Policy Evaluation with Variance Related Risk Criteria in Markov Decision Processes

In this paper we extend temporal difference policy evaluation algorithms to performance criteria that include the variance of the cumulative reward. Such criteria are useful for risk management, and are important in domains such as finance and process control. We propose both TD(0) and LSTD(lambda) variants with linear function approximation, prove their convergence, and demonstrate their utility in a 4-dimensional continuous state space problem

Variance-Constrained Actor-Critic Algorithms for Discounted and Average Reward MDPs

In many sequential decision-making problems we may want to manage risk by minimizing some measure of variability in rewards in addition to maximizing a standard criterion. Variance related risk measures are among the most common risk-sensitive criteria in finance and operations research. However, optimizing many such criteria is known to be a hard problem. In this paper, we consider both discounted and average reward Markov decision processes. For each formulation, we first define a measure of variability for a policy, which in turn gives us a set of risk-sensitive criteria to optimize. For each of these criteria, we derive a formula for computing its gradient. We then devise actor-critic algorithms that operate on three timescales - a TD critic on the fastest timescale, a policy gradient (actor) on the intermediate timescale, and a dual ascent for Lagrange multipliers on the slowest timescale. In the discounted setting, we point out the difficulty in estimating the gradient of the variance of the return and incorporate simultaneous perturbation approaches to alleviate this. The average setting, on the other hand, allows for an actor update using compatible features to estimate the gradient of the variance. We establish the convergence of our algorithms to locally risk-sensitive optimal policies. Finally, we demonstrate the usefulness of our algorithms in a traffic signal control application