Retrace is a Q-value estimation algorithm used in reinforcement learning. It works best when there are two policies, a target policy and a behavior policy, denoted as $\pi$ and $\beta$, respectively. The algorithm uses off-policy rollout for TD learning, meaning that it uses data generated by following one policy while trying to learn about another policy.
Importance Sampling
In Retrace, importance sampling is used for the update of Q-values. Importance sampling is a technique used in statistics and machine learning to estimate the expectation of a function with respect to one distribution, while only having samples from another distribution. In this case, the function we are interested in is the Q-value, which is used to estimate the expected total reward that is received when following a certain policy in a certain state.
The importance sampling weight for Retrace is:
$$ \prod\_{1\leq{\tau}\leq{t}}\frac{\pi\left(A\_{\tau}\mid{S\_{\tau}}\right)}{\beta\left(A\_{\tau}\mid{S\_{\tau}}\right)} $$
The product term above can lead to high variance, which can make the algorithm unstable and hard to optimize. To solve this issue, Retrace modifies the importance sampling weight by truncating it by a constant $c$. The modified importance sampling weight is:
$$ \prod\_{1\leq{\tau}\leq{t}}\min\left(c, \frac{\pi\left(A\_{\tau}\mid{S\_{\tau}}\right)}{\beta\left(A\_{\tau}\mid{S\_{\tau}}\right)}\right) $$
By truncating the importance sampling weight, Retrace can reduce the variance of the Q-value estimate and make the algorithm more stable.
Guaranteed Convergence
One of the advantages of Retrace is that it has guaranteed convergence for a target and behavior policy $\left(\pi, \beta\right)$. This means that as the algorithm runs for more iterations, the Q-value estimate will converge to the true Q-value under the given policies. This is an important property for reinforcement learning algorithms because it ensures that the algorithm will eventually find the optimal policy.
Retrace is an off-policy Q-value estimation algorithm used in reinforcement learning. It uses importance sampling for the update of Q-values, and the importance sampling weight is modified to reduce variance. The algorithm has guaranteed convergence under a given pair of policies, which ensures that the algorithm will eventually find the optimal policy.
