A Parallel Projection Method for Metric Constrained Optimization

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, metricconstrained 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 metricconstrained 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.

Learning Resolution Parameters for Graph Clustering

“…Computing the LambdaCC linear programming relaxation can be challenging due to the size of the constraint set. For our smaller graphs we apply Gurobi optimization software, and for larger problems we use recently developed memory-efficient projection methods [37,31]. For the local-flow objective we use a fast Julia implementation we developed in recent work [39].…”

“…We note for example that the resolution parameter corresponding to modularity is λ = 1/(2|E|), which is also inversely proportional to |E|. Computing all of the LP bounds is the bottleneck in our computations, and takes just under 2.5 hours using a recently developed parallel solver for the correlation clustering relaxation [31].…”

Metric Nearness Made Practical

“…A series of work (Brickell et al 2008;Duggal et al 2013;Boytsov and Naidan 2013;Gilbert and Jain 2017;Fan, Raichek, and Van Buskirk 2018;Gilbert and Sonthalia 2018;Veldt et al 2018;Ruggles, Veldt, and Gleich 2020) models the scenario of repairing a metric under different settings, among which we are interested in a specific metric nearness model in this paper. For a set of input dissimilarity values between pairs of data samples, the model aims to output a set of distances that satisfy the metric constraints while keeping the output distances as near as possible to the input dissimilarity according to a certain measure of nearness.…”

Correlation Clustering via Strong Triadic Closure Labeling: Fast Approximation Algorithms and Practical Lower Bounds

“…If a good LP lower bound can be computed, the output of fast heuristic methods can be compared against the LP lower bound to obtain a posteriori approximation guarantees that are often very good in practice [26,39,43,48]. However, despite some recent work on specialized solvers for these linear programs [33,37,44], these lower bounds can only be computed for medium-sized instances at best, and even then this can take a long time. There is therefore a need for faster approximation algorithms for correlation clustering, as well as an even more basic need to efficiently compute good lower bounds for the NP-hard objective in practice.…”