  We investigate the discounting mismatch in actor-critic algorithm
implementations from a representation learning perspective. Theoretically,
actor-critic algorithms usually have discounting for both actor and critic,
i.e., there is a $\gamma^t$ term in the actor update for the transition
observed at time $t$ in a trajectory and the critic is a discounted value
function. Practitioners, however, usually ignore the discounting ($\gamma^t$)
for the actor while using a discounted critic. We investigate this mismatch in
two scenarios. In the first scenario, we consider optimizing an undiscounted
objective $(\gamma = 1)$ where $\gamma^t$ disappears naturally $(1^t = 1)$. We
then propose to interpret the discounting in critic in terms of a
bias-variance-representation trade-off and provide supporting empirical
results. In the second scenario, we consider optimizing a discounted objective
($\gamma < 1$) and propose to interpret the omission of the discounting in the
actor update from an auxiliary task perspective and provide supporting
empirical results.
