Classifier Free Guidance

By Bayes’ Rule: $p(\mathbf{z | c}) = \frac{p(\mathbf{c | z})p(\mathbf{z})}{p(\mathbf{c)}}$ $\log p(\mathbf{z | c}) = \log p(\mathbf{c | z}) + \log p(\mathbf{z}) - \log p(\mathbf{c)}$ Taking the gradient with respect to $\mathbf{z}$, the third term is constant in $\mathbf{z}$ and goes to zero. $\nabla_z \log p(\mathbf{z | c}) = \nabla_z \log p(\mathbf{c | z}) + \nabla_z \log p(\mathbf{z})$

\[\nabla_z \log p(\mathbf{z | c}) = \color{red} \gamma \color{black} \nabla_z \log p(\mathbf{c | z}) + \nabla_z \log p(\mathbf{z})\]

Since we don’t have a classifier, computing

$\nabla_z \log p(\mathbf{c | z})$ Is not possible. But, we have $\nabla_z \log p(\mathbf{c | z}) = \nabla_z \log p(\mathbf{z | c}) - \nabla_z \log p(\mathbf{z})$ Substituting in: $\nabla_z \log p(\mathbf{z | c}) = \color{red} \gamma \color{black} \nabla_z \log p(\mathbf{c | z}) + \nabla_z \log p(\mathbf{z})$ $\nabla_z \log p(\mathbf{z | c}) = \nabla_z \log p(\mathbf{z}) + \color{red} \gamma \color{black} \left(\nabla_z \log p(\mathbf{z | c}) - \nabla_z \log p(\mathbf{z})\right)$ The above is \textbf{one} way people write CFG. We can also rewrite the formula this way: $\nabla_z \log p(\mathbf{z | c}) = (1 - \gamma) \nabla_z \log p(\mathbf{z}) + \gamma \color{black} \nabla_z \log p(\mathbf{z | c})$

Last Reviewed: 4/30/25