The Geometry of Memoryless Stochastic Policy Optimization in Infinite-Horizon POMDPs
This work addresses policy optimization in POMDPs, a core challenge in reinforcement learning, but appears incremental as it builds on existing polynomial optimization tools for a specific class of policies.
The paper tackles the problem of optimizing memoryless stochastic policies in infinite-horizon POMDPs by showing that state-action frequencies and rewards are rational functions of the policy, and reformulates it as a linear optimization with polynomial constraints to address combinatorial complexity, applying this to solve a navigation problem in a grid world.
We consider the problem of finding the best memoryless stochastic policy for an infinite-horizon partially observable Markov decision process (POMDP) with finite state and action spaces with respect to either the discounted or mean reward criterion. We show that the (discounted) state-action frequencies and the expected cumulative reward are rational functions of the policy, whereby the degree is determined by the degree of partial observability. We then describe the optimization problem as a linear optimization problem in the space of feasible state-action frequencies subject to polynomial constraints that we characterize explicitly. This allows us to address the combinatorial and geometric complexity of the optimization problem using recent tools from polynomial optimization. In particular, we estimate the number of critical points and use the polynomial programming description of reward maximization to solve a navigation problem in a grid world.