Replacing Eligibility Trace
Replacing Eligibility Trace
In a **Replacing Eligibility Trace**, each time the state is revisited, the trace is reset to $1$ regardless of the presence of a prior trace.. For the memory vector $\textbf{e}\_{t} \in \mathbb{R}^{b} \geq \textbf{0}$:

$$\mathbf{e\_{0}} = \textbf{0}$$

$$\textbf{e}\_{t} = \gamma\lambda{e}\_{t-1}\left(s\right) \text{ if } s \neq s\_{t}$$

$$\textbf{e}\_{t} = 1 \text{ if } s = s\_{t}$$

They can be seen as crude approximation to dutch traces, which have largely superseded them as they perform better than replacing traces and have a clearer theoretical basis. Accumulating traces remain of interest for nonlinear function approximations where dutch traces are not available.

Source: Sutton and Barto, Reinforcement Learning, 2nd Edition