Reward is enough for convex MDPs

Tom Zahavy, Brendan O'Donoghue, Guillaume Desjardins, Satinder Singh

Keywords: reinforcement-learning, convex optimisation

Abstract: Maximising a cumulative reward function that is Markov and stationary, i.e., defined over state-action pairs and independent of time, is sufficient to capture many kinds of goals in a Markov decision process (MDP). However, not all goals can be captured in this manner. In this paper we study convex MDPs in which goals are expressed as convex functions of the stationary distribution and show that they cannot be formulated using stationary reward functions. Convex MDPs generalize the standard reinforcement learning (RL) problem formulation to a larger framework that includes many supervised and unsupervised RL problems, such as apprenticeship learning, constrained MDPs, and so-called `pure exploration'. Our approach is to reformulate the convex MDP problem as a min-max game involving policy and cost (negative reward) `players', using Fenchel duality. We propose a meta-algorithm for solving this problem and show that it unifies many existing algorithms in the literature.

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TL;DR: Convex MDPs, which include RL, apprenticeship learning, constrained MDPs, and pure exploration are shown to be solved by an RL algorithm that maximizes the gradient of the convex objective as a non-stationary reward