Stanford CS229 Machine Learning | Spring 2026 | Lecture 11: Diffusion Models
Summary
The lecture introduces Diffusion Models as a predominant class of generative models used for tasks like image, video, and action generation. The core mechanism involves two processes: the forward process (gradually adding Gaussian noise to an original clean image $X_0$ over time steps $T$) and the reverse process (learning to iteratively denoise the noisy data back to a clean state). Training these models is achieved by applying the Evidence Lower Bound (ELBO) framework, which minimizes the Kullback-Leibler (KL) divergence between the true conditional distribution and the parameterized model's prediction.
Key takeaways
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Diffusion Models Overview
These models are generative frameworks that can be applied to images, videos, and robotics actions. They are noted as a superior approach compared to GANs and Variational Autoencoders (VAEs) for image generation.
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Forward vs. Reverse Process
4:05
The forward process is a fixed, known noising process that transforms $X_0$ to noise $X_T$. The reverse process is the learned component, parameterized by $ heta$, which predicts $p_ heta(X_{t-1}|X_t)$—the step-by-step denoising path.
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Training Objective (ELBO)
6:25
The goal is to maximize the marginal likelihood $ ext{log } p_ heta(X_0)$ by applying the Evidence Lower Bound (ELBO). This involves minimizing the KL divergence between the true posterior $q(X_{t-1}|X_t, X_0)$ and the parameterized model's distribution.
Technical details
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Forward Diffusion Process (Noising)
285s
The process is defined by adding Gaussian noise iteratively. The transition from $X_{t-1}$ to $X_t$ can be modeled as: $X_t = rac{1}{ ext{sqrt}(ar{eta}_t)} X_0 + ext{sqrt}(1 - ar{eta}_t) oldsymbol{ ilde{ u}}$, where $oldsymbol{ ilde{ u}}$ is standard normal Gaussian noise. The overall process maintains the scale and co-variance of $X_0$ (assuming normalization).
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Reverse Diffusion Process (Denoising)
340s
The reverse process is modeled as a Markov chain, predicting the distribution $p_ heta(X_{t-1}|X_t)$. This model is parameterized by $ heta$ and assumes a Gaussian form with mean $oldsymbol{ ilde{oldsymbol{ u}}}(X_t, t)$ and variance $ ext{sigma}_t^2$.
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Loss Function Derivation
425s
The training objective requires decomposing the KL divergence using a chain rule for KL. The final loss function minimizes the difference between two Gaussian distributions: $q(X_{t-1}|X_t, X_0)$ and $p_ heta(X_{t-1}|X_t)$. When assuming equal co-variance, the loss simplifies to minimizing a quadratic form based on the mean difference.
Mentioned resources
Channel & topics
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