Policy Newton methods for Distortion Riskmetrics

We consider the problem of risk-sensitive control in a reinforcement learning (RL) framework. In particular, we aim to find a risk-optimal policy by maximizing the distortion riskmetric (DRM) of the discounted reward in a finite horizon Markov decision process (MDP). DRMs are a rich class of risk measures that include several well-known risk measures as special cases. We derive a policy Hessian theorem for the DRM objective using the likelihood ratio method. Using this result, we propose a natural DRM Hessian estimator from sample trajectories of the underlying MDP. Next, we present a cubic-regularized policy Newton algorithm for solving this problem in an on-policy RL setting using estimates of the DRM gradient and Hessian. Our proposed algorithm is shown to converge to an $\epsilon$-second-order stationary point ($\epsilon$-SOSP) of the DRM objective, and this guarantee ensures the escaping of saddle points. The sample complexity of our algorithms to find an $\epsilon$-SOSP is $\mathcal{O}(\epsilon^{-3.5})$. Our experiments validate the theoretical findings. To the best of our knowledge, our is the first work to present convergence to an $\epsilon$-SOSP of a risk-sensitive objective, while existing works in the literature have either shown convergence to a first-order stationary point of a risk-sensitive objective, or a SOSP of a risk-neutral one.