Retrace

Retrace is an off-policy Q-value estimation algorithm which has guaranteed convergence for a target and behaviour policy $\left(\pi, \beta\right)$. With off-policy rollout for TD learning, we must use importance sampling for the update:

$\Delta{Q}^{\text{imp}}\left(S\_{t}, A\_{t}\right) = \gamma^{t}\prod\_{1\leq{\tau}\leq{t}}\frac{\pi\left(A\_{\tau}\mid{S\_{\tau}}\right)}{\beta\left(A\_{\tau}\mid{S\_{\tau}}\right)}\delta\_{t}$

This product term can lead to high variance, so Retrace modifies $\Delta{Q}$ to have importance weights truncated by no more than a constant $c$:

$\Delta{Q}^{\text{imp}}\left(S\_{t}, A\_{t}\right) = \gamma^{t}\prod\_{1\leq{\tau}\leq{t}}\min\left(c, \frac{\pi\left(A\_{\tau}\mid{S\_{\tau}}\right)}{\beta\left(A\_{\tau}\mid{S\_{\tau}}\right)}\right)\delta\_{t}$