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 $c$:

$$ \Delta{Q}^{\text{imp}}\left(S_{t}, A_{t}\right) = \gamma^{t}\prod_{1\leq{\tau}\leq{t}}\min\left(c, \frac{\pi\left(A_{\tau}\mid{S_{\tau}}\right)}{\beta\left(A_{\tau}\mid{S_{\tau}}\right)}\right)\delta_{t} $$