Last updated: 2020-10-27
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Knit directory: misc/
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Assume \(p\) is true distribution and \(q\) is the approximated one. The forward KL is \(KL(p||q)\) and reverse KL is \(KL(q||p)\).
What’s the difference between them? The forward KL is a sum of \(log\frac{p}{q}\), weighted by \(p\). We want to choose \(q\) to minimize \(KL(p||q)\), so \(q\) will try to cover everywhere \(p>0\), otherwise \(KL(p||q)\) will be large. On the other hand, the reverse KL is weighted by \(q\), so whenever \(q>0\), \(q\) and \(p\) should be close.
Forward KL is called zero avoiding while reverse KL is called zero forcing. A classical example is from GAN tutorial page 24 figure 14, by Ian.
Forward KL is mean seeking, and reverse KL is mode-seeking
Forward KL:
\[\begin{equation} \begin{split} argmin_q KL(p||q) &= argmin_q E_p(\log p) - E_p(\log q) \\&= argmax_q E_p(\log q) \end{split} \end{equation}\]
Reverse KL:
\[\begin{equation} \begin{split} argmin_q KL(q||p) &= argmin_q E_q(\log q) - E_q(\log p) \\&= argmax_q E_q(\log p) + H(q) \end{split} \end{equation}\]
Why variational inference uses reverse KL?
Because forward KL is intractable. https://ermongroup.github.io/cs228-notes/inference/variational/