Theoretical Guarantees of Fictitious Discount Algorithms for Episodic Reinforcement Learning and Global Convergence of Policy Gradient Methods

When designing algorithms for finite-time-horizon episodic reinforcement learning problems, a common approach is to introduce a fictitious discount factor and use stationary policies for approximations. Empirically, it has been shown that the fictitious discount factor helps reduce variance, and stationary policies serve to save the per-iteration computational cost. Theoretically, however, there is no existing work on convergence analysis for algorithms with this fictitious discount recipe. This paper takes the first step towards analyzing these algorithms. It focuses on two vanilla policy gradient (VPG) variants: the first being a widely used variant with discounted advantage estimations (DAE), the second with an additional fictitious discount factor in the score functions of the policy gradient estimators. Non-asymptotic convergence guarantees are established for both algorithms, and the additional discount factor is shown to reduce the bias introduced in DAE and thus improve the algorithm convergence asymptotically. A key ingredient of our analysis is to connect three settings of Markov decision processes (MDPs): the finite-time-horizon, the average reward and the discounted settings. To our best knowledge, this is the first theoretical guarantee on fictitious discount algorithms for the episodic reinforcement learning of finite-time-horizon MDPs, which also leads to the (first) global convergence of policy gradient methods for finite-time-horizon episodic reinforcement learning.