Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers

2024-10-17

arXiv.org

AI Summary

This work investigates theoretical performance guarantees for denoising diffusion models under score function mismatch, specifically in the zero-shot conditional sampling setting—where the target conditional distribution differs from the unconditional training distribution. Methodologically, it establishes the first explicit convergence bounds dependent on data dimensionality and conditional structure; quantitatively characterizes the asymptotic sampling bias in terms of accumulated score mismatch; and proposes a bias-optimal linear zero-shot conditional sampler. The theoretical analysis rigorously derives these results using probability measure convergence theory and linear inverse problem modeling, covering common target distributions such as those with bounded support and Gaussian mixtures. Numerical experiments demonstrate that the proposed sampler effectively suppresses distributional bias induced by score mismatch.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationComputer Vision: Diffusion Models for VisionReasoning under Uncertainty: Relational Probabilistic Models

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingUser Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The denoising diffusion model has recently emerged as a powerful generative technique, capable of transforming noise into meaningful data. While theoretical convergence guarantees for diffusion models are well established when the target distribution aligns with the training distribution, practical scenarios often present mismatches. One common case is in the zero-shot conditional diffusion sampling, where the target conditional distribution is different from the (unconditional) training distribution. These score-mismatched diffusion models remain largely unexplored from a theoretical perspective. In this paper, we present the first performance guarantee with explicit dimensional dependencies for general score-mismatched diffusion samplers, focusing on target distributions with finite second moments. We show that score mismatches result in an asymptotic distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions. This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise. Interestingly, the derived convergence upper bound offers useful guidance for designing a novel bias-optimal zero-shot sampler in linear conditional models that minimizes the asymptotic bias. For such bias-optimal samplers, we further establish convergence guarantees with explicit dependencies on dimension and conditioning, applied to several interesting target distributions, including those with bounded support and Gaussian mixtures. Our findings are supported by numerical studies.

Problem

• Analyzes score-mismatched diffusion models' theoretical performance.
• Explores zero-shot conditional sampling with distributional bias.
• Provides convergence guarantees for bias-optimal samplers.

Innovation

• Analyzing score-mismatched diffusion models theoretically
• Deriving convergence guarantees for zero-shot samplers
• Designing bias-optimal samplers for linear models