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.

Causal discovery is essential for understanding relationships among variables of interest in many scientific domains. In this paper, we focus on permutation-based methods for learning causal graphs in Linear Gaussian Acyclic Models (LiGAMs), where the permutation encodes a causal ordering of the variables. Existing methods in this setting do not scale due to their high computational complexity.

Proposition 1. Assume the robust loss function is non-decreasing in [0, 1 ], (0) = 0 and The proof of Proposition 1 is mainly adapted from the proof in Kim & Scott ( 2012). For any given function: R! We first introduce a few notations that are useful for stating this result. B> (2 +) |O| where is the failure probability. By adapting Lemma 1 in Nguyen et al. ( 2022c) to uniform concentration bound, ImageNet We use the full ImageNet dataset that contains 1 .