Fast approximation and learning of binary classification tasks in o-minimal structures using ReLU neural networks

2026-06-29 · Source: Machine Learning · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, quick

Summary

This research investigates binary classification problems where decision sets are defined within o-minimal expansions of the real field. The study introduces traceable sets as a classical proxy for these definable decision regions and analyzes their approximation capabilities using ReLU neural networks. Under specific conditions, including uniform bounds on connected components and suitable C^m extensions for boundary functions, the characteristic functions of traceable subsets of [-1/2,1/2]^n can be approximated in L^p to an accuracy ε>0. This approximation is achieved with ReLU networks of size ℴ(ε^-p(n-1)/m), depth independent of ε, and polynomially bounded weights. The approach also extends to a subclass of definable maps. Furthermore, by combining these approximation capabilities with entropy estimates, the study derives statistical learning rates for empirical risk minimization with hinge loss, showing that for N uniformly distributed samples, classifiers achieve an expected misclassification error of order N^-m/(m+pn-p) with minimal polynomial loss.

Key takeaway

For research scientists exploring the theoretical limits of neural network capabilities, this work demonstrates that ReLU networks can effectively approximate complex decision boundaries found in o-minimal structures. You should consider these quantitative approximation rates, specifically the ℴ(ε^-p(n-1)/m) network size and N^-m/(m+pn-p) misclassification error, when designing or analyzing models for tasks with geometrically intricate decision spaces. This insight could guide your architectural choices for achieving specific error bounds.

Key insights

ReLU neural networks can approximate complex decision regions in o-minimal structures, yielding quantitative approximation and statistical learning rates.

Principles

• Traceable sets proxy o-minimal definable decision regions.
• ReLU networks approximate these with ℴ(ε^-p(n-1)/m) size.

Method

The approach involves introducing traceable sets, analyzing their L^p approximation by ReLU neural networks, and then combining these approximation capabilities with entropy estimates for ReLU network classes to derive statistical learning rates for empirical risk minimization.

Topics

• Binary Classification
• o-minimal Structures
• ReLU Neural Networks
• Approximation Theory
• Statistical Learning
• Empirical Risk Minimization

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.