Parameterized MDPs and Reinforcement Learning Problems -- A Maximum Entropy Principle Based Framework

Abstract

We present a framework to address a class of sequential decision making problems. Our framework features learning the optimal control policy with robustness to noisy data, determining the unknown state and action parameters, and performing sensitivity analysis with respect to problem parameters. We consider two broad categories of sequential decision making problems modelled as infinite horizon Markov Decision Processes (MDPs) with (and without) an absorbing state. The central idea underlying our framework is to quantify exploration in terms of the Shannon Entropy of the trajectories under the MDP and determine the stochastic policy that maximizes it while guaranteeing a low value of the expected cost along a trajectory. This resulting policy enhances the quality of exploration early on in the learning process, and consequently allows faster convergence rates and robust solutions even in the presence of noisy data as demonstrated in our comparisons to popular algorithms such as Q-learning, Double Q-learning and entropy regularized Soft Q-learning. The framework extends to the class of parameterized MDP and RL problems, where states and actions are parameter dependent, and the objective is to determine the optimal parameters along with the corresponding optimal policy. Here, the associated cost function can possibly be non-convex with multiple poor local minima. Simulation results applied to a 5G small cell network problem demonstrate successful determination of communication routes and the small cell locations. We also obtain sensitivity measures to problem parameters and robustness to noisy environment data.

Time-Varying Parameters in Sequential Decision Making Problems

“…µ a|s p a ss ca,µ ss + γV µ βΥ (s ) + c 0 (s), (7) where µ a|s = µ(a|s), p a ss = p(s |s, a), ca,µ ss = c(s, a, s ) + γ β log p(s |s, a)+ γ β log µ a|s for simplicity in notation, and c 0 (s) depends on γ and β, and is independent of the policy µ and the parameters Υ. Without loss of generality, we ignore c 0 (s) in the subsequent calculations (see [23]). For proof of the above Bellman equation please see Theorem 1 in [12] (or detailed proof in [23]).…”

“…Without loss of generality, we ignore c 0 (s) in the subsequent calculations (see [23]). For proof of the above Bellman equation please see Theorem 1 in [12] (or detailed proof in [23]). The optimal policy µ * β is obtained by setting…”