Justification of Scaling in Classifier-Free Guidance in Diffusion Models

Background | Classifier-Free Guidance Derivation To summarize the derivation of Classifier-Free Guidance, looking at this paper (Page 21.), we can write classifier guidance as: $$\nabla_{x}\log p\left(x_{t}\mid y\right) =\nabla_{x}\log\left(\frac{p\left(x_{t}\right)\cdot p\left(y\mid x_{t}\right)}{p\left(y\right)}\right) =\nabla_{x}\log p\left(x_{t}\right)+\nabla_{x}\log p\left(y\mid x_{t}\right)-\nabla_{x}\log p\left(y\right) =\nabla_{x}\log p\left(x_{t}\right)+\nabla_{x}\log p\left(y\mid x_{t}\right)$$ and we can amplify the guidance by adding a factor $\gamma$ : $$\nabla_{x}\log p\left(x_{t}\mid y\right)=\nabla_{x}\log p\left(x_{t}\right)+\gamma\cdot\nabla_{x}\log p\left(y\mid x_{t}\right)$$ Then to derive classifier-free guidance all we do is rewrite the first equation we saw: $$\nabla_{x}\log p\left(y\mid x_{t}\right)=\nabla_{x}\log p\left(x_{t}\mid y\right)-\nabla_{x}\log p\left(x_{t}\right)$$ and substitue it into the 2nd equation we saw to get $$\nabla_{x}\log p\left(x_{t}\mid y\right)=\nabla_{x}\log p\left(x_{t}\right)+\gamma\cdot\left(\nabla_{x}\log p\left(x_{t}\mid y\right)-\nabla_{x}\log p\left(x_{t}\right)\right)$$ and this is classifier-free guidance. We can also rewrite it to be $$\nabla_{x}\log p\left(x_{t}\mid y\right)=\gamma\cdot\nabla_{x}\log p\left(x_{t}\mid y\right)-\left(\gamma-1\right)\cdot\nabla_{x}\log p\left(x_{t}\right)$$ The Question It seems reasonable to me that as we can amplify and play with the guidance factor, we could also add a factor $\be…