  We study the estimation of risk-sensitive policies in reinforcement learning
problems defined by a Markov Decision Process (MDPs) whose state and action
spaces are countably finite. Prior efforts are predominately afflicted by
computational challenges associated with the fact that risk-sensitive MDPs are
time-inconsistent. To ameliorate this issue, we propose a new definition of
risk, which we call caution, as a penalty function added to the dual objective
of the linear programming (LP) formulation of reinforcement learning. The
caution measures the distributional risk of a policy, which is a function of
the policy's long-term state occupancy distribution. To solve this problem in
an online model-free manner, we propose a stochastic variant of primal-dual
method that uses Kullback-Lieber (KL) divergence as its proximal term. We
establish that the number of iterations/samples required to attain
approximately optimal solutions of this scheme matches tight dependencies on
the cardinality of the state and action spaces, but differs in its dependence
on the infinity norm of the gradient of the risk measure. Experiments
demonstrate the merits of this approach for improving the reliability of reward
accumulation without additional computational burdens.
