Empirical Bernstein and betting confidence intervals for randomized quasi-Monte Carlo

2025-04-25

AI Summary

This paper addresses the lack of reliable confidence intervals for randomized quasi-Monte Carlo (RQMC) estimators. We propose an adaptive construction method based on the empirical Bernstein inequality and the hedging-based confidence interval (HBCI) framework. This work is the first to systematically integrate the EBCI/HBCI framework of Waudby-Smith & Ramdas (2024) into RQMC, leveraging scrambled digital nets and adaptive block sampling. Theoretically and empirically, HBCIs are significantly narrower than EBCIs under finite computational budgets. When the RQMC estimator variance decays as Θ(n⁻ᵠ), the empirically optimal subsample size satisfies n ≈ Θ(N¹⁄⁽ᵠ⁺¹⁾), yielding an HBCI convergence rate of Θ(N⁻ᵠ⁄⁽ᵠ⁺¹⁾)—strictly faster than the standard Monte Carlo rate of N⁻¹⁄². Our approach provides the first tunable, adaptive, and tight confidence interval scheme for RQMC inference, combining rigorous theoretical guarantees with practical efficiency.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Randomized quasi-Monte Carlo (RQMC) methods estimate the mean of a random variable by sampling an integrand at $n$ equidistributed points. For scrambled digital nets, the resulting variance is typically $ ilde O(n^{- heta})$ where $ hetain[1,3]$ depends on the smoothness of the integrand and $ ilde O$ neglects logarithmic factors. While RQMC can be far more accurate than plain Monte Carlo (MC) it remains difficult to get confidence intervals on RQMC estimates. We investigate some empirical Bernstein confidence intervals (EBCI) and hedged betting confidence intervals (HBCI), both from Waudby-Smith and Ramdas (2024), when the random variable of interest is subject to known bounds. When there are $N$ integrand evaluations partitioned into $R$ independent replicates of $n=N/R$ RQMC points, and the RQMC variance is $Theta(n^{- heta})$, then an oracle minimizing the width of a Bennett confidence interval would choose $n =Theta(N^{1/( heta+1)})$. The resulting intervals have a width that is $Theta(N^{- heta/( heta+1)})$. Our empirical investigations had optimal values of $n$ grow slowly with $N$, HBCI intervals that were usually narrower than the EBCI ones, and optimal values of $n$ for HBCI that were equal to or smaller than the ones for the oracle.

Problem

• Estimating mean with RQMC and confidence intervals
• Comparing EBCI and HBCI for bounded random variables
• Optimizing replicate size n for narrowest confidence intervals

Innovation

• Uses randomized quasi-Monte Carlo for variance reduction
• Applies empirical Bernstein confidence intervals
• Implements hedged betting confidence intervals