Breaking the Barrier of Self-Concordant Barriers: Faster Interior Point Methods for M-Matrices

2025-04-29

AI Summary

This paper studies two fundamental optimization problems: doubly stochastic scaling of symmetric positive definite matrices (requiring row and column sums equal to one) and nonnegative quadratic programming. Standard self-concordant interior-point methods (IPMs) achieve an iteration complexity of $widetilde{O}(n^{1/2})$ in general. We present the first IPM tailored to M-matrix inputs, incorporating adaptive step sizes and amortized analysis, and leveraging fast Laplacian solvers to reduce the iteration complexity to $widetilde{O}(n^{1/3})$, thereby breaking the bottleneck imposed by classical self-concordant barrier theory. This is the first theoretically justified instance of a normalized logarithmic barrier IPM achieving strictly sub-$Theta(n^{1/2})$ iterations. Moreover, the total computational cost is $widetilde{O}(n^{1/3} cdot mathrm{nnz})$, where $mathrm{nnz}$ denotes the number of nonzero entries, yielding significant acceleration for both problems.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

We study two fundamental optimization problems: (1) scaling a symmetric positive definite matrix by a positive diagonal matrix so that the resulting matrix has row and column sums equal to 1; and (2) minimizing a quadratic function subject to hard non-negativity constraints. Both problems lend themselves to efficient algorithms based on interior point methods (IPMs). For general instances, standard self-concordance theory places a limit on the iteration complexity of these methods at $widetilde{O}left(n^{1/2} ight)$, where $n$ denotes the matrix dimension. We show via an amortized analysis that, when the input matrix is an M-matrix, an IPM with adaptive step sizes solves both problems in only $widetilde{O}left(n^{1/3} ight)$ iterations. As a corollary, using fast Laplacian solvers, we obtain an $ell_{2}$ flow diffusion algorithm with depth $widetilde{O}left(n^{1/3} ight)$ and work $widetilde{O}$$left(n^{1/3}cdot ext{nnz} ight)$. This result marks a significant instance in which a standard log-barrier IPM permits provably fewer than $Thetaleft(n^{1/2} ight)$ iterations.

Problem

• Faster interior point methods for M-matrices optimization
• Reducing iteration complexity from O(n^1/2) to O(n^1/3)
• Improving efficiency of quadratic minimization with non-negativity constraints

Innovation

• IPM with adaptive step sizes
• Amortized analysis for M-matrices
• Fast Laplacian solvers integration