Discrepancy, Coresets, and Sketches in Machine Learning
Proceedings of the Thirty-Second Conference on Learning Theory, PMLR 99:1975-1993, 2019.
This paper defines the notion of class discrepancy for families of functions. It shows that low discrepancy classes admit small offline and streaming coresets. We provide general techniques for bounding the class discrepancy of machine learning problems. As corollaries of the general technique we bound the discrepancy of logistic regression, sigmoid activation loss, matrix covariance, kernel density and any analytic function of the dot product or the squared distance. Our result resolves a long-standing open problem regarding the coreset complexity of Gaussian kernel density estimation. We provide two more related but independent results. First, an exponential improvement of the widely used merge-and-reduce trick which gives improved streaming sketches for any low discrepancy problem. Second, an extremely simple deterministic algorithm for finding low discrepancy sequences (and therefore coresets) for any positive semi-definite kernel. This paper establishes some explicit connections between class discrepancy, coreset complexity, learnability, and streaming algorithms.