Stochastic Approximation for Risk-aware Markov Decision Processes

Abstract

We develop a stochastic approximation-type algorithm to solve finite state/action, infinite-horizon, risk-aware Markov decision processes. Our algorithm has two loops. The inner loop computes the risk by solving a stochastic saddle-point problem. The outer loop performs $Q$-learning to compute an optimal risk-aware policy. Several widely investigated risk measures (e.g. conditional value-at-risk, optimized certainty equivalent, and absolute semi-deviation) are covered by our algorithm. Almost sure convergence and the convergence rate of the algorithm are established. For an error tolerance $ϵ>0$ for the optimal $Q$-value estimation gap and learning rate $k\in (1/2,1]$, the overall convergence rate of our algorithm is $\Omega ((\ln (1/\delta ϵ)/{ϵ}^2{)}^{1/k}+(\ln (1/ϵ){)}^{1/(1-k)})$ with probability at least $1-\delta$.