Reward is enough for convex MDPs
This work addresses a foundational gap in reinforcement learning theory by extending the MDP framework to handle a broader class of goals, impacting researchers and practitioners in AI and machine learning.
The paper tackles the limitation of standard Markov decision processes (MDPs) in capturing goals expressed as convex functions of the stationary distribution, showing that these cannot be formulated with stationary reward functions. It proposes a min-max game reformulation using Fenchel duality and a meta-algorithm that unifies existing algorithms for convex MDPs, which generalize reinforcement learning to include problems like apprenticeship learning and constrained MDPs.
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.