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Matrix Moment and Concentration Inequalities for Martingales and Ergodic Markov Chains with Applications in Statistical Learning

Yang Peng, Yuchen Xin, Zhihua Zhang

TL;DR

This paper establishes novel Rosenthal-Burkholder inequalities for discrete-time matrix local martingales, Burkholder-Davis-Gundy inequality for continuous matrix local martingales, as well as matrix Rosenthal, Hoeffding, and Bernstein inequalities for ergodic Markov chains.

Abstract

In this paper, we study moment and concentration inequalities for the spectral norm of sums of dependent random matrices. We establish novel Rosenthal-Burkholder inequalities for discrete-time matrix local martingales, Burkholder-Davis-Gundy inequality for continuous matrix local martingales, as well as matrix Rosenthal, Hoeffding, and Bernstein inequalities for ergodic Markov chains. Compared with previous work on matrix concentration inequalities for Markov chains, which assume a non-zero absolute $L^2$-spectral gap or the stronger $ψ$-mixing condition, our results assume geometric ergodicity, a condition commonly used in statistical applications. Furthermore, our results have leading terms that match the Markov chain central limit theorem, rather than relying on suboptimal variance proxies. We also give dimension-free versions of the inequalities, which are independent of the ambient dimension $d$ and relies on the effective rank instead. This enables the generalization of our results to linear operators in infinite-dimensional Hilbert spaces. Our results have extensive applications in statistics and machine learning; in particular, we obtain improved bounds in covariance estimation and principal component analysis on Markovian data.

Matrix Moment and Concentration Inequalities for Martingales and Ergodic Markov Chains with Applications in Statistical Learning

TL;DR

This paper establishes novel Rosenthal-Burkholder inequalities for discrete-time matrix local martingales, Burkholder-Davis-Gundy inequality for continuous matrix local martingales, as well as matrix Rosenthal, Hoeffding, and Bernstein inequalities for ergodic Markov chains.

Abstract

In this paper, we study moment and concentration inequalities for the spectral norm of sums of dependent random matrices. We establish novel Rosenthal-Burkholder inequalities for discrete-time matrix local martingales, Burkholder-Davis-Gundy inequality for continuous matrix local martingales, as well as matrix Rosenthal, Hoeffding, and Bernstein inequalities for ergodic Markov chains. Compared with previous work on matrix concentration inequalities for Markov chains, which assume a non-zero absolute -spectral gap or the stronger -mixing condition, our results assume geometric ergodicity, a condition commonly used in statistical applications. Furthermore, our results have leading terms that match the Markov chain central limit theorem, rather than relying on suboptimal variance proxies. We also give dimension-free versions of the inequalities, which are independent of the ambient dimension and relies on the effective rank instead. This enables the generalization of our results to linear operators in infinite-dimensional Hilbert spaces. Our results have extensive applications in statistics and machine learning; in particular, we obtain improved bounds in covariance estimation and principal component analysis on Markovian data.

Paper Structure

This paper contains 29 sections, 21 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation.
  4. Organization.
  5. Main Results
  6. The Rosenthal-Burkholder Inequality for Matrix Martingales
  7. Discussion.
  8. Matrix Rosenthal and Concentration Inequalities for Markov Chains
  9. Results for Geometrically V-Ergodic Markov Chains
  10. Discussion.
  11. Applications
  12. Proofs
  13. Proof of the Rosenthal-Burkholder Inequality for Matrix Martingales
  14. Step 1: Conditionally Symmetric Martingales.
  15. Step 2: Generalize to Arbitrary Martingales
  16. ...and 14 more sections

Key Result

Theorem 2.1

For any $p\geq 2$, suppose that Then it holds that where $C_{1}=87$, $C_{2}=50$. When $p\geq 117$, the constants can be smaller: $C_{1}=64$, $C_{2}=28$. Furthermore, the following extensions hold:

Theorems & Definitions (35)

  • Theorem 2.1: Rosenthal-Burkholder Inequality
  • Theorem 2.2: Burkholder-Davis-Gundy Inequality
  • Theorem 2.3: Matrix Rosenthal Inequality for Uniformly Geometrically Ergodic Markov Chains
  • Theorem 2.4
  • Corollary 2.1: Matrix Hoeffding Inequalities for Uniformly Geometrically Ergodic Markov Chains
  • Corollary 2.2: Matrix Bernstein Inequalities for Uniformly Geometrically Ergodic Markov Chains
  • Theorem 2.5: Matrix Rosenthal Inequality for Geometrically $V$-Ergodic Markov Chains
  • Theorem 2.6
  • Definition 4.1: Conditionally Symmetric Martingales
  • Theorem 4.1: Good $\lambda$ Inequality
  • ...and 25 more