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$E_{\pi}[R_{t+1}|S_t=s,A_t=a] = E[R_{t+1}|S_t=s,A_t=a]$?

Question: Can I write it without the subscript? So $$E_{\pi}[R_{t+1}|S_t=s,A_t=a] = E[R_{t+1}|S_t=s,A_t=a]$$ Yes, your reasoning is sound, there is no need to condition the expectation on the policy, ...
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Does it make sense to provide a DQN with negative rewards for a network with relu and sigmoid activations?

A network with ReLU activation can predict negative values; we put ReLU between the hidden layers but return the output of the final layer without any activation function, or with a linear activation ...
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Why does the average-reward estimator for continuing tasks use the TD error?

Mystery solved thanks to Exercise 10.8 in the book. The reason is that we want the running mean to converge to the actual value of the average reward. With $\bar{R}_{t + 1} = \bar{R}_t + \beta \delta$...
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