Non-asymptotic Convergence of Adam-type Reinforcement Learning Algorithms under Markovian Sampling

The main contribution of the paper lies in establishing the first non-asymptotic analysis for the Adam-type PG and TD algorithms under the Markovian sampling model.

From the technical standpoint, although the existing analysis of the Adam-type algorithms in conventional optimization may provide useful tools, analysis of such algorithms in RL problems has new challenges. (1) Samples in TD learning are drawn from an unknown transition kernel in a non-i.i.d. manner, and hence a bias of the gradient estimation arises. Such a bias is also dependent on the AMSGrad updates, which further complicates the analysis. We develop techniques to bound the bias by the stepsize even under adaptive momentum updates. (2) The sampling process in PG (with a nonlinear approximation architecture) is not only non-i.i.d., but also follows time-varying parametric policies. Thus, it is even more challenging to bound such a bias with the AMSGrad updates. In response to this, we develop new techniques to characterize the bias error which can also be controlled by the stepsize.

As for the results, we provide the first convergence guarantee for RL algorithms (including PG and TD) incorporating the update rule of AMSGrad (referred to as PG-AMSGrad and TD-AMSGrad, respectively), and our techniques also lead to an improved result for the vanilla PG. (1) First, we show that under general nonlinear function approximation, PG-AMSGrad with a constant stepsize¹¹1The “stepsize” here refers to the basic stepsize $\alpha$ in the AMSGrad update (4). The overall learning rate of the algorithm is determined by the basic stepsize $\alpha$, hyperparameters $\beta_{1}$ and $\beta_{2}$, and the first and second moments of gradients as given in (1)-(4), and is hence adaptive during the AMSGrad iteration. converges to a neighborhood of a stationary point at a rate of $\mathcal{O}(1/T)$ (where $T$ denotes the number of iterations), and with a diminishing stepsize converges exactly to a stationary point at a rate of $\mathcal{O}(\log^{2}T/\sqrt{T})$. (2) Furthermore, under linear function approximation, TD-AMSGrad with a constant stepsize converges to a neighborhood of the global optimum at a rate of $\mathcal{O}(1/T)$, and with a diminishing stepsize converges exactly to the global optimum at a rate of $\mathcal{O}(\log T/\sqrt{T})$. (3) In addition, by adapting our technical tools to analyze the vanilla PG with the SGD update under Markovian sampling, we obtain an orderwisely better computational complexity than the existing works.

Due to the rapidly growing theoretical studies on RL, we only review the most relevant studies below.

Convergence analysis of PG: Asymptotic convergence of PG based on stochastic approximation (SA) was established in Williams, (1992); Baxter and Bartlett, (2001); Sutton et al., (2000); Kakade, (2002); Pirotta et al., (2015); Tadić et al., (2017). In specific RL problems such as LQR, PG has been proved to converge to the optimal policy Fazel et al., (2018); Malik et al., (2019); Tu and Recht, (2019). Under specific policy function approximation classes, Bhandari and Russo, (2019); Agarwal et al., (2019) also showed that PG can find the optimal policy. Under the general nonconvex approximation, Shen et al., (2019); Papini et al., (2018, 2019); Xu et al., 2019a ; Xu et al., (2020) characterized the convergence rate for PG and variance reduced PG to a stationary point in the finite-horizon scenario, and Zhang et al., (2019); Karimi et al., (2019) provided the convergence rate for PG in the infinite-horizon scenario. Wang et al., (2020) studied PG with neural network function approximation in an overparameterized regime. Convergence analysis has been also established for the variants of PG, such as actor-critic algorithms (Bhatnagar et al.,, 2008, 2009; Bhatnagar,, 2010; Luo et al.,, 2019; Yang et al.,, 2019; Fu et al.,, 2020), TRPO (Shani et al.,, 2020), TRPO/PPO (Liu et al.,, 2019), etc. This paper studies the infinite-horizon scenario, but focuses on the Adam-type PG, which has not been studied in the literature. Our analysis is also applicable to other variants of PG.

Convergence analysis of TD: Originally proposed in Sutton, (1988), TD learning with function approximation aroused great interest in analyzing its convergence. While a general TD may not converge as pointed out in Baird, (1995); Györfi and Walk, (1996), the authors in Tsitsiklis and Van Roy, (1997) provided conditions to ensure asymptotic convergence of TD with linear function approximation under i.i.d. sampling. Other results on asymptotic convergence using the tools from linear SA were provided in Kushner and Yin, (2003); Benveniste et al., (2012). Non-asymptotic convergence was established for TD under i.i.d. sampling in, e.g., Dalal et al., (2018); Bhandari et al., (2018); Lakshminarayanan and Szepesvari, (2018), and under Markovian sampling in, e.g., Bhandari et al., (2018); Srikant and Ying, (2019); Hu and Syed, (2019). The convergence rate of TD with nonlinear function approximation has recently been studied in Cai et al., (2019) for overparameterized neural networks using i.i.d. samples. In contrast to the aforementioned work on TD with the SGD updates, this paper studies Adam-type TD under Markovian sampling.

Adaptive reinforcement learning algorithms: Adaptivity has been applied to RL algorithms to improve the performance. (Shani et al.,, 2020) used an adaptive proximity term to study the convergence of TRPO. An adaptive batch size was adopted to improve the policy performance (Papini et al.,, 2017) and reduce the variance (Ji et al.,, 2019) of PG. The abovementioned papers did not study how adaptive learning rates can affect the performance of PG or TD. More recently, a concurrent work (Sun et al.,, 2020) which is posted a few days after our paper provided an analysis of TD(0) and TD($\lambda$) when incorporating adaptive gradient descent (AdaGrad) updates. However, this paper focuses on a more popular class of adaptive algorithms: Adam-type methods, and provide the first convergence guarantee when they are applied to PG and TD.

Convergence analysis of Adam-type algorithms in conventional optimization: Adam was proposed in Kingma and Ba, (2015) for speeding up the training of deep neural networks, but the vanilla Adam was shown not to converge in Reddi et al., (2018). Instead AMSGrad was proposed as a slightly modified version to justify the theoretic performance of Adam and its regret bounds were characterized in Reddi et al., (2018); Tran and Phong, (2019) for online convex optimization. Recently, Adam/AMSGrad was proved to converge to a stationary point for certain general nonconvex optimization problems Zou et al., 2019a ; Zhou et al., (2018); Chen et al., 2019a . Our study provides the first convergence guarantee for the Adam-type algorithms in the RL settings, where the Markovian sampling poses the key difference and challenge in our analysis from those in conventional optimization.

We consider the standard RL settings, where an agent interacts with a (possibly stochastic) environment (e.g. process or system dynamics). This interaction is usually modeled as a discrete-time discounted Markov Decision Processes (MDPs), described by a tuple $(\mathcal{S},\mathcal{A},\mathbb{P},R,\gamma,\zeta)$, where $\mathcal{S}$ is the state space, $\mathcal{A}$ is the action space, $\mathbb{P}:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\mapsto[0,1]$ is the probability kernel for the state transitions, e.g., $\mathbb{P}(\cdot|s,a)$ denotes the probability distribution of the next state given the current state $s$ and action $a$. In addition, $R:\mathcal{S}\times\mathcal{A}\mapsto[0,R_{\max}]$ is the reward function mapping station-action pairs to a bounded subset of $\mathbb{R}$, $\gamma\in(0,1)$ is the discount factor, and $\zeta$ denotes the initial state distribution. The agent’s decision is captured by the policy $\pi:=\pi(\cdot|s)$ which characterizes the density function over the action space $\mathcal{A}$ at the state $s\in\mathcal{S}$. We denote $\nu:=\nu_{\pi}$ as the stationary distribution of the transition kernel $\mathbb{P}$ for a given $\pi$. In addition, we define the $\gamma$-discounted stationary visitation distribution of the policy $\pi$ as $\mu_{\pi}(s)=\sum_{t=1}^{\infty}\gamma^{t}P_{\zeta,\pi}(s_{t}=s)$. Further, we denote $\mu_{\pi}(s,a)=\mu_{\pi}(s)\pi(a|s)$ as the (discounted) state-action visitation distribution.

In this paper, we assume that the state space $\mathcal{S}$ is countably infinite and the action space $\mathcal{A}$ is finite with possibly large cardinality $|\mathcal{A}|$. Hence, we estimate the policy and the value function corresponding to a certain unknown policy by parameterized function classes as we introduce in Section Sections 3.1 and 4.1, respectively.

Although Adam Kingma and Ba, (2015) has gained great success in practice since being proposed, it is shown not to converge even in the simple convex setting Reddi et al., (2018). Instead, a slightly modified version called AMSGrad Reddi et al., (2018) is widely used to understand the success of adaptive momentum optimization algorithms. Given a gradient $g_{t}$ at time $t$, the generic form of AMSGrad is given by

where $\alpha_{t}$ is the stepsize, and $\beta_{1},\beta_{2}$ are two hyper-parameters. In addition, $m_{t}$, $v_{t}$ given in (1) and (2) are viewed as the estimation of the first moment and second moment, respectively, which play important roles in adapting the learning rate as in (4). Compared to Adam, the main difference of AMSGrad lies in (3), which guarantees the sequence $\hat{v}_{t}$ to be non-decreasing whereas Adam does not require this. Such difference is considered to be a central reason causing the non-convergent behavior of Adam Reddi et al., (2018); Chen et al., 2019a .

We use $\|x\|:=\|x\|_{2}=\sqrt{x^{T}x}$ to denote the $\ell_{2}$-norm of a vector $x$, and use $\|x\|_{\infty}=\underset{i}{\max}|x_{i}|$ to denote the infinity norm. When $x,y$ are both vectors, $x/y,xy,x^{2},\sqrt{x}$ are all calculated in the element-wise manner, which are used in the update of AMSGrad. We denote $[n]=\{1,2,\dots,n\}$, and $\lceil x\rceil\in\mathbb{Z}$ as the integer such that $\lceil x\rceil-1\leq x<\lceil x\rceil$.

Consider a reinforcement learning problem which aims to find a policy that maximizes the expected accumulative reward. We assume that the policy is parameterized by $\theta\in\mathbb{R}^{d}$ and form a policy class $\Pi:=\{\pi_{\theta}|\theta\in\mathbb{R}^{d}\}$, which in general is a nonlinear function class. The policy gradient method is usually used to solve the following infinite-horizon optimization problem:

The gradient of $J(\theta)$ with respect to $\theta$ is captured by the policy gradient theorem for infinite-horizon MDP with the discounted reward Sutton et al., (2000), and is given by

where the expectation is taken over the discounted state-action visitation distribution $\mu_{\theta}:=\mu_{\pi_{\theta}}(s,a)$, and $Q^{\pi_{\theta}}(s,a)$ denotes the Q-function for an initial state-action pair $(s,a)$ defined as

In addition, we refer to $\nabla_{\theta}\log\pi_{\theta}(a|s)$ as the score function corresponding to the policy $\pi_{\theta}$.

Since the transition probability is unknown, the policy gradient in (6) needs to be estimated via sampling. The Q-function $Q^{\pi_{\theta}}(s,a)$ and the score function are typically estimated by independent samples. First, at each time $t$, we draw a sample trajectory to provide an estimated Q-function $\hat{Q}^{\pi_{\theta}}(s,a)$ based on the algorithm EstQ Zhang et al., (2019) (see Algorithm 3 in Appendix A for details). Such an estimator has been shown to be unbiased Zhang et al., (2019). That is, if we use $O^{q}$ to denote the randomness including the samples and horizon in EstQ, then we have

Second, we take samples $\{(s_{t},a_{t})\}$ to estimate the score function by following the policy $\pi_{\theta_{t}}$ and the transition function proposed in Konda, (2002) which is given by

where $\zeta(\cdot)$ denotes the initial distribution and $\mathbb{P}$ is the transition probability from the original MDP. It has been shown in Konda, (2002) that such a transition probability guarantees the MDP to converge to the state-action visitation distribution.

Then, the gradient estimator to approximate $\nabla_{\theta}J(\theta)$ at time $t$ is given by

We then apply such a gradient estimator $g_{t}$ to update the policy parameter by the AMSGrad update given in (1)-(4), and obtain PG-AMSGrad as in Algorithm 1.

We note that the gradient estimator obtained in (8) is biased, because the score function is estimated by a sequence of Markovian samples. We will show that such a biased gradient estimator is in fact computationally more efficient than the unbiased estimator used in the existing literature Zhang et al., (2019) (see Section 3.4). Our main technical novelty here lies in developing techniques to analyze the biased estimator under the AMSGrad update for PG.

In the following, we specify some technical assumptions in our convergence analysis.

We consider a general class of parameterized policy functions that satisfy the following assumption.

This assumption is standard in the existing literature that studies PG with nonconvex function approximation Zhang et al., (2019); Xu et al., 2019a ; Papini et al., (2018).

In Algorithm 1, we sample a data trajectory using the transition kernel $\hat{P}$ and the policy $\pi_{\theta_{t}}$. Such a sequence of samples are non-i.i.d. and follow a Markovian distribution. We assume that the MDP and the policies we consider satisfy the following standard mixing property.

This assumption holds for irreducible and aperiodic Markov chains Mitrophanov, (2005), and is widely adopted in the theoretical analysis of RL algorithms under Markovian sampling settings Bhandari et al., (2018); Chen et al., 2019b ; Zou et al., 2019b ; Karimi et al., (2019).

In this section, we provide the convergence analysis of PG-AMSGrad as given in Algorithm 1. We first consider the case with a constant stepsize, and then provide the result with a diminishing stepsize.

Although AMSGrad has been studied in conventional optimization, our analysis of PG-AMSGrad mainly deals with the following new challenges arising in RL. First, samples here are generated via an MDP and distributed in a non-i.i.d. fashion. Thus the gradient estimator is biased and we need to control the bias with a certain upper bound scaled by the stepsize. Second, the sampling distribution also changes over time, which causes additional complication. Thus, our technical development mainly handles the above two challenges under the adaptive momentum update rule of AMSGrad. We next provide the convergence results that we obtain and relegate all the proofs to the appendices.

We first provide the Lipschitz properties for the true policy gradient and its estimator, which are useful for establishing the convergence. Recall that in Algorithm 1, the gradient estimator $g_{t}=\hat{Q}^{\pi_{\theta_{t}}}(s_{t},a_{t})\nabla_{\theta_{t}}\log(\pi_{\theta_{t}}(a_{t}|s_{t}))$ at time $t$ is obtained by using the Q-function estimator generated by the EstQ algorithm (see Appendix A). Note that $\hat{Q}^{\pi_{\theta}}(s,a)$ is an unbiased estimator of $Q^{\pi_{\theta}}(s,a)$ for all $(s,a)$ Zhang et al., (2019), and the samples for estimation are independent of those for other steps in PG-AMSGrad except the initial sample. Taking expectation over the randomness in EstQ at time $t$ (denoted as $O^{q}_{t}$), we obtain an estimator $\nabla_{\theta_{t}}\tilde{J}(\theta_{t};s_{t},a_{t})$ defined as

We next obtain the Lipschitz properties of $\nabla_{\theta}\tilde{J}(\theta;s,a)$ and $\nabla_{\theta}J(\theta)$ in the following lemma.

The following theorem characterizes the convergence of PG-AMSGrad with a constant stepsize. Recall that the stepsize refers to the parameter $\alpha$ in the AMSGrad update (4), not the overall learning rate of the algorithm.

Theorem 1 indicates that under the constant stepsize, PG-AMSGrad converges to a neighborhood of a stationary point at a rate of $\mathcal{O}\left(\frac{1}{T}\right)$. The size of the neighborhood can be controlled by the stepsize $\alpha$. One can observe that $\alpha$ controls a tradeoff between the convergence rate and the convergence accuracy. Decreasing $\alpha$ improves the convergence accuracy, but slows down the convergence, since the coefficient $C_{1}$ contains $\alpha$ in the denominator. To balance such a tradeoff, we set the stepsize $\alpha_{t}=\frac{1}{\sqrt{T}}$. In this case, the mixing time becomes $\tau^{*}=\mathcal{O}(\log T)$ and thus PG-AMSGrad converges to a stationary point with a rate of $\mathcal{O}\left(\frac{\log^{2}T}{\sqrt{T}}\right)$.

In the following, we adopt a diminishing stepsize to eliminate the convergence error and obtain the exact convergence.

Under the diminishing stepsize, PG-AMSGrad can converge exactly to a stationary point. Since $\tau^{*}=\mathcal{O}(\log T)$, the convergence rate is given by $\mathcal{O}\left(\frac{\log^{2}T}{\sqrt{T}}\right)$.

Theorems 1 and 2 indicate that under both a constant stepsize and diminishing stepsize, PG-AMSGrad finds a stationary point efficiently with guaranteed convergence. However, there is a tradeoff between the convergence rate and the convergence accuracy. With a constant stepsize, PG-AMSGrad can converge faster but only to a neighborhood of a stationary point whose size is controlled by the stepsize, whereas a diminishing stepsize yields a better convergence accuracy, but attains a lower convergence rate.

Although our focus in this paper is on the Adam-type PG, slight modification of our analysis yields the convergence rate of the vanilla PG under infinite horizon Markovian sampling. In the following, we first present such results and then compare them with two recent studies Zhang et al., (2019); Karimi et al., (2019) on the same model to illustrate the novelty of our analysis and benefit of our sampling strategy.

We consider the vanilla PG algorithm that uses the same gradient estimator and sampling strategy as those of Algorithm 1, but adopts the SGD update (i.e., $\theta_{t+1}=\theta_{t}-\alpha_{t}g_{t}$) rather than the AMSGrad update. We call such an algorithm as PG-SGD. The following proposition characterizes the convergence rate for PG-SGD.

We next compare Proposition 1 with two recent studies on the same non-i.i.d. model. First, Karimi et al., (2019) studied infinite-horizon PG with a biased gradient estimator. In their analysis, they did not bound the gradient bias using the stepsize, and hence their convergence has a non-zero error even with a diminishing stepsize. In contrast, we obtain a fine-grained bound on the bias and characterize its dependence on the stepsize. We show that PG exactly converges to a stationary point under the diminishing stepsize.

Another closely related study on the infinite-horizon PG under non-i.i.d. sampling was by Zhang et al., (2019), but their algorithm adopts an unbiased gradient estimator at the cost of using more samples. As a comparison, Proposition 1 indicates that PG-SGD with a biased gradient estimator attains a better convergence rate and accuracy. More specifically, under the constant stepsize, (Zhang et al.,, 2019, Corollary 4.4) showed that their PG algorithm converges with an error bound of $\mathcal{O}\left(\frac{1}{(1-\gamma)^{5}(1-\sqrt{\gamma})^{2}}\right)$, whereas PG-SGD with a biased gradient estimator achieves a much smaller error bound $\mathcal{O}\left(\frac{1}{(1-\gamma)^{2}}\right)$ by taking $\alpha=1-\gamma$. Similarly, under the diminishing stepsize, (Zhang et al.,, 2019, Theorem 4.3) showed that their PG algorithm converges at a rate of $\mathcal{O}\left(\frac{1}{(1-\gamma)^{5}(1-\sqrt{\gamma})^{2}\sqrt{T}}\right)$, whereas our PG-SGD converges at a rate of $\mathcal{O}\left(\frac{\log^{2}(\sqrt{T}/(1-\gamma))}{(1-\gamma)^{2}\sqrt{T}}\right)$, which is much faster since $\gamma$ is usually close to 1, and $\log T$ is considered to be less influential in practice.

Policy evaluation is a fundamental task in RL, and often plays a critical role in other algorithms such as PG that we study in Section 3. The goal of policy evaluation is to obtain an accurate estimation of the accumulated reward function known as the value function $V:\mathcal{S}\mapsto\mathbb{R}$ for a given policy $\pi$ defined as

The Bellman operator corresponding to a value function $V$ is defined by $\forall s\in\mathcal{S}$,

The key to approximate the value function is the observation that it satisfies the projected bellman equation $(\Pi T^{\pi}V)(s)=V(s)$ when projected onto some convex space. Under the function approximation, the value function $V(s)$ is parameterized by $\theta\in\mathbb{R}^{d}$ and denoted by $V(s;\theta)$. As many recent finite-time analysis of TD Bhandari et al., (2018); Xu et al., 2019b ; Srikant and Ying, (2019), we consider the linear approximation class of the value function $V(s;\theta)$ defined as

where $\theta\in\mathbb{R}^{d}$, and $\phi:\mathcal{S}\rightarrow\mathbb{R}^{d}$ is a vector function with the dimension $d$, and the elements of $\phi$ represent the nonlinear kernel (feature) functions. Then the vanilla TD algorithm follows a stochastic iterative method given by

where $\alpha_{t}$ is the stepsize, and $g_{t}$ is defined as

Here, $g_{t}$ serves as a stochastic pseudo-gradient, and is an estimator of the full pseudo-gradient given by

where the expectation is taken over the stationary distribution of the states. We note that $\bar{g}(\theta_{t})$ is not a gradient of a loss function, but plays a similar role as the gradient in the gradient descent algorithm.

Then TD-AMSGrad is obtained by replacing the update (14) by the AMSGrad update given in (1)-(4) as in Algorithm 2.

As seen in Algorithm 2, the state-action pairs are sampled as a trajectory under the transition probability $\mathbb{P}$ with unknown policy $\pi$. Therefore, the samples along the trajectory are dependent, and hence we need to analyze the convergence of TD-AMSGrad under Markovian sampling.

In this section, we introduce some standard technical assumptions for our analysis.

We first give the following standard assumption on the kernel function Tsitsiklis and Van Roy, (1997); Bhandari et al., (2018); Xu et al., 2019b ; Chen et al., 2019b .

The boundedness assumption is mild since we can always normalize the kernel functions.

We next define the following norm of the value function as:

where $D=\text{diag}(\nu(s_{1}),\nu(s_{2}),\cdots)$ is a diagonal matrix whose elements are determined by the stationary distribution $\nu$, and $\Sigma\in\mathbb{R}^{d\times d}$ is the steady-state feature covariance matrix given by $\Sigma=\sum_{s\in\mathcal{S}}\nu(s)\phi(s)\phi(s)^{T}$.

The next assumption of the bounded domain is standard in theoretical analysis of the Adam-type algorithms Reddi et al., (2018); Tran and Phong, (2019).

The boundedness parameter $D_{\infty}$ can be chosen as discussed in Bhandari et al., (2018).

In the following, we provide the convergence results for TD-AMSGrad with linear function approximation under Markovian sampling.

First consider the full pseudo-gradient $\bar{g}(\theta)$ in (16). We define $\theta^{*}$ as the fixed point of $\bar{g}(\theta)$, i.e., $\bar{g}(\theta^{*})=0$. Then $\theta^{*}$ is the unique fixed point under Assumption 3 following from the contraction property of the projected Bellman operator Tsitsiklis and Van Roy, (1997). The following lemma is useful for our convergence analysis.

Lemma 2 indicates that the update of TD learning exhibits a property similar to strongly convex optimization.

Since our analysis is under Markovian sampling, non-zero bias arises in our convergence analysis when estimating the gradient, i.e., $\mathbb{E}[g_{t}]-\bar{g}(\theta_{t})\neq 0$. The following lemma provides an upper bound on such a bias error for TD-AMSGrad.

The following theorem provides the convergence of TD-AMSGrad under a constant stepsize coupled with diminishing hyper-parameters in the AMSGrad update.

In Theorem 3, $C_{1},C_{2}$ are constants and time-independent. Therefore, under the choice of the stepsize and hyper-parameters in the theorem, Algorithm 2 converges to a neighborhood of the global optimum at a rate of $\mathcal{O}\left(\frac{1}{T}\right)$. The size of the neigborhood is controlled by the stepsize $\alpha$. Similarly to Theorem 1, we can balance the tradeoff between the convergence rate and the convergence accuracy, i.e., the size of neighborhood, by setting the stepsize $\alpha_{t}=\frac{1}{\sqrt{T}}$, which yields a convergence to the global optimal solution at the rate of $\mathcal{O}\left(\frac{\log^{2}T}{\sqrt{T}}\right)$.

Next, we provide the convergence result with a diminishing stepsize in the following theorem.