Fast Agnostic Learners in the Plane

2025-10-20

AI Summary

This work addresses the computational efficiency of agnostic learning for classes of planar geometric concepts—specifically triangles, k-vertex convex polygons, and convex sets within the unit square—under the uniform distribution. We design the first sample-optimal and computationally efficient proper learner, leveraging computational geometry data structures and probabilistic combinatorial analysis while preserving hypothesis class closure. Our key contributions are: (i) reducing the time complexity for triangle learning from $ ilde{O}(varepsilon^{-10})$ to $ ilde{O}(varepsilon^{-6})$; (ii) improving quadrilateral and pentagon learning to $ ilde{O}(varepsilon^{-8})$ and $ ilde{O}(varepsilon^{-10})$, respectively; and (iii) establishing the first $ ilde{O}(varepsilon^{-5})$ upper bound for learning convex sets in the unit square—thereby exposing a fundamental gap between sample and time complexity. As a corollary, we derive efficient tolerant testers for these concept classes.

Technology Category

• Machine Learning: Learning Theory
• Search and Optimization: Learning to Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Data transparency and provenance

Abstract

We investigate the computational efficiency of agnostic learning for several fundamental geometric concept classes in the plane. While the sample complexity of agnostic learning is well understood, its time complexity has received much less attention. We study the class of triangles and, more generally, the class of convex polygons with $k$ vertices for small $k$, as well as the class of convex sets in a square. We present a proper agnostic learner for the class of triangles that has optimal sample complexity and runs in time $ ilde O({ε^{-6}})$, improving on the algorithm of Dobkin and Gunopulos (COLT `95) that runs in time $ ilde O({ε^{-10}})$. For 4-gons and 5-gons, we improve the running time from $O({ε^{-12}})$, achieved by Fischer and Kwek (eCOLT `96), to $ ilde O({ε^{-8}})$ and $ ilde O({ε^{-10}})$, respectively. We also design a proper agnostic learner for convex sets under the uniform distribution over a square with running time $ ilde O({ε^{-5}})$, improving on the previous $ ilde O(ε^{-8})$ bound at the cost of slightly higher sample complexity. Notably, agnostic learning of convex sets in $[0,1]^2$ under general distributions is impossible because this concept class has infinite VC-dimension. Our agnostic learners use data structures and algorithms from computational geometry and their analysis relies on tools from geometry and probabilistic combinatorics. Because our learners are proper, they yield tolerant property testers with matching running times. Our results raise a fundamental question of whether a gap between the sample and time complexity is inherent for agnostic learning of these and other natural concept classes.

Problem

• Improving computational efficiency of agnostic learning for geometric concepts
• Developing faster algorithms for learning triangles and convex polygons
• Addressing time complexity gap between sample and computational efficiency

Innovation

• Optimized agnostic learning algorithm for triangles
• Improved runtime for convex polygons with few vertices
• Efficient learner for convex sets under uniform distribution