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Expected star discrepancy based on stratified sampling

Xiaoda Xu, Jun Xian

TL;DR

This work strengthens the theoretical foundations of stratified sampling for high-dimensional integration by (i)提供 sharper expected star discrepancy bounds for jittered sampling through refined δ-covers and bracketing numbers, and (ii) proving the strong partition principle, which shows any equal-measure stratification yields strictly smaller expected star discrepancy than pure Monte Carlo. The approach combines variance reduction, discretization via δ-covers, and concentration inequalities to obtain rigorous, quantitative guarantees, complemented by extensive numerical validation. The results justify using stratified and jittered designs in RQMC and resolve open questions about their comparative performance, with practical impact in finance and uncertainty quantification. Overall, the paper offers both improved performance bounds and a broad, general justification for preferring stratified sampling in high-dimensional numerical integration.

Abstract

We present two main contributions to the expected star discrepancy theory. First, we derive a sharper expected upper bound for jittered sampling, improving the leading constants and logarithmic terms compared to the state-of-the-art [Doerr, 2022]. Second, we prove the strong partition principle for star discrepancy, showing that any equal-measure stratified sampling yields a strictly smaller expected discrepancy than simple random sampling, thereby resolving an open question in [Kiderlen and Pausinger, 2022]. Numerical simulations confirm our theoretical advances and illustrate the superiority of stratified sampling in low to moderate dimensions.

Expected star discrepancy based on stratified sampling

TL;DR

This work strengthens the theoretical foundations of stratified sampling for high-dimensional integration by (i)提供 sharper expected star discrepancy bounds for jittered sampling through refined δ-covers and bracketing numbers, and (ii) proving the strong partition principle, which shows any equal-measure stratification yields strictly smaller expected star discrepancy than pure Monte Carlo. The approach combines variance reduction, discretization via δ-covers, and concentration inequalities to obtain rigorous, quantitative guarantees, complemented by extensive numerical validation. The results justify using stratified and jittered designs in RQMC and resolve open questions about their comparative performance, with practical impact in finance and uncertainty quantification. Overall, the paper offers both improved performance bounds and a broad, general justification for preferring stratified sampling in high-dimensional numerical integration.

Abstract

We present two main contributions to the expected star discrepancy theory. First, we derive a sharper expected upper bound for jittered sampling, improving the leading constants and logarithmic terms compared to the state-of-the-art [Doerr, 2022]. Second, we prove the strong partition principle for star discrepancy, showing that any equal-measure stratified sampling yields a strictly smaller expected discrepancy than simple random sampling, thereby resolving an open question in [Kiderlen and Pausinger, 2022]. Numerical simulations confirm our theoretical advances and illustrate the superiority of stratified sampling in low to moderate dimensions.
Paper Structure (46 sections, 6 theorems, 58 equations, 3 figures, 2 tables)

This paper contains 46 sections, 6 theorems, 58 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. General equal-measure partition
  4. Jittered sampling.
  5. $\delta$-covers
  6. Bernstein inequality
  7. Improved expected star discrepancy bound for jittered sampling
  8. Strong Partition Principle for Star Discrepancy
  9. Introduction and Main Result
  10. Preliminary Results
  11. Variance Reduction Lemma
  12. Discretization via $\delta$-Covers
  13. Concentration Analysis
  14. Proof of Theorem \ref{['thm:strong-partition']}
  15. Step 1: Setting up the parameters
  16. ...and 31 more sections

Key Result

Lemma 2.2

FDX2007 Let $Z_1,\ldots,Z_N$ be independent random variables with expected values $\mathbb E (Z_j)=\mu_{j}$ and variances $\sigma_{j}^2$ for $j=1,\ldots,N$. Assume that $|Z_j-\mu_j|\le C$(C is a constant) for each $j$ and set $\Sigma^2:=\sum_{j=1}^N\sigma_{j}^2$; then for any $\lambda \ge 0$,

Figures (3)

  • Figure 1: Comparison of upper bounds under different dimensions $d$ and parameters $m$
  • Figure 2: Comparison of expected star discrepancy value under different $m$ and $d$.
  • Figure 3: Convergence Curve under different $d$.

Theorems & Definitions (15)

  • Definition 2.1
  • Lemma 2.2
  • Theorem 3.1
  • Remark 3.2
  • proof : Proof of Theorem 3.1
  • Theorem 4.1: Strong Partition Principle for Expected Star Discrepancy
  • Lemma 4.2: Strict Variance Reduction
  • proof
  • Lemma 4.3: Finite Covering
  • proof
  • ...and 5 more