Importance sampling is a technique in reinforcement learning that allows the agent to learn from experience generated by a policy that is different from the policy being learned. This is important in off-policy learning, where the agent must learn from experience generated by a different behavior policy than the target policy. However, it can be challenging because the distribution of the states/actions visited by the two policies can be very different, which creates a bias in the estimation of the return.
Importance sampling is a way to correct for this bias by reweighting the returns obtained under the behavior policy to make them compatible with the target policy. The basic idea is to estimate the ratio of the probabilities of the behavior and target policies in each step of the trajectory, and then multiply the return obtained under the behavior policy by the product of these ratios. This reweighting ensures that the returns obtained under the behavior policy are adjusted to match the distribution of the target policy.
For example, consider a case where the agent is trying to learn a policy that maximizes the reward in a grid-world environment, and the behavior policy is selecting actions randomly. The agent generates a set of trajectories using this policy, and then wishes to use these trajectories to update its target policy. The issue is that the behavior policy is very different from the target policy, which may result in the estimates of the return being biased. To address this, the agent can use importance sampling to reweight the returns obtained under the behavior policy, so that they are normalized to the target policy.
In summary, importance sampling is a powerful technique that helps to bridge the gap between off-policy and on-policy learning in reinforcement learning. It allows an agent to use experience generated by different policies, which can enhance the learning process, and it corrects for the bias that can result from using different policies for experience generation and policy evaluation.