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

Despite the wide applications of Adam in reinforcement learning (RL), the theoretical convergence of Adam-type RL algorithms has not been established. This paper provides the first such convergence analysis for two fundamental RL algorithms of policy gradient (PG) and temporal difference (TD) learning that incorporate AMSGrad updates (a standard alternative of Adam in theoretical analysis), referred to as PG-AMSGrad and TD-AMSGrad, respectively. Moreover, our analysis focuses on Markovian sampling for both algorithms. We show that under general nonlinear function approximation, PG-AMSGrad with a constant stepsize converges to a neighborhood of a stationary point at the rate of O(1/T ) (where T denotes the number of iterations), and with a diminishing stepsize converges exactly to a stationary point at the rate of O(log 2 T / √ T ). Furthermore, under linear function approximation, TD-AMSGrad with a constant stepsize converges to a neighborhood of the global optimum at the rate of O(1/T ), and with a diminishing stepsize converges exactly to the global optimum at the rate of O(log T / √ T ). Our study develops new techniques for analyzing the Adam-type RL algorithms under Markovian sampling. * equal contribution 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.,