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Intrinsic-dimension empirical Bernstein inequalities for bounded self-adjoint operators

NeurIPS 2026 · Learning theory and algorithms › Kernels and regression theory · approach: High-dimensional asymptotic generalization · 0 citations (OpenAlex)

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Derives the first fully empirical Bennett and Bernstein concentration inequalities for bounded self-adjoint Hilbert-Schmidt operators, with radii scaling by intrinsic dimension tr(Σ)/||Σ|| rather than ambient dimension, plus empirical confidence intervals for the intrinsic dimension.

Problem

Operator-valued Bennett/Bernstein concentration bounds require known variance and typically scale with ambient dimension, making them vacuous for infinite-dimensional or structured operators.

Approach

Builds empirical Bennett/Bernstein bounds by union-bounding oracle bounds with high-probability upper bounds on trace and operator norm of the variance, estimated via a U-statistic-style differencing of independent blocks; the oracle bound depends on intrinsic dimension tr(V_n)/||V_n|| rather than ambient dimension.

Key results

• Empirical Bernstein radius converges a.s. to the intrinsic oracle sqrt(2||Sigma|| log(2 tr(Sigma)/(delta ||Sigma||))) after scaling by sqrt(n) (sharpness theorem).
• For noncommuting rank-one updates in d=1000, OEB radius is smaller than both MEB baselines at effective ranks 3, 5, 10, 20; at rank 10 reductions are 27.6% and 17.5%.
• For bounded Rademacher-feature RKHS truncations, OEB changes only 0.6% from rank 32 to 256 while MEB 1 and MEB 2 increase by 15.0% and 13.9%; MEB 2 is tighter at ranks 32 and 64.

Surprising findings

• MEB 2 is tighter than the proposed bound at small ranks (32 and 64), so the method does not uniformly dominate at finite sample sizes.
• The new bound's advantage over the ambient-dimension baselines only appears for sufficiently large sample sizes in the covariance experiments.

Limitations

• Requires known almost-sure envelopes c and B for operator norm and HS norm; unbounded observations need truncation outside the scope.
• Does not uniformly dominate ambient-dimension baselines at small sample sizes due to additional trace and fourth-moment estimation terms.
• Restricted to independent (not martingale or dependent) observations.

Experimental setup

Metrics: ratio of empirical confidence radius to intrinsic-dimension reduction in radius relative to ambient baselines

Baselines: MEB 1 MEB 2 ambient-dimension matrix Bernstein oracle

Builds on: operator-valued Bennett inequality Tropp matrix Bernstein Maurer-Pontil scalar empirical Bernstein Lieb's concavity theorem matrix empirical Bernstein (Wang et al.)

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