Lower bounds for diffusion sample complexity in the tilted-sampling setting

Establish information-theoretic lower bounds on the sample complexity required by denoising diffusion probabilistic models trained on reweighted empirical samples to approximate an exponentially tilted target distribution μ_θ, measured via expected Wasserstein distance. Specifically, determine how the minimal number of base samples N (and its dependence on the tilt parameter θ, data dimension d, and other problem parameters) is bounded below to achieve a prescribed accuracy when using the plug-in self-normalized importance sampler μ_{N,θ} as input to the diffusion model.

Background

The paper studies generating samples from an exponentially tilted distribution ν(x) ∝ exp(θ^T g(x)) μ(x) using a two-step approach: (i) construct a reweighted empirical (plug-in) estimator μ{N,θ} via self-normalized importance weights, and (ii) run a diffusion sampler (DDPM) trained on these reweighted samples. The authors provide minimax justifications for the plug-in estimator and derive expected Wasserstein bounds between μ{N,θ} and the true μ_θ, giving sample-complexity upper bounds for reweighted sampling.

Under Lipschitz assumptions on score estimation error and with bounded training loss, they convert these Wasserstein guarantees into total variation accuracy guarantees for diffusion sampling, thereby obtaining overall upper bounds on the sample complexity needed to generate accurate tilted samples. Despite these upper bounds, no complementary lower bounds are known in this setting, leaving the fundamental limits of sample complexity unresolved.