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This lecture explores how eligibility traces emerge in policy gradient algorithms when optimizing return in a multi-step environment. The instructor sketches important mathematical steps, including the use of shadow variables and the update rule for parameters with eligibility trace. The concept of maximizing discounted return with policy gradient naturally leads to eligibility traces, making it easy to implement. The lecture also covers Actor-Critic with Eligibility Traces, emphasizing the estimation of state-value functions and policy parameterization. By updating eligibility traces while moving, the learning process becomes rapid and efficient.