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In chapter 10 of Sutton and Barto's book (2nd edition) is given the equation for TD(0) error with average reward (equation 10.10):

$$\delta_t = R_{t+1} - \bar{R} + \hat{v}(S_{t+1}, \mathbf{w}) - \hat{v}(S_{t}, \mathbf{w})$$

What is the intuition behind this equation? And how exactly is it derived?

Also, in chapter 13, section 6, is given the Actor-Critic algorithm, which uses the TD error. How can you use 1 error to update 3 distinct things - like the average reward, value function estimator (critic), and the policy function estimator (actor)?

Average Reward update rule: $\bar{R} \leftarrow \bar{R} + \alpha^{\bar{R}}\delta$

Critic weight update rule: $\mathbf{w} \leftarrow \mathbf{w} + \alpha^{\mathbf{w}}\delta\nabla \hat{v}(s,\mathbf{w})$

Actor weight update rule: $\mathbf{\theta} \leftarrow \mathbf{\theta} + \alpha^{\mathbf{\theta}}\delta\nabla ln \pi(A|S,\mathbf{\theta})$

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This is simply from definition of return in average reward setting (look at equation $10.9$). The "standard" TD error is defined as \begin{equation} TD_{\text{error}} = R_{t+1} + V(S_{t+1}) - V(S_t) \end{equation} In average reward setting, average reward $r(\pi)$ is subtracted from reward at $t$, $R_t$, so TD error in this case is \begin{equation} TD_{\text{error}} = R_{t+1} - \bar R_{t+1} + V(S_{t+1}) - V(S_t) \end{equation} where $\bar R_{t+1}$ is estimate of $r(\pi)$.

You can use $\delta_t$ in all 3 updates because neither of these updates depend on each other. For example if you update $\mathbf w$ you don't use that then to update $\mathbf \theta$, or if you update $\bar R$ you don't use updated version to update $\mathbf w$ or $\mathbf \theta$ so you're not introducing additional bias. In each separate update you also don't have $\delta_t$ present multiple times so that you require multiple sampling per timestep to get the unbiased update.

Additionally this is semi-gradient algorithm, it uses bootstrapped estimate $V_{t+1}$ but it doesn't calculate full derivative with respect to it aswell, only with respect to $V_t$ so the algorithm is biased by default, but works well enough in practice for linear case.

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