Selective Underfitting in Diffusion Models

TL;DR

Diffusion models learn a score via denoising score matching but do not uniformly approximate the empirical score across input space. The authors introduce selective underfitting, where supervision is confined to shells around training data and inference relies on extrapolation beyond these regions, shaping generalization and sample quality. They show that increasing supervision region can undermine generalization by reducing extrapolation freedom, and they propose a decomposed framework to assess generative performance, including a Perception-Aligned Training (PAT) principle that unifies several successful training strategies. The work provides a lens to understand how diffusion models generalize and generate novel samples, with practical guidance for crafting more efficient training recipes through extrapolation-aware design.

Abstract

Diffusion models have emerged as the principal paradigm for generative modeling across various domains. During training, they learn the score function, which in turn is used to generate samples at inference. They raise a basic yet unsolved question: which score do they actually learn? In principle, a diffusion model that matches the empirical score in the entire data space would simply reproduce the training data, failing to generate novel samples. Recent work addresses this question by arguing that diffusion models underfit the empirical score due to training-time inductive biases. In this work, we refine this perspective, introducing the notion of selective underfitting: instead of underfitting the score everywhere, better diffusion models more accurately approximate the score in certain regions of input space, while underfitting it in others. We characterize these regions and design empirical interventions to validate our perspective. Our results establish that selective underfitting is essential for understanding diffusion models, yielding new, testable insights into their generalization and generative performance.

Figures (11)

• Figure 1: Selective underfitting. (a) Training distribution concentrates in a small region of data space, where the model learns the empirical score function ${\mathbf{s}}_\star$. At inference, sampling trajectories go beyond this region, where predictions are not directly supervised and must be extrapolated. (b) The plotted distances reflect how far sampling trajectories are from the supervision region.
• Figure 2: Difference between conditional and unconditional scores, measured separately during training and inference. The gap is significant at inference but nearly zero during training.
• Figure 3: Quantitative verification of selective underfitting.$\|{\mathbf{s}}_\theta - {\mathbf{s}}_\star\|^2$ exhibits contrastive scaling behavior in supervision region vs. extrapolation region.
• Figure 4: Qualitative example of selective underfitting.Left: Sampling ($t \to 0$) from a noisy image $\textcolor{xblue}{{\mathbf{z}}_t}$ in the supervision region maps back to the original training image ${\mathbf{x}}^{(i)}$ for most time steps ($t < 0.8$). Right: The overlap coefficient $C(t)$ and memorization ratio. The memorization ratio exhibits the same trend quantitatively.
• Figure 5: Experimental verification of freedom of extrapolation on ImageNet. Deliberately limiting the freedom to extrapolate transforms the diffusion model from generalization to memorization.

Theorems & Definitions (5)

• Example 3.1: CFG Paradox
• Proposition 3.2: Supervision region
• Proposition 3.1: Supervision region
• proof
• proof : Equivalence Proof