[Paper Review] Experimental Analysis of a Generalized Stratified Sampling Algorithm for Hypercubes

Simon Wessing|arXiv (Cornell University)|May 10, 2017

Interconnection Networks and Systems3 citations

TL;DR

This paper proposes a generalized stratified sampling (GSS) algorithm that uses recursive binary partitioning to enable uniform sampling of any number of points in hypercubes, overcoming the exponential growth constraint of traditional paraxial stratification. The method ensures equal-sized hyperboxes, minimizes covering radius via linear-time computation, and improves discrepancy when combined with Latin hypercube sampling, outperforming state-of-the-art methods in numerical integration and optimization tasks across diverse dimensions and sample sizes.

ABSTRACT

Stratified sampling is a fast and simple method to generate point sets with uniform distribution in hypercubes. However, for the most common paraxial stratfication it has the prominent drawback that the number of sampled points in n dimensions has to be an n-th power of an integer number. This exponential growth makes its application unattractive or even infeasible in high dimensions. We present a stratification procedure that eliminates this problem by a recursive binary partitioning of the hypercube. The algorithm runs in linear time and tries to minimize the hyperboxes' deviation from the cubic shape. We analyze the properties of the algorithm using discrepancy and covering radius, and directly in practical applications, comparing it to quasirandom and other sampling methods. We also discuss a potential combination with Latin hypercube sampling, which positively affects discrepancy.

Motivation & Objective

• To address the limitation of traditional stratified sampling, which requires the number of points to be an n-th power of an integer, making it impractical in high dimensions.
• To develop a linear-time sampling algorithm that ensures uniform distribution across hypercubes regardless of the number of points.
• To improve sampling quality by minimizing hyperbox deviation from cubic shape and reducing covering radius.
• To evaluate the method’s performance against quasirandom and Latin hypercube sampling in numerical integration and worst-case optimization.
• To explore the trade-off between discrepancy and covering radius through distribution parameterization (e.g., Bates distribution).

Proposed method

• Uses recursive binary space partitioning to divide the hypercube into equal-volume hyperboxes, avoiding the need for n-th power constraints.
• Applies a greedy strategy to minimize hyperbox aspect ratio deviation from cubic shape during partitioning.
• Computes an upper bound for covering radius in linear time using the partitioned hyperboxes, enabling efficient minimization.
• Introduces a variant that samples each hyperbox using a Bates distribution, allowing trade-offs between low discrepancy and low covering radius.
• Combines the algorithm with Latin hypercube sampling to improve discrepancy while preserving uniformity.
• Employs bootstrapped confidence intervals and 2000 replications to assess statistical performance across dimensions and sample sizes.

Experimental results

Research questions

• RQ1 Can a stratified sampling method be designed to work with any number of points in high-dimensional hypercubes without exponential growth constraints?
• RQ2 How does minimizing covering radius through recursive binary partitioning affect optimization and integration performance?
• RQ3 To what extent can combining GSS with Latin hypercube sampling improve discrepancy compared to standard methods?
• RQ4 Is there a measurable trade-off between discrepancy and covering radius when using different sampling distributions within hyperboxes?
• RQ5 How does the proposed method compare to Korobov lattices and quasirandom sequences in terms of convergence speed and accuracy across varying dimensions and sample counts?

Key findings

• For dimensions n ≤ 4, the generalized stratified sampling (GSS) with b = ∞ (minimizing covering radius) regularly outperforms the Korobov lattice, which is considered optimal in terms of discrepancy.
• In high dimensions, GSS shows either excellent performance (on sphere and Rosenbrock functions) or poor performance (on double-sum and FP functions), indicating that covering radius is more critical in some problem classes.
• The variant of GSS that includes lines 6–7 (which refine partitioning to reduce covering radius) wins in 822 out of 1024 experiments in 2D, and maintains dominance up to 10D, with 712 wins in 7D and 517 in 8D.
• The distribution of results for GSS with b > 1 is more skewed than other methods, suggesting sensitivity to problem structure or ill-conditioning in certain subspaces.
• The upper bound for covering radius is computationally efficient and sufficient for minimizing covering radius, avoiding the prohibitive cost of exact Voronoi tessellation in high dimensions.
• Both the Korobov lattice and (A)LGSS (adaptive generalized stratified sampling) are strong general-purpose methods, with performance tunable via implementation choices.