It is shown that a soft $Q$-learning algorithm converges almost surely to the fixed point of a soft Bellman equation defined by the stationary averages that the behavioral policy and the chosen kernel induce, and what separates the resulting policy from an optimal one.
Abstract
We consider an agent acting against a general non-Markovian environment. The agent maintains its agent states, but is free to choose a transition kernel across those states and optimize its state-feedback control policies. We study the bi-level agent design problem that optimizes the transition kernel and the policy it induces, given said kernel with offline data of observations and actions obtained via a behavioral policy. For general environments, we show that a soft $Q$-learning algorithm converges almost surely to the fixed point of a soft Bellman equation defined by the stationary averages that the behavioral policy and the chosen kernel induce, and we delineate what separates the resulting policy from an optimal one. In partially observed Markov decision problems, we analyze convergence properties of parametrized transition kernel design via zero-th order and Bayesian optimization techniques.
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