Sharp empirical Bernstein bounds for the variance of bounded random variables

TL;DR

This work develops sharp empirical Bernstein-type bounds for the variance of bounded random variables under constant conditional mean and variance, without requiring independence. It constructs time-uniform, anytime-valid confidence sequences via a novel nonnegative supermartingale framework, yielding upper and lower variance bounds whose leading terms asymptotically match the oracle Bernstein rate $\sqrt{2 \mathbb{V}[(X-\mu)^2] \log (1/\alpha)}$ in the iid case. The results are instantiated for batch and sequential settings and extended to separable Hilbert spaces, with empirical evidence showing superior performance over the existing Maurer-Pontil inequalities. These tools enable robust, online variance inference in sequential decision-making contexts where variance estimation is crucial for uncertainty quantification.

Abstract

We develop novel empirical Bernstein inequalities for the variance of bounded random variables. Our inequalities hold under constant conditional variance and mean, without further assumptions like independence or identical distribution of the random variables, making them suitable for sequential decision making contexts. The results are instantiated for both the batch setting (where the sample size is fixed) and the sequential setting (where the sample size is a stopping time). Our bounds are asymptotically sharp: when the data are iid, our CI adpats optimally to both unknown mean $μ$ and unknown $\mathbb{V}[(X-μ)^2]$, meaning that the first order term of our CI exactly matches that of the oracle Bernstein inequality which knows those quantities. We compare our results to a widely used (non-sharp) concentration inequality for the variance based on self-bounding random variables, showing both the theoretical gains and improved empirical performance of our approach. We finally extend our methods to work in any separable Hilbert space.

Figures (3)

• Figure 1: Average confidence intervals over $100$ simulations for the std $\sigma$ for (I) the uniform distribution in $(0,1)$, (II) the beta distribution with parameters $(2, 6)$, and (III) the beta distribution with parameters $(5, 5)$. For each of the inequalities, the $0.95\%$-empirical quantiles are also displayed. The Maurer Pontil (MP) inequality maurer2009empirical is compared against our proposal (EB). We highlight the improved empirical performance of our methods in all scenarios.
• Figure 2: Average confidence intervals over $100$ simulations for the std $\sigma$ for (I) the uniform distribution in $(0,1)$, (II) the beta distribution with parameters $(2, 6)$, and (III) the beta distribution with parameters $(5, 5)$. For each of the inequalities, the $0.95\%$-empirical quantiles are also displayed. The decoupling approach (this appendix) is compared against EB (our proposal). EB clearly outperforms the decoupled approach in all the scenarios.
• Figure 3: Average confidence intervals over $100$ simulations for the std $\sigma$ for (I) the uniform distribution in $(0,1)$, (II) the beta distribution with parameters $(2, 6)$, and (III) the beta distribution with parameters $(5, 5)$. For each of the inequalities, the $0.95\%$-empirical quantiles are also displayed. The EB lower confidence intervals with the plug-ins from Section \ref{['section:lower_bound']} (our proposal) are compared against the EB lower confidence intervals with the plug-ins proposed in this appendix (alternative). Despite the expected similar outcomes, the plug-ins from Section \ref{['section:lower_bound']} lead to slightly sharper bounds.

Theorems & Definitions (47)

• Theorem 3.1: Ville's inequality
• Theorem 3.2: Empirical Bernstein inequality
• Theorem 3.3: Anytime valid Bennett's inequality
• Theorem 4.1
• Corollary 4.2
• Corollary 4.3: Upper empirical Bernstein for the variance
• Corollary 4.4: Lower empirical Bernstein for the variance
• Theorem 4.6
• Theorem 4.7
• Corollary 4.8: Sharpness