What does guidance do? A fine-grained analysis in a simple setting

TL;DR

This paper rigorously analyzes the effects of guidance in diffusion models, revealing that guidance does not produce the intended distribution and that large guidance distorts samples, with implications for practical use.

Contribution

It provides a detailed theoretical characterization of guidance dynamics in simple settings, clarifying misconceptions and offering practical insights.

Findings

Guidance causes samples to concentrate near the support boundary as guidance increases.

Large guidance levels lead to samples moving away from the true data support.

Theoretical results are validated through experiments on synthetic data.

Abstract

The use of guidance in diffusion models was originally motivated by the premise that the guidance-modified score is that of the data distribution tilted by a conditional likelihood raised to some power. In this work we clarify this misconception by rigorously proving that guidance fails to sample from the intended tilted distribution. Our main result is to give a fine-grained characterization of the dynamics of guidance in two cases, (1) mixtures of compactly supported distributions and (2) mixtures of Gaussians, which reflect salient properties of guidance that manifest on real-world data. In both cases, we prove that as the guidance parameter increases, the guided model samples more heavily from the boundary of the support of the conditional distribution. We also prove that for any nonzero level of score estimation error, sufficiently large guidance will result in sampling away from the support, theoretically justifying the empirical finding that large guidance results in distorted generations. In addition to verifying these results empirically in synthetic settings, we also show how our theoretical insights can offer useful prescriptions for practical deployment.