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fix ch6

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qiwang067
2022-02-24 09:12:11 +08:00
parent e0516ba953
commit 6a368c222e
1 changed files with 2 additions and 1 deletions
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docs/chapter6/chapter6.md
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@@ -232,7 +232,8 @@ $$
$$ $$
\begin{aligned} \begin{aligned}
V^{\pi}(s) &\le Q^{\pi}(s,\pi'(s)) \\ V^{\pi}(s) &\le Q^{\pi}(s,\pi'(s)) \\
&= E\left[r_{t}+Q^{\pi}\left(s_{t+1}, \pi^{\prime}\left(s_{t+1}\right)\right) | s_{t}=s, a_{t}=\pi^{\prime}\left(s_{t}\right)\right] \\ &=E\left[r_{t}+V^{\pi}\left(s_{t+1}\right) | s_{t}=s, a_{t}=\pi^{\prime}\left(s_{t}\right)\right]\\
&\le E\left[r_{t}+Q^{\pi}\left(s_{t+1}, \pi^{\prime}\left(s_{t+1}\right)\right) | s_{t}=s, a_{t}=\pi^{\prime}\left(s_{t}\right)\right] \\
&=E\left[r_{t}+r_{t+1}+V^{\pi}\left(s_{t+2}\right) |s_{t}=s, a_{t}=\pi^{\prime}\left(s_{t}\right)\right] \\ &=E\left[r_{t}+r_{t+1}+V^{\pi}\left(s_{t+2}\right) |s_{t}=s, a_{t}=\pi^{\prime}\left(s_{t}\right)\right] \\
& \le E\left[r_{t}+r_{t+1}+Q^{\pi}\left(s_{t+2},\pi'(s_{t+2}\right) | s_{t}=s, a_{t}=\pi^{\prime}\left(s_{t}\right)\right] \\ & \le E\left[r_{t}+r_{t+1}+Q^{\pi}\left(s_{t+2},\pi'(s_{t+2}\right) | s_{t}=s, a_{t}=\pi^{\prime}\left(s_{t}\right)\right] \\
& = E\left[r_{t}+r_{t+1}+r_{t+2}+V^{\pi}\left(s_{t+3}\right) |s_{t}=s, a_{t}=\pi^{\prime}\left(s_{t}\right)\right] \\ & = E\left[r_{t}+r_{t+1}+r_{t+2}+V^{\pi}\left(s_{t+3}\right) |s_{t}=s, a_{t}=\pi^{\prime}\left(s_{t}\right)\right] \\