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I am reading "Reinforcement Learning: An Introduction (2nd edition)" authored by Sutton and Barto. In Section 9, On-policy prediction with approximation, it first gives the mean squared value error objective function in (9.1):

$\bar{VE}(\boldsymbol{w}) = \sum_{s \in S} \mu(s)[v_{\pi}(s) - \hat{v}(s,\boldsymbol{w})]^2$. (9.1)

$\boldsymbol{w}$ is a vector of the parameterized function $\hat{v}(s,\boldsymbol{w})$ that approximates the value function $v_{\pi}(s)$. $\mu(s)$ is the fraction of time spent in $s$, which measures the "importance" of state $s$ in $\bar{VE}(\boldsymbol{w})$.

In (9.4), it states an update rule of $\boldsymbol{w}$ by gradient descent: $\boldsymbol{w}_{t+1} = \boldsymbol{w} -\frac{1}{2}\alpha \nabla[v_{\pi}(S_t) - \hat{v}(S_t,\boldsymbol{w})]^2$. (9.4)

I have two questions regarding (9.4).

  1. Why $\mu(s)$ is not in (9.4)?
  2. Why is it the "minus" instead of "+" in (9.4)? In other words, why is it $\boldsymbol{w} -\frac{1}{2}\alpha \nabla[v_{\pi}(S_t) - \hat{v}(S_t,\boldsymbol{w})]^2$ instead of $\boldsymbol{w} +\frac{1}{2}\alpha \nabla[v_{\pi}(S_t) - \hat{v}(S_t,\boldsymbol{w})]^2$?
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  • $\begingroup$ I think your equation number 9.4 is not correct? There is indeed + sign in Sutton's book and also $\nabla{V(s,w)}$ at the end. $\endgroup$ – Swakshar Deb Oct 7 '20 at 20:42
  • $\begingroup$ They are giving same importance to all the states, so there is no $\mu(s)$ $\endgroup$ – Swakshar Deb Oct 7 '20 at 20:45
  • $\begingroup$ @SwaksharDeb the + sign is in (9.5) because it expands the gradient equation. But why it is a - sign in the first place in (9.4)? $\endgroup$ – JYY Oct 7 '20 at 20:50
  • $\begingroup$ @SwaksharDeb So in (9.4) it "assumes" all the states are equally important, otherwise $\mu(s)$ will appear in the equation? $\endgroup$ – JYY Oct 7 '20 at 20:51
  • $\begingroup$ This is a stochastic gradient descent update. I think your problem is related to gradient descent. $\endgroup$ – Swakshar Deb Oct 7 '20 at 20:52
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  1. $\mu(s)$ is not in equation (9.4) because we are assuming that the examples by which we update our parameter $w$, i.e. the frequency of which we will observe the states during online training, is the same. That is, it is a constant with respect to $w$ and since we are differentiating it can be somewhat disregarded as a constant of proportionality -- it essentially can be 'absorbed' by $\alpha$.

  2. The minus is there because we are performing gradient descent. For more information on this, see e.g. the wikipedia page

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