  Existing work on risk-sensitive reinforcement learning - both for symmetric
and downside risk measures - has typically used direct Monte-Carlo estimation
of policy gradients. While this approach yields unbiased gradient estimates, it
also suffers from high variance and decreased sample efficiency compared to
temporal-difference methods. In this paper, we study prediction and control
with aversion to downside risk which we gauge by the lower partial moment of
the return. We introduce a new Bellman equation that upper bounds the lower
partial moment, circumventing its non-linearity. We prove that this proxy for
the lower partial moment is a contraction, and provide intuition into the
stability of the algorithm by variance decomposition. This allows
sample-efficient, on-line estimation of partial moments. For risk-sensitive
control, we instantiate Reward Constrained Policy Optimization, a recent
actor-critic method for finding constrained policies, with our proxy for the
lower partial moment. We extend the method to use natural policy gradients and
demonstrate the effectiveness of our approach on three benchmark problems for
risk-sensitive reinforcement learning.
