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Merge pull request #503 from huggingface/ThomasSimonini/UpdateUnit4

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Update pg-theorem.mdx
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Thomas Simonini
2024-03-05 10:49:34 +01:00
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parent 0e55db0106 72473f08a8
commit c929ba2e6e
1 changed files with 6 additions and 2 deletions
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units/en/unit4/pg-theorem.mdx
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@@ -27,9 +27,13 @@ We then multiply every term in the sum by \\(\frac{P(\tau;\theta)}{P(\tau;\theta
\\( = \sum_{\tau} \frac{P(\tau;\theta)}{P(\tau;\theta)}\nabla_\theta P(\tau;\theta)R(\tau) \\)
We can simplify further this since \\( \frac{P(\tau;\theta)}{P(\tau;\theta)}\nabla_\theta P(\tau;\theta) = P(\tau;\theta)\frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)} \\)
We can simplify further this since
\\(= \sum_{\tau} P(\tau;\theta) \frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)}R(\tau) \\)
\\( \frac{P(\tau;\theta)}{P(\tau;\theta)}\nabla_\theta P(\tau;\theta) = P(\tau;\theta)\frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)} \\)
\\( P(\tau;\theta)\frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)}= \sum_{\tau} P(\tau;\theta) \frac{\nabla_\theta P(\tau;\theta)}{P(\tau;\theta)}R(\tau) \\)
We can then use the *derivative log trick* (also called *likelihood ratio trick* or *REINFORCE trick*), a simple rule in calculus that implies that \\( \nabla_x log f(x) = \frac{\nabla_x f(x)}{f(x)} \\)