  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.
