Stochastic Approximation for Risk-aware Markov Decision Processes

TL;DR

This paper introduces a stochastic approximation algorithm for solving risk-aware Markov decision processes, combining saddle-point problem solving with Q-learning, and provides convergence guarantees for various risk measures.

Contribution

It presents a novel two-loop stochastic approximation algorithm that handles multiple risk measures in risk-aware MDPs with proven convergence properties.

Findings

Algorithm converges almost surely.

Convergence rate is explicitly characterized.

Applicable to multiple risk measures like CVaR and OCE.

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 $\epsilon>0$ for the optimal $Q$-value estimation gap and learning rate $k\in(1/2,\,1]$, the overall convergence rate of our algorithm is $\Omega((\ln(1/\delta\epsilon)/\epsilon^{2})^{1/k}+(\ln(1/\epsilon))^{1/(1-k)})$ with probability at least $1-\delta$.