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A Vector Bernstein Inequality for Self-Normalized Martingales

Ingvar Ziemann

TL;DR

The paper derives a Bernstein-type bound for vector-valued self-normalized martingales by reinterpreting the classical pseudo-maximization approach through the PAC-Bayesian inequality. It replaces Gaussian priors with uniform ellipsoidal priors to obtain analytically tractable, variance-sensitive bounds that depend on the true variance proxy $\sigma_{\mathrm{var}}^2$ rather than a sub-Gaussian surrogate. The main result provides a high-probability bound on $\|S_\tau\|_{(V_\tau+\Gamma)^{-1}}^2$ with explicit constants and sample-size dependent log-determinant terms, improving upon previous bounds in least-squares and linear bandit analyses. Auxiliary results establish the requisite mgf bounds, KL computations, and volume-based arguments that support the PAC-Bayesian derivations. Overall, the work sharpens concentration guarantees for sequential linear models by delivering a variance-aware Bernstein bound suitable for vector martingales and stopping times, with potential implications for regret analyses in linear bandits and autoregression tasks.

Abstract

We prove a Bernstein inequality for vector-valued self-normalized martingales. We first give an alternative perspective of the corresponding sub-Gaussian bound due to Abbasi-Yadkori et al. via a PAC-Bayesian argument with Gaussian priors. By instantiating this argument to priors drawn uniformly over well-chosen ellipsoids, we obtain a Bernstein bound.

A Vector Bernstein Inequality for Self-Normalized Martingales

TL;DR

The paper derives a Bernstein-type bound for vector-valued self-normalized martingales by reinterpreting the classical pseudo-maximization approach through the PAC-Bayesian inequality. It replaces Gaussian priors with uniform ellipsoidal priors to obtain analytically tractable, variance-sensitive bounds that depend on the true variance proxy rather than a sub-Gaussian surrogate. The main result provides a high-probability bound on with explicit constants and sample-size dependent log-determinant terms, improving upon previous bounds in least-squares and linear bandit analyses. Auxiliary results establish the requisite mgf bounds, KL computations, and volume-based arguments that support the PAC-Bayesian derivations. Overall, the work sharpens concentration guarantees for sequential linear models by delivering a variance-aware Bernstein bound suitable for vector martingales and stopping times, with potential implications for regret analyses in linear bandits and autoregression tasks.

Abstract

We prove a Bernstein inequality for vector-valued self-normalized martingales. We first give an alternative perspective of the corresponding sub-Gaussian bound due to Abbasi-Yadkori et al. via a PAC-Bayesian argument with Gaussian priors. By instantiating this argument to priors drawn uniformly over well-chosen ellipsoids, we obtain a Bernstein bound.
Paper Structure (7 sections, 5 theorems, 30 equations)

This paper contains 7 sections, 5 theorems, 30 equations.

Table of Contents

  1. Tail Bounds for Self-Normalized Martingales
  2. The Result
  3. Remarks on \ref{['thm:selfnormalizedbernstein']}:
  4. PAC-Bayesian Bounds
  5. Warm-up: Sub-Gaussian Deviation Bounds
  6. Variance Sensitive Deviation Bounds: Proof of \ref{['thm:selfnormalizedbernstein']}
  7. Auxiliary Results

Key Result

Theorem 1

Fix $\delta,\varepsilon,\nu \in (0,1)$, a stopping time $\tau$ with respect to $\mathcal{F}_{0:\infty}$, a positive semidefinite matrix $\Gamma \succeq 0$, a positive definite matrix $V\succ 0$ and assume that eq:boundedness holds. Define Then as long as $V_\tau +\Gamma \succeq e(1+\nu)^2V\succeq (1+\nu)^2 \varepsilon^{-1} (d+2) B_W^2 B_X^2$ we that with probability at least $1-\delta$:

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 1: PAC-Bayesian deviation bound
  • Remark 2.1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof : Proof of \ref{['lem:PACBAYES']}