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

Adrian Vladu

TL;DR

The paper shows that for two M-matrix optimization problems—Matrix Scaling and Quadratic Optimization—a standard log-barrier interior point method with adaptive step sizes achieves a surprising $\tilde{O}\left(n^{1/3}\right)$ iteration bound, rather than the classical $\tilde{O}\left(n^{1/2}\right)$ from self-concordance theory. Using an amortized analysis built around a potential function and an easily computable congestion vector, the authors demonstrate that long periods of small steps are compensated by phases with larger progress, and that M-matrix structure yields nonnegative Newton steps that enable stronger control of the path-following behavior. The practical payoff is a near-linear-time diffusion-like algorithm on graphs via fast Laplacian solvers, with depth $\tilde{O}\left(n^{1/3}\right)$ and work $\tilde{O}\left(n^{1/3}\cdot\text{nnz}\right)$, as well as nearly-linear runtimes for the underlying MS and QO problems. Overall, the work provides a significant instance where a standard IPM surpasses the usual $\Theta\left(n^{1/2}\right)$ iteration barrier in a structured setting, with broad implications for optimization on graphs and related linear-algebraic problems.

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 $\widetilde{O}\left(n^{1/2}\right)$, 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 $\widetilde{O}\left(n^{1/3}\right)$ iterations. As a corollary, using fast Laplacian solvers, we obtain an $\ell_{2}$ flow diffusion algorithm with depth $\widetilde{O}\left(n^{1/3}\right)$ and work $\widetilde{O}$$\left(n^{1/3}\cdot\text{nnz}\right)$. This result marks a significant instance in which a standard log-barrier IPM permits provably fewer than $Θ\left(n^{1/2}\right)$ iterations.

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

TL;DR

The paper shows that for two M-matrix optimization problems—Matrix Scaling and Quadratic Optimization—a standard log-barrier interior point method with adaptive step sizes achieves a surprising iteration bound, rather than the classical from self-concordance theory. Using an amortized analysis built around a potential function and an easily computable congestion vector, the authors demonstrate that long periods of small steps are compensated by phases with larger progress, and that M-matrix structure yields nonnegative Newton steps that enable stronger control of the path-following behavior. The practical payoff is a near-linear-time diffusion-like algorithm on graphs via fast Laplacian solvers, with depth and work , as well as nearly-linear runtimes for the underlying MS and QO problems. Overall, the work provides a significant instance where a standard IPM surpasses the usual iteration barrier in a structured setting, with broad implications for optimization on graphs and related linear-algebraic problems.

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 , where 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 iterations. As a corollary, using fast Laplacian solvers, we obtain an flow diffusion algorithm with depth and work . This result marks a significant instance in which a standard log-barrier IPM permits provably fewer than iterations.
Paper Structure (16 sections, 24 theorems, 121 equations, 2 algorithms)

This paper contains 16 sections, 24 theorems, 121 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Matrix Scaling (MS).
  3. Quadratic Optimization (QO).
  4. Our Results
  5. Notation, Terminology, and Overview of Our Interior Point Methods
  6. Preliminaries
  7. M-matrices
  8. Scaling M-matrices
  9. Setting up the IPM
  10. Following the central path, slowly
  11. Correction step
  12. Improved analysis using a potential function
  13. M-matrix Quadratic Optimization
  14. Discussion and Outlook
  15. Deferred Proofs
  16. ...and 1 more sections

Key Result

Theorem 1

Given an instance of the matrix scaling problem MS$(A,\varepsilon)$ consisting of symmetric M-matrix $A\in\mathbb{R}^{n\times n}$ and a scalar $\varepsilon>0$, an interior point method can compute a non-negative vector $x$ such that in $O\left(n^{1/3}\log\frac{\left\Vert A1-1\right\Vert _{2}}{\varepsilon}\right)$ predictor-corrector iterations.

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Lemma 5
  • proof
  • Definition 6: congestion vector
  • Lemma 7: correction lemma
  • proof
  • Lemma 8
  • proof
  • ...and 37 more