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Sharp Matrix Empirical Bernstein Inequalities
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Derives two sharp, closed-form matrix empirical Bernstein inequalities for symmetric matrices with bounded eigenvalues: a union-bound plug-in version via paired sample variance for i.i.d. data, and a self-normalized supermartingale version that also handles martingale dependence and time-uniform/stopped bounds.
Problem
Nonasymptotic confidence sets for the mean of bounded random symmetric matrices that adapt to unknown variance, matching the oracle matrix Bernstein inequality asymptotically including constants.
Approach
Generalizes scalar empirical Bernstein inequalities (union-bound Maurer-Pontil style with paired variance, and self-normalized supermartingale style) to symmetric matrices using the Lieb-Tropp trace-MGF supermartingale, with an imbalanced alpha split so the first-order term matches the oracle matrix Bernstein constant.
Key results
- Simulation with d=3 at n=1000: first matrix EB deviation is 1.589x the oracle matrix Bernstein term, second is 1.105x; both converge to 1 as n grows.
- At n=1,000,000 the ratios are 1.024 (first EB) and 1.002 (second EB).
- Both inequalities are shown analytically to satisfy sqrt(n) D_n -> sqrt(2 log(d/alpha) ||V||) almost surely, matching the oracle matrix Bernstein first-order term.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Matrix EB 2 instead of Matrix EB 1 | Matrix empirical Bernstein bounds · Matrix simulation (n=100) | relative confidence-set length: 2.612 -> 1.397 | Helped |
| Use Matrix EB 2 instead of Matrix EB 1 | Matrix empirical Bernstein bounds · Matrix simulation (n=1,000) | relative confidence-set length: 1.589 -> 1.105 | Helped |
| Use Matrix EB 2 instead of Matrix EB 1 | Matrix empirical Bernstein bounds · Matrix simulation (n=10,000) | relative confidence-set length: 1.214 -> 1.022 | Helped |
| Use Matrix EB 2 instead of Matrix EB 1 | Matrix empirical Bernstein bounds · Matrix simulation (n=100,000) | relative confidence-set length: 1.072 -> 1.007 | Helped |
| Use Matrix EB 2 instead of Matrix EB 1 | Matrix empirical Bernstein bounds · Matrix simulation (n=1,000,000) | relative confidence-set length: 1.024 -> 1.002 | Helped |
Surprising findings
- A balanced alpha split in the union bound yields an extra constant inside the logarithm (log(2d/alpha)), making the bound non-sharp; an imbalanced split restores sharpness.
- Applying a vector self-normalized EB bound to matrices via flattening gives a dimension-dependent, suboptimal result.
Limitations
- Requires eigenvalues in a known bounded interval such as [0,1]
- The paired-variance first inequality is empirically looser than the second
- Dimension-free and anytime-valid matrix Bernstein variants remain open
Experimental setup
Metrics: ratio of deviation term to oracle matrix Bernstein deviation
Baselines: matrix Bennett-Bernstein inequality (oracle) Tropp matrix Bernstein Minsker-type matrix Bernstein matrix Hoeffding
Builds on: Maurer-Pontil empirical Bernstein Lieb-Tropp matrix Bernstein Waudby-Smith and Ramdas empirical Bernstein Ville's inequality
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