Linear regression with partially mismatched data: local search with theoretical guarantees

TL;DR

This paper introduces a local search algorithm for linear regression with partially mismatched data, providing strong theoretical guarantees and demonstrating promising empirical performance, especially in noiseless scenarios.

Contribution

It proposes a simple greedy local search method with proven convergence guarantees for mismatched linear regression, scalable to large datasets.

Findings

Algorithm converges to nearly-optimal solutions under certain conditions.

In noiseless cases, the algorithm finds the global optimum with linear convergence.

Numerical experiments show improved performance over existing methods.

Abstract

Linear regression is a fundamental modeling tool in statistics and related fields. In this paper, we study an important variant of linear regression in which the predictor-response pairs are partially mismatched. We use an optimization formulation to simultaneously learn the underlying regression coefficients and the permutation corresponding to the mismatches. The combinatorial structure of the problem leads to computational challenges. We propose and study a simple greedy local search algorithm for this optimization problem that enjoys strong theoretical guarantees and appealing computational performance. We prove that under a suitable scaling of the number of mismatched pairs compared to the number of samples and features, and certain assumptions on problem data; our local search algorithm converges to a nearly-optimal solution at a linear rate. In particular, in the noiseless case, our algorithm converges to the global optimal solution with a linear convergence rate. Based on this result, we prove an upper bound for the estimation error of the parameter. We also propose an approximate local search step that allows us to scale our approach to much larger instances. We conduct numerical experiments to gather further insights into our theoretical results, and show promising performance gains compared to existing approaches.