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

Mathieu Even, Laurent Massoulié

TL;DR

The paper tackles the dimensionality bottleneck in learning with non-isotropic data by introducing an effective dimension $d_{\rm eff}(r)$ and proving non-asymptotic, ellipsoid-based metric-entropybased concentration bounds. Through a chaining framework, it derives uniform bounds for symmetric random tensors of rank 1 and connects these to ERM, Hessian concentration, and randomized smoothing, all while exploiting non-isotropic structure. The resulting results yield improved statistical preconditioning and smoothing performance when data exhibit low effective dimension, with infinite-dimensional extensions via spectral dimension. Practically, these insights enable more efficient optimization and learning in high-dimensional, anisotropic settings. The work provides a coherent pathway from rigorous probabilistic bounds to concrete algorithmic improvements across preconditioning and smoothing in ERM-like problems.

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.

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

TL;DR

The paper tackles the dimensionality bottleneck in learning with non-isotropic data by introducing an effective dimension and proving non-asymptotic, ellipsoid-based metric-entropybased concentration bounds. Through a chaining framework, it derives uniform bounds for symmetric random tensors of rank 1 and connects these to ERM, Hessian concentration, and randomized smoothing, all while exploiting non-isotropic structure. The resulting results yield improved statistical preconditioning and smoothing performance when data exhibit low effective dimension, with infinite-dimensional extensions via spectral dimension. Practically, these insights enable more efficient optimization and learning in high-dimensional, anisotropic settings. The work provides a coherent pathway from rigorous probabilistic bounds to concrete algorithmic improvements across preconditioning and smoothing in ERM-like problems.

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.

Paper Structure

This paper contains 49 sections, 21 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Theoretical Contributions
  3. Applications in Learning Problems and ERM
  4. Operator Norm Of Tensors
  5. Concentration of Hessians and Statistical Preconditioning
  6. Randomized Smoothing
  7. Organization of the paper
  8. Main Theoretical Results
  9. Concentration Bound With Centering
  10. Concentration Bound Without Centering
  11. Concentration of Non-Isotropic Random Tensors
  12. Results on Covering of Balls with Ellipsoids and Metric Entropy
  13. Metric Entropy of an Ellipsoid
  14. Coverings of the Unit Ball With Ellipsoids
  15. Ellipsoids in Infinite Dimension
  16. ...and 34 more sections

Key Result

Theorem 1

Let $r\ge2$ and $d,n\ge1$ integers. Let $\Sigma\in \mathbb{R}^{d\times d}$ a positive-definite matrix and $a,a_1,...,a_n$i.i.d.$\Sigma-$subgaussian random variables. Let $d_{ \rm eff}(s),s\in\mathbb{N}^*$ be defined as in eq:deff_r. Let $f_1,...,f_r$ be 1-Lipshitz continuous functions on $\mathbb{R} Let $B=B_1...B_k$. Define the following random variable: Then, for any $\lambda>0$ and for some un

Theorems & Definitions (39)

  • Definition 1: $\Sigma$-Subgaussian Random Vector
  • Definition 2: Effective Dimension $d_{ \rm eff}(r)$
  • Theorem 1: Concentration With Centering
  • Theorem 2: Concentration Without Centering
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 3: Tensor
  • Definition 4: Symmetric Random Tensor of Rank 1
  • Theorem 3: Non-Isotropic Concentration Bound on Random Tensors
  • ...and 29 more