ and the other comes from the inequality for bounding the variance term), ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/6a5c5ddf7c318525b0d4f1ee558d4ad96b16d7866c3a6c1a8f2cbe664f83a90c.jpg",
+ "text": "$$\n\\operatorname { R e g r e t } ( K ) = \\sum _ { k = 1 } ^ { K } ( V _ { 1 } ^ { \\dagger , \\nu ^ { k } } - V _ { 1 } ^ { \\mu ^ { k } , \\dagger } ) ( s _ { 1 } ) \\leq \\mathcal { O } ( \\sqrt { H ^ { 2 } S A B T \\iota } + H ^ { 3 } S ^ { 2 } A B \\iota ^ { 2 } ) .\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "Rescaling $p$ completes the proof. ",
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+ {
+ "type": "text",
+ "text": "G PROOF FOR SECTION 4 – REWARD-FREE LEARNING ",
+ "text_level": 1,
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "G.1 PROOF OF THEOREM 5 ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "In this section, we prove Theorem 5 for the single reward function case, i.e., $N = 1$ . The proof for multiple reward functions $N > 1$ ) simply follows from taking a union bound, that is, replacing the failure probability $p$ by $N p$ . ",
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+ "type": "text",
+ "text": "Let $( \\mu ^ { k } , \\nu ^ { k } )$ be an arbitrary Nash-equilibrium policy of $\\widehat { \\mathcal { M } } ^ { k } : = ( \\widehat { \\mathbb { P } } ^ { k } , \\widehat { r } ^ { k } )$ , where ${ \\widehat { \\mathbb { P } } } ^ { k }$ and $\\widehat { r } ^ { k }$ are our bempirical estimate of the transition and the reward at the beginning of the $k$ b’th episode in Algorithm 2, respectively. We use $N _ { h } ^ { k } ( s , a , b )$ to denote the number we have visited the state-action tuple $( s , a , b )$ at the $h$ -th step before the $k$ ’th episode. And the bonus used in the $k$ ’th episode can be written as ",
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+ "type": "equation",
+ "img_path": "images/c46e68f548a2977fa89857ffd4462408b47e65940c178b9ac59131651aef88a1.jpg",
+ "text": "$$\n\\beta _ { h } ^ { k } ( s , a , b ) : = C \\left( \\sqrt { \\frac { H ^ { 2 } \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { H ^ { 2 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\right) ,\n$$",
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+ "type": "text",
+ "text": "where $\\iota = \\log ( S A B T / p )$ and $C$ is some large absolute constant. ",
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+ "type": "text",
+ "text": "We use ${ \\widehat { Q } } ^ { k }$ and $\\widehat { V } ^ { k }$ to denote the empirical optimal value functions of ${ \\widehat { \\mathcal { M } } } ^ { k }$ as following. ",
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+ "type": "equation",
+ "img_path": "images/aaacff657754e52c239af3a88b7d69809316bc8817ebdd73fdaabcb135634417.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { \\widehat { Q } _ { h } ^ { k } ( s , a , b ) = ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widehat { V } _ { h + 1 } ) ( s , a , b ) + \\widehat { r } _ { h } ^ { k } ( s , a , b ) , } \\\\ { \\widehat { V } _ { h } ^ { k } ( s ) = \\displaystyle \\operatorname* { m a x } _ { \\mu } \\operatorname* { m i n } _ { \\nu } \\mathbb { D } _ { \\mu \\times \\nu } \\widehat { Q } _ { h } ^ { k } ( s ) . } \\end{array} \\right. } \\end{array}\n$$",
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+ },
+ {
+ "type": "text",
+ "text": "Since $( \\mu ^ { k } , \\nu ^ { k } )$ is a Nash-equilibrium policy of ${ \\widehat { \\mathcal { M } } } ^ { k }$ , we also have $\\widehat { V } _ { h } ^ { k } ( s ) = \\mathbb { D } _ { \\mu ^ { k } \\times \\nu ^ { k } } \\widehat { Q } _ { h } ^ { k } ( s )$ . ",
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+ "type": "text",
+ "text": "We begin with stating a useful property of matrix game that will be frequently used in our analysis. \nSince its proof is quite simple, we omit it here. ",
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+ "type": "text",
+ "text": "Lemma 24. Let $\\mathbf { X } , \\mathbf { Y } , \\mathbf { Z } \\in \\mathbb { R } ^ { A \\times B }$ and $\\Delta _ { d }$ be the $d$ -dimensional simplex. Suppose $| \\mathbf { X } - \\mathbf { Y } | \\leq \\mathbf { Z }$ where the inequality is entry-wise. Then ",
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+ "type": "equation",
+ "img_path": "images/c497df50cbf4cf0580ee2430cd1a79883d12b8a615f13ef401a20091869e8b97.jpg",
+ "text": "$$\n\\left| \\operatorname* { m a x } _ { \\mu \\in \\triangle _ { A } } \\operatorname* { m i n } _ { \\nu \\in \\triangle _ { B } } \\mu ^ { \\top } \\mathbf { X } \\nu - \\operatorname* { m a x } _ { \\mu \\in \\triangle _ { A } } \\operatorname* { m i n } _ { \\nu \\in \\triangle _ { B } } \\mu ^ { \\top } \\mathbf { Y } \\nu \\right| \\leq \\operatorname* { m a x } _ { i , j } \\mathbf { Z } _ { i j } .\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "Lemma 25. Let $c _ { 1 }$ be some large absolute constant such that $c _ { 1 } ^ { 2 } + c _ { 1 } \\leq C$ . Define event $E _ { 1 }$ to $b e$ : for all $h , s , a , b , s ^ { \\prime }$ and $k \\in [ K ]$ , ",
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+ {
+ "type": "equation",
+ "img_path": "images/1b007b2f2d38ff9ebba9ae9a32990195a540808b4f6245d9cf272c81e02fcf42.jpg",
+ "text": "$$\n\\begin{array}{c} \\begin{array} { r } { \\left\\{ \\vert \\left[ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) V _ { h + 1 } ^ { \\star } \\right] ( s , a , b ) \\vert \\le \\frac { c _ { 1 } } { 1 0 } \\sqrt { \\frac { H ^ { 2 } \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } , \\right.} \\\\ { \\left. \\left( \\widehat { \\boldsymbol { r } } _ { h } ^ { k } - \\boldsymbol { r } _ { h } \\right) ( s , a , b ) \\right. \\le \\frac { c _ { 1 } } { 1 0 } \\sqrt { \\frac { H ^ { 2 } \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } , } \\\\ { \\left. \\left( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } \\right) ( s ^ { \\prime } \\mid s , a , b ) \\right. \\le \\frac { c _ { 1 } } { 1 0 } \\left( \\sqrt { \\frac { \\widehat { \\mathbb { P } } _ { h } ^ { k } ( s ^ { \\prime } \\mid s , a , b ) \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } + \\frac { \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } \\right) . } \\end{array} \\end{array}\n$$",
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+ "type": "text",
+ "text": "We have $\\mathbb { P } ( E _ { 1 } ) \\geq 1 - p .$ ",
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+ {
+ "type": "text",
+ "text": "Proof. The proof is standard and folklore: apply standard concentration inequalities and then take a union bound. For completeness, we provide the proof of the third one here. ",
+ "bbox": [
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+ "type": "text",
+ "text": "Consider a fixed $( s , a , b , h )$ tuple. ",
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+ "type": "text",
+ "text": "Let’s consider the following equivalent random process: (a) before the agent starts, the environment samples $\\{ s ^ { ( 1 ) } , s ^ { ( 2 ) } , \\ldots , s ^ { ( \\bar { K } ) } \\}$ independently from $\\mathbb { P } _ { h } ( \\cdot \\mid s , a , b )$ ; (b) during the interaction between the agent and environment, the $i ^ { \\mathrm { t h } }$ time the agent reaches $( s , a , b , h )$ , the environment will make the agent transit to $s ^ { ( i ) }$ . Note that the randomness induced by this interaction procedure is exactly the same as the original one, which means the probability of any event in this context is the same as in the original problem. Therefore, it suffices to prove the target concentration inequality in this ’easy’ context. Denote by $\\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ( \\cdot \\mid s , a , b )$ the empirical estimate of $\\mathbb { P } _ { h } ( \\cdot \\mid s , a , b )$ calculated using $\\{ s ^ { ( 1 ) } , s ^ { ( 2 ) } , \\ldots , s ^ { ( t ) } \\}$ . For a fixed $t$ and $s ^ { \\prime }$ , by the empirical Bernstein inequality, we have with probability at least $1 - p \\big / S ^ { 2 } A B T$ , ",
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+ "type": "text",
+ "text": "",
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+ "type": "equation",
+ "img_path": "images/b76e8a0352ebf0eb26979416483847c4aa80cc28717b5776fb4d9f7b328f2025.jpg",
+ "text": "$$\n\\vert ( \\mathbb { P } _ { h } - \\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ) ( s ^ { \\prime } \\mid s , a , b ) \\vert \\leq \\mathcal { O } \\left( \\sqrt { \\frac { \\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ( s ^ { \\prime } \\mid s , a , b ) \\iota } { t } } + \\frac { \\iota } { t } \\right) .\n$$",
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+ "type": "text",
+ "text": "Now we can take a union bound over all $s , a , b , h , s ^ { \\prime }$ and $t \\in [ K ]$ , and obtain that with probability at least $1 - p$ , for all $s , a , b , h , s ^ { \\prime }$ and $t \\in [ K ]$ , ",
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+ "type": "equation",
+ "img_path": "images/03486516d2e34e3a46766befda7dfe71cb2b823d1872b46beddc4371d3119cb8.jpg",
+ "text": "$$\n\\vert ( \\mathbb { P } _ { h } - \\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ) ( s ^ { \\prime } \\mid s , a , b ) \\vert \\leq \\mathcal { O } \\left( \\sqrt { \\frac { \\widehat { \\mathbb { P } } _ { h } ^ { ( t ) } ( s ^ { \\prime } \\mid s , a , b ) \\iota } { t } } + \\frac { \\iota } { t } \\right) .\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "Note that the agent can reach each $( s , a , b , h )$ for at most $K$ times, this directly implies that the third inequality also holds with probability at least $1 - p$ . □ ",
+ "bbox": [
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+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "The following lemma states that the empirical optimal value functions are close to the true optimal ones, and their difference is controlled by the exploration value functions calculated in Algorithm 2. ",
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+ "type": "text",
+ "text": "Lemma 26. Suppose event $E _ { 1 }$ (defined in Lemma 25) holds. Then for all $h , s , a , b$ and $k \\in [ K ]$ , we have, ",
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+ "type": "equation",
+ "img_path": "images/f8efd015c5c3b2ce9b8ce2ba83545ddae661e3917253246c516ddf0fe606a8a0.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { \\Big | \\widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \\star } ( s , a , b ) \\Big | \\le \\widetilde { Q } _ { h } ^ { k } ( s , a , b ) , } \\\\ { \\Big | \\widehat { V } _ { h } ^ { k } ( s ) - V _ { h } ^ { \\star } ( s ) \\Big | \\le \\widetilde { V } _ { h } ^ { k } ( s ) . } \\end{array} \\right. } \\end{array}\n$$",
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+ "bbox": [
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+ {
+ "type": "text",
+ "text": "Proof. Let’s prove by doing backward induction on $h$ . The case of $h = H + 1$ holds trivially. ",
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+ "type": "text",
+ "text": "Assume the conclusion hold for $( h + 1 )$ ’th step. For $h$ ’th step, ",
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+ "type": "equation",
+ "img_path": "images/3580352f6c642498dad50853a1ef68c81c7358c46774c25647328cc0228ac274.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\quad \\Bigl | \\widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \\star } ( s , a , b ) \\Bigr | } \\\\ & { \\le \\operatorname* { m i n } \\Big \\{ \\bigl | [ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - { \\mathbb { P } } _ { h } ) V _ { h + 1 } ^ { \\star } ] ( s , a , b ) \\bigr | + | ( \\widehat { r } _ { h } ^ { k } - r _ { h } ) ( s , a , b ) | + \\bigl | [ \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \\star } ) ] ( s , a , b ) \\bigr | , H \\Bigr \\} } \\\\ & { \\overset { ( i ) } { \\le } \\operatorname* { m i n } \\Big \\{ \\beta _ { h } ^ { k } ( s , a , b ) + ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ) ( s , a , b ) , H \\Big \\} \\overset { ( i i ) } { = } \\widetilde { Q } _ { h } ^ { k } ( s , a , b ) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "where $( i )$ follows from the induction hypothesis and event $E _ { 1 }$ , and $( i i )$ follows from the definition of $\\widetilde { Q } _ { h } ^ { k }$ . By Lemma 24, we immediately obtain $| \\widehat { V } _ { h } ^ { k } ( s ) - V _ { h } ^ { \\star } ( s ) | \\le \\widetilde { V } _ { h } ^ { k } ( s )$ . ",
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+ "type": "text",
+ "text": "Now, we are ready to establish the key lemma in our analysis using Lemma 26. ",
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+ "type": "text",
+ "text": "Lemma 27. Suppose event $E _ { 1 }$ (defined in Lemma 25) holds. Then for all $h , s , a , b$ and $k \\in [ K ]$ , we have ",
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+ "type": "equation",
+ "img_path": "images/77ecd1a422c142d4385d797a06eeba9ba89dfb68796e26cdf6bd48483710c8fd.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { \\vert \\widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \\dagger , \\nu ^ { k } } ( s , a , b ) \\vert \\le \\alpha _ { h } \\widetilde { Q } _ { h } ^ { k } ( s , a , b ) , } \\\\ { \\vert \\widehat { V } _ { h } ^ { k } ( s ) - V _ { h } ^ { \\dagger , \\nu ^ { k } } ( s ) \\vert \\le \\alpha _ { h } \\widetilde { V } _ { h } ^ { k } ( s ) , } \\end{array} \\right. } \\end{array}\n$$",
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+ "type": "text",
+ "text": "and ",
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+ "type": "equation",
+ "img_path": "images/858e05f09b76aa3ec7e8f7a5ba2be52c94725caf4a5ff3bacf1bd5f04307e39b.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { \\vert \\widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \\mu ^ { k } , \\dagger } ( s , a , b ) \\vert \\le \\alpha _ { h } \\widetilde { Q } _ { h } ^ { k } ( s , a , b ) , } \\\\ { \\vert \\widehat { V } _ { h } ^ { k } ( s ) - V _ { h } ^ { \\mu ^ { k } , \\dagger } ( s ) \\vert \\le \\alpha _ { h } \\widetilde { V } _ { h } ^ { k } ( s ) , } \\end{array} \\right. } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where $\\alpha _ { H + 1 } = 0$ and $\\begin{array} { r } { \\alpha _ { h } = [ ( 1 + \\frac { 1 } { H } ) \\alpha _ { h + 1 } + \\frac { 1 } { H } ] \\leq 4 } \\end{array}$ . ",
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+ },
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+ "type": "text",
+ "text": "Proof. We only prove the first set of inequalities. The second one follows exactly the same. Again, the proof is by performing backward induction on $h$ . It is trivial to see the conclusion holds for ",
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+ "type": "text",
+ "text": "$( H + 1 ) ^ { \\prime }$ ’th step with $\\alpha _ { H + 1 } = 0$ . Now, assume the conclusion holds for $( h + 1 )$ ’th step. For $h$ ’th step, ",
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+ "type": "equation",
+ "img_path": "images/8a9a441a08aa2bde7cd1960eebcdc22f4537ea2d02f4b11591eca483f8382593.jpg",
+ "text": "$$\n\\begin{array} { r l } & { | \\widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \\dagger , \\nu } ( s , a , b ) | } \\\\ & { \\leq \\operatorname* { m i n } \\bigg \\{ | [ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) ( V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } - V _ { h + 1 } ^ { \\ast } ) ] ( s , a , b ) | + | ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) V _ { h + 1 } ^ { \\star } ( s , a , b ) | } \\\\ & { \\quad + | ( \\widehat { r } _ { h } ^ { k } - r _ { h } ) ( s , a , b ) | + | [ \\widehat { \\mathbb { P } } _ { h } ( \\widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } ) ] ( s , a , b ) | , H \\bigg \\} } \\\\ & { \\leq \\operatorname* { m i n } \\bigg \\{ \\underbrace { | [ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } ) ( V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } - V _ { h + 1 } ^ { \\ast } ) ] ( s , a , b ) | } _ { ( T _ { 1 } ^ { k } ) } + c _ { 1 } \\sqrt { \\frac { H ^ { 2 } \\iota } { \\operatorname* { m a x } \\{ N _ { h + 1 } ^ { k } ( s , a , b ) , 1 \\} } } } \\\\ & { \\qquad + \\underbrace { | [ \\widehat { \\mathbb { P } } _ { h } ( \\widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } ) ] ( s , a , b ) | } _ { ( T _ { 2 } ^ { k } ) } , H \\bigg \\} , } \\end{array}\n$$",
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+ {
+ "type": "text",
+ "text": "where the second inequality follows from the definition of event $E _ { 1 }$ . ",
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+ "type": "text",
+ "text": "We can control the term $( T _ { 1 } )$ by combining Lemma 26 and the induction hypothesis to bound $| V _ { h + 1 } ^ { \\dag , \\nu ^ { k } } - V _ { h + 1 } ^ { \\star } |$ , and then applying the third inequality in event $E _ { 1 }$ : ",
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+ "type": "equation",
+ "img_path": "images/ddab21d302b330fcff26874981c4f14f62dcb12de905752407eec0b49ada4f10.jpg",
+ "text": "$$\n\\begin{array} { r l } & { ( T _ { 1 } ) \\leq \\displaystyle \\sum _ { s ^ { \\prime } } | \\widehat { \\mathbb { P } } _ { h } ^ { k } ( s ^ { \\prime } \\mid s , a , b ) - { \\mathbb { P } } _ { h } ( s ^ { \\prime } \\mid s , a , b ) | | V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } - V _ { h + 1 } ^ { \\star } ( s ^ { \\prime } ) | } \\\\ & { \\quad \\leq \\displaystyle \\sum _ { s ^ { \\prime } } | \\widehat { \\mathbb { P } } _ { h } ^ { k } ( s ^ { \\prime } \\mid s , a , b ) - { \\mathbb { P } } _ { h } ( s ^ { \\prime } \\mid s , a , b ) | \\Big ( | V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } - \\widehat { V } _ { h + 1 } ^ { k } ( s ^ { \\prime } ) | + | \\widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \\star } ( s ^ { \\prime } ) | \\Big ) } \\\\ & { \\quad \\leq \\displaystyle \\sum _ { s ^ { \\prime } } | \\widehat { \\mathbb { P } } _ { h } ^ { k } ( s ^ { \\prime } \\mid s , a , b ) - { \\mathbb { P } } _ { h } ( s ^ { \\prime } \\mid s , a , b ) | ( \\alpha _ { h + 1 } + 1 ) \\widetilde { V } _ { h + 1 } ^ { k } } \\\\ & { \\quad \\leq \\displaystyle \\frac { ( \\alpha _ { h + 1 } + 1 ) } { H } ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ) ( s , a , b ) + \\displaystyle \\frac { c _ { 1 } ^ { 2 } ( \\alpha _ { h + 1 } + 1 ) H ^ { 2 } S _ { L } } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "The term $( T _ { 2 } )$ is bounded by directly applying the induction hypothesis ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/ce19ed3ae56cddc5ccd2be95d2d2195fed582906e66624bb0a0a8973a86fd3a3.jpg",
+ "text": "$$\n| [ \\widehat { \\mathbb { P } } _ { h } ( \\widehat { V } _ { h + 1 } ^ { k } - V _ { h + 1 } ^ { \\dagger , \\nu ^ { k } } ) ] ( s , a , b ) | \\leq \\alpha _ { h + 1 } [ \\widehat { \\mathbb { P } } _ { h } \\widetilde { V } _ { h + 1 } ^ { k } ] ( s , a , b ) .\n$$",
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+ "type": "text",
+ "text": "Plugging (29) and (30) into (28), we obtain ",
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+ },
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+ "type": "equation",
+ "img_path": "images/5d473549c55909ff076cc3f9b276d6fa6df07f1b9d9c7db236dfe57b001adae1.jpg",
+ "text": "$$\n\\begin{array} { r l } & { ~ \\left. \\widehat { Q } _ { h } ^ { k } ( s , a , b ) - Q _ { h } ^ { \\dagger , s ^ { \\prime } } ( s , a , b ) \\right. } \\\\ & { \\leq \\operatorname* { m i n } \\bigg \\{ ( 1 + \\frac { 1 } { H } ) \\alpha _ { h + 1 } + \\frac { 1 } { H } [ \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ] ( s , a , b ) + c _ { 1 } \\sqrt { \\frac { H ^ { 2 } \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } } } \\\\ & { ~ + \\frac { c _ { 1 } ^ { 2 } ( \\alpha _ { h + 1 } + 1 ) H ^ { 2 } S \\iota } { \\operatorname* { m a x } \\{ N _ { h } ^ { k } ( s , a , b ) , 1 \\} } , H \\bigg \\} } \\\\ & { \\overset { ( i ) } { \\leq } \\operatorname* { m i n } \\bigg \\{ \\bigg ( ( 1 + \\frac { 1 } { H } ) \\alpha _ { h + 1 } + \\frac { 1 } { H } \\bigg ) [ \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ] ( s , a , b ) + \\beta _ { h } ^ { k } ( s , a , b ) , H \\bigg \\} } \\\\ & { \\overset { ( i i ) } { \\leq } \\bigg ( ( 1 + \\frac { 1 } { H } ) \\alpha _ { h + 1 } + \\frac { 1 } { H } \\bigg ) \\widehat { Q } _ { h } ^ { k } ( s , a , b ) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "type": "text",
+ "text": "where $( i )$ follows from the definition of $\\beta _ { h } ^ { k }$ , and $( i i )$ follows from the definition of $\\widetilde Q _ { h } ^ { k }$ . Therefore, by (31), choosing $\\begin{array} { r } { \\alpha _ { h } = [ ( 1 + \\frac { 1 } { H } ) \\alpha _ { h + 1 } + \\frac { \\top } { H } ] } \\end{array}$ suffices for the purpose of induction. ",
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+ {
+ "type": "text",
+ "text": "Now, let’s prove the inequality for $V$ functions. ",
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+ "type": "equation",
+ "img_path": "images/0005b60042febbab3bb9542d34ec7cd7ff64b5de7a11db731c1c34f370762b60.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\lvert ( \\widehat { V } _ { h } ^ { k } - V _ { h } ^ { \\dagger , \\nu ^ { k } } ) ( s ) \\rvert \\stackrel { ( i ) } { = } \\lvert \\displaystyle \\operatorname* { m a x } _ { \\mu \\in \\triangle _ { A } } ( { \\mathbb D } _ { \\mu , \\nu ^ { k } } \\widehat { Q } _ { h } ^ { k } ) ( s ) - \\displaystyle \\operatorname* { m a x } _ { \\mu \\in \\triangle _ { A } } ( { \\mathbb D } _ { \\mu , \\nu ^ { k } } Q _ { h } ^ { \\dagger , \\nu ^ { k } } ) ( s ) \\rvert } \\\\ & { \\qquad \\stackrel { ( i i ) } { \\leq } \\displaystyle \\operatorname* { m a x } _ { a , b } \\Big [ \\alpha _ { h } \\widetilde { Q } _ { h } ^ { k } ( s , a , b ) \\Big ] = \\alpha _ { h } \\widetilde { V } _ { h } ^ { k } ( s ) , } \\end{array}\n$$",
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+ "type": "text",
+ "text": "where $( i )$ follows from the definition of $\\widehat { V } _ { h } ^ { k }$ and $V _ { h } ^ { \\dag , \\nu ^ { k } }$ , and $( i i )$ uses (31) and Lemma 24. ",
+ "bbox": [
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+ "type": "text",
+ "text": "Theorem 28 (Guarantee for UCB-VI from Azar et al. (2017)). For any $p \\in \\mathsf { \\Gamma } ( 0 , 1 ]$ , choose the exploration bonus $\\beta _ { t }$ in Algrothm $2 \\ : a s$ (20). Then, with probability at least $1 - p$ , ",
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+ "type": "equation",
+ "img_path": "images/cacfc2a6fb1683ada14cb7180994fe63c8d05b430f3ea052a57e5955b016abc1.jpg",
+ "text": "$$\n\\sum _ { k = 1 } ^ { K } \\widetilde { V } _ { 1 } ^ { k } ( s _ { 1 } ) \\leq \\mathcal { O } ( \\sqrt { H ^ { 4 } S A K \\iota } + H ^ { 3 } S ^ { 2 } A \\iota ^ { 2 } ) .\n$$",
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+ "bbox": [
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+ "type": "text",
+ "text": "Proof of Theorem 5. Recall that o ${ \\mathfrak { x } } = \\arg \\operatorname* { m i n } _ { k \\in [ K ] } { \\widetilde { V } } _ { h } ^ { k } ( s )$ . By Lemma 27 and Theorem 28, with probability at least $1 - 2 p$ , ",
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+ "type": "equation",
+ "img_path": "images/0e24d3c8cd3ecac9c30aaf44235514f686f3f6417f4bb4808013984ef4fdf8a1.jpg",
+ "text": "$$\n\\begin{array} { r l } { { V _ { h } ^ { \\dagger , \\nu ^ { \\mathrm { o u t } } } ( s ) - V _ { h } ^ { \\mu ^ { \\mathrm { o u t } } , \\dagger } ( s ) \\le | V _ { h } ^ { \\dagger , \\nu ^ { \\mathrm { o u t } } } ( s ) - \\widehat { V } _ { h } ^ { \\mathrm { o u t } } ( s ) | + | \\widehat { V } _ { h } ^ { \\mathrm { o u t } } ( s ) - V _ { h } ^ { \\mu ^ { \\mathrm { o u t } } , \\dagger } ( s ) | } } \\\\ & { \\le 8 \\widetilde { V } _ { h } ^ { \\mathrm { o u t } } ( s ) \\le \\mathcal { O } ( \\sqrt { \\frac { H ^ { 4 } S A \\iota } { K } } + \\frac { H ^ { 3 } S ^ { 2 } A \\iota ^ { 2 } } { K } ) . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
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+ "type": "text",
+ "text": "Rescaling $p$ completes the proof. ",
+ "bbox": [
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+ ],
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+ },
+ {
+ "type": "text",
+ "text": "G.2 VANILLA NASH VALUE ITERATION ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 28
+ },
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+ "type": "text",
+ "text": "Here, we provide one optional algorithm, Vanilla Nash VI, for computing the Nash equilibrium policy for a known model. Its only difference from the value iteration algorithm for MDPs is that the maximum operator is replaced by the minimax operator in Line 7. We remark that the Nash equilibrium for a two-player zero-sum game can be computed in polynomial time. ",
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+ },
+ {
+ "type": "text",
+ "text": "Algorithm 5 Vanilla Nash Value Iteration ",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "1: Input: model $\\widehat { \\mathcal { M } } = ( \\widehat { \\mathbb { P } } , \\widehat { r } )$ . \n2: Initialize: for all $( s , a , b )$ , $V _ { H + 1 } ( s , a , b ) 0$ . \n3: for step $h = H , H - 1 , \\ldots , 1$ do \n4: for $( s , a , b ) \\in \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { B }$ do \n5: $\\begin{array} { r } { Q _ { h } ( s , a , b ) \\gets [ \\widehat { \\mathbb { P } } _ { h } V _ { h + 1 } ] ( s , a , b ) + \\widehat { r } _ { h } ( s , a , b ) . } \\end{array}$ \n6: for $s \\in S$ do \n7: $\\big ( \\hat { \\mu } _ { h } ( \\cdot \\vert \\ s ) , \\hat { \\nu } _ { h } ( \\cdot \\ \\vert \\ s ) \\big ) \\gets \\mathrm { N A S H - Z E R O - S U M } ( Q _ { h } ( s , \\cdot , \\cdot ) )$ . \n8: $V _ { h } ( s ) \\gets \\hat { \\mu } _ { h } ( \\cdot \\mid s ) ^ { \\top } Q _ { h } ( s , \\cdot , \\cdot ) \\hat { \\nu } _ { h } ( \\cdot \\mid s )$ . \n9: Output $( \\hat { \\mu } , \\hat { \\nu } ) \\gets \\{ ( \\hat { \\mu } _ { h } ( \\cdot { | } s ) , \\hat { \\nu } _ { h } ( \\cdot { | } s ) ) \\} _ { ( h , s ) \\in [ H ] \\times { \\mathcal { S } } } .$ . ",
+ "bbox": [
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+ },
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+ "type": "text",
+ "text": "By recalling the definition of best responses in Appendix D, one can directly see that the output policy $( \\hat { \\mu } , \\hat { \\nu } )$ is a Nash equilibrium for $\\widehat { \\mathcal { M } }$ . ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "G.3 PROOF OF THEOREM 6 ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "In this section, we first prove a $\\Theta ( A B / \\epsilon ^ { 2 } )$ lower bound for reward-free matrix games, i.e., $S =$ $H = 1$ , and then generalize it to $\\Theta ( S A \\dot { B } H ^ { 2 } / \\epsilon ^ { 2 } )$ for the Markov games setting. ",
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+ },
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+ "type": "text",
+ "text": "G.3.1 REWARD-FREE MATRIX GAMES ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "In the matrix game, let the max-player pick row and the min-player pick column. We consider the following family of Bernoulli matrix games: ",
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+ "img_path": "images/b1df05e423aadb30b5df34081745e61b13e918b7cdbe86243525215f1b51ec7c.jpg",
+ "text": "$$\n\\mathfrak { M } ( \\epsilon ) = \\left\\{ \\mathcal { M } \\in \\mathbb { R } ^ { A \\times B } \\mathrm { ~ w i t h ~ } \\mathcal { M } _ { a b } = \\frac { 1 } { 2 } + ( 1 - 2 \\cdot \\mathbf { 1 } \\{ a \\neq a ^ { \\star } \\mathfrak { E } b = b ^ { \\star } \\} ) \\epsilon \\colon ( a ^ { \\star } , b ^ { \\star } ) \\in [ A ] \\times [ B ] \\right\\} ,\n$$",
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+ "type": "text",
+ "text": "where in matrix game $\\mathcal { M }$ , the reward is sampled from Bernoulli $( \\mathcal { M } _ { a b } )$ if the max-player picks the $a$ ’th row and the min-player picks the $b ^ { \\mathrm { : } }$ ’th column. ",
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+ "type": "text",
+ "text": "Min-player ",
+ "text_level": 1,
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+ "type": "equation",
+ "img_path": "images/faf1578b9e776e37ac7ed43bc40d794fe666db122e643fd247f92f6959842620.jpg",
+ "text": "$$\n\\begin{array} { r l } { \\mathrm { ~ a c t i o n } \\quad 1 } & { \\dots \\quad b ^ { \\star } - 1 \\quad b ^ { \\star } \\quad b ^ { \\star } + 1 \\quad \\dots \\quad B } \\\\ { \\mathrm { ~ 1 } \\quad + \\quad \\dots \\quad + \\quad - \\quad - \\quad + \\quad \\dots \\quad + } & { \\dots \\quad + } \\\\ { \\vdots \\quad \\vdots \\quad \\ddots \\quad } & { \\vdots \\quad \\vdots \\quad \\vdots \\quad \\ddots \\quad \\vdots } \\\\ { \\mathrm { ~ 2 } \\quad a ^ { \\star } - 1 \\quad + \\quad \\dots \\quad + \\quad - \\quad - } & { + \\quad \\dots \\quad + } \\\\ { \\mathrm { ~ a ^ { \\star } ~ - p l a y e r } \\quad a ^ { \\star } \\quad + \\quad \\dots \\quad + \\quad + \\quad + \\quad + \\quad \\dots \\quad + } \\\\ { \\ a ^ { \\star } + 1 \\quad + \\quad \\dots \\quad + \\quad - \\quad + \\quad - \\quad + \\quad \\dots \\quad + } \\\\ { \\vdots \\quad \\vdots \\quad \\vdots \\quad \\ddots \\quad } & { \\vdots \\quad \\vdots \\quad \\vdots \\quad \\ddots \\quad \\vdots } \\\\ { \\quad \\ A \\quad + \\quad \\dots \\quad + \\quad - \\quad - \\quad + \\quad \\dots \\quad + \\quad \\dots \\quad + } \\end{array}\n$$",
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+ "type": "text",
+ "text": "Above, we visualize the hard instance by using $^ +$ and − to represent $1 / 2 + \\epsilon$ and $1 / 2 { - \\epsilon }$ , respectively. It is direct to see that the optimal policy for the max-player is always picking the $a ^ { \\star }$ ’th row and the optimal policy for the min-player is always picking the $b ^ { \\star }$ ’th column. If the max-player picks the $a ^ { \\star }$ ’th row with probability smaller than $2 / 3$ , it is at least $\\epsilon / 1 0$ suboptimal. ",
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+ "type": "text",
+ "text": "Lemma 29. For any fixed matrix game $\\mathcal { M }$ from $\\mathfrak { M } ( \\epsilon )$ and $N \\in \\mathbb { N }$ , if an algorithm $\\mathcal { A }$ can output a policy that is at most $\\epsilon / 1 0$ suboptimal with probability at least $p$ using at most $N$ samples, then there exists an algorithm $\\hat { A }$ that can identify the best row in $\\mathcal { M }$ with probability at least $p$ using at most $N$ samples. ",
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+ "type": "text",
+ "text": "Proof. We simply define $\\hat { A }$ as running algorithm $\\mathcal { A }$ and choosing the most played row by its outputted policy as the guess for the best row. By simple calculation, one can show $\\hat { A }$ will output the best row in $\\mathcal { M }$ with probability at least $p$ . □ ",
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+ "type": "text",
+ "text": "Lemma 29 directly implies that in order to prove the desired lower bound for matrix games: ",
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+ "type": "text",
+ "text": "Claim 30. for any algorithm $\\mathcal { A }$ using at most $N = A B / ( 1 0 ^ { 3 } \\epsilon ^ { 2 } )$ samples, there exists a matrix game $\\mathcal { M }$ in $\\mathfrak { M } ( \\epsilon )$ such that when running $\\mathcal { A }$ on $\\mathcal { M }$ , it will output a policy that is at least $\\epsilon / 1 0$ suboptimal for the max-player with probability at least $1 / 4$ , ",
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+ "type": "text",
+ "text": "it suffices to prove the following claim: ",
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+ "type": "text",
+ "text": "Claim 31. for any algorithm $\\hat { A }$ using at most $N = A B / ( 1 0 ^ { 3 } \\epsilon ^ { 2 } )$ samples, there exists a matrix game $\\mathcal { M }$ in $\\mathfrak { M } ( \\epsilon )$ such that when running $\\hat { A }$ on $\\mathcal { M }$ , it will fail to identify the optimal row with probability at least $1 / 4$ . ",
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+ "type": "text",
+ "text": "Proof of Claim 31. WLOG, we assume $\\hat { A }$ is deterministic. Since this is the reward-free setting, being deterministic means that algorithm $\\hat { A }$ will always pull each arm $( a , b )$ for some fixed $n ( a , b )$ times and then output a guess for $a ^ { \\star }$ which is a function of the reward revealed. ",
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+ },
+ {
+ "type": "text",
+ "text": "Denote by $L$ the reward revealed after algorithm $\\hat { A }$ ’s pulling. Denote by $\\mathbb { P } _ { \\star }$ the probability induced by picking $\\mathcal { M }$ uniformly at random from $\\mathfrak { M } ( \\epsilon )$ and running $\\hat { A }$ on $\\mathcal { M }$ . Denote by $\\mathbb { P } _ { a , b }$ the probability induced by running $\\hat { A }$ on $\\mathcal { M }$ , whose indices of the special row and the special column are $a$ and $b$ , respectively. Denote by $\\mathbb { P } _ { 0 , b }$ the probability induced by running $\\mathcal { A }$ on $\\mathcal { M }$ , whose $b '$ th column are all $1 / \\bar { 2 } - \\epsilon$ and other columns are all $1 / 2 + \\epsilon$ . We want to mention that the $\\mathcal { M }$ we use to define $\\mathbb { P } _ { 0 , b }$ does not belong to $\\mathfrak { M } ( \\epsilon )$ . ",
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+ },
+ {
+ "type": "text",
+ "text": "We have ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/e0e1e6ed65d6ecef961821d116b25bddcc00c749c4c15f067e125361217c0d2d.jpg",
+ "text": "$$\n\\begin{array} { l } { \\displaystyle \\mathbb { P } _ { \\star } ( \\hat { A } ( L ) \\neq a ^ { \\star } ) \\geq \\frac { 1 } { A B } \\sum _ { u , b } \\mathbb { P } _ { 0 , b } ( \\hat { A } ( L ) \\neq a ) - \\frac { 1 } { A B } \\sum _ { a , b } \\| \\mathbb { P } _ { a , b } - \\mathbb { P } _ { 0 , b } \\| _ { 1 } } \\\\ { \\displaystyle \\geq 1 - \\frac { 1 } { A } - \\frac { 1 } { A B } \\sum _ { u , b } \\sqrt { 2 \\mathrm { K I } ( \\mathbb { P } _ { 0 , b } \\| \\mathbb { P } _ { a , b } ) } } \\\\ { \\displaystyle = 1 - \\frac { 1 } { A } - \\frac { 1 } { A B } \\sum _ { a , b } \\sqrt { 2 n ( a , b ) [ ( \\frac { 1 } { 2 } - \\epsilon ) \\log \\frac { \\frac { 1 } { 2 } - \\epsilon } { 2 } + ( \\frac { 1 } { 2 } + \\epsilon ) \\log \\frac { \\frac { 1 } { 2 } + \\epsilon } { 2 } ] } } \\\\ { \\displaystyle \\geq 1 - \\frac { 1 } { A } - \\frac { 1 0 } { A B } \\sum _ { a , b } \\sqrt { n ( a , b ) \\epsilon ^ { 2 } } } \\\\ { \\displaystyle \\geq 1 - \\frac { 1 } { A } - \\sqrt { \\frac { 1 0 0 N \\epsilon ^ { 2 } } { a B } } . } \\end{array}\n$$",
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+ },
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+ "type": "text",
+ "text": "Choosing $N = A B / ( 1 0 ^ { 3 } \\epsilon ^ { 2 } )$ concludes the proof. ",
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "G.3.2 REWARD-FREE MARKOV GAMES ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Now let’s generalize the $\\Theta ( A B / \\epsilon ^ { 2 } )$ lower bound to $\\Theta ( S A B H ^ { 2 } / \\epsilon ^ { 2 } )$ for reward-free Markov games. We define the following family of MDPs: ",
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+ "text": "$$\n\\Im ( \\epsilon ) : = \\left\\{ \\mathcal { I } ( \\boldsymbol { a } ^ { \\star } , \\boldsymbol { b } ^ { \\star } ) : ( \\boldsymbol { a } ^ { \\star } , \\boldsymbol { b } ^ { \\star } ) \\in [ A ] ^ { H \\times S } \\times [ B ] ^ { H \\times S } \\right\\} ,\n$$",
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+ "type": "text",
+ "text": "where MDP $\\mathcal { I } ( a ^ { \\star } , b ^ { \\star } )$ is defined as below: ",
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+ "type": "text",
+ "text": "• States and actions: $\\mathcal { I } ( a ^ { \\star } , b ^ { \\star } )$ is a finite-horizon MDP with $S + 1$ states of length $H + 1$ . There is a fixed initial state $s _ { 0 }$ in the first step, $S$ states $\\{ s _ { 1 } , \\ldots , s _ { S } \\}$ for the remaining steps. The two players have $A$ and $B$ actions, respectively. ",
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+ "type": "text",
+ "text": "• Rewards: there is no reward in the first step. For the remaining steps $h \\in \\{ 2 , \\dots , H + 1 \\}$ , if the agent takes action $( a , b )$ at state $s _ { i }$ in the $h ^ { \\mathrm { t h } }$ step, it will receive a binary reward sampled from ",
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+ "type": "equation",
+ "img_path": "images/90565ce7d449baa5d943cf123c308c2fed6b5477cca4d4c48c95028527c403ec.jpg",
+ "text": "$$\n\\operatorname { B e r n o u l l i } { \\left( { \\frac { 1 } { 2 } } + ( 1 - 2 \\cdot \\mathbf { 1 } \\{ a \\neq a _ { h - 1 , i } ^ { \\star } \\& b = b _ { h - 1 , i } ^ { \\star } \\} ) { \\frac { \\epsilon } { H } } \\right) }\n$$",
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+ "type": "text",
+ "text": "• Transitions: Regardless of the current state, actions and index of steps, the agent will always transit to one of $s _ { 1 } , \\ldots , s _ { S }$ uniformly at random. ",
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+ "type": "text",
+ "text": "It is direct to see that $\\mathcal { I } ( a ^ { \\star } , b ^ { \\star } )$ is a collection of $S H$ independent matrix games from $\\mathfrak { M } ( \\epsilon / H )$ . Therefore, the optimal policy for the max player is to always pick action $\\pmb { a } _ { h - 1 , i } ^ { \\star }$ whenever it reaches state $s _ { i }$ in the $h ^ { \\mathrm { t h } }$ step $\\left( h \\geq 2 \\right)$ ). In other words, $\\pmb { a } _ { h - 1 , i } ^ { \\star }$ is the unique optimal action for the step-state pair $( h , i )$ . ",
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+ "type": "text",
+ "text": "At a high level, in order to find an $\\epsilon$ -optimal policy for the above Markov game, we need to identify at least half of the entries of $\\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { \\delta } \\mathbf { a } \\left( \\frac { \\mathcal { \\nu } } { \\mathcal { \\delta } } \\right) \\delta \\mathbf { \\delta } \\left( \\frac { \\mathcal { \\nu } } { \\mathcal { \\delta } } \\right) .$ . Therefore, the number of episodes should be at least ",
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+ "text": "$$\n\\Theta \\bigg ( \\frac { A B } { ( \\epsilon / H ) ^ { 2 } } \\bigg ) \\times \\frac { S } { 2 } = \\Theta \\bigg ( \\frac { A B S H ^ { 2 } } { \\epsilon ^ { 2 } } \\bigg ) .\n$$",
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+ "type": "text",
+ "text": "Below we provide a formal proof of this argument, which is almost the same as that for the setting of reward-free matrix games. ",
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+ "type": "text",
+ "text": "We start by proving an analogue of Lemma 29. ",
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+ "type": "text",
+ "text": "Lemma 32. For any fixed matrix game $\\mathcal { I } ( a ^ { \\star } , b ^ { \\star } )$ from $\\Im ( \\epsilon )$ and $N \\in \\mathbb { N }$ , if an algorithm $\\mathcal { A }$ can output a policy that is at most $\\epsilon / 1 0 ^ { 3 }$ suboptimal with probability at least $p$ using at most $N$ samples, then there exists an algorithm $\\hat { A }$ that can correctly identify at least $S H - \\lfloor S H / 5 0 0 \\rfloor$ entries of $\\mathbf { \\pmb { a } } ^ { \\star }$ with probability at least $p$ using at most $N$ samples. ",
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+ "text": "Proof. Denote by $\\pi$ the output policy for the max player. Denote by $Z$ the collection of $( h , i )$ ’s in $[ H ] \\times [ S ]$ such that $\\pi _ { h + 1 } ( \\boldsymbol { a } _ { h , i } ^ { \\star } \\mid s _ { i } ) \\le 2 / 3$ . ",
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+ "type": "text",
+ "text": "Observe that each time the max player picks a suboptimal action, it will incur an $2 \\epsilon / H$ suboptimality in expectation. As a result, if $\\pi$ is at most $\\epsilon / 1 0 ^ { 3 }$ -suboptimal, we must have ",
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+ "text": "$$\n\\frac { 1 } { S } \\sum _ { ( h , i ) \\in \\cal Z } ( 1 - \\pi _ { h + 1 } ( { \\pmb a } _ { h , i } ^ { \\star } \\mid s _ { i } ) ) \\times \\frac { 2 \\epsilon } { H } \\leq \\frac { \\epsilon } { 1 0 ^ { 3 } } ,\n$$",
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+ "type": "text",
+ "text": "which implies $| Z | \\le S H / 5 0 0$ , that is, for at most $\\lfloor S H / 5 0 0 \\rfloor$ different $i$ ’s, $\\pi ( \\mathbf { a } _ { i } ^ { \\star } \\mid s _ { i } ) \\ \\leq \\ 2 / 3$ . Therefore, we can simply pick $\\operatorname { a r g m a x } _ { a } \\pi _ { h + 1 } ( a \\mid s _ { i } )$ as the guess for $\\pmb { a } _ { h , i } ^ { \\star }$ . Since policy $\\pi$ is at most $\\epsilon / 1 0 ^ { 3 }$ suboptimal with probability at least $p$ , we can correctly identify the optimal actions for at least $S H - \\lfloor S H / 5 0 0 \\rfloor$ different $( s , h )$ pairs also with probability no smaller than $p$ . □ ",
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+ "type": "text",
+ "text": "Lemma 32 directly implies that in order to prove the desired lower bound for reward-free Markov games: ",
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+ "type": "text",
+ "text": "Claim 33. for any algorithm $\\mathcal { A }$ interacting with the environment for at most $K = A B S H ^ { 2 } / ( 1 0 ^ { 4 } \\epsilon ^ { 2 } )$ episodes, there exists $\\mathcal { I } \\in \\Im ( \\epsilon )$ such that when running $\\mathcal { A }$ on $\\mathcal { I }$ , it will output a policy that is at least $\\epsilon / 1 0 ^ { 3 }$ suboptimal for the max-player with probability at least $1 / 4$ , ",
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+ "type": "text",
+ "text": "it suffices to prove the following claim: ",
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+ "type": "text",
+ "text": "Claim 34. for any algorithm $\\hat { A }$ interacting with the environment for at most $K = A B S H ^ { 2 } / ( 1 0 ^ { 4 } \\epsilon ^ { 2 } )$ episodes, there exists $\\mathcal { I } \\in \\Im ( \\epsilon )$ such that when running $\\hat { A }$ on $\\mathcal { I }$ , it will fail to identify the optimal actions for at least $\\lfloor S H / 5 0 0 \\rfloor + 1$ different $( s , h )$ pairs with probability at least $1 / 4$ . ",
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+ "type": "text",
+ "text": "Proof of Claim 34. Denote by $\\mathbb { P } _ { \\star }$ $( \\mathbb { E } _ { \\star } )$ the probability (expectation) induced by picking $\\mathcal { I } ( a ^ { \\star } , b ^ { \\star } )$ uniformly at random from $\\Im ( \\epsilon )$ and running $\\hat { A }$ on $\\mathcal { I }$ . Denote by $n _ { \\mathrm { w r o n g } }$ the number of $( s , h )$ pairs for which $\\hat { A }$ fails to identify the optimal actions. Denote by $\\mathrm { e r r o r } _ { h , i }$ the indicator function of the event that $\\hat { A }$ fails to identify the optimal action for $( h + 1 , i )$ . ",
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+ "type": "text",
+ "text": "We prove by contradiction. Suppose for any $\\mathcal { I } \\in \\Im ( \\epsilon )$ , $\\hat { A }$ can identify the optimal actions for at least $S H - \\lfloor S H / 5 0 0 \\rfloor$ different $( s , h )$ pairs with probability larger than $3 / 4$ . Then we have ",
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+ "text": "$$\n\\mathbb { E } _ { \\star } [ n _ { \\mathrm { w r o n g } } ] \\leq \\frac { 1 } { 4 } \\times S H + \\frac { 3 } { 4 } \\times \\left\\lfloor \\frac { S H } { 5 0 0 } \\right\\rfloor \\leq \\frac { 1 0 1 S H } { 4 0 0 } .\n$$",
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+ "type": "text",
+ "text": "Since $\\begin{array} { r } { \\sum _ { ( h , i ) \\in [ H ] \\times [ S ] } \\mathbb { E } _ { \\star } [ \\mathrm { e r r o r } _ { h , i } ] = \\mathbb { E } _ { \\star } [ n _ { \\mathrm { w r o n g } } ] } \\end{array}$ , there must exists $( h ^ { \\prime } , i ^ { \\prime } ) \\in [ H ] \\times [ S ]$ such that $\\mathbb { E } _ { \\star } [ \\mathrm { e r r o r } _ { h ^ { \\prime } , i ^ { \\prime } } ] \\le 1 0 1 / 4 0 0$ . However, in the following, we show that for every $( h , i ) \\in [ H ] \\times [ S ] , \\hat { \\mathcal { A } }$ fails to identify the optimal action for the step-state pair $( h + 1 , i )$ with probability at least $1 / 3$ , which directly implies $\\mathbb { E } _ { \\star } [ \\mathrm { e r r o r } _ { h , i } ] \\ge 1 / 3$ for all $( h , i ) \\in [ H ] \\times [ S ]$ . As a result, we obtain a contraction and Claim 34 holds. ",
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+ "type": "text",
+ "text": "Now, let us prove that for every $( h , i ) \\in [ H ] \\times [ S ]$ , $\\mathbb { E } _ { \\star } [ \\mathrm { e r r o r } _ { h , i } ] \\ge 1 / 6$ . WLOG, we assume $\\hat { A }$ is deterministic and it runs for exactly $K = A B S H ^ { 2 } / ( 1 0 ^ { 3 } \\epsilon ^ { 2 } )$ episodes. In the following, we consider a fixed $( h ^ { \\prime } , i ^ { \\prime } )$ pair. For technical reason, we define MDP $\\mathcal { T } _ { - ( h ^ { \\prime } , i ^ { \\prime } ) } ( \\boldsymbol { a } ^ { \\star } , \\boldsymbol { b } ^ { \\star } )$ as below: ",
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+ "type": "text",
+ "text": "• States, actions and transitions: same as $\\mathcal { I } ( a ^ { \\star } , b ^ { \\star } )$ . ",
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+ "type": "text",
+ "text": "• Rewards: there is no reward in the first step. For the remaining steps $h \\in \\{ 2 , \\dots , H + 1 \\}$ , if the agent takes action $( a , b )$ at state $s _ { i }$ in the $h ^ { \\mathrm { t h } }$ step such that $( h - 1 , i ) \\neq ( h ^ { \\prime } , i ^ { \\prime } )$ , it will receive a binary reward sampled from ",
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+ "text": "$$\n\\operatorname { B e r n o u l l i } \\Big ( \\frac { 1 } { 2 } + ( 1 - 2 \\cdot \\mathbf { 1 } \\{ a \\neq a _ { h - 1 , i } ^ { \\star } \\& b = b _ { h - 1 , i } ^ { \\star } \\} ) \\frac { \\epsilon } { H } \\Big ) ,\n$$",
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+ "type": "text",
+ "text": "otherwise it will receive a binary reward sampled from ",
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+ "text": "$$\n\\operatorname { B e r n o u l l i } \\Big ( \\frac { 1 } { 2 } + ( 1 - 2 \\cdot { \\bf 1 } \\{ b = b _ { h - 1 , i } ^ { \\star } \\} ) \\frac { \\epsilon } { H } \\Big ) .\n$$",
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+ "type": "text",
+ "text": "Intuitively, $\\mathcal { T } _ { - ( h ^ { \\prime } , i ^ { \\prime } ) } ( \\boldsymbol { a } ^ { \\star } , \\boldsymbol { b } ^ { \\star } )$ is the same as $\\mathcal { I } ( a ^ { \\star } , b ^ { \\star } )$ except that for the max player, all its actions at state $s _ { i ^ { \\prime } }$ in the $h ^ { \\prime } ^ { \\mathrm { t h } }$ step are equivalent. In other words, $\\mathcal { T } _ { - ( h ^ { \\prime } , i ^ { \\prime } ) } ( \\boldsymbol { a } ^ { \\star } , \\boldsymbol { b } ^ { \\star } )$ is independent of $\\pmb { a } _ { h ^ { \\prime } , i ^ { \\prime } } ^ { \\star }$ . To proceed, we need to define the following notations: denote by $n ( a , b )$ the number of times $\\hat { A }$ picks action $( a , b )$ at state $s _ { i ^ { \\prime } }$ in the $\\left( h ^ { \\prime } + 1 \\right) ^ { \\mathrm { t h } }$ step; denote by $\\mathbb { P } ( \\cdot \\mid \\mathcal { I } ( a ^ { \\star } , b ^ { \\star } ) ) ( \\mathbb { E } [ \\cdot \\mid \\mathcal { I } ( a ^ { \\star } , b ^ { \\star } ) ] )$ the probability (expectation) induced by running algorithm $\\hat { A }$ on $\\mathcal { I } ( a ^ { \\star } , b ^ { \\star } )$ ; similarly, we define $\\mathbb { P } ( \\cdot \\mid$ $\\mathcal { T } _ { - ( h ^ { \\prime } , i ^ { \\prime } ) } ( { \\pmb a } ^ { \\star } , { \\pmb b } ^ { \\star } ) ) \\ ( \\mathbb { E } [ \\cdot \\ \\vert \\ { \\mathcal { T } } _ { - ( h ^ { \\prime } , i ^ { \\prime } ) } ( { \\pmb a } ^ { \\star } , { \\pmb b } ^ { \\star } ) ] )$ ; also recall that we denote by $\\mathbb { P } _ { \\star }$ $( \\mathbb { E } _ { \\star } )$ the probability (expectation) induced by picking $\\mathcal { I } ( a ^ { \\star } , b ^ { \\star } )$ uniformly at random from $\\Im ( \\epsilon )$ and running $\\hat { A }$ on $\\mathcal { I }$ ; denote by $L$ the whole trajectory of states, actions and rewards produced by algorithm $\\hat { A }$ in $N$ episodes; with slight abuse of notation, denote by $\\hat { \\mathcal { A } } ( L )$ the guess of $\\hat { A }$ for $\\pmb { a } _ { h ^ { \\prime } , i ^ { \\prime } } ^ { \\star }$ based on $L$ . ",
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+ "type": "text",
+ "text": "First, note that for any $( a , b ) ~ \\in ~ [ A ] \\times [ B ] , \\mathbb { E } [ n ( s , a ) ~ | ~ \\mathcal { J } _ { - ( h ^ { \\prime } , i ^ { \\prime } ) } ( { \\pmb a } ^ { \\star } , { \\pmb b } ^ { \\star } ) ]$ is independent of $( h ^ { \\prime } , i ^ { \\prime } , a ^ { \\star } , b ^ { \\star } )$ because the agent cannot observe any reward when interacting with the environment and the transition dynamics of different $\\mathcal { T } _ { - ( h ^ { \\prime } , i ^ { \\prime } ) } ( \\boldsymbol { a } ^ { \\star } , \\boldsymbol { b } ^ { \\star } )$ ’s are the same. For simplicity of notation, we denote this expectation by $m ( a , b )$ . Note that $\\begin{array} { r } { \\sum _ { a , b } m ( a , b ) = K / S } \\end{array}$ because the agent always reach state $s _ { i ^ { \\prime } }$ in the $\\left( h ^ { \\prime } + 1 \\right) ^ { \\mathrm { t h } }$ step with probability $1 / S$ regardless of the actions taken. ",
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+ "type": "text",
+ "text": "We have ",
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+ "type": "equation",
+ "img_path": "images/bc1cc65850330115728b1f20c0d81e29fc7c097a5fc5e19d0fa32f6c9ebcd1fc.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\qquad \\quad - [ - \\sum _ { j \\in \\mathbb { R } _ { + } } ( a _ { j } ^ { \\top } - \\sum _ { j \\in \\mathbb { R } _ { + } } ( a _ { j } ^ { \\top } - \\sum _ { j } - \\lfloor ( j \\cdot \\theta ) ^ { \\top } + k _ { j } ^ { \\top } ) \\lfloor \\frac { 1 } { 2 } \\rfloor ) } \\\\ { = } & { \\frac { 1 } { 3 } ( \\underbrace { \\sin ^ { 2 } ( \\theta ) } _ { \\mathrm { { \\mathbb { R } _ { + } } } } \\cos ( \\theta ) ) \\cos ( \\theta ) } \\\\ & { \\qquad \\quad \\sin ^ { 2 } ( \\theta ) } \\\\ & { \\le } & { \\frac { 1 } { 3 } ( \\underbrace { \\sin ^ { 2 } ( \\theta ) } _ { \\mathrm { { \\mathbb { R } _ { + } } } } \\cos ( \\theta ) \\cos ( \\theta ) - \\sin ^ { 2 } ( \\theta ) ( \\theta ) \\cos ( \\theta ) ) } \\\\ & { \\qquad \\quad - \\frac { 1 } { 3 } ( \\sin ^ { 2 } ( \\theta ) \\cos ( \\theta ) ) } \\\\ & { = } & { \\frac { 1 } { 3 } ( \\frac { 1 } { 3 } ( \\sin ^ { 2 } ( \\theta ) - \\sin ^ { 2 } ( \\theta ) ) ) \\cos ( \\theta ) } \\\\ & { \\qquad \\cos ( \\theta ) \\cos ( \\theta ) } \\\\ & { = } & { \\frac { 1 } { 3 } ( \\frac { 1 } { 3 } ( \\sin ^ { 2 } ( \\theta ) \\cos ( \\theta ) ) ) \\cos ( \\theta ) \\sin ^ { 2 } ( \\theta ) \\sin ( \\theta ) \\sin ^ { 2 } ( \\theta ) \\sin ^ { 3 } ( \\theta ) } \\\\ & { \\qquad \\quad \\sin ^ { 2 } ( \\theta ) } \\\\ & { = } & { \\frac { 1 } { 3 } ( \\frac { 1 } { 3 } ( \\sin ^ { 2 } ( \\theta ) \\cos ( \\theta ) ) \\cos ( \\theta ) \\sin ^ { 2 } ( \\theta ) + \\frac { 1 } { 3 } ( \\sin ^ { 2 } ( \\theta ) ( \\theta ) \\sin ^ { 2 } ( \\theta ) ) } \\\\ & { \\qquad \\quad \\sin ^ { 2 } ( \\theta ) \\cos ( \\theta ) } \\\\ & { \\qquad \\quad \\sin ^ { 2 } ( \\theta ) \\cos ( \\theta ) \\sin ^ { 2 } ( \\theta ) \\sin ^ { 2 } ( \\theta ) \\sin ^ { 2 } \\theta ) } \\\\ & \\qquad \\quad \\sin ^ { 2 } ( \\theta ) \\cos \\end{array}\n$$",
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+ "type": "text",
+ "text": "Plugging in $K = S A B H ^ { 2 } / ( 1 0 ^ { 4 } \\epsilon ^ { 2 } )$ completes the proof. ",
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+ {
+ "type": "text",
+ "text": "H PROOF FOR APPENDIX C – MULTI-PLAYER GENERAL-SUM MARKOV GAMES ",
+ "text_level": 1,
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "H.1 PROOF OF THEOREM 15 ",
+ "text_level": 1,
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+ {
+ "type": "text",
+ "text": "H.1.1 NE VERSION ",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "In this section, we prove Theorem 15 (NE version). As before, we begin with proving the optimistic estimations are indeed upper bounds of corresponding value and Q-value functions. ",
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+ "type": "text",
+ "text": "Lemma 35. With probability $1 - p ,$ , for any $( s , a , h , k , i )$ : ",
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+ "type": "equation",
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+ "text": "$$\n\\begin{array} { r l } & { \\overline { { Q } } _ { h , i } ^ { k } \\left( \\boldsymbol { s } , \\boldsymbol { a } \\right) \\geq Q _ { h , i } ^ { \\dag , \\pi _ { - i } ^ { k } } \\left( \\boldsymbol { s } , \\boldsymbol { a } \\right) , \\underline { { Q } } _ { h , i } ^ { k } \\left( \\boldsymbol { s } , \\boldsymbol { a } \\right) \\leq Q _ { h , i } ^ { \\pi ^ { k } } \\left( \\boldsymbol { s } , \\boldsymbol { a } \\right) , } \\\\ & { \\qquad \\overline { { V } } _ { h , i } ^ { k } \\left( \\boldsymbol { s } \\right) \\geq V _ { h , i } ^ { \\dag , \\pi _ { - i } ^ { k } } \\left( \\boldsymbol { s } \\right) , \\underline { { V } } _ { h , i } ^ { k } \\left( \\boldsymbol { s } \\right) \\leq V _ { h , i } ^ { \\pi ^ { k } } \\left( \\boldsymbol { s } \\right) . } \\end{array}\n$$",
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+ "bbox": [
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+ "type": "text",
+ "text": "Proof. know a d - $k$ , we step, $h = H + 1$ to um $h = 1$ . For base case, weequality (40) holds $( H + 1 )$ $V _ { H + 1 , i } ^ { k } \\left( s \\right) = \\overline { { { V } } } _ { H + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s \\right) = 0$ $( h + 1 )$ -th step, for the $h$ -th step, by definition of the $Q$ functions, ",
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+ "type": "equation",
+ "img_path": "images/b417ccb24295ab3e83126ee6e0209928b8b830ba0391d3a985b695bf2d590f00.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\overline { { Q } } _ { h , i } ^ { k } \\left( s , \\boldsymbol { a } \\right) - Q _ { h , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s , \\boldsymbol { a } \\right) = \\left[ \\widehat { \\mathbb { P } } _ { h } ^ { k } \\overline { { V } } _ { h + 1 , i } ^ { k } \\right] \\left( s , \\boldsymbol { a } \\right) - \\left[ \\mathbb { P } _ { h } V _ { h + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\right] \\left( s , \\boldsymbol { a } \\right) + \\beta _ { t } } \\\\ & { \\qquad = \\underbrace { \\widehat { \\mathbb { P } } _ { h } ^ { k } \\left( \\overline { { V } } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\right) \\left( s , \\boldsymbol { a } \\right) } _ { ( A ) } + \\underbrace { \\left( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } \\right) V _ { h + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s , \\boldsymbol { a } \\right) } _ { ( B ) } + \\beta _ { t } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "By induction hypothesis, for any $s ^ { \\prime }$ , $\\left( \\overline { { V } } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\right) ( s ^ { \\prime } ) \\geq 0$ , and thus $( A ) \\geq 0$ . By uniform concentration (e.g., Lemma 12 in Bai & Jin (2020)), $( B ) \\leq C \\sqrt { S H ^ { 2 } \\iota / N _ { h } ^ { k } ( s , \\pmb { a } ) } = \\beta _ { t }$ . Putting everything together we have $Q _ { h , i } ^ { k } \\left( s , { \\pmb a } \\right) - Q _ { h , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s , { \\pmb a } \\right) \\geq 0$ . The second inequality can be proved similarly. ",
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+ "page_idx": 33
+ },
+ {
+ "type": "text",
+ "text": "Now assume inequality (39) holds for the $h$ -th step, by definition of value functions and Nash equilibrium, ",
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+ "type": "equation",
+ "img_path": "images/0bcc0fd1ae32ab17a81e1fc7fbbb9fb1d62017ee5e2ee296ada5ff081e565141.jpg",
+ "text": "$$\n\\overline { { V } } _ { h , i } ^ { k } \\left( s \\right) = { \\mathbb D } _ { \\pi ^ { k } } \\overline { { Q } } _ { h , i } ^ { k } \\left( s \\right) = \\operatorname* { m a x } _ { \\mu } { \\mathbb D } _ { \\mu \\times \\pi _ { - i } ^ { k } } \\overline { { Q } } _ { h , i } ^ { k } \\left( s \\right) .\n$$",
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+ "bbox": [
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+ {
+ "type": "text",
+ "text": "By Bellman equation, ",
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+ "page_idx": 33
+ },
+ {
+ "type": "equation",
+ "img_path": "images/c7c003cc069542f000811cee3e2ba5fe379cf18112d5421e04b465a234c67e49.jpg",
+ "text": "$$\nV _ { h , i } ^ { \\dag , \\pi _ { - i } ^ { k } } \\left( s \\right) = \\operatorname* { m a x } _ { \\mu } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } Q _ { h , i } ^ { \\dag , \\pi _ { - i } ^ { k } } \\left( s \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 33
+ },
+ {
+ "type": "text",
+ "text": "Since by induction hypothesis, for any $( s , \\pmb { a } ) , \\overline { { Q } } _ { h , i } ^ { k } \\left( s , \\pmb { a } \\right) \\geq Q _ { h , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s , \\pmb { a } \\right)$ . As a result, we also have $\\overline { { V } } _ { h , i } ^ { k } \\left( s \\right) \\geq V _ { h , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s \\right)$ , which is exactly inequality (40) for the $h$ -th step. The second inequality can be proved similarly. ",
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+ {
+ "type": "text",
+ "text": "Proof of Theorem 15. Let us focus on the $i$ -th player and ignore the subscript when there is no confusion. To bound ",
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+ "type": "equation",
+ "img_path": "images/02bbfbb457acfcd0bd594cf9ca7aeb5d7dc008a14e04b10aa756bbec58254aff.jpg",
+ "text": "$$\n\\operatorname* { m a x } _ { i } \\left( V _ { 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } - V _ { 1 , i } ^ { \\pi ^ { k } } \\right) \\left( s _ { h } ^ { k } \\right) \\leq \\operatorname* { m a x } _ { i } \\left( \\overline { { V } } _ { 1 , i } ^ { k } - \\underline { { V } } _ { 1 , i } ^ { k } \\right) \\left( s _ { h } ^ { k } \\right) ,\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "we notice the following propogation: ",
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+ "type": "equation",
+ "img_path": "images/af7d90dbb80e77fd6cfaa93f685c95735fa2942d503482d0e0acdebe8a3a595c.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { ( \\overline { { Q } } _ { h , i } ^ { k } - \\underline { { Q } } _ { h , i } ^ { k } ) ( s , { \\mathbf { a } } ) \\leq \\widehat { \\mathbb { P } } _ { h } ^ { k } ( \\overline { { V } } _ { h + 1 , i } ^ { k } - \\underline { { V } } _ { h + 1 , i } ^ { k } ) ( s , { \\mathbf { a } } ) + 2 \\beta _ { h } ^ { k } ( s , { \\mathbf { a } } ) , } \\\\ { ( \\overline { { V } } _ { h , i } - \\underline { { V } } _ { h , i } ) ( s ) = [ \\mathbb { D } _ { \\pi _ { h } } ( \\overline { { Q } } _ { h , i } ^ { k } - \\underline { { Q } } _ { h , i } ^ { k } ) ] ( s ) . } \\end{array} \\right. } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "We can define $\\widetilde { Q } _ { h } ^ { k }$ and $\\widetilde { V } _ { h } ^ { k }$ recursively by $\\widetilde V _ { H + 1 } ^ { k } = 0$ and ",
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "equation",
+ "img_path": "images/36481d12a441a27a3bf0a4718a17db759400d6c12d37ec6ae4a8bb3e2865daf4.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { \\widetilde { Q } _ { h } ^ { k } ( s , \\pmb { a } ) = \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ( s , \\pmb { a } ) + 2 \\beta _ { h } ^ { k } ( s , \\pmb { a } ) , } \\\\ { \\widetilde { V } _ { h } ^ { k } ( s ) = [ \\mathbb { D } _ { \\pi _ { h } } \\widetilde { Q } _ { h } ^ { k } ] ( s ) . } \\end{array} \\right. } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "text",
+ "text": "Then we can prove inductively that for any $k , h , s$ and $\\textbf { \\em a }$ we have ",
+ "bbox": [
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+ "page_idx": 34
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+ "type": "equation",
+ "img_path": "images/aeefaa7bdce1b309dc604b08e83ccdc82067cf1d8c8ad80ec0a6530aefa35ed0.jpg",
+ "text": "$$\n\\left\\{ \\begin{array} { l l } { \\operatorname* { m a x } _ { i } ( \\overline { { Q } } _ { h , i } ^ { k } - \\underline { { Q } } _ { h , i } ^ { k } ) ( s , \\pmb { a } ) \\leq \\widetilde { Q } _ { h } ^ { k } ( s , \\pmb { a } ) , } \\\\ { \\operatorname* { m a x } _ { i } ( \\overline { { V } } _ { h , i } - \\underline { { V } } _ { h , i } ) ( s ) \\leq \\widetilde { V } _ { h } ^ { k } ( s ) . } \\end{array} \\right.\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "text",
+ "text": "Thus we only need to bound $\\textstyle \\sum _ { k = 1 } ^ { K } { \\widetilde { V } } _ { 1 } ^ { k } ( s )$ . Define the shorthand notation ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/5ed91bfe6f9826fb907c62b7cd620ea2972e7865c5ba82e16325c22e77ef556f.jpg",
+ "text": "$$\n\\left\\{ \\begin{array} { l l } { \\beta _ { h } ^ { k } : = \\beta _ { h } ^ { k } ( s _ { h } ^ { k } , \\boldsymbol { a } _ { h } ^ { k } ) , } \\\\ { \\Delta _ { h } ^ { k } : = \\widetilde { V } _ { h } ^ { k } ( s _ { h } ^ { k } ) , } \\\\ { \\zeta _ { h } ^ { k } : = \\mathbb { D } _ { \\pi ^ { k } } \\widetilde { Q } _ { h } ^ { k } \\left( s _ { h } ^ { k } \\right) - \\widetilde { Q } _ { h } ^ { k } ( s _ { h } ^ { k } , \\boldsymbol { a } _ { h } ^ { k } ) , } \\\\ { \\xi _ { h } ^ { k } : = \\mathbb { P } _ { h } \\widetilde { V } _ { h } ^ { k } ( s _ { h } ^ { k } , \\boldsymbol { a } _ { h } ^ { k } ) - \\Delta _ { h + 1 } ^ { k } . } \\end{array} \\right.\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "text",
+ "text": "We can check $\\zeta _ { h } ^ { k }$ and $\\xi _ { h } ^ { k }$ are martingale difference sequences. As a result, ",
+ "bbox": [
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+ "page_idx": 34
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+ "type": "equation",
+ "img_path": "images/40d40c6ed8c13de7e7c0192ed2d8f9245859c67d4d82a9c167b25887643ed47f.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\Delta _ { h } ^ { k } = \\mathbb { D } _ { \\pi ^ { k } } \\widetilde { Q } _ { h } ^ { k } \\left( s _ { h } ^ { k } \\right) } \\\\ & { \\quad = \\zeta _ { h } ^ { k } + \\widetilde { Q } _ { h } ^ { k } \\left( s _ { h } ^ { k } , \\pmb { a } _ { h } ^ { k } \\right) } \\\\ & { \\quad = \\zeta _ { h } ^ { k } + 2 \\beta _ { h } ^ { k } + \\mathbb { \\widehat { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } \\left( s _ { h } ^ { k } , \\pmb { a } _ { h } ^ { k } \\right) } \\\\ & { \\quad \\le \\zeta _ { h } ^ { k } + 3 \\beta _ { h } ^ { k } + \\mathbb { P } _ { h } \\widetilde { V } _ { h + 1 } ^ { k } \\left( s _ { h } ^ { k } , \\pmb { a } _ { h } ^ { k } \\right) } \\\\ & { \\quad = \\zeta _ { h } ^ { k } + 3 \\beta _ { h } ^ { k } + \\xi _ { h } ^ { k } + \\Delta _ { h + 1 } ^ { k } . } \\end{array}\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "Recursing this argument for $h \\in [ H ]$ and taking the sum, ",
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+ "page_idx": 34
+ },
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+ "type": "equation",
+ "img_path": "images/1c28383a2a7c44b03532fa936f1e7ae09b538742cf932f1f2b2980413a7532cb.jpg",
+ "text": "$$\n\\sum _ { k = 1 } ^ { K } \\Delta _ { 1 } ^ { k } \\leq \\sum _ { k = 1 } ^ { K } \\left( \\zeta _ { h } ^ { k } + 3 \\beta _ { h } ^ { k } + \\xi _ { h } ^ { k } \\right) \\leq O \\left( S \\sqrt { H ^ { 3 } T \\iota \\prod _ { i = 1 } ^ { M } A _ { i } } \\right) .\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "H.1.2 CCE VERSION ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "text",
+ "text": "The proof is very similar to the NE version. Specifically, the only part that uses the properties of NE there is Lemma 35. We prove a counterpart here. ",
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+ "page_idx": 34
+ },
+ {
+ "type": "text",
+ "text": "Lemma 36. With probability $1 - p ,$ , for any $( s , a , h , k , i )$ : ",
+ "bbox": [
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+ "page_idx": 34
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+ "type": "equation",
+ "img_path": "images/d74500c8976ce10fa5d014747f5a837c3843ec8f7619367fbb1cdc476a559c5e.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\overline { { Q } } _ { h , i } ^ { k } \\left( \\boldsymbol { s } , \\boldsymbol { a } \\right) \\geq Q _ { h , i } ^ { \\dag , \\pi _ { - i } ^ { k } } \\left( \\boldsymbol { s } , \\boldsymbol { a } \\right) , \\underline { { Q } } _ { h , i } ^ { k } \\left( \\boldsymbol { s } , \\boldsymbol { a } \\right) \\leq Q _ { h , i } ^ { \\pi ^ { k } } \\left( \\boldsymbol { s } , \\boldsymbol { a } \\right) , } \\\\ & { \\qquad \\overline { { V } } _ { h , i } ^ { k } \\left( \\boldsymbol { s } \\right) \\geq V _ { h , i } ^ { \\dag , \\pi _ { - i } ^ { k } } \\left( \\boldsymbol { s } \\right) , \\underline { { V } } _ { h , i } ^ { k } \\left( \\boldsymbol { s } \\right) \\leq V _ { h , i } ^ { \\pi ^ { k } } \\left( \\boldsymbol { s } \\right) . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "text",
+ "text": "Proof. know a d - $k$ , we step, $h = H + 1$ to um $h = 1$ . For base case, weequality (40) holds $( H + 1 )$ $V _ { H + 1 , i } ^ { k } \\left( s \\right) = \\overline { { { V } } } _ { H + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s \\right) = 0$ $( h + 1 )$ $h$ $Q$ ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/6aa5c44bc7011179f7ab1f3cd624e7f8ad9fef0765f3a62e5de338626d02e275.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\overline { { Q } } _ { h , i } ^ { k } \\left( s , \\boldsymbol { a } \\right) - Q _ { h , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s , \\boldsymbol { a } \\right) = \\left[ \\widehat { \\mathbb { P } } _ { h } ^ { k } \\overline { { V } } _ { h + 1 , i } ^ { k } \\right] \\left( s , \\boldsymbol { a } \\right) - \\left[ \\mathbb { P } _ { h } V _ { h + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\right] \\left( s , \\boldsymbol { a } \\right) + \\beta _ { t } } \\\\ & { \\qquad = \\underbrace { \\widehat { \\mathbb { P } } _ { h } ^ { k } \\left( \\overline { { V } } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\right) \\left( s , \\boldsymbol { a } \\right) } _ { ( A ) } + \\underbrace { \\left( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } \\right) V _ { h + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s , \\boldsymbol { a } \\right) } _ { ( B ) } + \\beta _ { t } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 34
+ },
+ {
+ "type": "text",
+ "text": "By induction hypothesis, for any $s ^ { \\prime }$ , $\\left( \\overline { { V } } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\right) ( s ^ { \\prime } ) \\geq 0$ , and thus $( A ) \\geq 0$ . By uniform concentration, $( B ) \\leq C \\sqrt { S H ^ { 2 } \\iota / N _ { h } ^ { k } ( s , \\pmb { a } ) } = \\beta _ { t }$ . Putting everything together we have $Q _ { h , i } ^ { k } \\left( s , \\pmb { a } \\right) -$ $Q _ { h , i } ^ { \\dag , \\pi _ { - i } ^ { k } } \\left( s , { \\pmb a } \\right) \\geq 0$ . The second inequality can be proved similarly. ",
+ "bbox": [
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+ ],
+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "Now assume inequality (45) holds for the $h$ -th step, by definition of value functions and CCE, ",
+ "bbox": [
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+ ],
+ "page_idx": 35
+ },
+ {
+ "type": "equation",
+ "img_path": "images/7abb1d8035e57f5e5b2a705bf2a33d4879694aeba80738fbd693557fe641d629.jpg",
+ "text": "$$\n\\overline { { V } } _ { h , i } ^ { k } \\left( s \\right) = { \\mathbb D } _ { \\pi ^ { k } } \\overline { { Q } } _ { h , i } ^ { k } \\left( s \\right) = \\operatorname* { m a x } _ { \\mu } { \\mathbb D } _ { \\mu \\times \\pi _ { - i } ^ { k } } \\overline { { Q } } _ { h , i } ^ { k } \\left( s \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "By Bellman equation, ",
+ "bbox": [
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+ ],
+ "page_idx": 35
+ },
+ {
+ "type": "equation",
+ "img_path": "images/46a9fe44f142edf81a587aa366bdfd6ecb6fca064bc087035b53b3812a376bfc.jpg",
+ "text": "$$\nV _ { h , i } ^ { \\dag , \\pi _ { - i } ^ { k } } \\left( s \\right) = \\operatorname* { m a x } _ { \\mu } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } Q _ { h , i } ^ { \\dag , \\pi _ { - i } ^ { k } } \\left( s \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "Since by induction hypothesis, for any $( s , \\pmb { a } ) , \\overline { { Q } } _ { h , i } ^ { k } \\left( s , \\pmb { a } \\right) \\geq Q _ { h , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s , \\pmb { a } \\right)$ . As a result, we also have $\\overline { { V } } _ { h , i } ^ { k } \\left( s \\right) \\geq V _ { h , i } ^ { \\dagger , \\pi _ { - i } ^ { k } } \\left( s \\right)$ , which is exactly inequality (40) for the $h$ -th step. The second inequality can be proved similarly. □ ",
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+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "H.1.3 CE VERSION ",
+ "text_level": 1,
+ "bbox": [
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+ "type": "text",
+ "text": "The proof is very similar to the NE version. Specifically, the only part that uses the properties of NE there is Lemma 35. We prove a counterpart here. ",
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+ "type": "text",
+ "text": "Lemma 37. With probability $1 - p ,$ for any $( s , a , h , k , i )$ : ",
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+ "page_idx": 35
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+ "type": "equation",
+ "img_path": "images/6f5f7b0b4b771518915eab1fbf6ec66f2a33eff3fc3b45008b500ec26cbebd5c.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\overline { { Q } } _ { h , i } ^ { k } \\left( s , \\pmb { a } \\right) \\geq \\operatorname* { m a x } _ { \\phi } { Q } _ { h , i } ^ { \\phi \\diamond \\pi ^ { k } } \\left( s , \\pmb { a } \\right) , \\underline { { Q } } _ { h , i } ^ { k } \\left( s , \\pmb { a } \\right) \\leq { Q } _ { h , i } ^ { \\pi ^ { k } } \\left( s , \\pmb { a } \\right) , } \\\\ & { } \\\\ & { \\overline { { V } } _ { h , i } ^ { k } \\left( s \\right) \\geq \\operatorname* { m a x } _ { \\phi } { V } _ { h , i } ^ { \\phi \\diamond \\pi ^ { k } } \\left( s \\right) , \\underline { { V } } _ { h , i } ^ { k } \\left( s \\right) \\leq { V } _ { h , i } ^ { \\pi ^ { k } } \\left( s \\right) . } \\end{array}\n$$",
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+ "bbox": [
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+ "type": "text",
+ "text": "Proof. For each fixed $k$ , we prove this by induction from $h = H + 1$ to $h = 1$ . For base case, we know at the $( H + 1 )$ -th step, $\\overline { { { V } } } _ { H + 1 , i } ^ { k } \\left( s \\right) = \\underset { \\phi } { \\operatorname* { m a x } } \\overline { { { V } } } _ { H + 1 , i } ^ { \\phi \\diamond \\pi ^ { k } } \\left( s \\right) = 0$ . Now, assume the inequality (40) holds for the $( h + 1 )$ -th step, for the $h$ -th step, by definition of the $Q$ functions, ",
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+ "type": "equation",
+ "img_path": "images/eaab53f0e53aaf8836a332715c249d31244c815324222dc82aefdbac6346d34f.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\overline { { Q } } _ { h , i } ^ { k } \\left( s , \\mathfrak { a } \\right) - \\underset { \\phi } { \\operatorname* { m a x } } Q _ { h , i } ^ { \\phi \\phi \\pi ^ { k } } \\left( s , \\mathfrak { a } \\right) } \\\\ & { = \\left[ \\widehat { { \\mathbb { P } } } _ { h } ^ { k } \\overline { { V } } _ { h + 1 , i } ^ { k } \\right] \\left( s , \\mathfrak { a } \\right) - \\left[ \\mathbb { P } _ { h } \\underset { \\phi } { \\operatorname* { m a x } } V _ { h + 1 , i } ^ { \\phi \\varphi \\pi ^ { k } } \\right] \\left( s , \\mathfrak { a } \\right) + \\beta _ { t } } \\\\ & { = \\underbrace { \\widehat { { \\mathbb { P } } } _ { h } ^ { k } \\left( \\overline { { V } } _ { h + 1 , i } ^ { k } - \\underset { \\phi } { \\operatorname* { m a x } } V _ { h + 1 , i } ^ { \\phi \\varphi \\pi ^ { k } } \\right) \\left( s , \\mathfrak { a } \\right) } _ { ( A ) } + \\underbrace { \\left( \\widehat { { \\mathbb { P } } } _ { h } ^ { k } - \\mathbb { P } _ { h } \\right) \\underset { \\phi } { \\operatorname* { m a x } } V _ { h + 1 , i } ^ { \\phi \\varphi \\pi ^ { k } } \\left( s , \\mathfrak { a } \\right) } _ { ( B ) } + \\beta _ { t } . } \\end{array}\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "By induction hypothesis, for any $s ^ { \\prime }$ , $\\left( \\overline { { V } } _ { h + 1 , i } ^ { k } - \\underset { \\phi } { \\operatorname* { m a x } } V _ { h + 1 , i } ^ { \\phi \\diamond \\pi ^ { k } } \\right) ( s ^ { \\prime } ) \\geq 0$ , and thus $( A ) ~ \\geq ~ 0$ . By uniform concentration, $( B ) \\leq C \\sqrt { S H ^ { 2 } \\iota / N _ { h } ^ { k } ( s , \\pmb { a } ) } = \\beta _ { t }$ . Putting everything together we have $\\overline { { Q } } _ { h , i } ^ { k } \\left( s , { \\pmb a } \\right) - \\operatorname* { m a x } _ { \\phi } Q _ { h , i } ^ { \\phi \\diamond \\pi ^ { k } } \\left( s , { \\pmb a } \\right) \\geq 0$ . The second inequality can be proved similarly. ",
+ "bbox": [
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+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "Now assume inequality (47) holds for the $h$ -th step, by definition of value functions and CE, ",
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+ "page_idx": 35
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+ "type": "equation",
+ "img_path": "images/49e83da51a1bf51f9a1577f52ff89caaf9d64cee0a8907828b226af647d46ec0.jpg",
+ "text": "$$\n\\overline { { V } } _ { h , i } ^ { k } \\left( s \\right) = { \\mathbb D } _ { \\pi ^ { k } } \\overline { { Q } } _ { h , i } ^ { k } \\left( s \\right) = \\operatorname* { m a x } _ { \\phi } { \\mathbb D } _ { \\phi \\circ \\pi ^ { k } } \\overline { { Q } } _ { h , i } ^ { k } \\left( s \\right) .\n$$",
+ "text_format": "latex",
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+ },
+ {
+ "type": "text",
+ "text": "By Bellman equation, ",
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+ "page_idx": 35
+ },
+ {
+ "type": "equation",
+ "img_path": "images/6726a858db580b21b2cc745784e277560b86dc3647b1e26505852a889e444631.jpg",
+ "text": "$$\n\\displaystyle \\operatorname* { m a x } _ { \\phi } V _ { h , i } ^ { \\phi \\diamond \\pi ^ { k } } \\left( s \\right) = \\displaystyle \\operatorname* { m a x } _ { \\phi } \\mathbb { D } _ { \\phi \\diamond \\pi ^ { k } } \\displaystyle \\operatorname* { m a x } _ { \\phi ^ { \\prime } } Q _ { h , i } ^ { \\phi ^ { \\prime } \\diamond \\pi ^ { k } } \\left( s \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 35
+ },
+ {
+ "type": "text",
+ "text": "Since by induction hypothesis, for any ${ \\mathfrak { s } } , \\pmb { a } ) , \\overline { { Q } } _ { h , i } ^ { k } \\left( { \\mathfrak { s } } , \\pmb { a } \\right) \\geq \\operatorname* { m a x } _ { \\phi } { Q _ { h , i } ^ { \\phi \\diamond \\pi ^ { k } } \\left( { \\mathfrak { s } } , \\pmb { a } \\right) }$ . As a result, we also have V kh,i (s) ≥ maxV φ\u0005πkh,i , which is exactly inequality (40) for the $h$ -th step. The second inequality can be proved similarly. □ ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "H.2 PROOF OF THEOREM 16 ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "In this section, we prove each theorem for the single reward function case, i.e., $N = 1$ . The proof for the case of multiple reward functions $N > 1 \\AA$ ) simply follows from taking a union bound, that is, replacing the failure probability $p$ by $N p$ . ",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "H.2.1 NE VERSION ",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "Let $( \\mu ^ { k } , \\nu ^ { k } )$ be an arbitrary Nash-equilibrium policy of $\\widehat { \\mathcal { M } } ^ { k } : = ( \\widehat { \\mathbb { P } } ^ { k } , \\widehat { r } ^ { k } )$ , where $\\widehat { \\mathbb { P } } ^ { k }$ and $\\widehat { r } ^ { k }$ are our bempirical estimate of the transition and the reward at the beginning of the $k$ b’th episode in Algorithm 4. Given an arbitrary Nash equilibrium $\\pi ^ { k }$ of ${ \\widehat { \\mathcal { M } } } ^ { k }$ , we use $\\widehat { Q } _ { h , i } ^ { k }$ and $\\widehat { V } _ { h , i } ^ { k }$ to denote its value functions of the $\\because$ ’th player at the $h$ ’th step in ${ \\widehat { \\mathcal { M } } } ^ { k }$ . ",
+ "bbox": [
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "We prove the following two lemmas, which together imply the conclusion about Nash equilibriums in Theorem 16 as in the proof of Theorem 5. ",
+ "bbox": [
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "Lemma 38. With probability $1 - p ,$ , for any $( h , s , \\pmb { a } , i , k )$ , we have ",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/7cc703c7b535becbe5980569a5497d269e322746033054aa8e26fda5f37cc11e.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { | \\widehat { Q } _ { h , i } ^ { k } ( s , \\pmb { a } ) - Q _ { h , i } ^ { \\pi ^ { k } } ( s , \\pmb { a } ) | \\leq \\widetilde { Q } _ { h } ^ { k } ( s , \\pmb { a } ) , } \\\\ { | \\widehat { V } _ { h , i } ^ { k } ( s ) - V _ { h , i } ^ { \\pi ^ { k } } ( s ) | \\leq \\widetilde { V } _ { h } ^ { k } ( s ) . } \\end{array} \\right. } \\end{array}\n$$",
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+ "bbox": [
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "Proof. For eachwe know at the $k$ $h = H + 1$ $h = 1$ . For base case,Now, assume the $( H + 1 )$ $\\widehat { V } _ { H + 1 , i } ^ { k } = V _ { H + 1 , i } ^ { \\pi ^ { k } } = \\widehat { Q } _ { H + 1 , i } ^ { k } = Q _ { H + 1 , i } ^ { \\pi ^ { k } } = 0$ $( h + 1 ) ^ { \\ }$ $h$ $Q$ ",
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+ },
+ {
+ "type": "equation",
+ "img_path": "images/c6399ea2b1e72e7a46b186761da8f8c958b4db07c17e1ea1eda84086b403b0fe.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\quad \\left| \\widehat { Q } _ { h , i } ^ { k } \\left( s , a \\right) - Q _ { h , i } ^ { \\pi ^ { k } } \\left( s , a \\right) \\right| } \\\\ & { \\leq \\left| \\left[ \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widehat { V } _ { h + 1 , i } ^ { k } \\right] \\left( s , a \\right) - \\left[ \\mathbb { P } _ { h } V _ { h + 1 , i } ^ { \\pi ^ { k } } \\right] \\left( s , a \\right) \\right| + \\left| r _ { h } ( s , a ) - \\widehat { r } _ { h } ^ { k } ( s , a ) \\right| } \\\\ & { \\leq \\left| \\underbrace { \\widehat { \\mathbb { P } } _ { h } ^ { k } \\left( \\widehat { V } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \\pi ^ { k } } \\right) \\left( s , a \\right) } _ { ( A ) } \\right| + \\underbrace { \\left| \\left( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } \\right) V _ { h + 1 , i } ^ { \\pi ^ { k } } \\left( s , a \\right) \\right| + \\left| r _ { h } ( s , a ) - \\widehat { r } _ { h } ^ { k } ( s , a ) \\right| } _ { ( B ) } } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "By the induction hypothesis, ",
+ "bbox": [
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+ ],
+ "page_idx": 36
+ },
+ {
+ "type": "equation",
+ "img_path": "images/c5ea5481bad1001742dc7932d3064857c5d01a322c6cacbe9c548e1c1c7eecb7.jpg",
+ "text": "$$\n\\begin{array} { r } { ( A ) \\leq \\widehat { \\mathbb { P } } _ { h } ^ { k } \\left| \\widehat { V } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \\pi ^ { k } } \\right| ( s , \\pmb { a } ) \\leq ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ) ( s , \\pmb { a } ) . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "By uniform concentration (e.g., Lemma 12 in Bai & Jin (2020)), $( B ) \\leq \\sqrt { S H ^ { 2 } \\iota / N _ { h } ^ { k } ( s , \\pmb { a } ) } = \\beta _ { t } .$ Putting everything together we have ",
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+ "page_idx": 36
+ },
+ {
+ "type": "equation",
+ "img_path": "images/434e2a4be60af84c77f1195b50b8d5676063fa10ad73269afbf95ead5cabfba9.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left| Q _ { h , i } ^ { \\pi ^ { k } } \\left( s , \\pmb { a } \\right) - \\widehat { Q } _ { h , i } ^ { k } \\left( s , \\pmb { a } \\right) \\right| \\leq \\operatorname* { m i n } \\left\\{ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ) ( s , \\pmb { a } ) + \\beta _ { t } , H \\right\\} = \\widetilde { Q } _ { h } ^ { k } ( s , \\pmb { a } ) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "which proves the first inequality in (49). The inequality for $V$ functions follows directly by noting that the value functions are computed using the same policy $\\pi ^ { k }$ . □ ",
+ "bbox": [
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "Lemma 39. With probability $1 - p ,$ , for any $( h , s , \\pmb { a } , i , k )$ , we have ",
+ "bbox": [
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+ "type": "equation",
+ "img_path": "images/0e11f492b100aa694c719e898ed87d9c54143bea6c59b04ab926d7202dce76b0.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ \\begin{array} { l l } { | \\widehat { Q } _ { h , i } ^ { k } ( s , \\pmb { a } ) - Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } ( s , \\pmb { a } ) | \\leq \\widetilde { Q } _ { h } ^ { k } ( s , \\pmb { a } ) , } \\\\ { | \\widehat { V } _ { h , i } ^ { k } ( s ) - V _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } ( s ) | \\leq \\widetilde { V } _ { h } ^ { k } ( s ) . } \\end{array} \\right. } \\end{array}\n$$",
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+ "page_idx": 36
+ },
+ {
+ "type": "text",
+ "text": "Proof. For eachwe know at the $k$ we pro-th step, $h = H + 1$ $h = 1$ . For base case,Now, assume the $( H + 1 )$ $\\widehat { V } _ { H + 1 , i } ^ { k } = V _ { H + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } = \\widehat { Q } _ { H + 1 , i } ^ { k } = Q _ { H + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } = 0 .$ conclusion holds for the ’th step, for the $h$ ’th step, by definition of the $Q$ functions, ",
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+ "type": "equation",
+ "img_path": "images/96a80fc3cd24aec22d3787f69150f2a3f03bbb180e40fa6e90268783f82f97f6.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\quad \\left| \\widehat { Q } _ { h , i } ^ { k } \\left( s , a \\right) - Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } \\left( s , a \\right) \\right| } \\\\ & { = \\left| \\left[ \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widehat { V } _ { h + 1 , i } ^ { k } \\right] \\left( s , a \\right) - \\left[ \\mathbb { P } _ { h } V _ { h + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } \\right] \\left( s , a \\right) \\right| + \\left| r _ { h } ( s , a ) - \\widehat { r } _ { h } ^ { k } ( s , a ) \\right| } \\\\ & { \\leq \\left| \\widehat { \\mathbb { P } } _ { h } ^ { k } \\left( \\widehat { V } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } \\right) \\left( s , a \\right) \\right| + \\left| \\left( \\widehat { \\mathbb { P } } _ { h } ^ { k } - \\mathbb { P } _ { h } \\right) V _ { h + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } \\left( s , a \\right) \\right| + \\left| r _ { h } ( s , a ) - \\widehat { r } _ { h } ^ { k } ( s , a ) \\right| } \\\\ & { \\quad \\qquad ( A ) } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "By the induction hypothesis, ",
+ "bbox": [
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+ "page_idx": 37
+ },
+ {
+ "type": "equation",
+ "img_path": "images/1feadc376a8afa9889dac74f6b36a80fd44b7677a1d421507834132b9369a095.jpg",
+ "text": "$$\n\\begin{array} { r } { ( A ) \\leq \\widehat { \\mathbb { P } } _ { h } ^ { k } \\left| \\widehat { V } _ { h + 1 , i } ^ { k } - V _ { h + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } \\right| ( s , \\pmb { a } ) \\leq ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ) ( s , \\pmb { a } ) . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "By uniform concentration, $( B ) \\leq \\sqrt { S H ^ { 2 } \\iota / N _ { h } ^ { k } ( s , \\pmb { a } ) } = \\beta _ { t }$ . Putting everything together we have ",
+ "bbox": [
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+ "page_idx": 37
+ },
+ {
+ "type": "equation",
+ "img_path": "images/523fc3d3e4bc423850e261b97307b17e475fa3e7c23a49dd7cb7ed83cec27563.jpg",
+ "text": "$$\n\\left| Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s , \\pmb { a } \\right) - \\widehat { Q } _ { h , i } ^ { k } \\left( s , \\pmb { a } \\right) \\right| \\leq \\operatorname* { m i n } \\left\\{ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ) ( s , \\pmb { a } ) + \\beta _ { t } , H \\right\\} = \\widetilde { Q } _ { h } ^ { k } ( s , \\pmb { a } ) ,\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "which proves the first inequality in (50). It remains to show the inequality for $V$ functions also hold in $h$ ’th step. ",
+ "bbox": [
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+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "Since $\\pi ^ { k }$ is a Nash-equilibrium policy, we have ",
+ "bbox": [
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+ "page_idx": 37
+ },
+ {
+ "type": "equation",
+ "img_path": "images/3d9b003ff7e286d8975811437b9202b5ed41591542c6e83976ae2764ac38c897.jpg",
+ "text": "$$\n\\widehat V _ { h , i } ^ { k } \\left( s \\right) = \\operatorname* { m a x } _ { \\mu } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } \\widehat Q _ { h , i } ^ { k } \\left( s \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "By Bellman equation, ",
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+ "page_idx": 37
+ },
+ {
+ "type": "equation",
+ "img_path": "images/55414c026317373914223387d5ec7fa8cf95766a4d0620059bfefa2bfeaaf22d.jpg",
+ "text": "$$\nV _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s \\right) = \\operatorname* { m a x } _ { \\mu } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s \\right) .\n$$",
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+ {
+ "type": "text",
+ "text": "Combining the two equations above, and utilizing the bound we just proved for $Q$ functions, we obtain ",
+ "bbox": [
+ 173,
+ 597,
+ 825,
+ 626
+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "equation",
+ "img_path": "images/89c7a2a4d14114cf0bdc8fd2f8bff1456c4c8b951dee3829eb8328cc452975f2.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\left| \\widehat { V } _ { h , i } ^ { k } \\left( s \\right) - V _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s \\right) \\right| \\leq \\left| \\underset { \\mu } { \\operatorname* { m a x } } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } \\widehat { Q } _ { h , i } ^ { k } \\left( s \\right) - \\underset { \\mu } { \\operatorname* { m a x } } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s \\right) \\right| } \\\\ & { \\qquad \\leq \\underset { a } { \\operatorname* { m a x } } \\widetilde { Q } _ { h } ^ { k } ( s , a ) = \\widetilde { V } _ { h } ^ { k } ( s ) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 251,
+ 630,
+ 745,
+ 691
+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "which completes the whole proof. ",
+ "bbox": [
+ 174,
+ 696,
+ 398,
+ 712
+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "H.2.2 CCE VERSION ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 728,
+ 333,
+ 743
+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "The proof is almost the same as that for Nash equilibriums. We will reuse Lemma 38 and prove an analogue of Lemma 39. The conclusion for CCEs will follow directly by combining the two lemmas as in the proof of Theorem 5. ",
+ "bbox": [
+ 176,
+ 752,
+ 823,
+ 795
+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "Lemma 40. With probability $1 - p ,$ for any $( h , s , \\pmb { a } , i , k )$ , we have ",
+ "bbox": [
+ 171,
+ 799,
+ 614,
+ 815
+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "equation",
+ "img_path": "images/eb5c0acddce9d42aa4e82f79af249cea2abc3d92758bcb549f06e2c82136c5fe.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } ( s , \\pmb { a } ) - \\widehat { Q } _ { h , i } ^ { k } ( s , \\pmb { a } ) \\leq \\widetilde { Q } _ { h } ^ { k } ( s , \\pmb { a } ) , \\right. } \\\\ { \\left. V _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } ( s ) - \\widehat { V } _ { h , i } ^ { k } ( s ) \\leq \\widetilde { V } _ { h } ^ { k } ( s ) . \\right. } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 361,
+ 821,
+ 635,
+ 875
+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "Proof. For eachwe know at the $k$ we pro-th step, $h = H + 1$ $h = 1$ . For base case,Now, assume the $( H + 1 )$ $\\widehat { V } _ { H + 1 , i } ^ { k } = V _ { H + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } = \\widehat { Q } _ { H + 1 , i } ^ { k } = Q _ { H + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } = 0$ ",
+ "bbox": [
+ 173,
+ 888,
+ 825,
+ 928
+ ],
+ "page_idx": 37
+ },
+ {
+ "type": "text",
+ "text": "conclusion holds for the $( h + 1 ) ^ { \\dagger }$ th step, for the $h$ ’th step, by definition of the $Q$ functions, ",
+ "bbox": [
+ 169,
+ 102,
+ 767,
+ 119
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "equation",
+ "img_path": "images/9c77adc16907abc452f6f18d031bb5082c87b3d8f6e34bbfe9c36932af56d98a.jpg",
+ "text": "$$\n\\begin{array} { r l } & { \\quad Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } \\left( s , a \\right) - \\widehat { Q } _ { h , i } ^ { k } \\left( s , a \\right) } \\\\ & { \\le \\left[ \\mathbb { P } _ { h } V _ { h + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } \\right] \\left( s , a \\right) - \\left[ \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widehat { V } _ { h + 1 , i } ^ { k } \\right] \\left( s , a \\right) + \\left| r _ { h } ( s , a ) - \\widehat { r } _ { h } ^ { k } ( s , a ) \\right| } \\\\ & { \\le \\underbrace { \\widehat { \\mathbb { P } } _ { h } ^ { k } \\left( V _ { h + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } - \\widehat { V } _ { h + 1 , i } ^ { k } \\right) \\left( s , a \\right) } _ { ( A ) } + \\underbrace { \\left( \\mathbb { P } _ { h } - \\widehat { \\mathbb { P } } _ { h } ^ { k } \\right) V _ { h + 1 , i } ^ { \\pi _ { - i } ^ { k } , \\dagger } \\left( s , a \\right) + \\left| r _ { h } ( s , a ) - \\widehat { r } _ { h } ^ { k } ( s , a ) \\right| } _ { ( B ) } . } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 220,
+ 125,
+ 779,
+ 242
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "By the induction hypothesis, $( A ) \\leq ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ) ( s , \\pmb { a } )$ . ",
+ "bbox": [
+ 173,
+ 256,
+ 522,
+ 276
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "By uniform concentration, $( B ) \\leq \\sqrt { S H ^ { 2 } \\iota / N _ { h } ^ { k } ( s , \\pmb { a } ) } = \\beta _ { t }$ . Putting everything together we have ",
+ "bbox": [
+ 171,
+ 284,
+ 803,
+ 306
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "equation",
+ "img_path": "images/8e212173b458942f4f80ab2499815f54e4ade62589113070a127a1cab259142f.jpg",
+ "text": "$$\n\\begin{array} { r } { Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s , \\boldsymbol { a } \\right) - \\widehat { Q } _ { h , i } ^ { k } \\left( s , \\boldsymbol { a } \\right) \\leq \\operatorname* { m i n } \\left\\{ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ) ( s , \\boldsymbol { a } ) + \\beta _ { t } , H \\right\\} = \\widetilde { Q } _ { h } ^ { k } ( s , \\boldsymbol { a } ) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 248,
+ 315,
+ 746,
+ 343
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "which proves the first inequality in (51). It remains to show the inequality for $V$ functions also hold in $h$ ’th step. ",
+ "bbox": [
+ 171,
+ 348,
+ 823,
+ 377
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "Since $\\pi ^ { k }$ is a CCE, we have ",
+ "bbox": [
+ 173,
+ 383,
+ 359,
+ 398
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "equation",
+ "img_path": "images/9fd200265fa1ca6a91c90892bdb42605e9367f0483818750bd4032035d824a9e.jpg",
+ "text": "$$\n\\widehat V _ { h , i } ^ { k } \\left( s \\right) \\geq \\operatorname* { m a x } _ { \\mu } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } \\widehat Q _ { h , i } ^ { k } \\left( s \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 390,
+ 396,
+ 607,
+ 421
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "Observe that $V _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag }$ obeys the Bellman optimality equation, so we have ",
+ "bbox": [
+ 173,
+ 434,
+ 645,
+ 457
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "equation",
+ "img_path": "images/b0ef373dad078b1446ba46f6af0c4f667c5dc088b2a85e686d311bf5bbc8082f.jpg",
+ "text": "$$\nV _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s \\right) = \\operatorname* { m a x } _ { \\mu } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 372,
+ 464,
+ 625,
+ 494
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "Combining the two equations above, and utilizing the bound we just proved for $Q$ functions, we obtain ",
+ "bbox": [
+ 173,
+ 508,
+ 825,
+ 537
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "equation",
+ "img_path": "images/80ecb95ede9b72fbf8be66d75a95e762409f19d8d79b9c0a4c03cdf67fa0e68d.jpg",
+ "text": "$$\n\\begin{array} { r l } & { V _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s \\right) - \\widehat { V } _ { h , i } ^ { k } \\left( s \\right) \\leq \\underset { \\mu } { \\operatorname* { m a x } } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } Q _ { h , i } ^ { \\pi _ { - i } ^ { k } , \\dag } \\left( s \\right) - \\underset { \\mu } { \\operatorname* { m a x } } \\mathbb { D } _ { \\mu \\times \\pi _ { - i } ^ { k } } \\widehat { Q } _ { h , i } ^ { k } \\left( s \\right) } \\\\ & { \\qquad \\leq \\underset { a } { \\operatorname* { m a x } } \\widetilde { Q } _ { h } ^ { k } ( s , a ) = \\widetilde { V } _ { h } ^ { k } ( s ) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 263,
+ 541,
+ 735,
+ 599
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "which completes the whole proof. ",
+ "bbox": [
+ 174,
+ 604,
+ 398,
+ 619
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "H.2.3 CE VERSION ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 635,
+ 321,
+ 650
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "The proof is almost the same as that for Nash equilibriums. We will reuse Lemma 38 and prove an analogue of Lemma 39. The conclusion for CEs will follow directly by combining the two lemmas as in the proof of Theorem 5. ",
+ "bbox": [
+ 173,
+ 659,
+ 825,
+ 702
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "Lemma 41. With probability $1 - p ,$ for any $( h , s , \\pmb { a } , i , k )$ and strategy modification $\\phi$ for player $i$ , we have ",
+ "bbox": [
+ 169,
+ 705,
+ 823,
+ 733
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "equation",
+ "img_path": "images/f3e221bb52e11a6084096d83ef9ec12578bb3795da453f6cf4346229f73643da.jpg",
+ "text": "$$\n\\begin{array} { r } { \\left\\{ Q _ { h , i } ^ { \\phi \\diamond \\pi ^ { k } } ( s , \\pmb { a } ) - \\widehat { Q } _ { h , i } ^ { k } ( s , \\pmb { a } ) \\leq \\widetilde { Q } _ { h } ^ { k } ( s , \\pmb { a } ) , \\right. } \\\\ { \\left. V _ { h , i } ^ { \\phi \\diamond \\pi ^ { k } } ( s ) - \\widehat { V } _ { h , i } ^ { k } ( s ) \\leq \\widetilde { V } _ { h } ^ { k } ( s ) . \\right. } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 362,
+ 729,
+ 635,
+ 780
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "we know at the Proof. For each fixed $( H + 1 )$ $k$ , we prove this by induction from -th step, $\\widehat { V } _ { H + 1 , i } ^ { k } \\ : = V _ { H + 1 , i } ^ { \\phi \\diamond \\pi ^ { k } } \\ : = \\widehat { Q } _ { H + 1 , i } ^ { k } \\ : = Q _ { H + 1 , i } ^ { \\phi \\diamond \\pi ^ { k } } \\ : = \\ : 0$ $h = H + 1$ to $h = 1$ . Now, assume the . For base case, conclusion holds for the $( h + 1 )$ ’th step, for the $h$ ’th step, following exactly the same argument as Lemma 40, we can show ",
+ "bbox": [
+ 173,
+ 792,
+ 826,
+ 857
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "equation",
+ "img_path": "images/a107b8fc17611bf0cb8acda5bae134199357b52929923df4d212396f798cd4c8.jpg",
+ "text": "$$\n\\begin{array} { r } { Q _ { h , i } ^ { \\phi \\circ \\pi ^ { k } } ( s , a ) - \\widehat { Q } _ { h , i } ^ { k } ( s , a ) \\leq \\operatorname* { m i n } \\left\\{ ( \\widehat { \\mathbb { P } } _ { h } ^ { k } \\widetilde { V } _ { h + 1 } ^ { k } ) ( s , a ) + \\beta _ { t } , H \\right\\} = \\widetilde { Q } _ { h } ^ { k } ( s , a ) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 250,
+ 862,
+ 745,
+ 890
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "which proves the first inequality in (52). It remains to show the inequality for $V$ functions also hold in $h$ ’th step. ",
+ "bbox": [
+ 171,
+ 895,
+ 823,
+ 925
+ ],
+ "page_idx": 38
+ },
+ {
+ "type": "text",
+ "text": "Since $\\pi ^ { k }$ is a CE, we have ",
+ "bbox": [
+ 174,
+ 103,
+ 348,
+ 118
+ ],
+ "page_idx": 39
+ },
+ {
+ "type": "equation",
+ "img_path": "images/42195b06a7ff58662d58e4c66add4768c9d175bb073c751449c3616588e26e7c.jpg",
+ "text": "$$\n\\widehat { V } _ { h , i } ^ { k } \\left( s \\right) = \\operatorname* { m a x } _ { \\widetilde { \\phi } _ { h , s } } \\mathbb { D } _ { \\widetilde { \\phi } _ { h , s } \\diamond \\pi ^ { k } } \\widehat { Q } _ { h , i } ^ { k } \\left( s \\right) ,\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 387,
+ 114,
+ 609,
+ 145
+ ],
+ "page_idx": 39
+ },
+ {
+ "type": "text",
+ "text": "where the maximum is take over all possible injective functions from $\\mathbf { \\mathcal { A } } _ { i }$ to itself. ",
+ "bbox": [
+ 173,
+ 147,
+ 705,
+ 164
+ ],
+ "page_idx": 39
+ },
+ {
+ "type": "text",
+ "text": "Observe that V φ\u0005πk obeys the Bellman optimality equation, so we have ",
+ "bbox": [
+ 174,
+ 170,
+ 645,
+ 189
+ ],
+ "page_idx": 39
+ },
+ {
+ "type": "equation",
+ "img_path": "images/aa59210532637773b05e7b9d3a6690725e8463c4b7f553982c4db59f171ae7a2.jpg",
+ "text": "$$\nV _ { h , i } ^ { \\phi \\diamond \\pi ^ { k } } \\left( s \\right) = \\operatorname* { m a x } _ { \\widetilde { \\phi } _ { h , s } } \\mathbb { D } _ { \\widetilde { \\phi } _ { h , s } \\diamond \\pi ^ { k } } Q _ { h , i } ^ { \\phi \\diamond \\pi ^ { k } } \\left( s \\right) .\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 370,
+ 196,
+ 625,
+ 229
+ ],
+ "page_idx": 39
+ },
+ {
+ "type": "text",
+ "text": "Combining the two equations above, and utilizing the bound we just proved for $Q$ functions, we obtain ",
+ "bbox": [
+ 171,
+ 243,
+ 825,
+ 272
+ ],
+ "page_idx": 39
+ },
+ {
+ "type": "equation",
+ "img_path": "images/9795e71665817e6ee9b06934d80dc55b1aa0db3dbc6baf0333cefcbd4206b979.jpg",
+ "text": "$$\n\\begin{array} { r l } & { V _ { h , i } ^ { \\phi \\circ \\pi ^ { k } } \\left( s \\right) - \\widehat { V } _ { h , i } ^ { k } \\left( s \\right) = \\underset { \\bar { \\phi } _ { h , s } } { \\operatorname* { m a x } } \\mathbb { D } _ { \\widetilde { \\phi } _ { h , s } \\circ \\pi ^ { k } } Q _ { h , i } ^ { \\phi \\circ \\pi ^ { k } } \\left( s \\right) - \\underset { \\bar { \\phi } _ { h , s } } { \\operatorname* { m a x } } \\mathbb { D } _ { \\widetilde { \\phi } _ { h , s } \\circ \\pi ^ { k } } \\widehat { Q } _ { h , i } ^ { k } \\left( s \\right) } \\\\ & { \\qquad \\leq \\underset { a } { \\operatorname* { m a x } } \\widetilde { Q } _ { h } ^ { k } ( s , a ) = \\widetilde { V } _ { h } ^ { k } ( s ) , } \\end{array}\n$$",
+ "text_format": "latex",
+ "bbox": [
+ 258,
+ 276,
+ 738,
+ 335
+ ],
+ "page_idx": 39
+ },
+ {
+ "type": "text",
+ "text": "which completes the whole proof. ",
+ "bbox": [
+ 173,
+ 340,
+ 398,
+ 356
+ ],
+ "page_idx": 39
+ }
+]
\ No newline at end of file
diff --git a/parse/train/9Y7_c5ZAd5i/9Y7_c5ZAd5i_middle.json b/parse/train/9Y7_c5ZAd5i/9Y7_c5ZAd5i_middle.json
new file mode 100644
index 0000000000000000000000000000000000000000..c8d1cefcf56cf58252a1b27261fd46e905384056
--- /dev/null
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+ "html": "| 1: Initialize: for any (s,a,b,h),Qn(s,a,b) ← H,Q,(s,a,b) ← 0,△ ← H,Nn(s,a,b) ←0. | |
| 2: | for episodek=1,...,Kdo for steph=H,H-1,...,1do |
| 3: | |
| 4: | for(s,a,b) ∈S× A× Bdo |
| 5: | t ←Nn(s,a,b). |
| 6: | if t>O then |
| 7: | β ← BoNUs(t,Vn[(Vh+1 +Vh+1)/2](s,a,b)). |
| 8: | γ ← (c/H)Ph(Vh+1 -Vh+1)(s,a,b). |
| 9: | Qn(s,a,b)←min{(rh +PnVh+1)(s,a,b)+γ+β,H}. |
| 10: | Q(s,a,b)←max{(rh+PhVh+1)(s,a,b)-γ-β,0}. |
| 11: | fors∈Sdo |
| 12: | Th(·,·|s) ← CCE(Qn(s,·,),Q(s,·, )). |
| 13: | Vn(s)← (DπnQn)(s)；Vn(s) ←(DπnQn)(s). |
| 14: 15: | if(V1-V1)(s1)<△ then △←(V1-V1)(s1) and πout ←π. |
| 16: | for step h =1,...,H do |
| 17: | take action (an, br) ~ πh(*,|sh),observe reward rh and next state Sh+1· |
| |
| 18: | add1 to Nn(sh,ah,bh) and Nh(sh,ah,bh,Sh+1). |
| 19: | Ph(-Ish,ah,bh) ← Nn(Sh,ah,bh,:)/Nn(sh,ah,bh). |
| 20: | Output (μout,vout) that are the marginal policies of πout. |
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+ "content": "\\mathbb { E } _ { ( a , b ) \\sim \\pi } \\overline { { Q } } ( a , b ) \\geq \\operatorname* { m a x } _ { a ^ { \\star } } \\mathbb { E } _ { ( a , b ) \\sim \\pi } \\overline { { Q } } ( a ^ { \\star } , b ) , \\qquad \\mathbb { E } _ { ( a , b ) \\sim \\pi } \\underline { { Q } } ( a , b ) \\leq \\operatorname* { m i n } _ { b ^ { \\star } } \\mathbb { E } _ { ( a , b ) \\sim \\pi } \\underline { { Q } } ( a , b ^ { \\star } ) .",
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+ "content": "is also a CCE and a Nash equilibrium always exists. Furthermore, a CCE can be computed by",
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+ "html": "| 1: Initialize: for any (s,a,b,h),Qn(s,a,b) ← H,Q,(s,a,b) ← 0,△ ← H,Nn(s,a,b) ←0. | |
| 2: | for episodek=1,...,Kdo for steph=H,H-1,...,1do |
| 3: | |
| 4: | for(s,a,b) ∈S× A× Bdo |
| 5: | t ←Nn(s,a,b). |
| 6: | if t>O then |
| 7: | β ← BoNUs(t,Vn[(Vh+1 +Vh+1)/2](s,a,b)). |
| 8: | γ ← (c/H)Ph(Vh+1 -Vh+1)(s,a,b). |
| 9: | Qn(s,a,b)←min{(rh +PnVh+1)(s,a,b)+γ+β,H}. |
| 10: | Q(s,a,b)←max{(rh+PhVh+1)(s,a,b)-γ-β,0}. |
| 11: | fors∈Sdo |
| 12: | Th(·,·|s) ← CCE(Qn(s,·,),Q(s,·, )). |
| 13: | Vn(s)← (DπnQn)(s)；Vn(s) ←(DπnQn)(s). |
| 14: 15: | if(V1-V1)(s1)<△ then △←(V1-V1)(s1) and πout ←π. |
| 16: | for step h =1,...,H do |
| 17: | take action (an, br) ~ πh(*,|sh),observe reward rh and next state Sh+1· |
| |
| 18: | add1 to Nn(sh,ah,bh) and Nh(sh,ah,bh,Sh+1). |
| 19: | Ph(-Ish,ah,bh) ← Nn(Sh,ah,bh,:)/Nn(sh,ah,bh). |
| 20: | Output (μout,vout) that are the marginal policies of πout. |
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| 2: | for episodek=1,...,Kdo for steph=H,H-1,...,1do |
| 3: | |
| 4: | for(s,a,b) ∈S× A× Bdo |
| 5: | t ←Nn(s,a,b). |
| 6: | if t>O then |
| 7: | β ← BoNUs(t,Vn[(Vh+1 +Vh+1)/2](s,a,b)). |
| 8: | γ ← (c/H)Ph(Vh+1 -Vh+1)(s,a,b). |
| 9: | Qn(s,a,b)←min{(rh +PnVh+1)(s,a,b)+γ+β,H}. |
| 10: | Q(s,a,b)←max{(rh+PhVh+1)(s,a,b)-γ-β,0}. |
| 11: | fors∈Sdo |
| 12: | Th(·,·|s) ← CCE(Qn(s,·,),Q(s,·, )). |
| 13: | Vn(s)← (DπnQn)(s)；Vn(s) ←(DπnQn)(s). |
| 14: 15: | if(V1-V1)(s1)<△ then △←(V1-V1)(s1) and πout ←π. |
| 16: | for step h =1,...,H do |
| 17: | take action (an, br) ~ πh(*,|sh),observe reward rh and next state Sh+1· |
| |
| 18: | add1 to Nn(sh,ah,bh) and Nh(sh,ah,bh,Sh+1). |
| 19: | Ph(-Ish,ah,bh) ← Nn(Sh,ah,bh,:)/Nn(sh,ah,bh). |
| 20: | Output (μout,vout) that are the marginal policies of πout. |
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+# PNAT: NON-AUTOREGRESSIVE TRANSFORMER BY POSITION LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Non-autoregressive models are promising on various text generation tasks. Previous work hardly considers to explicitly model the positions of generated words. However, the position modeling is an essential problem in non-autoregressive text generation. In this study, we propose PNAT, which incorporates positions as a latent variable into the text generative process. Experimental results show that PNAT achieves top results on machine translation and paraphrase generation tasks, outperforming several strong baselines.
+
+# 1 INTRODUCTION
+
+Transformer (Vaswani et al., 2017) has been widely used in many text generation tasks, which is first proposed in neural machine translation, achieving great success for its promising performance. Nevertheless, the auto-regressive property of Transformer has been a bottleneck. Specifically, the decoder of Transformer generates words sequentially, and the latter words are conditioned on previous ones in a sentence. Such bottleneck prevents the decoder from higher efficiency in parallel computation, and imposes strong constrains in text generation, with which the generation order has to be left to right (or right to left) (Shaw et al., 2018; Vaswani et al., 2017).
+
+Recently, many researches (Gu et al., 2018; Lee et al., 2018; Wang et al., 2019; Wei et al., 2019) are devoted to break the auto-regressive bottleneck by introducing non-autoregressive Transformer (NAT) for neural machine translation, where the decoder generates all words simultaneously instead of sequentially. Intuitively, NAT abandons feeding previous predicted words into decoder state at the next time step, but directly copy encoded representation at source side to the decoder inputs (Gu et al., 2018). However, without the auto-regressive constrain, the search space of the output sentence becomes larger (Wei et al., 2019), which brings the performance gap (Lee et al., 2018) between NAT and auto-regressive Transformer (AT). Related works propose to include some inductive priors or learning techniques to boost the performance of NAT. But most of previous work hardly consider explicitly modeling the position of output words during text generation.
+
+We argue that position prediction is an essential problem of NAT. Current NAT approaches do not explicitly model the position of output words, and may ignore the reordering issue in generating output sentences. Compared to machine translation, the reorder problem is much more severe in tasks such as table-to-text (Liu et al., 2018) and dialog generations (Shen et al., 2017). Additionally, it is straightforward to explicitly model word positions in output sentences, as position embeddings are used in Transformer, which is natively non-autoregressive, to include the order information. Intuitively, if output positions are explicitly modeled, the predicted position combined with Transformer to realize non-autoregressive generation would become more natural.
+
+In this paper, we propose non-autoregressive transformer by position learning (PNAT). PNAT is simple yet effective, which explicitly models positions of output words as latent variables in the text generation. Specifically, we introduce a heuristic search process to guide the position learning, and max sampling is adopted to inference the latent model. The proposed PNAT is motivated by learning syntax position (also called syntax distance). Shen et al. (2018) show that syntax position of words in a sentence could be predicted by neural networks in a non-autoregressive fashion, which even obtains top parsing accuracy among strong parser baselines. Given the observations above, we try to directly predict the positions of output words to build a NAT model for text generation.
+
+Our proposed PNAT takes following advantages:
+
+• We propose PNAT, which first includes positions of output words as latent variables for text generation. Experiments show that PNAT achieves very top results in non-autoregressive NMT, outperforming many strong baselines. PNAT also obtains better results than AT in paraphrase generation task. Further analysis shows that PNAT has great potentials. With the increase of position prediction accuracy, performances of PNAT could increase significantly. The observations may shed light on the future direction of NAT. Thanks to the explicitly modeling of position, we could control the generation by facilitating the position latent variable, which may enable interesting applications such as controlling one special word left to another one. We leave this as future work.
+
+# 2 BACKGROUND
+
+# 2.1 AUTOREGRESSIVE DECODING
+
+A target sequence $Y { = } y _ { 1 : M }$ is decomposed into a series of conditional probabilities autoregressively, each of which is parameterized using neural networks. This approach has become a de facto standard in language modeling(Sundermeyer et al., 2012), and has been also applied to conditional sequence modeling $p ( Y | X )$ by introducing an additional conditional variable $X { = } x _ { 1 : N }$ :
+
+$$
+p ( \boldsymbol { Y } | \boldsymbol { X } ) = \prod _ { t = 1 } ^ { M } p ( y _ { t } | y _ { < t } , \boldsymbol { X } ; \theta )
+$$
+
+With different choices of neural network architectures such as recurrent neural networks (RNNs) (Bahdanau et al., 2014; Cho et al., 2014), convolutional neural networks (CNNs) (Krizhevsky et al., 2012; Gehring et al., 2017), as well as self-attention based transformer (Vaswani et al., 2017), the autoregressive decoding has achieved great success in tasks such as machine translation (Bahdanau et al., 2014), paraphrase generation (Gupta et al., 2018), speech recognition (Graves et al., 2013), etc.
+
+# 2.2 NON-AUTOREGRESSIVE DECODING
+
+Autoregressive model suffers from the issue of slow decoding in inference, because tokens are generated sequentially and each of them depends on previous ones. As a solution to this issue, Gu et al. (2018) proposed Non-Autoregressive Transformer (denoted as NAT) for machine translation, breaking the dependency among the target tokens through time by decoding all the tokens simultaneously. Put simply, NAT (Gu et al., 2018) factorizes the conditional distribution over a target sequence into a series of conditionally independent distributions with respect to time:
+
+$$
+p ( { \cal Y } | { \cal X } ) = p _ { L } ( { \cal M } | { \cal X } : \theta ) \cdot \prod _ { t = 1 } ^ { M } p ( y _ { t } | { \cal X } )
+$$
+
+which allows trivially finding the most likely target sequence by arg $\operatorname* { m a x } _ { Y }$ $p ( Y | X )$ for each timestep $t$ , effectively bypassing computational overhead and sub-optimality in decoding from an autoregressive model.
+
+Although non-autoregressive models achieves $1 5 \times$ speedup in machine translation compared with autoregressive models, it comes at the expense of potential performance degradation (Gu et al., 2018). The degradation results from the removal of conditional dependencies within the decoding sentence $y _ { t }$ depend on $y _ { < t , }$ ). Without such dependencies, the decoder is hard to leverage the inherent sentence structure in prediction.
+
+# 2.3 LATENT VARIABLES FOR NON-AUTOREGRESSIVE DECODING
+
+A non-autoregressive model could be incorporated with conditional dependency as latent variable to alleviate the degradation resulted from the absence of dependency:
+
+$$
+P ( Y | X ) = \int _ { z } P ( z | X ) \prod _ { t = 1 } ^ { M } P ( y _ { t } | z , X ) d z
+$$
+
+For example, NAT-FT (Gu et al., 2018) models the inherent sentence structure with a latent fertility variable, which represents how many target tokens that a source token would translate to. Lee et al. (2018) introduces $L$ intermediate predictions $Y ^ { 1 : L }$ as random variables , and to refine the predictions from $Y ^ { 1 }$ to $Y ^ { L }$ in a iterative manner.
+
+# 3 PNAT: POSITION-BASED NON-AUTOREGRESSIVE TRANSFORMER
+
+We propose position-based non-autoregressive transformer (PNAT), an extension to transformer incorporated with non auto-regressive decoding and position learning.
+
+# 3.1 MODELING POSITION WITH LATENT VARIABLES
+
+Languages are usually inconsistent with each other in word order. Thus reordering is usually required when translating a sentence from a language to another. In NAT family, words representations or encoder states at source side are copied to the target side to feed into decoder as its input. Previously, Gu et al. (2018) utilizes positional attention which incorporates positional encoding into decoder attention to perform local reordering. But such implicitly reordering mechanism by position attention may cause a repeated generation problem, because position learning module is not optimized directly, and is likely to be misguided by target supervision.
+
+To tackle with this problem, we propose to explicitly model the position as a latent variable. We rewrite the target sequence $Y$ with its corresponding position latent variable $z = z _ { 1 : M }$ as a set $Y _ { z } =$ $y _ { z _ { 1 } : z _ { M } }$ . The conditional probability $P ( { Y \vert { X } } )$ is factorized with respect to the position latent variable:
+
+$$
+P ( Y | X ) = \sum _ { z \in \pi ( M ) } P ( z | X ) \cdot P ( Y | z , X )
+$$
+
+where $\pi ( M )$ is a set consisting of permutations with $M$ elements. At decoding time, the factorization allows us to decode sentences in parallel by pre-predicting the corresponding position variables $z$ .
+
+# 3.2 MODEL ARCHITECTURE
+
+
+attention over the output of the encoder stack. Similar to the encoder, we employ residual connectionsaround each of the sub-layers, followed by layer normalization. We also modify the self-attention around each of the sub-layers, followed by layer normalization. We also modify the self-attentionFigure 1: Illustration of the proposed model, where the black solid arrows represent differentiable sub-layer in the decoder stack to prevent positions from attending to subsequent positions. Tmasking, combined with fact that the output embeddings are offset by one position, ensures thatsub-layer in the decoder stack to prevent positions frommasking, combined with fact that the output embeddings aconnections and the dashed arrows are non-differentiable operations.
+
+As shown in Figure 1, PNAT is composed of four modules: an encoder stack, a bridge block, a 3.2 Attention 3.2 Attentionposition predictor as well as a decoder stack. Before detailing each component of PNAT model, we overview the architecture for a brief understanding.
+
+Like most sequence-to-sequence models, PNAT first encodes a source sequence $X { = } x _ { 1 : N }$ into its contextual word representations $\scriptstyle { E = e _ { 1 : N } }$ with the encoder stack. With generated contextual word representation $E$ at source side, the bridge block is leveraged to computed the target length $M$ as well as the corresponding features $D { = } d _ { 1 : { M } }$ , which is fed into the decoder as its input. It is worth noting that the decoder inputs $D$ is computed without reordering. Thus the position predictor is introduced to deal with this issue by predicting a permutation $z { = } z _ { 1 : { M } }$ over $D$ . Finally, PNAT generates the target sequence from the decoder input $D$ and its permutation $_ z$ .
+
+Encoder and Decoder Given a source sentence $X$ with length $N$ , PNAT encoder produces its contextual word representations $E$ . The contextual word representations $E$ are further used in computing target length $M$ and decoder initial states $D$ , and are also used as memory of attention at decoder side.
+
+Generally, PNAT decoder can be considered as a transformer with a broader vision, because it leverages future word information that is blind to the autoregressive transformer. Intuitively, we use relative position encoding in self-attention(Shaw et al., 2018), rather than absolute one that is more likely to cause position errors. Following Shaw et al. (2018) with a clipping distance $d$ (usually $d \geq 2 \AA$ ) set for relative positions, we preserve $d = 4$ relations.
+
+Bridge The bridge module predicts the target length $M$ , and initializes the decoder inputs $D$ from the source representations $E$ . The target length $M$ could be estimated from the source encoder representation:
+
+$$
+M = N + \arg \operatorname* { m a x } \phi ( E )
+$$
+
+where $\phi ( \cdot )$ produces a categorical distribution ranged in $[ - B , B ]$ $B = 2 0$ ). It is notable that we use the predicted length at inference stage, although during training, we simply use the length of each reference target sequence. Then, we adopt the method proposed by Li et al. (2019) to compute $D$ . Given the source representation $E$ and the estimated target length $M$ , we linearly combine the embeddings of the neighboring source tokens to generate $D$ as follows:
+
+$$
+d _ { j } = \sum _ { i } w _ { j i } \cdot e _ { i }
+$$
+
+$$
+w _ { j i } = \mathrm { s o f t m a x } ( - | j - i | / \tau )
+$$
+
+where $w _ { j i }$ is a normalized weight that reflects the contribution of $e _ { i }$ to $d _ { j }$ , and $\tau$ is a hyperparameter indicating the sharpness of the weight distribution.
+
+Position Predictor For the proposed PNAT, we model position permutations with a position predictor. As shown in Figure 1, the position predictor takes the decoder inputs $D$ and the source representation $E$ to predict a permutation $_ z$ . The position predictor has a sub-encoder which stacks multiple layers of encoder units to predict its predicted input $R { = } r _ { 1 : { M } }$ .
+
+With the predicted inputs $R$ , we conduct an autoregressive position predictor, denoted as AR-Predictor. The AR-Predictor searches a permutation $_ { z }$ with:
+
+$$
+P ( \boldsymbol { z } | D , E ) = \prod _ { t = 1 } ^ { M } p _ { ( } z _ { t } | \boldsymbol { z } _ { < t } , D , E ; \boldsymbol { \theta } )
+$$
+
+where $\theta$ is the parameter of AR-Predictor, which includes a RNN-based model incorporated with a pointer network (Vinyals et al., 2015).
+
+To purse the efficiency of decoding, we also explore a non-autoregressive version for the position predictor, denoted as NAR-Predictor, to model the position permutation probabilities with:
+
+$$
+P ( z | D , E ) = \prod _ { t = 1 } ^ { M } p ( z _ { t } | D , E ; \theta )
+$$
+
+To obtain the permutation $_ { z }$ , AR-Predictor performs greedy search whereas NAR-Predictor performs direct arg max. We chose the AR-Predictor as our mainly position module in PNAT, and we also analyze the effectiveness of position modeling in Sec. 4.4.
+
+# 3.3 TRAINING
+
+Training requires maximizing the marginalized likelihood in Eqn. 4. However, this is intractable since we need to enumerate all the $M !$ permutations of tokens. We therefore optimize this objective by Monte Carlo sampling method with a heuristic search algorithm.
+
+Heuristic Search for Positions Intuitively, each target token should have a corresponding decoder input, and meanwhile each decoder input should be assigned to a target token. Based on this idea, we design a heuristic search algorithm to allocate positions. Given the decoder inputs and its target tokens, we first estimate the similarity between each pair of the decoder input $d _ { i }$ and the target token embedding $y _ { j }$ , which is also the weights of the target word classifier:
+
+$$
+\mathrm { s i m } _ { i , j } = \mathrm { c o s i n e } \left( d _ { i } , y _ { j } \right)
+$$
+
+Based on the cosine similarity matrix, HSP is designed to find a perfect matching between decoder inputs and target tokens:
+
+$$
+\mathrm { H S P } ( z ) = \underset { z } { \arg \operatorname* { m a x } } \sum _ { i = 0 } ^ { M } ( \ s i \mathrm { m } _ { i , z _ { i } } )
+$$
+
+Here we apply a greedy algorithm to select the pair with the highest similarity score iteratively until a permutation $z _ { \mathrm { r e f } }$ is generated. More details are provided in Appendix A.
+
+The intuition behind is that, if the decoder input $d _ { i }$ is already the most similar one to a target word, it would be easier to keep and even reinforce this association in learning the model. We also analyze the effectiveness of the HSP in the Sec. 4.4.
+
+Objective Function With the heuristically discovered positions as reference positions $z _ { \mathrm { r e f } }$ , the position predictor could be trained with a position loss:
+
+$$
+\mathcal { L } _ { \mathrm { p } } = - \log P ( z _ { \mathrm { r e f } } | D , E )
+$$
+
+Grounding on the referenced positions, the generative process of target sequences is optimized by:
+
+$$
+\mathcal { L } _ { \mathrm { g } } = - \sum _ { t = 1 } ^ { M } \log P ( Y | z _ { \mathrm { r e f } } ; X )
+$$
+
+Finally, combining two loss functions mentioned above, a full-fledged loss is derived as
+
+$$
+{ \mathcal { L } } = { \mathcal { L } } _ { \mathrm { g } } + \alpha { \mathcal { L } } _ { \mathrm { p } }
+$$
+
+The length predictor is a classifier that follows the previous settings. We also follow the previous practice (Gu et al., 2018; Wei et al., 2019) and perform an extra training process for the length predictor after the model trained and do not tune the parameter of the encoder.
+
+# 3.4 INFERENCE
+
+We follow the common choice of approximating decoding algorithms (Gu et al., 2018; Lee et al., 2018) to reduce the search space of latent variable model.
+
+Argmax Decoding Following Gu et al. (2018), one simple and effective method is to select the best sequence by choosing the highest-probability latent sequence $z$ :
+
+$$
+\begin{array} { c } { { z ^ { * } = \arg \operatorname* { m a x } _ { z } P ( z | D , E ) } } \\ { { } } \\ { { Y ^ { * } = \arg \operatorname* { m a x } _ { y } P ( Y | z ^ { * } , X ) } } \end{array}
+$$
+
+where identifying $Y ^ { * }$ only requires independently maximizing the local probability for each output position.
+
+Length Parallel Decoding We also consider the common practice of noisy parallel decoding (Gu et al., 2018), which generates a number of decoding candidates in parallel and selects the best via re-scoring using a pre-trained autoregressive model. For PNAT, we first predict the target length as $\hat { M }$ , then generate output sequence with argmax decoding for each target length candidate $M \in$ $[ \hat { M } - \Delta M , \hat { M } + \Delta M ]$ ( $M = 4$ in our experiments), which was called length parallel decoding (LPD). Then we use the pre-trained autoregressive model to rank these sequences and identify the best overall output as the final output.
+
+# 4 EXPERIMENTS
+
+We test PNAT on several benchmark sequence generation tasks. We first describe the experimental setting and implementation details and then present the main results, followed by some deep studies.
+
+# 4.1 EXPERIMENTAL SETTING
+
+To show the generation ability of PNAT, we conduct experiments on the popular machine translation and paraphrase generation tasks. These sequence generation task evaluation models from different perspectives. Translation tasks test the ability of semantic transforming across bilingual corpus. While paraphrase task focuses on substitution between the same languages while keeping the semantics.
+
+Machine Translation We valid the effectiveness of PNAT on the most widely used benchmarks for machine translation — WMT14 EN-DE(4.5M pairs) and IWSLT16 DE-EN(196K pairs). The dataset is processed with Moses script (Koehn et al., 2007), and the words are segmented into subword units using byte-pair encoding (Sennrich et al., 2016, BPE). For both WMT datasets, the source and target languages share the same set of subword embeddings while for IWSLT we use separate embeddings.
+
+Paraphrase Generation We conduct experiments following previous work (Miao et al., 2019) for paraphrase generation. We make use of the established Quora dataset 1 to evaluate on the paraphrase generation task. We consider the supervised paraphrase generation and split the Quora dataset in the standard setting. We sample $1 0 0 \mathrm { k }$ pairs sentence as training data, and holds out 3k, 30k for validation and testing, respectively.
+
+# 4.2 IMPLEMENTATION DETAILS
+
+Module Setting For machine translation, we follow the settings from Gu et al. (2018). In the case of IWSLT task, we use a small setting $( d _ { \mathrm { m o d e l } } = 2 7 8$ , $d _ { \mathrm { h i d d e n } } = 5 0 7$ , $p _ { \mathrm { d r o p o u t } } = 0 . 1$ , $n _ { \mathrm { l a y e r } } = 5$ and nhead $= 2$ ) suggested by Gu et al. (2018) for Transformer and NAT models. For WMT task, we use the base setting of the Vaswani et al. (2017) $\dot { d } _ { \mathrm { m o d e l } } = 5 1 2$ , $d _ { \mathrm { h i d d e n } } = 5 1 2$ , $p _ { \mathrm { d r o p o u t } } = 0 . 1$ , $n _ { \mathrm { l a y e r } } = 6 $ ).
+
+For paraphrase generation, we follow the settings from Miao et al. (2019), and set the 300-dimensional GRU with 2 layer for Seq-to-Seq (GRU). We empirically select a Transformer and NAT models with hyperparameters $\dot { d } _ { \mathrm { m o d e l } } = 4 0 0$ , $d _ { \mathrm { h i d d e n } } = 8 0 0$ , $p _ { \mathrm { d r o p o u t } } = 0 . 1$ , $n _ { \mathrm { l a y e r } } = 3$ and $n _ { \mathrm { h e a d } } = 4$ ).
+
+Optimization We optimize the parameter with the Adam optimizer (Kingma & Ba, 2014). The hyperparameter $\alpha$ used in Eqn. 14 was be set to 1.0 for WMT, 0.3 for IWSLT and Quora. We also use inverse square root learning rate scheduling (Vaswani et al., 2017) for the WMT, and using linear annealing (from $3 e - 4$ to $1 e - 5$ , suggested by Lee et al. (2018)) for the IWSLT and Quora. Each mini-batch consists of approximately 2K tokens for IWSLT and Quora, 32K tokens for WMT.
+
+Knowledge Distillation Sequence-level knowledge distillation is applied to alleviate multimodality problem while training, using Transformer as a teacher (Hinton et al., 2015). Previous studies on non-autoregressive generation (Gu et al., 2018; Lee et al., 2018; Wei et al., 2019) have used translations produced by a pre-trained Transformer model as the training data, which significantly improves the performance. We follow this setting in translation tasks.
+
+Table 1: Performance on the newstest-2014 for WMT14 EN-DE and test2013 for IWSLT EN-DE. ‘-’ denotes same numbers as above. ‘\*’ indicates our implementation. The decoding speed is measured sentence-by-sentence and the speedup is computed by comparing with Transformer.
+
+| Model | WMT14 EN-DE DE-EN | IWSLT16 DE-EN | | Speedup |
| Autoregressive Methods | | |
| Transformer-base (Vaswani et al., 2017) *Transformer(Beam=4) | 27.30 27.40 | 1 31.33 | 34.81 | 1.0× |
| Non-Autoregressive Methods | | |
| Flowseq (Ma et al., 2019) | 18.55 | 23.36 | | |
| *NAT-base | / | 11.02 | | / |
| *PNAT | 19.73 NAT w/ Knowledge Distillation | 24.04 | | / |
| NAT-FT (Gu et al., 2018) LT (Kaiser et al., 2018) IR-NAT (Lee et al., 2018) | 17.69 19.80 13.91 | 21.47 / | / / 27.68 | 15.6× 5.8× |
| ENAT (Guo et al., 2019) NAT-REG (Wang et al., 2019) imitate-NAT (Wei et al.,2019) Flowseq (Ma et al.,2019) | 20.65 20.65 22.44 | 16.77 23.02 24.77 25.67 | / / 1 | 9.0× 24.3× 1 18.6× |
| *NAT-base | 21.45 1 23.05 | 26.16 16.69 | 一 | 1.1× 13.5x |
| *PNAT 27.18 NATw/Reranking orIterative Refinments | 31.23 | 7.3× |
| NAT-FT (rescoring 10 candidates) LT (rescoring 10 candidates) IR-NAT (refinement 10) | 18.66 22.50 | 22.42 / | / 一 | 7.7× / |
| ENAT (rescoring 9 candidates) | 21.61 24.28 | 25.48 26.10 | 32.31 / | 1.3× 12.4× |
| NAT-REG (rescoring 9 candidates) imitate-NAT (rescoring 9 candidates) | 24.61 24.15 | 28.90 27.28 | / / | 1 9.7× |
| Flowseq (rescoring 30 candidates) *PNAT (LPD n=9,△M=4) | 23.48 24.48 | 28.40 29.16 | / 32.60 | / 3.7x |
+
+# 4.3 MAIN RESULTS
+
+Machine Translation We compare the PNAT with strong NAT baselines, including the NAT with fertility (Gu et al., 2018, NAT-FT), the NAT with iterative refinement (Lee et al., 2018, IR-NAT), the NAT with regularization (Wang et al., 2019, NAT-REG), the NAT with enhanced decoder input (Guo et al., 2019, ENAT), the NAT with learning from auto-regressive model (Wei et al., 2019, imitateNAT), the NAT build on latent variables (Kaiser et al., 2018, LT), and the flow-based NAT model (Ma et al., 2019, Flowseq).
+
+The results are shown in Table 1. We basically compare the proposed PNAT against the autoregressive counterpart both in terms of generation quality, which is measured with BLEU (Papineni et al., 2002) and inference speedup. For all our tasks, we obtain the performance of competitors by either directly using the performance figures reported in the previous works if they are available or producing them by using the open source implementation of baseline algorithms on our datasets.2 Clearly, PNAT achieves a comparable or better result to previous NAT models on both WMT and IWSLT tasks.
+
+We list the result of the NAT models trained without using knowledge distillation in the second block of the Table 1. The PNAT achieves significant improvements (more than 13.0 BLEU points) over the naive baselines, which indicate that position learning greatly contributes to improve the model capability of NAT model. The PNAT also achieves a better result than the Flowseq around 1.0 BLEU, which demonstrates the effectiveness of PNAT in modeling dependencies between the target outputs.
+
+As shown in the third block of the Table 1, without using reranking techniques, the PNAT outperforms all the competitors with a large margin, achieves a balance between performance and efficiency. In particular, the previous state-of-the-art(WMT14 DE-EN) Flowseq achieves good performance with the slow speed $( 1 . 1 \times )$ , while PNAT goes beyond Flowseq in both respects.
+
+Our best results are obtained with length parallel decoding which employ autoregressive model to rerank the multiple generation candidates of different target length. Specifically, on the large scale WMT14 DE-EN task, PNAT $( + \mathrm { L P D } )$ surpass the NAT-REG by 0.76 BLEU score. Without reranking, the gap has increased to 2.4 BLEU score (27.18 v.s. 24.77). The experiments shows the power of explicitly position modeling which reduces the gap between non-autoregressive and the autoregressive models.
+
+Paraphrase Generation Given a sentence, paraphrase generation aims to synthesize another sentence that is different from the given one, but conveys the same meaning. Comparing with translation task, paraphrase generation prefers a more similar order between source and target sentence, which possibly learn a trivial position model. PNAT can potentially yield better results with the position model to infer the relatively ordered alignment relationship.
+
+Table 2: Results on validation set and test set of Quora.
+
+| Source | bei dem deutschen Gesetz geht es um die Zuweisung bei der Geburt. |
| Reference Heuristic Searched Position(HSP) PNATw/HSP | German law is about assigning it at birth . 3,6, 1, 2, 10, 0, 5,4, 7, 8, 9 German law is about assigning them at birth . |
| PredictedPosition PNAT w/Predicted Postion | 3, 6,1, 2, 10, 0, 5, 4, 7, 8, 9 German law is about assigning them at birth . |
| Source | weiB er über das Telefon @-@ Hacking Bescheid ? |
| Reference Heuristic Searched Position(HSP) PNATw/HSP | does he know about phone hacking ? 2, 1, 3,4, 8,5,6, 0, 7,9 does he know the telephone hacking ? |
| PredictedPosition PNAT w/Predicted Postion | 1, 0, 3,4, 8,5, 6,2, 7, 9 he know about the telephone hacking ? |
| Source | was CCAAbedeutet,mochte eineBesucherin wissen. |
| Reference Heuristic Searched Position(HSP) | one visitor wants to know what CCAA means . 5,6, 7,8,9,3,2, 0,1, 11,4, 10 |
| PNATw/HSP Predicted Position | a visitor wants to know what CCAA means . 5,0, 1, 2,3, 7,4, 8, 9,11,6, 10 |
| PNAT w/Predicted Postion Source | CCAA means wants to know to a visitor . eines von 2O Kindern in den Vereinigten |
| Staaten hat inzwischen eine Lebensmittelal- |
| Reference | lergie . one in 20 children in the United States now |
| Heuristic Searched Position(HSP) | have food allergies . 14,1,2, 3,4,5,6,7,9,8,0,10,11, 12,13 |
| PNAT w/HSP Predicted Position | one of 2O children in theUnited Statesnowhas food allergy . 14, 0, 1, 2, 3,4,5, 6, 8,7, 9, 10, 11, 12, 13 |
\ No newline at end of file
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new file mode 100644
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| Source | weiB er über das Telefon @-@ Hacking Bescheid ? |
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| Staaten hat inzwischen eine Lebensmittelal- |
| Reference | lergie . one in 20 children in the United States now |
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| PredictedPosition PNAT w/Predicted Postion | 3, 6,1, 2, 10, 0, 5, 4, 7, 8, 9 German law is about assigning them at birth . |
| Source | weiB er über das Telefon @-@ Hacking Bescheid ? |
| Reference Heuristic Searched Position(HSP) PNATw/HSP | does he know about phone hacking ? 2, 1, 3,4, 8,5,6, 0, 7,9 does he know the telephone hacking ? |
| PredictedPosition PNAT w/Predicted Postion | 1, 0, 3,4, 8,5, 6,2, 7, 9 he know about the telephone hacking ? |
| Source | was CCAAbedeutet,mochte eineBesucherin wissen. |
| Reference Heuristic Searched Position(HSP) | one visitor wants to know what CCAA means . 5,6, 7,8,9,3,2, 0,1, 11,4, 10 |
| PNATw/HSP Predicted Position | a visitor wants to know what CCAA means . 5,0, 1, 2,3, 7,4, 8, 9,11,6, 10 |
| PNAT w/Predicted Postion Source | CCAA means wants to know to a visitor . eines von 2O Kindern in den Vereinigten |
| Staaten hat inzwischen eine Lebensmittelal- |
| Reference | lergie . one in 20 children in the United States now |
| Heuristic Searched Position(HSP) | have food allergies . 14,1,2, 3,4,5,6,7,9,8,0,10,11, 12,13 |
| PNAT w/HSP Predicted Position | one of 2O children in theUnited Statesnowhas food allergy . 14, 0, 1, 2, 3,4,5, 6, 8,7, 9, 10, 11, 12, 13 |
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diff --git a/parse/train/H1egcgHtvB/H1egcgHtvB.md b/parse/train/H1egcgHtvB/H1egcgHtvB.md
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+# RAT-SQL: RELATION-AWARE SCHEMA ENCODING AND LINKING FOR TEXT-TO-SQL PARSERS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+When translating natural language questions into SQL queries to answer questions from a database, contemporary semantic parsing models struggle to generalize to unseen database schemas. The generalization challenge lies in (a) encoding the database relations in an accessible way for the semantic parser, and (b) modeling alignment between database columns and their mentions in a given query. We present a unified framework, based on the relation-aware self-attention mechanism, to address schema encoding, schema linking, and feature representation within a text-to-SQL encoder. On the challenging Spider dataset this framework boosts the exact match accuracy to $5 3 . 7 \%$ , compared to $4 7 . 4 \%$ for the state-of-the-art model unaugmented with BERT embeddings. In addition, we observe qualitative improvements in the model’s understanding of schema linking and alignment.
+
+# 1 INTRODUCTION
+
+The ability to effectively query databases with natural language has the potential to unlock the power of large datasets to the vast majority of users who are not proficient in query languages. As such, a large body of research has focused on the task of translating natural language questions into queries that existing database software can execute.
+
+The release of large annotated datasets containing questions and the corresponding database SQL queries has catalyzed progress in the field, by enabling the training of supervised learning models for the task. In contrast to prior semantic parsing datasets (Finegan-Dollak et al., 2018), new tasks such as WikiSQL (Zhong et al., 2017) and Spider (Yu et al., 2018b) pose the real-life challenge of generalization to unseen database schemas. Every query is conditioned on a multi-table database schema, and the databases do not overlap between the train and test sets.
+
+Schema generalization is challenging for three interconnected reasons. First, any text-to-SQL semantic parsing model must encode a given schema into column and table representations suitable for decoding a SQL query that might involve any of the given columns or tables. Second, these representations should encode all the information about the schema, including its column types, foreign key relations, and primary keys used for database joins. Finally, the model must recognize natural language used to refer to database columns and tables, which might differ from the referential language seen in training. The latter challenge is known as schema linking – aligning column/table references in the question to the corresponding schema columns/tables.
+
+
+Figure 1: A challenging text-to-SQL task from the Spider dataset.
+
+While the question of schema encoding has been studied in recent literature (Bogin et al., 2019b), schema linking has been relatively less explored. Consider the example in Figure 1. It illustrates the challenge of ambiguity in linking: while “model” in the question refers to car_names.model rather than model_list.model, “cars” actually refers to both cars_data and car_names (but not car_makers) for the purpose of table joining. To resolve the column/table references properly, the semantic parser must take into account both the known schema relations (e.g. foreign keys) and the question context.
+
+Prior work (Bogin et al., 2019b) addressed the schema representation problem by encoding the directed graph of foreign key relations among the columns with a graph neural network. While effective, this approach has two important shortcomings. First, it does not contextualize schema encoding with the question, thus making it difficult for the model to reason about schema linking after both the column representations and question word representations have been built. Second, it limits information propagation during schema encoding to predefined relations in the schema such as foreign keys. The advent of self-attentional mechanisms in natural language processing (Vaswani et al., 2017) shows that global reasoning is crucial to building effective representations of relational structures. However, we would like any global reasoning to also take into account the aforementioned predefined schema relations.
+
+In this work, we present a unified framework, called RAT-SQL,1 for encoding relational structure in the database schema and a given question. It uses relation-aware self-attention to combine global reasoning over the schema entities and question words with structured reasoning over predefined schema relations. We then apply RAT-SQL to the problems of schema encoding and schema linking. As a result, we obtain $5 3 . 7 \%$ exact match accuracy on the Spider test set. At the time of writing, this result is the state of the art among models unaugmented with pretrained BERT embeddings. In addition, we experimentally demonstrate that RAT-SQL enables the model to build more accurate internal representations of the question’s true alignment with schema columns and tables.
+
+# 2 RELATED WORK
+
+Semantic parsing of natural language to SQL queries recently surged in popularity thanks to the creation of two new multi-table datasets with the challenge of schema generalization – WikiSQL (Zhong et al., 2017) and Spider (Yu et al., 2018b). Schema encoding is not as challenging in WikiSQL as in Spider thanks to the lack of multi-table relations. Schema linking is relevant for both tasks but also more challenging in Spider due to the richer natural language expressiveness and less restricted SQL grammar observed in it. Indeed, the state of the art semantic parser on WikiSQL (He et al., 2019) achieves a test set accuracy of $9 1 . 8 \%$ , significantly higher than the state of the art on Spider.
+
+The recent state-of-the-art models evaluated on Spider use various attentional architectures for question/schema encoding and AST-based structural architectures for query decoding. IRNet (Guo et al., 2019) encodes the question and schema separately with LSTM and self-attention respectively, augmenting them with custom type vectors for schema linking. They further use the AST-based decoder of Yin and Neubig (2017) to decode a query in an intermediate representation (IR) that exhibits higher-level abstraction structure than SQL. Bogin et al. (2019b) encode the schema with a graph neural network and a similar grammar-based decoder. Both approaches highlight the importance of schema encoding and schema linking, but design separate feature engineering techniques to augment word vectors (as opposed to relations between words and columns) to resolve it. In contrast, the relational framework of RAT-SQL provides a unified way to encode arbitrary relational information among the inputs.
+
+Concurrently with this work, Bogin et al. (2019a) published Global-GNN, a different approach to schema linking for Spider which applies global reasoning between question words and schema columns/tables. Global reasoning is implemented by gating the graph neural network that computes the representation of schema elements using question token representations. This conceptually differs from RAT-SQL in two important ways: (a) question word representations influence the schema representations but not vice versa, and (b) like in other GNN-based encoding approaches, message propagation is limited to the schema-induced edges such as foreign key relations. In contrast, our relation-aware transformer mechanism allows encoding arbitrary relations between question words and schema elements explicitly, and these representations are computed jointly using self-attention.
+
+
+Figure 2: An illustration of an example schema as a graph. We do not depict all edges and label types of Table 1 to reduce clutter.
+
+We use the same formulation of relation-aware self-attention as Shaw et al. (2018). However, that work only applied it to sequences of words in the context of machine translation, and as such, their set of relation types only encoded the relative distance between two words. We extend their work and show that relation-aware self-attention can effectively encode more complex relationships that exist within an unordered sets of elements (in this case, columns and tables within a database schema as well as relations between the schema and the question). To the best of our knowledge, this is the first application of relation-aware self-attention to joint representation learning with both predefined and softly induced relations in the input structure.
+
+# 3 RAT-SQL
+
+We now describe the RAT-SQL framework and its application to the problems of schema encoding and linking. First, we formally define the text-to-SQL semantic parsing problem and its components. Then, we introduce the relation-aware self-attention mechanism, our framework for jointly encoding relational structure between the question and the schema. Finally, we present our implementation of schema linking in the RAT-SQL framework.
+
+# 3.1 PROBLEM DEFINITION
+
+Given a natural language question $Q$ and a schema $s = \langle \mathcal { C } , \mathcal { T } \rangle$ for a relational database, our goal is to generate the corresponding SQL $P$ . Here the question $Q = q _ { 1 } \ldots q _ { | Q | }$ is a sequence of words, and the schema consists of columns $\mathcal { C } = \{ c _ { 1 } , \ldots , c _ { | \mathcal { C } | } \}$ and tables $\mathcal { T } = \left\{ t _ { 1 } , \dots , t _ { | T | } \right\}$ . Each column name $c _ { i }$ contains words $c _ { i , 1 } , \ldots , c _ { i , \left| \boldsymbol { c } _ { i } \right| }$ and each table name $t _ { i }$ contains words $t _ { i , 1 } , \ldots , t _ { i , | t _ { i } | }$ . The desired program $P$ is represented as an abstract syntax tree $T$ in the context-free grammar of SQL.
+
+Some columns in the schema are primary keys, used for uniquely indexing the corresponding table, and some are foreign keys, used to reference a primary key column in a different table. As described in Section 1, we would like to softly bias our schema encoding mechanism toward these predefined relations. In addition, each column has a type $\tau$ such as number or text.
+
+Schema linking aims at finding the alignment between question words and mentioned columns or tables. It’s a crucial step for a parser to generate the right columns and tables in SQL. We model the latent alignment explicitly using an alignment matrix (Section 3.6), which is softly biased towards some string-match based relations, as inspired by Guo et al. (2019).
+
+# 3.2 ENCODING THE SCHEMA AS A GRAPH
+
+To support reasoning about relationships between schema elements in the encoder, we begin by representing the database schema using a directed graph $\mathcal { G }$ , where each node and edge has a label. We represent each table and column in the schema as a node in this graph, labeled with the words in the name; for columns, we prepend the type of the column to the label. For each pair of nodes $x$ and $y$ in the graph, Table 1 describes when there exists an edge from $x$ to $y$ and the label it should have. Figure 2 illustrates an example graph (although not all edges and labels are shown).
+
+
+Figure 3: Overview of the stages of our approach.
+
+# 3.3 INITIAL ENCODING OF THE INPUT
+
+We now obtain an initial representation for each of the nodes in the graph, as well as for the words in the input question. For the graph nodes, we use a bidirectional LSTM (BiLSTM) over the words contained in the label. We concatenate the output of the initial and final time steps of this LSTM to form the embedding for the node. For the question, we also use a bidirectional LSTM over the words:
+
+$$
+\begin{array} { r l } & { c _ { i , 0 } ^ { \mathrm { f w d } } , c _ { i , 0 } ^ { \mathrm { r e v } } ) \cdot \cdot \cdot , ( c _ { i , | c _ { i } | } ^ { \mathrm { f w d } } , c _ { i , | c _ { i } | } ^ { \mathrm { r e v } } ) = \mathrm { B i L S T M } _ { \mathrm { C o h m m } } ( c _ { i } ^ { \mathrm { t p e } } , c _ { i , 1 } , \cdot \cdot , c _ { i , | c _ { i } | } ) ; \quad c _ { i } ^ { \mathrm { i n t } } = \mathrm { C o n c a t } ( c _ { i , | c _ { i } | } ^ { \mathrm { f w d } } , c _ { i , 0 } ^ { \mathrm { r e v } } ) } \\ & { ( t _ { i , 1 } ^ { \mathrm { f w d } } , t _ { i , 1 } ^ { \mathrm { r e v } } ) \cdot \cdot \cdot , ( t _ { i , | c _ { i } | } ^ { \mathrm { f w d } } , t _ { i , | i _ { i } | } ^ { \mathrm { r e v } } ) = \mathrm { B i L S T M } _ { \mathrm { T a b s } } ( t _ { i , 1 } , \cdot \cdot , t _ { i , | t _ { i } | } ) ; \quad t _ { i } ^ { \mathrm { i n t } } = \mathrm { C o n c a t } ( t _ { i , | c _ { i } | } ^ { \mathrm { f w d } } , t _ { i , 1 } ^ { \mathrm { r e v } } ) } \\ & { ( q _ { 1 } ^ { \mathrm { f w d } } , q _ { 1 } ^ { \mathrm { r e v } } ) , \cdot \cdot \cdot , ( q _ { | Q | } ^ { \mathrm { f w d } } , q _ { | Q | } ^ { \mathrm { r e v } } ) = \mathrm { B i L S T M } _ { \mathrm { Q u e s i o n } } ( q _ { 1 } , \cdot \cdot \cdot , q _ { | Q | } ) ; \quad q _ { i } ^ { \mathrm { i n t } } = \mathrm { C o n c a t } ( q _ { i } ^ { \mathrm { f w d } } , q _ { i } ^ { \mathrm { r e v } } ) } \end{array}
+$$
+
+where each of the BiLSTM functions first lookup word embeddings for each of the input tokens. The LSTMs do not share any parameters.
+
+# 3.4 RELATION-AWARE TRANSFORMER
+
+At this point, we have representations $c _ { i } ^ { \mathrm { i n i t } } , t _ { i } ^ { \mathrm { i n i t } }$ , and $\pmb q _ { i } ^ { \mathrm { i n i t } }$ . Similar to encoders used in some previous papers, these initial representations are independent of each other (uninfluenced by which other columns or tables are present). Now, we would like to imbue these representations with the information in the schema graph. We use a form of self-attention (Vaswani et al., 2017) that is relation-aware (Shaw et al., 2018) to achieve this goal.
+
+In one step of relation-aware self-attention, we begin with an input $_ { \textbf { \em x } }$ of $n$ elements (where $x _ { i } \in \mathbb { R } ^ { d _ { x } } .$ ) and transform each $x _ { i }$ into $y _ { i } \in \mathbb { R } ^ { d _ { x } }$ . We follow the formulation described in Shaw et al. (2018):
+
+$$
+\begin{array} { r l } & { e _ { i j } ^ { ( h ) } = \cfrac { x _ { i } W _ { Q } ^ { ( h ) } ( x _ { j } W _ { K } ^ { ( h ) } + r _ { i j } ^ { K } ) ^ { T } } { \sqrt { d _ { z } / H } } ; \quad \alpha _ { i j } ^ { ( h ) } = \cfrac { \exp ( e _ { i j } ^ { ( h ) } ) } { \sum _ { l = 1 } ^ { n } \exp ( e _ { i l } ^ { ( h ) } ) } } \\ & { z _ { i } ^ { ( h ) } = \displaystyle \sum _ { j = 1 } ^ { n } \alpha _ { i j } ^ { ( h ) } ( x _ { j } W _ { V } ^ { ( h ) } + r _ { i j } ^ { V } ) ; \quad z _ { i } = \mathrm { C o n c a t } ( z _ { i } ^ { ( 1 ) } , \cdots , z _ { i } ^ { ( H ) } ) } \\ & { \tilde { y } _ { i } = \mathrm { L a y e r N o r m } ( x _ { i } + z _ { i } ) ; \quad y _ { i } = \mathrm { L a y e r N o r m } ( \tilde { y } _ { i } + \mathrm { F C } ( \mathrm { R e L U } ( \mathrm { F C } ( \tilde { y } _ { i } ) ) ) } \end{array}
+$$
+
+where FC is a fully-connected layer, $1 \leq h \leq H$ , and $W _ { Q } ^ { ( h ) } , W _ { K } ^ { ( h ) } , W _ { V } ^ { ( h ) } \in \mathbb { R } ^ { d _ { x } \times ( d _ { x } / H ) }$ . The $r _ { i j }$ terms encode the relationship between the two elements $x _ { i }$ and $x _ { j }$ in the input. We explain how we obtain $r _ { i j }$ in the next part.
+
+Application Within Our Encoder At the start, we construct the input $x$ of $| c | + | t | + | q |$ elements using $c _ { i } ^ { \mathrm { i n i t } } , t _ { i } ^ { \mathrm { i n i t } }$ , and $\pmb q _ { i } ^ { \mathrm { i n i t } }$ :
+
+$$
+\boldsymbol { x } = ( \boldsymbol { c } _ { 1 } ^ { \mathrm { { i n i t } } } , \cdot \cdot \cdot , \boldsymbol { c } _ { | \mathcal { C } | } ^ { \mathrm { { i n i t } } } , t _ { 1 } ^ { \mathrm { { i n i t } } } , \cdot \cdot \cdot , t _ { | \mathcal { T } | } ^ { \mathrm { { i n i t } } } , \boldsymbol { q } _ { 1 } ^ { \mathrm { { i n i t } } } , \cdot \cdot \cdot , \boldsymbol { q } _ { | \mathcal { Q } | } ^ { \mathrm { { i n i t } } } ) .
+$$
+
+We then apply a stack of $N$ relation-aware self-attention layers, where $N$ is a hyperparameter. The weights of the encoder layers are not tied; each layer has its own set of weights. After processing through the stack of $N$ encoder layers, we obtain
+
+$$
+\begin{array} { r } { ( c _ { 1 } ^ { \mathrm { f i n a l } } , \cdot \cdot \cdot , c _ { | \mathcal { C } | } ^ { \mathrm { f i n a l } } , t _ { 1 } ^ { \mathrm { f i n a l } } , \cdot \cdot \cdot , t _ { | \mathcal { T } | } ^ { \mathrm { f i n a l } } , q _ { 1 } ^ { \mathrm { f i n a l } } , \cdot \cdot \cdot , q _ { | \mathcal { Q } | } ^ { \mathrm { f i n a l } } ) = y . } \end{array}
+$$
+
+Table 1: Description of edge types present in the directed graph created to represent the schema. An edge exists from source node $x \in S$ to target node $y \in S$ if the pair fulfills one of the descriptions listed in the table, with the corresponding label. Otherwise, no edge exists from $x$ to $y$ .
+
+| Model | log p(x) ≥ |
| Models with autoregressive (AR) or coupling(C) components | |
| VLAE(Chen et al.,2017) | -79.03 |
| Pixel RNN(van den Oord et al.,2016b) | -79.20 |
| RQ-NSF(C) (Durkan et al., 2019) | -79.63 |
| Pixel VAE (Gulrajani et al., 2017) | -79.66 |
| RQ-NSF (AR) (Durkan et al., 2019) | -79.71 |
| IAF VAE (Kingma et al., 2016) | -79.88 |
| DRAW (Gregor et al., 2015) Pixel CNN (van den Oord et al.,2016a) | -80.97 |
| -81.30 |
| Models without autoregressive or coupling components SeRe-VAE | -79.50 |
| BIVA (Maalpe et al.,2019) | -80.47 |
| Discrete VAE (Rolfe,2017) | -81.01 |
+
+# 4.2 CIFAR10 NATURAL IMAGES
+
+# 4.2.1 ABLATION STUDY
+
+In this section, we study the effect of the different couplings between the layers of the architecture presented in Figure 1 on CIFAR-10 images which have dimension (32, 32, 3). We consider a 16- layer architecture with each layer generating a $( 8 , 8 , 3 )$ patch of the image when partitioned in a spatial checkerboard pattern. We use $( 8 , 8 , 2 )$ latent spaces per layer. For the decoder, we use the mixture of discretized logistic distributions (Salimans et al., 2017). In particular, we investigate three different architectures:
+
+• case 1: there are no couplings (no vertical edges) between the layers and each patch is independently generated from the others, • case 2: there are couplings only between the decoders $( \pmb { x } _ { l - 1 } \pmb { x } _ { l }$ edges), • case 3: there is feedback from the previous inference layer both in the observed space ${ \bf { x } } _ { l - 1 } { \bf { x } } _ { l }$ edges) and the latent space $z _ { l - 1 } \to \epsilon _ { l }$ , and $z _ { l - 1 } z _ { l }$ edges).
+
+In all of the above cases, we consider joint bijective layers between the inference and generative network. One observation that we would like to make and turned out to be critical , when we tested our architecture on more complex regimes such as CIFAR-10, and in order to obtain significant predictive benefits from case 3 compared to case 2 was that we had to consider a lower bound for the variance in the prior layers. In other words, a deep probabilistic should self-reflect by obtaining information from the previous inference layers but without being overly confident in its prior assumptions. This can also be mathematically corroborated by examining the KL-divergence in the VAE objective of equation 2 for the residual parametrization introduced in section 3.4.2:
+
+$$
+\mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } } \mathtt \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \beta } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \beta } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \beta } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \beta } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \beta } } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \beta } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \beta } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \beta } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } } \mathtt { \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt { \mathtt \beta } } } } } } } \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt \beta } } } } } } \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt \beta } } } } } } \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt \mathtt { \mathtt { \mathtt \beta } } } } } } \mathtt \mathtt \mathtt \mathtt \mathtt \mathtt \mathtt \mathtt
+$$
+
+As it can be seen, bounding $\pmb { \sigma } ( z )$ from below prevents making the first term of the $\mathrm { K L }$ arbitrarily large. In these experiments, we take a unit lower bound.
+
+| Model | log p(x) ≥ |
| Models with autoregressive (AR) or coupling(C) components | |
| VLAE(Chen et al.,2017) | -79.03 |
| Pixel RNN(van den Oord et al.,2016b) | -79.20 |
| RQ-NSF(C) (Durkan et al., 2019) | -79.63 |
| Pixel VAE (Gulrajani et al., 2017) | -79.66 |
| RQ-NSF (AR) (Durkan et al., 2019) | -79.71 |
| IAF VAE (Kingma et al., 2016) | -79.88 |
| DRAW (Gregor et al., 2015) Pixel CNN (van den Oord et al.,2016a) | -80.97 |
| -81.30 |
| Models without autoregressive or coupling components SeRe-VAE | -79.50 |
| BIVA (Maalpe et al.,2019) | -80.47 |
| Discrete VAE (Rolfe,2017) | -81.01 |
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| Models with autoregressive (AR) or coupling(C) components | |
| VLAE(Chen et al.,2017) | -79.03 |
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| Models with autoregressive (AR) or coupling(C) components | |
| VLAE(Chen et al.,2017) | -79.03 |
| Pixel RNN(van den Oord et al.,2016b) | -79.20 |
| RQ-NSF(C) (Durkan et al., 2019) | -79.63 |
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