We prove that conditional diffusion models whose reverse kernels are finite Gaussian mixtures with ReLU-network logits can approximate suitably regular target distributions arbitrarily well in context-averaged conditional KL divergence, up to an irreducible terminal mismatch that typically vanishes with increasing diffusion horizon. A path-space decomposition reduces the output error to this mismatch plus per-step reverse-kernel errors; assuming each reverse kernel factors through a finite-dimensional feature map, each step becomes a static conditional density approximation problem, solved by composing Norets' Gaussian-mixture theory with quantitative ReLU bounds. Under exact terminal matching the resulting neural reverse-kernel class is dense in conditional KL.

Estimating the density of a distribution from samples is a fundamental problem in statistics. In many practical settings, the Wasserstein distance is an appropriate error metric for density estimation. For example, when estimating population densities in a geographic region, a small Wasserstein distance means that the estimate is able to capture roughly where the population mass is. In this work we study differentially private density estimation in the Wasserstein distance. We design and analyze instance-optimal algorithms for this problem that can adapt to easy instances.

We study the problem of how to repeatedly sell to a buyer running a no-regret,mean-based algorithm. Previous work [Braverman et al., 2018] shows that it ispossible to design effective mechanisms in such a setting that extract almost allof the economic surplus, but these mechanisms require the buyer's values each

The regularizer P λi|b|(i) is a sorted `1-norm (denoted asJλ(b) henceforth), which isnon-separabledue to the sorting operation involved in its calculation.