Negative curvature obstructs the existence of good barriers for interior-point methods

TL;DR

This paper demonstrates that in hyperbolic and certain Hadamard spaces, natural domains have barrier parameters that grow polynomially with size, limiting the efficiency of interior-point methods for large-scale problems.

Contribution

It proves that in hyperbolic space, natural domains have polynomially growing barrier parameters, revealing fundamental limitations for interior-point methods in these settings.

Findings

Barrier parameter grows polynomially with domain diameter in hyperbolic space.

Interior-point methods cannot efficiently solve certain large-scale scaling problems.

Results extend to positive-definite matrices and symmetric Hadamard spaces.

Abstract

Interior-point methods (IPMs) are a cornerstone of Euclidean convex optimization, due to their strong theoretical guarantees and practical performance. Motivated by scaling problems, recent work by Hirai and the last two authors (FOCS'23) extended IPMs to geodesically convex optimization on Hadamard manifolds. Crucially, the complexity of IPMs (both in Euclidean and Hadamard spaces) is governed by the \emph{barrier parameter} of the domain. Here we prove that already in hyperbolic space, several natural domains -- including geodesic balls and triangles -- have a barrier parameter that grows polynomially with the domain's diameter. By extension, the same holds for the positive-definite matrices and other symmetric Hadamard spaces. This growth implies a fundamental limitation: interior-point methods relying on barriers for a ball cannot efficiently solve challenging scaling problems, such as tensor scaling, where the domain's diameter can be exponentially large in the input size. Our results are partially inspired by, and complement, lower bounds on the condition number of geodesically convex functions established by Hamilton and Moitra (NeurIPS'21).