On Statistical Rates of Conditional Diffusion Transformers: Approximation, Estimation and Minimax Optimality

TL;DR

This paper analyzes the statistical approximation and estimation rates of conditional diffusion transformers, establishing their minimax optimality under various data assumptions and providing insights for improving DiT models.

Contribution

It offers a comprehensive theoretical analysis of conditional DiTs, including approximation bounds, estimation rates, and minimax optimality, under multiple data assumptions.

Findings

Conditional DiTs achieve minimax optimality under certain data assumptions.

Latent DiTs outperform conditional DiTs in approximation and estimation.

Theoretical bounds guide the development of more efficient DiT models.

Abstract

We investigate the approximation and estimation rates of conditional diffusion transformers (DiTs) with classifier-free guidance. We present a comprehensive analysis for ``in-context'' conditional DiTs under four common data assumptions. We show that both conditional DiTs and their latent variants lead to the minimax optimality of unconditional DiTs under identified settings. Specifically, we discretize the input domains into infinitesimal grids and then perform a term-by-term Taylor expansion on the conditional diffusion score function under H\"older smooth data assumption. This enables fine-grained use of transformers' universal approximation through a more detailed piecewise constant approximation and hence obtains tighter bounds. Additionally, we extend our analysis to the latent setting under the linear latent subspace assumption. We not only show that latent conditional DiTs achieve lower bounds than conditional DiTs both in approximation and estimation, but also show the minimax optimality of latent unconditional DiTs. Our findings establish statistical limits for conditional and unconditional DiTs, and offer practical guidance toward developing more efficient and accurate DiT models.