Concentration of Non-Isotropic Random Tensors with Applications to Learning and Empirical Risk Minimization

Mathieu Even , Laurent Massoulie

[Proceedings link] [PDF]

Session: Generalization and PAC-Learning 2 (A)

Session Chair: Steve Hanneke

Poster: Poster Session 4

Abstract: Dimension is an inherent bottleneck to some modern learning tasks, where optimization methods suffer from the size of the data. In this paper, we study non-isotropic distributions of data and develop tools that aim at reducing these dimensional costs by a dependency on an effective dimension rather than the ambient one. Based on non-asymptotic estimates of the metric entropy of ellipsoids -that prove to generalize to infinite dimensions- and on a chaining argument, our uniform concentration bounds involve an effective dimension instead of the global dimension, improving over existing results. We show the importance of taking advantage of non-isotropic properties in learning problems with the following applications: i) we improve state-of-the-art results in statistical preconditioning for communication-efficient distributed optimization, ii) we introduce a non-isotropic randomized smoothing for non-smooth optimization. Both applications cover a class of functions that encompasses empirical risk minization (ERM) for linear models.

Summary presentation

Full presentation