Sample Complexity Bounds for Estimating Probability Divergences under Invariances
This work addresses sample efficiency in machine learning for data with symmetries, such as graphs and images, providing new theoretical bounds that are foundational for invariant models.
The paper tackles the problem of estimating probability divergences for group-invariant distributions, showing that invariances reduce sample complexity by a multiplicative factor related to group size or volume and improve convergence rates for groups of positive dimension.
Group-invariant probability distributions appear in many data-generative models in machine learning, such as graphs, point clouds, and images. In practice, one often needs to estimate divergences between such distributions. In this work, we study how the inherent invariances, with respect to any smooth action of a Lie group on a manifold, improve sample complexity when estimating the 1-Wasserstein distance, the Sobolev Integral Probability Metrics (Sobolev IPMs), the Maximum Mean Discrepancy (MMD), and also the complexity of the density estimation problem (in the $L^2$ and $L^\infty$ distance). Our results indicate a two-fold gain: (1) reducing the sample complexity by a multiplicative factor corresponding to the group size (for finite groups) or the normalized volume of the quotient space (for groups of positive dimension); (2) improving the exponent in the convergence rate (for groups of positive dimension). These results are completely new for groups of positive dimension and extend recent bounds for finite group actions.