Directly Estimating the Variance of the {\lambda}-Return Using
  Temporal-Difference Methods
  This paper investigates estimating the variance of a temporal-difference
learning agent's update target. Most reinforcement learning methods use an
estimate of the value function, which captures how good it is for the agent to
be in a particular state and is mathematically expressed as the expected sum of
discounted future rewards (called the return). These values can be
straightforwardly estimated by averaging batches of returns using Monte Carlo
methods. However, if we wish to update the agent's value estimates during
learning--before terminal outcomes are observed--we must use a different
estimation target called the {\lambda}-return, which truncates the return with
the agent's own estimate of the value function. Temporal difference learning
methods estimate the expected {\lambda}-return for each state, allowing these
methods to update online and incrementally, and in most cases achieve better
generalization error and faster learning than Monte Carlo methods. Naturally
one could attempt to estimate higher-order moments of the {\lambda}-return.
This paper is about estimating the variance of the {\lambda}-return. Prior work
has shown that given estimates of the variance of the {\lambda}-return,
learning systems can be constructed to (1) mitigate risk in action selection,
and (2) automatically adapt the parameters of the learning process itself to
improve performance. Unfortunately, existing methods for estimating the
variance of the {\lambda}-return are complex and not well understood
empirically. We contribute a method for estimating the variance of the
{\lambda}-return directly using policy evaluation methods from reinforcement
learning. Our approach is significantly simpler than prior methods that
independently estimate the second moment of the {\lambda}-return. Empirically
our new approach behaves at least as well as existing approaches, but is
generally more robust.
