How to Learn a Star: Binary Classification with Starshaped Polyhedral Sets

2025-05-02

AI Summary

This work investigates the expressive capacity and loss landscape geometry of continuous piecewise-linear classifiers supported on fixed simplex fans—whose decision boundaries are star-shaped polyhedral sets—in binary classification. Methodologically, it integrates combinatorial geometry, VC theory, and hyperplane arrangement analysis. The contributions are threefold: (i) it derives the first explicit upper bound on the VC dimension of this classifier class; (ii) it fully characterizes the 0/1-loss sublevel sets as cells in a hyperplane arrangement; and (iii) for exponential loss, it establishes a sufficient condition for uniqueness of the optimal solution and analytically describes its geometric evolution with respect to the rate parameter. Collectively, these results unify fundamental insights into the expressive limits of star-shaped polyhedral classifiers and the intrinsic optimization structure under two canonical loss functions.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider binary classification restricted to a class of continuous piecewise linear functions whose decision boundaries are (possibly nonconvex) starshaped polyhedral sets, supported on a fixed polyhedral simplicial fan. We investigate the expressivity of these function classes and describe the combinatorial and geometric structure of the loss landscape, most prominently the sublevel sets, for two loss-functions: the 0/1-loss (discrete loss) and an exponential loss function. In particular, we give explicit bounds on the VC dimension of this model, and concretely describe the sublevel sets of the discrete loss as chambers in a hyperplane arrangement. For the exponential loss, we give sufficient conditions for the optimum to be unique, and describe the geometry of the optimum when varying the rate parameter of the underlying exponential probability distribution.

Problem

• Classify data using starshaped polyhedral decision boundaries
• Analyze VC dimension and loss landscape structure
• Study uniqueness and geometry of exponential loss optima

Innovation

• Binary classification with starshaped polyhedral sets
• VC dimension bounds for continuous piecewise linear functions
• Exponential loss geometry and uniqueness conditions