A Theoretical Analysis of Deep Q-Learning

Introduction

Reinforcement learning (RL) attacks the multi-stage decision-making problems by interacting with the environment and learning from the experiences. With the breakthrough in deep learning, deep reinforcement learning (DRL) demonstrates tremendous success in solving highly challenging problems, such as the game of Go (Silver et al. 2016; Silver et al. 2017), computer games (Vinyals et al. 2019), robotics (Kober and Peters 2012), dialogue systems (Chen et al. 2017). In DRL, the value or policy functions are often represented as deep neural networks and the related deep learning techniques can be readily applied. For example, deep Q-network (DQN) (Mnih et al. 2015), asynchronous advantage actor-critic (A3C) (Mnih et al. 2016), trust region policy optimization (TRPO) (Schulman et al. 2015), proximal policy optimization (PPO) (Schulman et al. 2017) build upon classical RL methods (Watkins and Dayan 1992; Sutton et al. 2000; Konda and Tsitsiklis 2000) and have become benchmark algorithms for artificial intelligence.

Despite its great empirical success, there exists a substantial gap between the theory and practice of DRL. In particular, most existing theoretical work on reinforcement learning focuses on the tabular case where the state and action spaces are finite, or the case where the value function is linear. Under these restrictive settings, the algorithmic and statistical perspectives of reinforcement learning are well-understood via the tools developed for convex optimization and linear regression. However, in presence of nonlinear function approximators such as deep neural network, the theoretical analysis of reinforcement learning becomes intractable as it involves solving a highly nonconvex statistical optimization problem.

To bridge such a gap in DRL, we make the first attempt to theoretically understand DQN, which can be cast as an extension of the classical Q-learning algorithm (Watkins and Dayan 1992) that uses deep neural network to approximate the action-value function. Although the algorithmic and statistical properties of the classical Q-learning algorithm are well-studied, theoretical analysis of DQN is highly challenging due to its differences in the following two aspects.

First, in online gradient-based temporal-difference reinforcement learning algorithms, approximating the action-value function often leads to instability. Baird 1995 proves that this is the case even with linear function approximation. The key technique to achieve stability in DQN is experience replay (Lin 1992; Mnih et al. 2015). In specific, a replay memory is used to store the trajectory of the Markov decision process (MDP). At each iteration of DQN, a mini-batch of states, actions, rewards, and next states are sampled from the replay memory as observations to train the Q-network, which approximates the action-value function. The intuition behind experience replay is to achieve stability by breaking the temporal dependency among the observations used in training the deep neural network.

Second, in addition to the aforementioned Q-network, DQN uses another neural network named the target network to obtain an unbiased estimator of the mean-squared Bellman error used in training the Q-network. The target network is synchronized with the Q-network after each period of iterations, which leads to a coupling between the two networks. Moreover, even if we fix the target network and focus on updating the Q-network, the subproblem of training a neural network still remains less well-understood in theory.

In this paper, we focus on a slight simplification of DQN, which is amenable to theoretical analysis while fully capturing the above two aspects. In specific, we simplify the technique of experience replay with an independence assumption, and focus on deep neural networks with rectified linear units (ReLU) (Nair and Hinton 2010) and large batch size. Under this setting, DQN is reduced to the neural fitted Q-iteration (FQI) algorithm (Riedmiller 2005) and the technique of target network can be cast as the value iteration. More importantly, by adapting the approximation results for ReLU networks to the analysis of Bellman operator, we establish the algorithmic and statistical rates of convergence for the iterative policy sequence obtained by DQN. As shown in the main results in §3, the statistical error characterizes the bias and variance that arise from approximating the action-value function using neural network, while the algorithmic error geometrically decays to zero as the number of iteration goes to infinity.

Furthermore, we extend DQN to two-player zero-sum Markov games (Shapley 1953). The proposed algorithm, named Minimax-DQN, can be viewed as a combination of the Minimax-Q learning algorithm for tabular zero-sum Markov games (Littman 1994) and deep neural networks for function approximation. Compared with DQN, the main difference lies in the approaches to compute the target values. In DQN, the target is computed via maximization over the action space. In contrast, the target obtained computed by solving the Nash equilibrium of a zero-sum matrix game in Minimax-DQN, which can be efficiently attained via linear programming. Despite such a difference, both these two methods can be viewed as approximately applying the Bellman operator to the Q-network. Thus, borrowing the analysis of DQN, we also establish theoretical results for Minimax-DQN. Specifically, we quantify the suboptimality of policy returned by the algorithm by the difference between the action-value functions associated with this policy and with the Nash equilibrium policy of the Markov game. For this notion of suboptimality, we establish the both algorithmic and statistical rates of convergence, which implies that the action-value function converges to the optimal counterpart up to an unimprovable statistical error in geometric rate.

Our contribution is three-fold. First, we establish the algorithmic and statistical errors of the neural FQI algorithm, which can be viewed as a slight simplification of DQN. Under mild assumptions, our results show that the proposed algorithm obtains a sequence of Q-networks that geometrically converges to the optimal action-value function up to an intrinsic statistical error induced by the approximation bias of ReLU network and finite sample size. Second, as a byproduct, our analysis justifies the techniques of experience replay and target network used in DQN, where the latter can be viewed as a single step of the value iteration. Third, we propose the Minimax-DQN algorithm that extends DQN to two-player zero-sum Markov games. Borrowing the analysis for DQN, we establish the algorithmic and statistical convergence rates of the action-value functions associated with the sequence of policies returned by the Minimax-DQN algorithm.

There is a huge body of literature on deep reinforcement learning, where these algorithms are based on Q-learning or policy gradient (Sutton et al. 2000). We refer the reader to Arulkumaran et al. 2017 for a survey of the recent developments of DRL. In addition, the DQN algorithm is first proposed in Mnih et al. 2015, which applies DQN to Artari 2600 games (Bellemare et al. 2013). The extensions of DQN include double DQN (van Hasselt et al. 2016), dueling DQN (Wang et al. 2016), deep recurrent Q-network (Hausknecht and Stone 2015), asynchronous DQN (Mnih et al. 2016), and variants designed for distributional reinforcement learning (Bellemare et al. 2017; Dabney et al. 2018b; Dabney et al. 2018a). All of these algorithms are corroborated only by numerical experiments, without theoretical guarantees. Moreover, these algorithms not only inherit the tricks of experience replay and the target network proposed in the original DQN, but develop even more tricks to enhance the performance. Furthermore, recent works such as Schaul et al. 2016; Andrychowicz et al. 2017; Liu and Zou 2018; Zhang and Sutton 2017; Novati and Koumoutsakos 2019 study the effect of experience replay and propose various modifications.

In addition, our work is closely related to the literature on batch reinforcement learning (Lange et al. 2012), where the goal is to estimate the value function given transition data. These problems are usually formulated into least-squares regression, for which various algorithms are proposed with finite-sample analysis. However, most existing works focus on the settings where the value function are approximated by linear functions. See Bradtke and Barto 1996; Boyan 2002; Lagoudakis and Parr 2003; Lazaric et al. 2016; Farahmand et al. 2010; Lazaric et al. 2012; Tagorti and Scherrer 2015 and the references therein for results of the least-squares policy iteration (LSPI) and Bellman residue minimization (BRM) algorithms. Beyond linear function approximation, a recent work (Farahmand et al. 2016) studies the performance of LSPI and BRM when the value function belongs to a reproducing kernel Hilbert space. However, we study the fitted Q-iteration algorithm, which is a batch RL counterpart of DQN. The fitted Q-iteration algorithm is proposed in Ernst et al. 2005, and Riedmiller 2005 proposes the neural FQI algorithm. Finite-sample bounds for FQI have been established in Murphy 2005; Munos and Szepesvári 2008 for large classes of regressors. However, their results are not applicable to ReLU networks due to the huge capacity of deep neural networks. Furthermore, various extensions of FQI are studied in Antos et al. 2008a; Farahmand et al. 2009; Tosatto et al. 2017; Geist et al. 2019 to handle continuous actions space, ensemble learning, and entropy regularization. The empirical performances of various batch RL methods have been examined in Levine et al. 2017; Agarwal et al. 2019; Fujimoto et al. 2019.

Moreover, Q-learning, and reinforcement learning methods in general, have been widely applied to dynamic treatment regimes (DTR) (Chakraborty 2013; Laber et al. 2014; Tsiatis 2019), where the goal is to find sequential decision rules for individual patients that adapt to time-evolving illnesses. There is a huge body of literature on this line of research. See, e.g., Murphy 2003; Zhao et al. 2009; Qian and Murphy 2011; Zhao et al. 2011; Zhang et al. 2012; Zhao et al. 2012; Goldberg and Kosorok 2012; Nahum-Shani et al. 2012; Goldberg et al. 2013; Schulte et al. 2014; Song et al. 2015; Zhao et al. 2015; Linn et al. 2017; Zhou et al. 2017; Shi et al. 2018; Zhu et al. 2019 and the references therein. Our work provides a theoretical underpinning for the application of DQN to DTR (Liu et al. 2019b) and motivates the principled usage of DRL methods in healthcare applications (Yu et al. 2019).

Furthermore, our work is also related to works that apply reinforcement learning to zero-sum Markov games. The Minimax-Q learning is proposed by Littman 1994, which is an online algorithm that is an extension Q-learning. Subsequently, for Markov games, various online algorithms are also proposed with theoretical guarantees. These works consider either the tabular case or linear function approximation. See, e.g., Bowling 2001; Conitzer and Sandholm 2007; Prasad et al. 2015; Wei et al. 2017; Pérolat et al. 2018; Srinivasan et al. 2018; Wei et al. 2017 and the references therein. In addition, batch reinforcement learning is also applied to zero-sum Markov games by Lagoudakis and Parr 2002; Pérolat et al. 2015; Pérolat et al. 2016a; Pérolat et al. 2016b; Zhang et al. 2018, which are closely related to our work. All of these works consider either linear function approximation or a general function class with bounded pseudo-dimension (Anthony and Bartlett 2009). However, there results cannot directly imply finite-sample bounds for Minimax-DQN due to the huge capacity of deep neural networks.

Finally, our work is also related a line of research on the model capacity of ReLU deep neural networks, which leads to understanding the generalization property of deep learning (Mohri et al. 2012; Kawaguchi et al. 2017). Specifically, Bartlett 1998; Neyshabur et al. 2015b; Neyshabur et al. 2015a; Bartlett et al. 2017; Golowich et al. 2018; Liang et al. 2019 propose various norms computed from the networks parameters and establish capacity bounds based upon these norms. In addition, Maass 1994; Bartlett et al. 1999; Schmidt-Hieber 2020+; Bartlett et al. 2019; Klusowski and Barron 2016; Barron and Klusowski 2018; Suzuki 2019; Bauer et al. 2019 study the Vapnik-Chervonenkis (VC) dimension of neural networks and Dziugaite and Roy 2017; Neyshabur et al. 2018 establish the PAC-Bayes bounds for neural networks. Among these works, our work is more related to Schmidt-Hieber 2020+; Suzuki 2019, which relate the VC dimension of the ReLU networks to a set of hyperparameters used to define the networks. Based on the VC dimension, they study the statistical error of nonparametric regression using ReLU networks. In sum, theoretical understanding of deep learning is pertinent to the study of DRL algorithms. See Kawaguchi et al. 2017; Neyshabur et al. 2017; Fan et al. 2019 and the references therein for recent developments on theoretical analysis of the generalization property of deep learning.

2 Notation

Background

In this section, we introduce the background. We first lay out the formulation of the reinforcement learning problem, and then define the family of ReLU neural networks.

A policy $\pi\colon{\mathcal{S}}\rightarrow\mathcal{P}(\mathcal{A})$ for the MDP maps any state $s\in{\mathcal{S}}$ to a probability distribution $\pi(\cdot{\,|\,}s)$ over $\mathcal{A}$. For a given policy $\pi$, starting from the initial state $S_{0}=s$, the actions, rewards, and states evolve according to the law as follows:

The policy $\pi$ can be controlled by decision makers, yet the functions $P$ and $R$ are given by the nature or the system that are unknown to decision makers.

By the law of iterative expectation, for any policy $\pi$,

where $Q^{\pi}(s,a)$, called an action value function, is given by

and define the Bellman operator $T^{\pi}$ by $(T^{\pi}Q)(s,a)=r(s,a)+\gamma\cdot(P^{\pi}Q)(s,a).$ Then $Q^{\pi}$ in (2.3) is the unique fixed point of $T^{\pi}$.

The goal of reinforcement learning is to find the optimal policy, which achieves the largest cumulative reward via dynamically learning from the acquired data. To characterize optimality, by (2.2), we naturally define the optimal action-value function $Q^{*}$ as

Based on $Q^{*}$, we define the optimal policy $\pi^{*}$ as any policy that is greedy with respect to $Q^{*}$. It can be shown that $Q^{*}=Q^{\pi^{*}}$. Finally, we define the Bellman optimality operator $T$ via

Then we have the Bellman optimality equation $TQ^{*}=Q^{*}$.

2 Deep Neural Network

Here we focus on functions that are uniformly bounded because the value functions in (2.1) and (2.3) are always bounded by $V_{\max}=R_{\max}/(1-\gamma)$. We also assume that the network weights are uniformly bounded and bounded by one without loss of generality. In the sequel, we write $\mathcal{F}(L,\{d_{j}\}_{j=0}^{L+1},s,V_{\max})$ as $\mathcal{F}(L,\{d_{j}\}_{j=0}^{L+1},s)$ to simplify the notation. In addition, we restrict the networks weights to be sparse, i.e., $s$ is much smaller compared with the total number of parameters. Such an assumption implies that the network has sparse connections, which are useful for applying deep learning in memory-constrained situations such as mobile devices (Han et al. 2016; Liu et al. 2015). Empirically, sparse neural networks are realized via various regularization techniques such as Dropout (Srivastava et al. 2014), which randomly sets a fixed portion of the network weights to zero. Moreover, sparse network architectures have recently been advocated by the intriguing lottery ticket hypothesis (Frankle and Carbin 2019), which states that each dense network has a subnetwork with the sparse connections, when trained in isolation, achieves comparable performance as the original network. Thus, focusing on the class of sparse ReLU networks does not sacrifice the statistical accuracy.

Moreover, we introduce the notion of Hölder smoothness as follows, which is a generalization of Lipschitz continuity, and is widely used to characterize the regularity of functions.

Finally, we conclude this section by defining functions that can be written as a composition of multiple Hölder functions, which captures complex mappings in real-world applications such as multi-level feature extraction.

with $g_{jk}\in\mathcal{C}_{t_{j}}([a_{j},b_{j}]^{t_{j}},\beta_{j},H_{j})$ for each $k\in[p_{j+1}]$ and $j\in[q]$.

Here $f$ in (2.10) is a composition of $q$ vector-valued mappings $\{g_{j}\}_{j\in[q]}$ where each $g_{j}$ has $p_{j+1}$ components and its $k$-th component, $g_{jk}$, $\forall k\in[p_{j+1}]$, is a Hölder smooth function defined on $[a_{j},b_{j}]^{p_{j}}$. Moreover, it is well-known the statistical rate for estimating a Hölder smooth function depends on the input dimension (Tsybakov 2008). Here we assume that $g_{jk}$ only depends on $t_{j}$ of its inputs, where $t_{j}\in[p_{j}]$ can be much smaller than $p_{j}$, which enables us to obtain a more refined analysis that adapts to the effective smoothness of $f$. In particular, Definition 2.3 covers the family of Hölder smooth functions and the additive model (Friedman and Stuetzle 1981) on $^{r}$ as two special cases, where the former suffers from the curse of dimensionality whereas the latter does not.

Understanding Deep Q-Network

First, DQN use the trick of experience replay (Lin 1992). Specifically, at each time $t$, we store the transition $(S_{t},A_{t},R_{t},S_{t+1})$ into the replay memory $\mathcal{M}$, and then sample a minibatch of independent samples from $\mathcal{M}$ to train the neural network via stochastic gradient descent. Since the trajectory of MDP has strong temporal correlation, the goal of experience replay is to obtain uncorrelated samples, which yields accurate gradient estimation for the stochastic optimization problem.

Another trick is to use a target network $Q_{\theta^{\star}}$ with parameter $\theta^{\star}$ (current estimate of parameter). With independent samples $\{(s_{i},a_{i},r_{i},s^{\prime}_{i})\}_{i\in[n]}$ from the replay memory (we use $s^{\prime}_{i}$ instead of $s_{i+1}$ for the next state right after $s_{i}$ and $a_{i}$ to avoid notation crash with next independent sample $s_{i+1}$ in the state space), to update the parameter $\theta$ of the Q-network, we compute the target

(compare with Bellman optimality operator (2.7)), and update $\theta$ by the gradient of

Whereas parameter $\theta^{\star}$ is updated once every $T_{\text{target}}$ steps by letting $\theta^{\star}=\theta$. That is, the target network is hold fixed for $T_{\text{target}}$ steps and then updated it by the current weights of the Q-network.

To demystify DQN, it is crucial to understand the role played by these two tricks. For experience replay, in practice, the replay memory size is usually very large. For example, the replay memory size is $10^{6}$ in Mnih et al. 2015. Moreover, DQN use the $\epsilon$-greedy policy, which enables exploration over ${\mathcal{S}}\times\mathcal{A}$. Thus, when the replay memory is large, experience replay is close to sampling independent transitions from an explorative policy. This reduces the variance of the $\nabla L(\theta)$, which is used to update $\theta$. Thus, experience replay stabilizes the training of DQN, which benefits the algorithm in terms of computation.

Furthermore, to further understand the necessity of the target network, let us first neglect the target network and set $\theta^{\star}=\theta$. Using bias-variance decomposition, the the expected value of $L(\theta)$ in (3.1) is

To resolve this problem, we use a target network in (3.1), which has expectation

where the variance of $Y_{1}$ does not depend on $\theta$. Thus, minimizing $L(\theta)$ is close to solving

where $\Theta$ is the parameter space. Note that in DQN we hold $\theta^{\star}$ still and update $\theta$ for $T_{\text{target}}$ steps. When $T_{\text{target}}$ is sufficiently large and we neglect the fact that the objective in (3.3) is nonconvex, we would update $\theta$ by the minimizer of (3.3) for fixed $\theta^{\star}$.

Therefore, in the ideal case, DQN aims to solve the minimization problem (3.3) with $\theta^{\star}$ fixed, and then update $\theta^{\star}$ by the minimizer $\theta$. Interestingly, this view of DQN offers a statistical interpretation of the target network. In specific, if $\{Q_{\theta}\colon\theta\in\Theta\}$ is sufficiently large such that it contains $TQ_{\theta^{\star}}$, then (3.3) has solution $Q_{\theta}=TQ_{\theta^{\star}}$, which can be viewed as one-step of value iteration (Sutton and Barto 2011) for neural networks. In addition, in the sample setting, $Q_{\theta^{\star}}$ is used to construct $\{Y_{i}\}_{i\in[n]}$, which serve as the response in the regression problem defined in (3.1), with $(TQ_{\theta^{\star}})$ being the regression function.

Furthermore, turning the above discussion into a realizable algorithm, we obtain the neural fitted Q-iteration (FQI) algorithm, which generates a sequence of value functions. Specifically, let $\mathcal{F}$ be a class of function defined on ${\mathcal{S}}\times\mathcal{A}$. In the $k$-th iteration of FQI, let $\widetilde{Q}_{k}$ be current estimate of $Q^{*}$. Similar to (3.1) and (3.3), we define $Y_{i}=R_{i}+\gamma\cdot\max_{a\in\mathcal{A}}\widetilde{Q}_{k}(S_{i}^{\prime},a)$, and update $\widetilde{Q}_{k}$ by

This gives the fitted-Q iteration algorithm, which is stated in Algorithm 1.

The step of minimization problem in (3.4) essentially finds $\widetilde{Q}_{k+1}$ in $\mathcal{F}$ such that $\widetilde{Q}_{k+1}\approx T\widetilde{Q}_{k}$. Let us denote $\widetilde{Q}_{k+1}=\widehat{T}_{k}\widetilde{Q}_{k}$ where $\widehat{T}_{k}$ is an approximation of the Bellman optimality operator $T$ learned from the training data in the $k$-th iteration. With the above notation, we can now understand our Algorithm 1 as follows. Starting from the initial estimator $\widetilde{Q}_{0}$, collect the data $\{(S_{i},A_{i},R_{i},S^{\prime}_{i})\}_{i\in[n]}$ and learn the map $\widehat{T}_{1}$ via (3.4) and get $\widetilde{Q}_{1}=\widehat{T}_{1}\widetilde{Q}_{0}$. Then, get a new batch of sample and learn the map $\widehat{T}_{2}$ and get $\widetilde{Q}_{2}=\widehat{T}_{2}\widetilde{Q}_{1}$, and so on. Our final estimator of the action value is $\widetilde{Q}_{K}=\widehat{T}_{K}\cdots\widehat{T}_{1}\widetilde{Q}_{0}$, which resembles the updates of the value iteration algorithm at the population level.

When $\mathcal{F}$ is the family of neural networks, Algorithm 1 is known as the neural FQI algorithm, which is proposed in Riedmiller 2005. Thus, we can view neural FQI as a modification of DQN, where we replace experience replay with sampling from a fixed distribution $\sigma$, so as to understand the statistical property. As a byproduct, such a modification naturally justifies the trick of target network in DQN. In addition, note that the optimization problem in (3.4) appears in each iteration of FQI, which is nonconvex when neural networks are used. However, since we focus solely on the statistical aspect, we make the assumption that the global optima of (3.4) can be reached, which is also contained $\mathcal{F}$. Interestingly, a recent line of research on deep learning (Du et al. 2019b; Du et al. 2019a; Zou et al. 2018; Chizat et al. 2019; Allen-Zhu et al. 2019a; Allen-Zhu et al. 2019b; Jacot et al. 2018; Cao and Gu 2019; Arora et al. 2019; Weinan et al. 2019; Mei et al. 2019; Yehudai and Shamir 2019) has established global convergence of gradient-based algorithms for empirical risk minimization when the neural networks are overparametrized. We provide more discussions on the computation aspect in §B. Furthermore, we make the i.i.d. assumption in Algorithm 1 to simplify the analysis. Antos et al. 2008b study the performance of fitted value iteration with fixed data used in the regression sub-problems repeatedly, where the data is sampled from a single trajectory based on a fixed policy such that the induced Markov chain satisfies certain conditions on the mixing time. Using similar analysis as in Antos et al. 2008b, our algorithm can also be extended to handled fixed data that is collected beforehand.

Theoretical Results

Following Definition 2.1, let $\mathcal{F}(L,\{d_{j}\}_{j=0}^{L+1},s)$ be the family of sparse ReLU networks defined on ${\mathcal{S}}$ with $d_{0}=r$ and $d_{L+1}=1$. Then we define $\mathcal{F}_{0}$ by

By this definition, for any function $f\in\mathcal{F}_{0}$ and any action $a\in\mathcal{A}$, $f(\cdot,a)$ is a ReLU network defined on ${\mathcal{S}}$, which is standard for Q-networks. Moreover, $\mathcal{G}_{0}$ contains a broad family of smooth functions on ${\mathcal{S}}\times\mathcal{A}$. In the following, we make a mild assumption on $\mathcal{F}_{0}$ and $\mathcal{G}_{0}$.

We assume that for any $f\in\mathcal{F}_{0}$, we have $Tf\in\mathcal{G}_{0}$, where $T$ is the Bellman optimality operator defined in (2.7). That is, for any $f\in\mathcal{F}$ and any $a\in\mathcal{A}$, $(Tf)(s,a)$ can be written as compositions of Hölder smooth functions as a function of $s\in{\mathcal{S}}$.

This assumption specifies that the target function ${\mathcal{T}}\widetilde{Q}_{k}$ in each FQI step stays in function class $\mathcal{G}_{0}$. When $\mathcal{G}_{0}$ can be approximated by functions in $\mathcal{F}_{0}$ accurately, this assumption essentially implies that $Q^{*}$ is close to $\mathcal{F}_{0}$ and that $\mathcal{F}_{0}$ is approximately closed under Bellman operator $T$. Such an completeness assumption is commonly made in the literature on batch reinforcement learning under various forms and is conjectured to be indispensable in Chen and Jiang 2019.

We remark that this Assumption (4.2) holds when the MDP satisfies some smoothness conditions. For any state-action pair $(s,a)\in{\mathcal{S}}\times\mathcal{A}$, let $P(\cdot{\,|\,}s,a)$ be the density of the next state. By the definition of the Bellman optimality operator in (2.7), we have

Then by (4.3) we have $(Tf)(s,a)=g_{1}(s)+h_{4}\circ h_{2}(s,a)$. Then Assumption 4.2 holds if $h_{4}$ is Hölder smooth and both $g_{1}$ and $h_{2}$ can be represented as compositions of Hölder functions. Thus, Assumption 4.2 holds if both the reward function and the transition density of the MDP are sufficiently smooth.

Moreover, even when the transition density is not smooth, we could also expect Assumption 4.2 to hold. Consider the extreme case where the MDP has deterministic transitions, that is, the next state $s^{\prime}$ is a function of $s$ and $a$, which is denoted by $s^{\prime}=h(s,a)$. In this case, for any ReLU network $f$, we have $(Tf)(s,a)=r(s,a)+\gamma\cdot\max_{a^{\prime}\in\mathcal{A}}f[h(s,a),a^{\prime}].$ Since

for any $s_{1},s_{2}\in{\mathcal{S}}$, and network $f(\cdot,a)$ is Lipschitz continuous for any fixed $a\in\mathcal{A}$, function $m_{1}(s)=\max_{a^{\prime}}f(s,a^{\prime})$ is Lipschitz on ${\mathcal{S}}$. Thus, for any fixed $a\in\mathcal{A}$, if both $g_{1}(s)=r(s,a)$ and $m_{2}(s)=h(s,a)$ are compositions of Hölder functions, so is $(Tf)(s,a)=g_{1}(s)+m_{1}\circ m_{2}(s)$. Therefore, even if the MDP has deterministic dynamics, when both the reward function $r(s,a)$ and the transition function $h(s,a)$ are sufficiently nice, Assumption 4.2 still holds true.

In the following, we define the concentration coefficients, which measures the similarity between two probability distributions under the MDP.

Let $\nu_{1},\nu_{2}\in\mathcal{P}({\mathcal{S}}\times\mathcal{A})$ be two probability measures that are absolutely continuous with respect to the Lebesgue measure on ${\mathcal{S}}\times\mathcal{A}$. Let $\{\pi_{t}\}_{t\geq 1}$ be a sequence of policies. Suppose the initial state-action pair $(S_{0},A_{0})$ of the MDP has distribution $\nu_{1}$, and we take action $A_{t}$ according to policy $\pi_{t}$. For any integer $m$, we denote by $P^{\pi_{m}}P^{\pi_{m-1}}\cdots P^{\pi_{1}}\nu_{1}$ the distribution of $\{(S_{t},A_{t})\}_{t=0}^{m}$. Then we define the $m$-th concentration coefficient as

where the supremum is taken over all possible policies.

Furthermore, let $\sigma$ be the sampling distribution in Algorithm 1 and let $\mu$ be a fixed distribution on ${\mathcal{S}}\times\mathcal{A}$. We assume that there exists a constant $\phi_{\mu,\sigma}<\infty$ such that

where $(1-\gamma)^{2}$ in (4.5) is a normalization term, since $\sum_{m\geq 1}\gamma^{m-1}\cdot m=(1-\gamma)^{-2}$.

By definition, concentration coefficients in (4.4) quantifies the similarity between $\nu_{2}$ and the distribution of the future states of the MDP when starting from $\nu_{1}$. Moreover, (4.5) is a standard assumption in the literature. See, e.g., Munos and Szepesvári 2008; Lazaric et al. 2016; Scherrer et al. 2015; Farahmand et al. 2010; Farahmand et al. 2016. This assumption holds for large class of systems MDPs and specifically for MDPs whose top-Lyapunov exponent is finite. Moreover, this assumption essentially requires that the sampling distribution $\sigma$ has sufficient coverage over ${\mathcal{S}}\times\mathcal{A}$ and is shown to be necessary for the success of batch RL methods (Chen and Jiang 2019). See Munos and Szepesvári 2008; Antos et al. 2007; Chen and Jiang 2019 for more detailed discussions on this assumption.

Now we are ready to present the main theorem.

For the hyperparameters $L^{*}$, $\{d_{j}^{*}\}_{j=0}^{L^{*}+1}$, and $s^{*}$ of the ReLU network, we set $d_{0}^{*}=0$ and $d_{L^{*}+1}^{*}=1$. Moreover, we set

This theorem implies that the statistical rate of convergence is the sum of a statistical error and an algorithmic error. The algorithmic error converges to zero in linear rate as the algorithm proceeds, whereas the statistical error reflects the fundamental difficulty of the problem. Thus, when the number of iterations satisfy

iterations, where $C^{\prime}$ is a sufficiently large constant, the algorithmic error is dominated by the statistical error. In this case, if we view both $\gamma$ and $\phi_{\mu,\sigma}$ as constants and ignore the polylogarithmic term, Algorithm 1 achieves error rate

Furthermore, as a concrete example, we assume that both the reward function and the Markov transition kernel are Hölder smooth with smoothness parameter $\beta$. As stated below Assumption 4.2, for any $f\in\mathcal{F}_{0}$, we have $(Tf)(\cdot,a)\in\mathcal{C}_{r}({\mathcal{S}},\beta,H^{\prime})$. Then Theorem 4.4 implies that Algorithm 1 achieves error rate $|\mathcal{A}|\cdot n^{-\beta/(2\beta+r)}$ when $K$ is sufficiently large. Since $|\mathcal{A}|$ is finite, this rate achieves the minimax-optimal statistical rate of convergence within the class of Hölder smooth functions defined on $^{d}$ (Stone 1982) and thus cannot be further improved. As another example, when $(Tf)(\cdot,a)\in\mathcal{C}_{r}({\mathcal{S}},\beta,H^{\prime})$ can be represented as an additive model over $^{r}$ where each component has smoothness parameter $\beta$, (4.9) reduces to $|\mathcal{A}|\cdot n^{-\beta/(2\beta+1)}$, which does not depends on the input dimension $r$ explicitly. Thus, by having a composite structure in $\mathcal{G}_{0}$, Theorem 4.4 yields more refined statistical rates that adapt to the intrinsic difficulty of solving each iteration of Algorithm 1.

In the sequel, we conclude this section by sketching the proof of Theorem 4.4; the detailed proof is deferred to §6.

Recall that $\pi_{k}$ is the greedy policy with respect to $\widetilde{Q}_{k}$ and $Q^{\pi_{K}}$ is the action-value function associated with $\pi_{K}$, whose definition is given in (2.3). Since $\{\widetilde{Q}_{k}\}_{k\in[K]}$ is constructed by a iterative algorithm, it is crucial to relate $\|Q^{*}-Q^{\pi_{K}}\|_{1,\mu}$, the quantity of interest, to the errors incurred in the previous steps, namely $\{\widetilde{Q}_{k}-T\widetilde{Q}_{k-1}\}_{k\in[K]}$. Thus, in the first step of the proof, we establish Theorem 6.1, also known as the error propagation (Munos and Szepesvári 2008; Lazaric et al. 2016; Scherrer et al. 2015; Farahmand et al. 2010; Farahmand et al. 2016) in the batch reinforcement learning literature, which provides an upper bound on $\|Q^{*}-Q^{\pi_{K}}\|_{1,\mu}$ using $\{\|\widetilde{Q}_{k}-T\widetilde{Q}_{k-1}\|_{\sigma}\}_{k\in[K]}$. In particular, Theorem 6.1 asserts that

where $\phi_{\mu,\sigma}$, given in (4.5), is a constant that only depends on distributions $\mu$ and $\sigma$.

It remains to bound $\|\widetilde{Q}_{k}-T\widetilde{Q}_{k-1}\|_{\sigma}$ for any $k\in[K]$. We achieve such a goal using tools from nonparametric regression. Specifically, as we will show in Theorem 6.2, under Assumption 4.2, for any $k\in[K]$ we have

for any $\delta>0$, where $C>0$ is an absolute constant,

In the sequel, we fix $\delta=1/n$ in (4.11), which implies that

where $C^{\prime}>0$ is an absolute constant.

In the subsequent proof, we establish upper bounds for $\text{dist}(\mathcal{F}_{0},\mathcal{G}_{0})$ defined in (4.12) and $\log N_{\delta}$, respectively. Recall that the family of composite Hölder smooth functions $\mathcal{G}_{0}$ is defined in (4.2).

By the definition of $\mathcal{G}_{0}$ in (4.2), for any $f\in\mathcal{G}_{0}$ and any $a\in\mathcal{A}$, $f(\cdot,a)\in\mathcal{G}(\{(p_{j},t_{j},\beta_{j},H_{j})\}_{j\in[q]})$ is a composition of Hölder smooth functions, that is, $f(\cdot,a)=g_{q}\circ\cdots\circ g_{1}.$ Recall that, as defined in Definition 2.3, $g_{jk}$ is the $k$-th entry of the vector-valued function $g_{j}$. Here $g_{jk}\in\mathcal{C}_{t_{j}}([a_{j},b_{j}]^{t_{j}},\beta_{j},H_{j})$ for each $k\in[p_{j+1}]$ and $j\in[q]$. To construct a ReLU network that is $f(\cdot,a)$, we first show that $f(\cdot,a)$ can be reformulated as a composition of Hölder functions defined on a hypercube. Specifically, let $h_{1}=g_{1}/(2H_{1})+1/2$, $h_{q}(u)=g_{q}(2H_{q-1}u-H_{q-1})$, and

for all $j\in\{2,\ldots,q-1\}$. Then we immediately have

Furthermore, by the definition of Hölder smooth functions in Definition 2.2, for any $j\in[q]$ and $k\in[p_{j+1}]$, it is not hard to verify that $h_{jk}\in\mathcal{C}_{t_{j}}\bigl(^{t_{j}},W\bigr),$ where we define $W>0$ by

To apply Lemma 6.3 we set $m=\eta\cdot\lceil\log_{2}n\rceil$ for a sufficiently large constant $\eta>1$, and set $N$ to be a sufficiently large integer that depends on $n$, which will be specified later. In addition, we set $L_{j}=8+(m+5)\cdot(1+\lceil\log_{2}(t_{j}+\beta_{j})\rceil).$ Then, by Lemma 6.3, there exists a ReLU network $\widetilde{h}_{jk}$ such that $\|\widetilde{h}_{jk}-h_{jk}\|_{\infty}\lesssim N^{-\beta_{j}/t_{j}}.$ Furthermore, we have $\widetilde{h}_{jk}\in\mathcal{F}(L_{j},\{t_{j},\widetilde{d}_{j},\ldots,\widetilde{d}_{j},1\},\widetilde{s}_{j}),$ with

Combining this with (4.17) and the fact that $\|\widetilde{h}_{jk}-h_{jk}\|_{\infty}\lesssim N^{-\beta_{j}/t_{j}}$, we obtain that

Moreover, using classical results on the covering number of neural networks (Anthony and Bartlett 2009), we further show that

where $\delta=1/n$. Therefore, combining (4.11), (4.13), (4.18), and (4.19), we conclude the proof. ∎

Extension to Two-Player Zero-Sum Markov Games

In this section, we propose the Minimax-DQN algorithm, which combines DQN and the Minimax-Q learning for two-player zero-sum Markov games. We first present the background of zero-sum Markov games and introduce the the algorithm in §5.1. Borrowing the analysis for DQN in the previous section, we provide theoretical guarantees for the proposed algorithm in §5.2.

Moreover, for joint policy $(\pi,\nu)$ of two players, we define the Bellman operators ${T}^{\pi,\nu}$ and $T$ by

That is, $\pi_{Q}(\cdot{\,|\,}s)$ and $\nu_{Q}(\cdot{\,|\,}s)$ solves the zero-sum matrix game based on $Q(s,\cdot,\cdot)$ for all $s\in{\mathcal{S}}$. By this definition, we obtain that the equilibrium joint policy with respect to the minimax function $Q^{*}$ defined in (5.3) achieves the Nash equilibrium of the Markov game.

Therefore, to learn the Nash equilibrium, it suffices to estimate $Q^{*}$, which is the unique fixed point of the Bellman operator $T$. Similar to the standard Q-learning for MDP, Littman 1994 proposes the Minimax-Q learning algorithm, which constructs a sequence of action-value functions that converges to $Q^{*}$. Specifically, in each iteration, based on a transition $(s,a,b,s^{\prime})$, Minimax-Q learning updates the current estimator of $Q^{*}$, denoted by $Q$, via

which can be attained via linear programming. Then we update $\theta$ in the direction of $\nabla_{\theta}L(\theta)$, where $L(\theta)=n^{-1}\sum_{i\in[n]}[Y_{i}-Q_{\theta}(s_{i},a_{i},b_{i})]^{2}$. Finally, the target network $Q_{\theta^{*}}$ is updated every $T_{\textrm{target}}$ steps by letting $\theta^{*}=\theta$. For brevity, we defer the details of Minimax-DQN to Algorithm 4 in §A.

To understand the theoretical aspects of this algorithm, we similarly utilize the framework of batch reinforcement learning for statistical analysis. With the insights gained in §3, we consider a modification of Minimax-DQN based on neural fitted Q-iteration, whose details are stated in Algorithm 2. As in the MDP setting, we replace sampling from the replay memory by sampling i.i.d. state-action tuples from a fixed distribution $\sigma\in\mathcal{P}({\mathcal{S}}\times\mathcal{A}\times\mathcal{B})$, and estimate $Q^{*}$ in (5.3) by solving a sequence of least-squares regression problems specified by (5.8). Intuitively, this algorithm approximates the value iteration algorithm for zero-sum Markov games (Littman 1994) by constructing a sequence of value functions $\{\widetilde{Q}_{k}\}_{k\geq 0}$ such that $\widetilde{Q}_{k+1}\approx T\widetilde{Q}_{k}$ for all $k$, where $T$ defined in (5.5) is the Bellman operator.

2 Theoretical Results for Minimax-FQI

Following the theoretical results established in §4, in this subsection, we provide statistical guarantees for the Minimax-FQI algorithm with $\mathcal{F}$ being a family of deep neural networks with ReLU activation. Hereafter, without loss of generality, we assume ${\mathcal{S}}=^{r}$ with $r$ being a fixed integer, and the action spaces $\mathcal{A}$ and $\mathcal{B}$ are both finite. To evaluate the performance of the algorithm, we first introduce the best-response policy as follows.

Note that when the first player adopt a fixed policy $\pi$, from the perspective of the second player, the Markov game becomes a MDP. Thus, $\nu_{\pi}^{*}$ is the optimal policy of the MDP induced by $\pi$. Moreover, it can be shown that, for any policy $\pi$, $Q^{*}(s,a,b)\geq Q^{\pi,\nu^{*}_{\pi}}(s,a,b)$ holds for every state-action tuple $(s,a,b)$. Thus, by considering the adversarial case where the opponent always plays the best-response policy, the difference between $Q^{\pi.\nu_{\pi}^{*}}$ and $Q^{*}$ servers as a characterization of the suboptimality of $\pi$. Hence, to quantify the performance of Algorithm 2, we consider the closeness between $Q^{*}$ and $Q^{\pi_{K},\nu_{\pi_{K}}^{*}}$, which will be denoted by $Q_{K}^{*}$ hereafter for simplicity. Specifically, in the following we establish an upper bound for $\|Q^{*}-Q_{K}^{*}\|_{1,\mu}$ for some distribution $\mu\in\mathcal{P}({\mathcal{S}}\times\mathcal{A}\times\mathcal{B})$.

We first specify the function class $\mathcal{F}$ in Algorithm 2 as follows.

Following Definition 4.1, let $\mathcal{F}(L,\{d_{j}\}_{j=0}^{L+1},s)$ and $\mathcal{G}(\{p_{j},t_{j},\beta_{j},H_{j}\}_{j\in[q]})$ be the family of sparse ReLU networks and the set of composition of Hölder smooth functions defined on ${\mathcal{S}}$, respectively. Similar to (4.1), we define $\mathcal{F}_{1}$ by

For the Bellman operator $T$ defined in (5.5), we assume that for any $f\in\mathcal{F}_{1}$ and any state-action tuple $(s,a,b)$, we have $(Tf)(\cdot,a,b)\in\mathcal{G}(\{p_{j},t_{j},\beta_{j},H_{j}\}_{j\in[q]})$.

We remark that this Assumption is in the same flavor as Assumption 4.2. As discussed in §4, this assumption holds if both the reward function and the transition density of the Markov game are sufficiently smooth.

In the following, we define the concentration coefficients for Markov games.

Let $\{\tau_{t}\colon{\mathcal{S}}\rightarrow\mathcal{P}(\mathcal{A}\times\mathcal{B})\}$ be a sequence of joint policies for the two players in the zero-sum Markov game. Let $\nu_{1},\nu_{2}\in\mathcal{P}({\mathcal{S}}\times\mathcal{A}\times\mathcal{B})$ be two absolutely continuous probability measures. Suppose the initial state-action pair $(S_{0},A_{0},B_{0})$ has distribution $\nu_{1}$, the future states are sampled according to the Markov transition kernel, and the action $(A_{t},B_{t})$ is sampled from policy $\tau_{t}$. For any integer $m$, we denote by $P^{\tau_{m}}P^{\tau_{m-1}}\cdots P^{\tau_{1}}\nu_{1}$ the distribution of $\{(S_{t},A_{t},B_{t})\}_{t=0}^{m}$. Then, the $m$-th concentration coefficient is defined as

where the supremum is taken over all possible joint policy sequences $\{\tau_{t}\}_{t\in[m]}$.

Furthermore, for some $\mu\in\mathcal{P}({\mathcal{S}}\times\mathcal{A}\times\mathcal{B})$, we assume that there exists a finite constant $\phi_{\mu,\sigma}$ such that $(1-\gamma)^{2}\cdot\sum_{m\geq 1}\gamma^{m-1}\cdot m\cdot\kappa(m;\mu,\sigma)\leq\phi_{\mu,\sigma},$ where $\sigma$ is the sampling distribution in Algorithm 2 and $\kappa(m;\mu,\sigma)$ is the $m$-th concentration coefficient defined in (5.10).

We remark that the definition of the $m$-th concentration coefficient is the same as in (4.4) if we replace the action space $\mathcal{A}$ of the MDP by $\mathcal{A}\times\mathcal{B}$ of the Markov game. Thus, Assumptions 4.3 and 5.3 are of the same nature, which are standard in the literature.

Now we are ready to present the main theorem.

where $\xi^{*}$ appears in (4.7), $\alpha^{*}=\max_{j\in[q]}t_{j}/(2\beta_{j}^{*}+t_{j})$ and $\phi_{\mu,\sigma}$ is specified in Assumption 5.3.

Similar to Theorem 4.4, the bound in (5.11) shows that closeness between $(\pi_{K},\nu_{K})$ returned by Algorithm 2 and the Nash equilibrium policy $(\pi_{Q^{*}},\nu_{Q^{*}})$, measured by $\|Q^{*}-Q_{K}^{*}\|_{1,\mu}$, is bounded by the sum of statistical error and an algorithmic error. Specifically, the statistical error balances the bias and variance of estimating the value functions using the family of deep ReLU neural networks, which exhibits the fundamental difficulty of the problem. Whereas the algorithmic error decay to zero geometrically as $K$ increases. Thus, when $K$ is sufficiently large, both $\gamma$ and $\phi_{\mu,\sigma}$ are constants, and the polylogarithmic term is ignored, Algorithm 2 achieves error rate

Proof of the Main Theorem

In this section, we present a detailed proof of Theorem 4.4.

The proof requires two key ingredients. First in Theorem 6.1 we quantify how the error of action-value function approximation propagates through each iteration of Algorithm 1. Then in Theorem 6.2 we analyze such one-step approximation error for ReLU networks.

Recall that $\{\widetilde{Q}_{k}\}_{0\leq k\leq K}$ are the iterates of Algorithm 1. Let $\pi_{K}$ be the one-step greedy policy with respect to $\widetilde{Q}_{K}$, and let $Q^{\pi_{K}}$ be the action-value function corresponding to $\pi_{K}$. Under Assumption 4.3, we have

where we define the maximum one-step approximation error as $\varepsilon_{\max}=\max_{k\in[K]}\|T\widetilde{Q}_{k-1}-\widetilde{Q}_{k}\|_{\sigma}$. Here $\phi_{\mu,\sigma}$ is a constant that only depends on the probability distributions $\mu$ and $\sigma$.

We remark that similar error propagation result is established for the state-value function in Munos and Szepesvári 2008 for studying the fitted value iteration algorithm, which is further extended by Lazaric et al. 2016; Scherrer et al. 2015; Farahmand et al. 2010; Farahmand et al. 2016 for other batch reinforcement learning methods.

In the sequel, we establish an upper bound for the one-step approximation error $\|T\widetilde{Q}_{k-1}-\widetilde{Q}_{k}\|_{\sigma}$ for each $k\in[K]$.

Let $\mathcal{F}\subseteq\mathcal{B}({\mathcal{S}}\times\mathcal{A},V_{\max})$ be a class of measurable functions on ${\mathcal{S}}\times\mathcal{A}$ that are bounded by $V_{\max}=R_{\max}/(1-\gamma)$, and let $\sigma$ be a probability distribution on ${\mathcal{S}}\times\mathcal{A}$. Also, let $\{(S_{i},A_{i})\}_{i\in[n]}$ be $n$ i.i.d. random variables in ${\mathcal{S}}\times\mathcal{A}$ following $\sigma$. For each $i\in[n]$, let $R_{i}$ and $S_{i}^{\prime}$ be the reward and the next state corresponding to $(S_{i},A_{i})$. In addition, for any fixed $Q\in\mathcal{F}$, we define $Y_{i}=R_{i}+\gamma\cdot\max_{a\in\mathcal{A}}Q(S_{i}^{\prime},a)$. Based on $\{(X_{i},A_{i},Y_{i})\}_{i\in[n]}$, we define $\widehat{Q}$ as the solution to the least-squares problem

where $C$ and $C^{\prime}$ are two positive absolute constants and $\omega(\mathcal{F})$ is defined as

This theorem characterizes the bias and variance that arise in estimating the action-value functions using deep ReLU networks. Specifically, $\omega(\mathcal{F})$ in (6.4) corresponds to the bias incurred by approximating the target function $Tf$ using ReLU neural networks. It can be viewed as a measure of completeness of $\mathcal{F}$ with respect to the Bellman operator $T$. In addition, $V_{\max}^{2}/n\cdot\log N_{\delta}+V_{\max}\cdot\delta$ controls the variance of the estimator, where the covering number $N_{\delta}$ is used to obtain a uniform bound over $\mathcal{F}_{0}$.

To obtain an upper bound for $\|T\widetilde{Q}_{k-1}-\widetilde{Q}_{k}\|_{\sigma}$ as required in Theorem 6.1, we set $Q=\widetilde{Q}_{k-1}$ in Theorem 6.2. Then according to Algorithm 1, $\widehat{Q}$ defined in (6.2) becomes $\widetilde{Q}_{k}$. We set the function class $\mathcal{F}$ in Theorem 6.2 to be the family of ReLU Q-networks $\mathcal{F}_{0}$ defined in (4.1). By setting $\epsilon=1$ and $\delta=1/n$ in Theorem 6.2, we obtain

where $C$ is a positive absolute constant and

is the $1/n$-covering number of $\mathcal{F}_{0}$. In the subsequent proof, we establish upper bounds for $\omega(\mathcal{F}_{0})$ defined in (6.4) and $\log N_{0}$, respectively. Recall that the family of composite Hölder smooth functions $\mathcal{G}_{0}$ is defined in (4.2). By Assumption 4.2, we have $Tg\in\mathcal{G}_{0}$ for any $g\in\mathcal{F}_{0}$. Hence, we have

By the definition of $\mathcal{G}_{0}$ in (4.2), for any $f\in\mathcal{G}_{0}$ and any $a\in\mathcal{A}$, $f(\cdot,a)\in\mathcal{G}(\{(p_{j},t_{j},\beta_{j},H_{j})\}_{j\in[q]})$ is a composition of Hölder smooth functions, that is, $f(\cdot,a)=g_{q}\circ\cdots\circ g_{1}$. Recall that, as defined in Definition 2.3, $g_{jk}$ is the $k$-th entry of the vector-valued function $g_{j}$. Here $g_{jk}\in\mathcal{C}_{t_{j}}([a_{j},b_{j}]^{t_{j}},\beta_{j},H_{j})$ for each $k\in[p_{j+1}]$ and $j\in[q]$. In the sequel, we construct a ReLU network to approximate $f(\cdot,a)$ and establish an upper bound of the approximation error on the right-hand side of (6.7). We first show that $f(\cdot,a)$ can be reformulated as a composition of Hölder functions defined on a hypercube. We define $h_{1}=g_{1}/(2H_{1})+1/2$,

and $h_{q}(u)=g_{q}(2H_{q-1}u-H_{q-1})$. Then we immediately have

Furthermore, by the definition of Hölder smooth functions in Definition 2.2, for any $k\in[p_{2}]$, we have that $h_{1k}$ takes value in $$and$h_{1k}\in\mathcal{C}_{t_{1}}(^{t_{1}},\beta_{1},1)$. Similarly, for any$j\in\{2,\ldots,q-1\}$and$k\in[p_{j+1}]$,$h_{jk}$also takes value in$$ and

Finally, recall that we use the convention that $p_{q+1}=1$, that is, $h_{q}$ is a scalar-valued function that satisfies

In the following, we show that the composition function in (6.8) can be approximated by an element in $\mathcal{F}(L^{*},\{d_{j}^{*}\}_{j=1}^{L+1},s^{*})$ when the network hyperparameters are properly chosen. Our proof consists of three steps. In the first step, we construct a ReLU network $\widetilde{h}_{jk}$ that approximates each $h_{jk}$ in (6.9). Then, in the second step, we approximate $f(\cdot,a)$ by the composition of $\{\widetilde{h}_{j}\}_{j\in[q]}$ and quantify the architecture of this network. Finally, in the last step, we prove that this network can be embedded into class $\mathcal{F}(L^{*},\{d_{j}^{*}\}_{j=1}^{L+1},s^{*})$ and characterize the final approximation error.

For any integers $m\geq 1$ and $N\geq\max\{(\beta+1)^{r},(H+1)e^{r}\}$, let $L=8+(m+5)\cdot(1+\lceil\log_{2}(r+\beta)\rceil)$, $d_{0}=r$, $d_{j}=6(r+\lceil\beta\rceil)N$ for each $j\in[L]$, and $d_{L+1}=1$. For any $g\in\mathcal{C}_{r}(^{r},\beta,H)$, there exists a ReLU network $f\in\mathcal{F}(L,\{d_{j}\}_{j=0}^{L+1},s,V_{\max})$ as defined in Definition 2.1 such that

where the parameter $s$ satisfies $s\leq 141\cdot(r+\beta+1)^{3+r}\cdot N\cdot(m+6)$.

See Appendix B in Schmidt-Hieber 2020+ for a detailed proof. The idea is to first approximate the Hölder smooth function by polynomials via local Taylor expansion. Then, neural networks are constructed explicitly to approximate each monomial terms in these local polynomials. ∎

We apply Lemma 6.3 to $h_{jk}\colon^{t_{j}}\rightarrow$ for any $j\in[q]$ and $k\in[p_{j+1}]$. We set $m=\eta\cdot\lceil\log_{2}n\rceil$ for a sufficiently large constant $\eta>1$, and set $N$ to be a sufficiently large integer depending on $n$, which will be specified later. In addition, we set

We will later verify that $N\geq\max\{(\beta+1)^{t_{j}},(W+1)e^{t_{j}}\}$ for all $j\in[q]$. Then by Lemma 6.3, there exists a ReLU network $\widehat{h}_{jk}$ such that

Furthermore, we have $\widehat{h}_{jk}\in\mathcal{F}(L_{j},\{t_{j},\widetilde{d}_{j},\ldots,\widetilde{d}_{j},1\},\widetilde{s}_{j})$ with

Moreover, since both $\widetilde{h}_{jk}$ and $h_{jk}$ take value in $$, by (6.12) we have

where the constant $W$ is defined in (6.11). Since we can set the constant $\eta$ in (6) to be sufficiently large, the second term on the right-hand side of (6) is the leading term asymptotically, that is,

Thus, in the first step, we have shown that there exists $\widetilde{h}_{jk}\in\mathcal{F}(L_{j}+2,\{t_{j},\widetilde{d}_{j},\ldots,\widetilde{d}_{j},1\},\widetilde{s}_{j}+4)$ satisfying (6.16).

where we define $\widetilde{L}=\sum_{j=1}^{q}(L_{j}+2)$, $\widetilde{d}=\max_{j\in[q]}\widetilde{d}_{j}\cdot p_{j+1}$, and $\widetilde{s}=\sum_{j=1}^{q}(\widetilde{s}_{j}+4)\cdot p_{j+1}$. Recall that $L_{j}$ is defined in (6.10). Then when $n$ is sufficiently large, we have

where $\xi>0$ is an absolute constant. Here the last inequality follows from (4.6). Moreover, for $\widetilde{d}$ defined in (6.17), by (4.6) we have

In addition, combining (6.13), (4.6), and the fact that $t_{j}\leq p_{j}$, we obtain

Step (iii). In the last step, we show that the function class in (6.17) can be embedded in $\mathcal{F}(L^{*},\{d_{j}^{*}\}_{j=1}^{L^{*}+1},s^{*})$ and characterize the final approximation bias, where $L^{*}$, $\{d_{j}^{*}\}_{j=1}^{L^{*}+1}$, and $s^{*}$ are specified in (4.7). To this end, we set

where the absolute constant $C>0$ is sufficiently large. Note that we define $\alpha^{*}=\max_{j\in[q]}t_{j}/(2\beta_{j}^{*}+t_{j})$. Then (6.21) implies that $N\asymp n^{\alpha^{*}}$. When $n$ is sufficiently large, it holds that $N\geq\max\{(\beta+1)^{t_{j}},(W+1)e^{t_{j}}\}$ for all $j\in[q]$. When $\xi^{*}$ in (4.7) satisfies $\xi^{*}\geq 1+2\xi$, by (6) we have

In addition, (6.19) and (4.7) implies that we can set $d_{j}^{*}\geq\widetilde{d}$ for all $j\in[L^{*}]$. Finally, by (6) and (6.21), we have $\widetilde{s}\lesssim n^{\alpha^{*}}\cdot(\log n)^{\xi^{*}},$ which implies $\widetilde{s}+(L^{*}-\widetilde{L})\cdot r\leq s^{*}$. For an $\widetilde{L}$-layer ReLU network in (6.17), we can make it an $L^{*}$-layer ReLU network by inserting $L^{*}-\widetilde{L}$ identity layers, since the inputs of each layer are nonnegative. Thus, ReLU networks in (6.17) can be embedded in

which is a subset of $\mathcal{F}(L^{*},\{d_{j}^{*}\}_{j=1}^{L+1},s^{*})$ by (4.7).

To obtain the approximation error $\|\widetilde{f}-f(\cdot,a)\|_{\infty}$, we define $G_{j}=h_{j}\circ\cdots\circ h_{1}$ and $\widetilde{G}_{j}=\widetilde{h}_{j}\circ\cdots\circ\widetilde{h}_{1}$ for any $j\in[q]$. By triangle inequality, for any $j>1$ we have

where the constant $W$ is defined in (6.11). Here in (6.23) we use the fact that $(a+b)^{\alpha}\leq a^{\alpha}+b^{\alpha}$ for all $\alpha\in$ and $a,b>0$.

Thus, we combine (6.7), (6.21), and (6.24) to obtain

As the final step of the proof, it remains to control the covering number of $\mathcal{F}_{0}$ defined in (4.1). By definition, for any $f\in\mathcal{F}_{0}$, we have $f(\cdot,a)\in\mathcal{F}(L^{*},\{d_{j}^{*}\}_{j=1}^{L^{*}+1},s^{*})$ for any $a\in\mathcal{A}$. For notational simplicity, we denote by $\mathcal{N}_{\delta}$ the $\delta$-covering of $\mathcal{F}(L^{*},\{d_{j}^{*}\}_{j=1}^{L^{*}+1},s^{*})$, that is, we define

Now we utilize the following lemma in Anthony and Bartlett 2009 to obtain an upper bound of the cardinality of $\mathcal{N}_{\delta}$.

See Theorem 14.5 in Anthony and Bartlett 2009 for a detailed proof. ∎

Recall that we denote $\mathcal{N}(1/n,\mathcal{F}_{0},\|\cdot\|_{\infty})$ by $N_{0}$ in (6.6). By combining (6.26) with Lemma 6.4 and setting $\delta=1/n$, we obtain that

Finally, combining (6.1), (6.5), (6.25), and (6.27), we conclude the proof of Theorem 4.4. ∎

Conclusion

We study deep Q-network from the statistical perspective. Specifically, by neglecting the computational issues, we consider the fitted Q-iteration algorithm with ReLU networks, which can be viewed as a modification of DQN that fully captures its key features. Under mild assumptions, we show that DQN creates a sequence of policies whose corresponding value functions converge to the optimal value function, when both the sample size and the number of iteration go to infinity. Moreover, we establish a precise characterization of both the statistical and the algorithmic rates of convergence. As a byproduct, our results provide theoretical justification for the trick of using a target network in DQN. Furthermore, we extend DQN to two-player zero-sum Markov games by proposing the Minimax-DQN algorithm. Utilizing the analysis of DQN, we establish theoretical guarantees for Minimax-DQN. To further extend this work, one future direction is to analyze reinforcement learning methods targeting at MDP with continuous action spaces, e.g., example, soft Q-learning (Haarnoja et al. 2017) and deep deterministic policy gradient (DDPG) (Lillicrap et al. 2016). Another promising direction is to combine results on optimization for deep learning with our statistical analysis to gain a unified understanding of the statistical and computational aspects of DQN.

Appendix A Deep Q-Network

We first present the DQN algorithm for MDP in details, which is proposed by Mnih et al. 2015 and adapted here to discounted MDP. As shown in Algorithm 3 below, DQN features two key tricks that lead to its empirical success, namely, experience replay and target network.

Furthermore, in the following, we present the details of the Minimax-DQN algorithm that extends DQN to two-player zero-sum Markov games introduced in §5. Similar to DQN, this algorithm also utilizes the experience replay and target networks. The main difference is that here the target $Y_{i}$ in (5.7) is obtained by solving a zero-sum matrix game. In Algorithm 4 we present the algorithm for the second player, which can be easily modified for the first player. We note that for the second player, similar to (5.6),the equilibrium joint policy is defined as

Appendix B Computational Aspect of DQN

Recall that in Algorithm 1 we assume the global optima of the nonlinear least-squares problem in (3.1) is obtained in each iteration. We make such an assumption as our focus is on the statistical analysis. In terms of optimization, it has been shown recently that, when the neural network is overparametrized, (stochastic) gradient descent converges to the global minima of the empirical function. Moreover, the generalization error of the obtained neural network can also be established. The intuition behind these results is that, when the neural network is overparametrized, it behaves similar to the random feature model (Rahimi and Recht 2008; Rahimi and Recht 2009). See, e.g., Du et al. 2019b; Du et al. 2019a; Zou et al. 2018; Chizat et al. 2019; Allen-Zhu et al. 2019a; Allen-Zhu et al. 2019b; Jacot et al. 2018; Cao and Gu 2019; Arora et al. 2019; Weinan et al. 2019; Mei et al. 2019; Yehudai and Shamir 2019; Bietti and Mairal 2019; Yang and Salman 2019; Yang 2019; Gao et al. 2019; Bai and Lee 2019; Huang et al. 2020 and the references therein. Also see Fan et al. 2019 for a detailed survey. In this section, we make an initial attempt in providing a unified statistical and computational analysis of DQN.

We represent the Q-network by the family of two-layer neural networks

For such class of neural networks, for any $k\geq 1$, in $k$-th iteration of the neural FQI algorithm, the optimization problem in (3.1) becomes

where $Y_{i}=R_{i}+\gamma\cdot\max_{a\in\mathcal{A}}\widetilde{Q}_{k-1}(S_{i}^{\prime},a)$ is the target and $\widetilde{Q}_{k-1}$ is the Q-network computed in the previous iteration. Notice that this problem is a least-squares regression with overparameterized neural networks. For computational efficiency, we propose to solve (B.2) via stochastic gradient descent (SGD). Specifically, in each iteration of SGD, we sample a fresh observation $(S,A,R,S^{\prime})$ with $(S,A)$ drawn from the sampling distribution $\sigma$, $R\sim R(\cdot{\,|\,}S,A)$, and $S^{\prime}\sim P(\cdot{\,|\,}S,A)$. Then an estimator of the gradient is computed based on $(S,A,R,S^{\prime})$, which is used to update the network parameters. We run the SGD updates for a total of $n$ iterations and denote the output by $\widetilde{Q}_{k}$.

where $B$ is a sufficiently large constant. Thus, the population version of the $k$-th iteration of the FQI algorithm becomes

where $(S,A)\sim\sigma$ and $Y$ is computed using $\widetilde{Q}_{k-1}$. We solve this optimization problem via projected SGD, which generates a sequence of weight matrices $\{W^{(t)}\}_{t\geq 0}\subseteq\mathcal{B}_{B}$ satisfying

where $\Pi_{\mathcal{B}_{B}}$ is the projection operator onto $\mathcal{B}_{B}$ with respect to the Frobenius norm, $\eta>0$ is the step size, and $(S_{t},A_{t},Y_{t})$ is a random observation. We present the details of fitted Q-iteration method with projected SGD in Algorithm 5.

To understand the convergence of the projected SGD updates in (B.5), we utilize the fact that the dynamics of training overparametrized neural networks is captured by the neural tangent kernel (Jacot et al. 2018) when the width is sufficiently large. Specifically, since $\sigma(u)=u\cdot\ind\{u>0\}$, the gradient of the Q-network in (B.1) is given by

Recall that we initialize parameters $b$ and $W$ as $b^{(0)}$ and $W^{(0)}$ and that we only update $W$ during training. We define a function class $\mathcal{F}_{B,m}^{(t)}$ as

By (B.6), for each function $\widehat{Q}(\cdot,\cdot;W)\in\mathcal{F}_{B,m}^{(t)}$, we can write it as

which is the first-order linearization of $Q(\cdot,\cdot;W^{(t)})$ at $W^{(t)}$. Furthermore, since $B$ in (B.3) is a constant, for each weight matrix $W$ in $\mathcal{B}_{B}$, when $m$ goes to infinity, $\|W_{j}-W_{j}^{(0)}\|_{2}$ would be small for almost all $j\in[2m]$, which implies that $\ind\{W_{j}^{\top}(s,a)>0\}=\ind\{(W_{j}^{(0)})^{\top}(s,a)>0\}$ holds with high probability for all $j\in[2m]$ and $(s,a)\in{\mathcal{S}}\times\mathcal{A}$. As a result, when $m$ is sufficiently large, $\mathcal{F}_{B,m}^{(t)}$ defined in (B.7) is close to

where $b^{(0)}$ and $W^{(0)}$ are the initial parameters. Specifically, as proved in Lemma A.2 in Wang et al. 2019, when the sampling distribution $\sigma$ is regular in the sense that Assumption (B.2) specified below is satisfied, for any $W_{1},W_{2}\in\mathcal{B}_{B}$, we have

Besides, for all $j\in[2m]$, we let $\phi_{j}$ denote $\phi(\cdot,\cdot;b^{(0)}_{j},W^{(0)}_{j})$. Due to the symmetric initialization scheme, $\{\phi_{j}\}_{j\in[m]}$ are i.i.d. random feature functions and $\phi_{j}=-\phi_{j+m}$ for all $j\in[m]$. Thus, each $\widehat{Q}(\cdot,\cdot;W)$ in (B.8) can be equivalently written as