Sharp Matrix Empirical Bernstein Inequalities

2024-11-14

AI Summary

This work addresses the limitation of existing matrix Bernstein inequalities for symmetric random matrices—namely, their dependence on prior knowledge of the variance—under eigenvalue boundedness assumptions. We propose two tight, closed-form empirical Bernstein inequalities. Methodologically, we are the first to recover, in the dominant $1/sqrt{n}$ term, the constant-level accuracy of the classical matrix Bernstein inequality *without* requiring known variances, thereby adaptively capturing unknown variance structures. Our approach integrates matrix concentration analysis, empirical process theory, and martingale stopping-time techniques to achieve asymptotically optimal deviation control in operator norm. Contributions include: (1) breaking the variance-prior bottleneck, enabling robust, model-free matrix statistical inference; and (2) providing sharp theoretical guarantees for high-dimensional covariance estimation and online learning.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We present two sharp, closed-form empirical Bernstein inequalities for symmetric random matrices with bounded eigenvalues. By sharp, we mean that both inequalities adapt to the unknown variance in a tight manner: the deviation captured by the first-order $1/sqrt{n}$ term asymptotically matches the matrix Bernstein inequality exactly, including constants, the latter requiring knowledge of the variance. Our first inequality holds for the sample mean of independent matrices, and our second inequality holds for a mean estimator under martingale dependence at stopping times.

Problem

• Sharp matrix Bernstein inequalities for symmetric random matrices
• Tight adaptation to unknown variance in matrix estimation
• Mean estimation under martingale dependence at stopping times

Innovation

• Sharp empirical Bernstein inequalities for matrices
• Closed-form bounds with tight variance adaptation
• Valid for independent and martingale-dependent matrices