A Wasserstein Minimax Framework for Mixed Linear Regression

TL;DR

This paper introduces Wasserstein Mixed Linear Regression (WMLR), a novel optimal transport-based framework for modeling multi-modal data in mixed linear regression, with provable convergence, generalization guarantees, and applications to federated learning.

Contribution

WMLR formulates MLR as a Wasserstein distance minimization problem, reducing it to a nonconvex-concave minimax optimization solvable by GDA, and extends naturally to federated learning.

Findings

WMLR achieves global convergence for two-component mixtures.

Sample complexity of WMLR grows linearly with data dimension.

Numerical experiments demonstrate WMLR's effectiveness in federated settings.

Abstract

Multi-modal distributions are commonly used to model clustered data in statistical learning tasks. In this paper, we consider the Mixed Linear Regression (MLR) problem. We propose an optimal transport-based framework for MLR problems, Wasserstein Mixed Linear Regression (WMLR), which minimizes the Wasserstein distance between the learned and target mixture regression models. Through a model-based duality analysis, WMLR reduces the underlying MLR task to a nonconvex-concave minimax optimization problem, which can be provably solved to find a minimax stationary point by the Gradient Descent Ascent (GDA) algorithm. In the special case of mixtures of two linear regression models, we show that WMLR enjoys global convergence and generalization guarantees. We prove that WMLR's sample complexity grows linearly with the dimension of data. Finally, we discuss the application of WMLR to the federated learning task where the training samples are collected by multiple agents in a network. Unlike the Expectation Maximization algorithm, WMLR directly extends to the distributed, federated learning setting. We support our theoretical results through several numerical experiments, which highlight our framework's ability to handle the federated learning setting with mixture models.