Loss Functions for Finite Sets

TL;DR

This paper introduces new loss functions for finite sets that are efficient, have no spurious local minima in certain cases, and can be constructed from samples, with demonstrated practical effectiveness.

Contribution

It provides sum-of-square loss functions of minimal degree for finite sets, ensuring no spurious minimizers in key cases, and offers a method to derive such functions from data samples.

Findings

Loss functions with no spurious minimizers for vertex sets of simplexes.

Construction of loss functions from sample data via quadratic optimization.

Numerical experiments demonstrating the efficiency of the proposed loss functions.

Abstract

This paper studies loss functions for finite sets. For a given finite set $S$, we give sum-of-square type loss functions of minimum degree. When $S$ is the vertex set of a standard simplex, we show such loss functions have no spurious minimizers (i.e., every local minimizer is a global one). Up to transformations, we give similar loss functions without spurious minimizers for general finite sets. When $S$ is approximately given by a sample set $T$, we show how to get loss functions by solving a quadratic optimization problem. Numerical experiments and applications are given to show the efficiency of these loss functions.