# A Parallel Projection Method for Metric Constrained Optimization

## TL;DR

This paper introduces a parallel projection method that significantly accelerates solving large-scale metric-constrained optimization problems, enabling efficient handling of trillions of constraints in clustering applications.

## Contribution

The paper proposes a novel parallel execution schedule for projection methods, improving convergence speed without conflicts, and demonstrates its effectiveness on large-scale correlation clustering problems.

## Key findings

- Successfully solved problems with up to 2.9 trillion constraints
- Achieved faster convergence compared to traditional projection methods
- Enabled large-scale metric-constrained optimization in practical time

## Abstract

Many clustering applications in machine learning and data mining rely on solving metric-constrained optimization problems. These problems are characterized by $O(n^3)$ constraints that enforce triangle inequalities on distance variables associated with $n$ objects in a large dataset. Despite its usefulness, metric-constrained optimization is challenging in practice due to the cubic number of constraints and the high-memory requirements of standard optimization software. Recent work has shown that iterative projection methods are able to solve metric-constrained optimization problems on a much larger scale than was previously possible, thanks to their comparatively low memory requirement. However, the major limitation of projection methods is their slow convergence rate. In this paper we present a parallel projection method for metric-constrained optimization which allows us to speed up the convergence rate in practice. The key to our approach is a new parallel execution schedule that allows us to perform projections at multiple metric constraints simultaneously without any conflicts or locking of variables. We illustrate the effectiveness of this execution schedule by implementing and testing a parallel projection method for solving the metric-constrained linear programming relaxation of correlation clustering. We show numerous experimental results on problems involving up to 2.9 trillion constraints.