Learning Half-Spaces from Perturbed Contrastive Examples

2026-02-02

AI Summary

This work addresses the problem of efficiently learning halfspace classifiers in the presence of perturbed contrastive samples. It proposes a two-step contrastive mechanism wherein each sample is paired with a contrastive counterpart of opposite label, and the magnitude of perturbation is determined by a non-decreasing noise function based on the sample’s distance to the decision boundary. This formulation more realistically captures practical contrastive learning scenarios. The paper provides the first analysis of both active and passive sample complexity for halfspaces under this framework. Under the assumption of a uniform data distribution, it establishes that, for both fixed and random perturbations, contrastive samples can significantly reduce asymptotic and expected query complexity under certain noise conditions, thereby enhancing learning efficiency.

Technology Category

• Machine Learning: Active Learning
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We study learning under a two-step contrastive example oracle, as introduced by Mansouri et. al. (2025), where each queried (or sampled) labeled example is paired with an additional contrastive example of opposite label. While Mansouri et al. assume an idealized setting, where the contrastive example is at minimum distance of the originally queried/sampled point, we introduce and analyze a mechanism, parameterized by a non-decreasing noise function $f$, under which this ideal contrastive example is perturbed. The amount of perturbation is controlled by $f(d)$, where $d$ is the distance of the queried/sampled point to the decision boundary. Intuitively, this results in higher-quality contrastive examples for points closer to the decision boundary. We study this model in two settings: (i) when the maximum perturbation magnitude is fixed, and (ii) when it is stochastic. For one-dimensional thresholds and for half-spaces under the uniform distribution on a bounded domain, we characterize active and passive contrastive sample complexity in dependence on the function $f$. We show that, under certain conditions on $f$, the presence of contrastive examples speeds up learning in terms of asymptotic query complexity and asymptotic expected query complexity.

Problem

• contrastive learning
• half-spaces
• perturbed examples
• sample complexity
• noise function

Innovation

• contrastive learning
• perturbed examples
• sample complexity
• half-spaces
• noise function