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Dimension is an inherent bottleneck to some modern learning tasks, whereoptimization methods suffer from the size of the data. In this paper, we studynon-isotropic distributions of data and develop tools that aim at reducingthese dimensional costs by a dependency on an effective dimension rather thanthe ambient one. Based on non-asymptotic estimates of the metric entropy ofellipsoids -- that prove to generalize to infinite dimensions -- and on achaining argument, our uniform concentration bounds involve an effectivedimension instead of the global dimension, improving over existing results. Weshow the importance of taking advantage of non-isotropic properties in learningproblems with the following applications: i) we improve state-of-the-artresults in statistical preconditioning for communication-efficient distributedoptimization, ii) we introduce a non-isotropic randomized smoothing fornon-smooth optimization. Both applications cover a class of functions thatencompasses empirical risk minization (ERM) for linear models.

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