In this post, I’ll describe a simple method on how to condition diffusion models called classifier-free guidance. I won’t go into the SDE side of things because I don’t understand it yet.

Score function

The score function for an arbitrary step along our markov chain \(\textbf{x}_t\) is defined as \(s_\theta(\textbf{x}_t, t) = \nabla_{\textbf{x}_t} \log q(\textbf{x}_t)\). Intuitively, this quantity tells us how to change the noisy \(\textbf{x}_t\) to make it more likely under the true data distribution.

In diffusion models, we have a forward diffusion distribution \(q(\textbf{x}_t \mid \textbf{x}_0) = \mathcal{N}(\sqrt{\bar{\alpha_t}} \textbf{x}_0, (1- \bar{\alpha_t}) \textbf{I})\). The gradient of the \(\log\) pdf of a gaussian \(\mathcal{N}(\textbf{x}; \mu, \sigma^2)\) can be computed as:

\[\begin{align*} & \nabla_{\textbf{x}} p(\textbf{x}) = \nabla_\textbf{x} \left ( - \frac{1}{2 \sigma^2} (\textbf{x} - \mathbf{\mu})^2 \right) = -\frac{\textbf{x} - \mathbf{\mu}}{\sigma^2}\\ &= -\frac{\mathbf{\epsilon}}{\sigma} \hspace{2 cm} (\textbf{x} = \mathbf{\mu} + \sigma \odot \mathbf{\epsilon}, \mathbf{\epsilon} \sim \mathcal{N}(\textbf{0, I})) \end{align*}\]

Therefore, we can express the score function of a sample \(\textbf{x}_t\) as:

\[\begin{align*} &\mathbf{s}_\theta(\mathbf{x}_t, t) \approx \nabla_{\mathbf{x}_t} \log q(\mathbf{x}_t) = - \frac{\boldsymbol{\epsilon}_\theta(\mathbf{x}_t, t)}{\sqrt{1 - \bar{\alpha}_t}} \end{align*}\]

I simply plugged the variance of \(q(\textbf{x}_t \mid \textbf{x}_0)\) and the diffusion model we train \(\mathbf{\epsilon}_{\theta} (\textbf{x}_t, t)\) is supposed to match the noise \(\mathbf{\epsilon}\), so I substitute that in as well.

Therefore, I now have an equivalence of the score function in terms of our diffusion model’s U-Net. Learning the diffusion model as stated in DDPM is the same thing as learning the score function.

Classifier Guidance

Given our equivalence of the score function and noise prediction network, we can intuitively understand conditioning.

If we have some auxillary input \(y\) that we want to condition on, we the need to model the score function \(\nabla_{\textbf{x}_t} \log q(\textbf{x}_t \mid \textbf{y})\). Hence, using bayes rule we can write this as:

\[\begin{align*} & q(\textbf{x}_t \mid \textbf{y}) = \frac{q(\textbf{y} \mid \textbf{x}_t) q(\textbf{x}_t )}{q(\textbf{y})} \\ & \implies \log q(\textbf{x}_t \mid \textbf{y}) = \log q(\textbf{y} \mid \textbf{x}_t) + \log q(\textbf{x}_t ) - \log q(\textbf{y}) \\ & \implies \nabla_{\textbf{x}_t} \log q(\textbf{x}_t \mid \textbf{y}) = \nabla_{\textbf{x}_t} \log q(\textbf{y} \mid \textbf{x}_t) + \nabla_{\textbf{x}_t} \log q(\textbf{x}_t ) \end{align*}\]

It’s evident here that \(\nabla_{\textbf{x}_t} \log q(\textbf{y} \mid \textbf{x}_t)\) can be computed using a differentiable approximator, such as a softmax classifier (in the case of labels). We can add a hyperparameter \(s\) (called “guidance”), which controls how much influence this classifier has on our final prediction.

\[\nabla_{\textbf{x}_t} \log q(\textbf{x}_t \mid \textbf{y}) = \nabla_{\textbf{x}_t} \log q(\textbf{x}_t ) + s \cdot \nabla_{\textbf{x}_t} \log q(\textbf{y} \mid \textbf{x}_t)\]

The issue is, our \(\textbf{x}_t\) can be arbitrarily noisy and our classifier will not be able to be accurate at high levels of noise.

Classifier-Free Guidance

Hence, we seek to eliminate our dependence on a classifier, so we use bayes rule once again in the other direction:

\[\begin{align*} & q(\textbf{y} \mid \textbf{x}_t) = \frac{q(\textbf{x}_t \mid \textbf{y}) q(\textbf{y})}{q(\textbf{x}_t)} \\ & \implies \log q(\textbf{y} \mid \textbf{x}_t) = \log q(\textbf{x}_t \mid \textbf{y}) + \log q(\textbf{y}) - \log q(\textbf{x}_t) \\ & \implies \nabla_{\textbf{x}_t} \log q(\textbf{y} \mid \textbf{x}_t) = \nabla_{\textbf{x}_t} \log q(\textbf{x}_t \mid \textbf{y}) - \nabla_{\textbf{x}_t} \log q(\textbf{x}_t) \\ \end{align*}\]

Plugging this back into our equation from Classifier Guidance:

\[\begin{align*} \nabla_{\textbf{x}_t} \log q(\textbf{x}_t \mid \textbf{y}) &= \nabla_{\textbf{x}_t} \log q(\textbf{x}_t ) + s \cdot (\nabla_{\textbf{x}_t} \log q(\textbf{x}_t \mid \textbf{y}) - \nabla_{\textbf{x}_t} \log q(\textbf{x}_t)) \\ &= (1-s) \cdot \nabla_{\textbf{x}_t} \log q(\textbf{x}_t ) + s \cdot \nabla_{\textbf{x}_t} \log q(\textbf{x}_t \mid \textbf{y}) \end{align*}\]

Ultimately, using our discussion of score functions earlier, we can equate this to learning a diffusion model as:

\[\begin{align*} \hat{\epsilon}(\textbf{x}_t, \textbf{y}, t) &= -\frac{1}{\sqrt{1 - \bar{\alpha}_t}} \left ( (1-s) \cdot \boldsymbol{\epsilon}_\theta(\mathbf{x}_t, t) + s \cdot \boldsymbol{\epsilon}_\theta(\mathbf{x}_t, \mathbf{y}, t) \right) \end{align*}\]

We could just model \(\boldsymbol{\epsilon}_\theta(\mathbf{x}_t, \mathbf{y}, t)\) directly, however this formulation allows us to have more fine grained control over how much our learned conditional distribution affects the final generated sample.