Stochastic convergence of a class of greedy-type algorithms for Configuration Optimization Problems

TL;DR

The paper develops a probabilistic framework for greedy configuration optimization by treating greedy updates as a continuous-time Markov process on finite point configurations. It establishes convergence in expectation and probability for both local and global error measures under mild structural assumptions, and provides explicit rates for global functionals via an improvement-factor condition. A pedagogical 1D interpolation example demonstrates exponential convergence of the $L^1$ interpolation error for $C^2$ functions, and the Randomized Polytope Division Method (R-PDM) is introduced as a variance-reducing randomized variant with strong practical performance. Collectively, the results offer a probabilistic alternative to classical width-based analyses and yield scalable, convergent algorithms for COPs with provable guarantees.

Abstract

Greedy Sampling Methods (GSMs) are widely used to construct approximate solutions of Configuration Optimization Problems (COPs), where a loss functional is minimized over finite configurations of points in a compact domain. While effective in practice, deterministic convergence analyses of greedy-type algorithms are often restrictive and difficult to verify. We propose a stochastic framework in which greedy-type methods are formulated as continuous-time Markov processes on the space of configurations. This viewpoint enables convergence analysis in expectation and in probability under mild structural assumptions on the error functional and the transition kernel. For global error functionals, we derive explicit convergence rates, including logarithmic, polynomial, and exponential decay, depending on an abstract improvement condition. As a pedagogical example, we study stochastic greedy sampling for one-dimensional piecewise linear interpolation and prove exponential convergence of the $L^1$-interpolation error for $C^2$-functions. Motivated by this analysis, we introduce the Randomized Polytope Division Method (R-PDM), a randomized variant of the classical Polytope Division Method, and demonstrate its effectiveness and variance reduction in numerical experiments

Figures (9)

• Figure 1: Depiction of the steps in R-PDM for the $2$-dimensional parameter case and split domain via facet linking. (1) Sample first parameter and divide $P$ via facet linking (2) Compute barycenters. (3) Select a point based on the transition kernel (4) Mark the polytope containing this point (5) Select parameters in other polytope with error function below tolerance (6) Sample new points to replace these barycenters. (7) Update polytope division. (8) Compute the new barycenters.
• Figure 2: Example $1$
• Figure 3: Comparing variances of $L^1$-error for Examples $1$ and $2$
• Figure 4: Pointwise error and variance of Example $1$
• Figure 5: Example $2$

Theorems & Definitions (44)

• Remark 1.2
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Remark 2.5
• Proposition 2.6
• Definition 2.7
• Remark 3.2
• Theorem 3.3
• Remark 3.4