Homotopic Policy Mirror Descent: Policy Convergence, Implicit Regularization, and Improved Sample Complexity

We propose the homotopic policy mirror descent (HPMD) method for solving discounted, infinite horizon MDPs with finite state and action space, and study its policy convergence. We report several properties that seem to be new in the literature of policy gradient methods: (1) HPMD exhibits global linear convergence of the value optimality gap, and local superlinear convergence of the policy to the set of optimal policies with order $\gamma^{-2}$. The superlinear convergence of the policy takes effect after no more than $\mathcal{O}(\log(1/\Delta^*))$ number of iterations, where $\Delta^*$ is defined via a gap quantity associated with the optimal state-action value function; (2) HPMD also exhibits last-iterate convergence of the policy, with the limiting policy corresponding exactly to the optimal policy with the maximal entropy for every state. No regularization is added to the optimization objective and hence the second observation arises solely as an algorithmic property of the homotopic policy gradient method; (3) The last-iterate convergence of HPMD holds for a much broader class of decomposable distance-generating functions, including the $p$-th power of $\ell_p$-norm and the negative Tsallis entropy. As a byproduct of the analysis, we also discover the finite-time exact convergence of HPMD with these divergences, and show that HPMD continues converging to the limiting policy even if the current policy is already optimal; (4) For the stochastic HPMD method, we further demonstrate that a better than $\mathcal{O}(|\mathcal{S}| |\mathcal{A}| / \epsilon^2)$ sample complexity for small optimality gap $\epsilon$ holds with high probability, when assuming a generative model for policy evaluation.