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Conference paper
A Wasserstein Minimax Framework for Mixed Linear Regression
Proceedings of Machine Learning Research, Vol.139, pp.2697-2706
38th International Conference on Machine Learning (ICML 2021), 38th (Virtual, 18/Jul/2021 - 24/Jul/2021)
Jul/2021
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.
Details
- Title
- A Wasserstein Minimax Framework for Mixed Linear Regression
- Creators
- Theo Diamandis (Corresponding Author) - Massachusetts Institute of TechnologyYonina Eldar - 972WIS_INST___83Alireza Fallah (Corresponding Author) - Massachusetts Institute of TechnologyFarzan Farnia (Corresponding Author) - Massachusetts Institute of TechnologyAsuman Ozdaglar - Massachusetts Institute of Technology
- Resource Type
- Conference paper
- Publication Details
- Proceedings of Machine Learning Research, Vol.139, pp.2697-2706
- Number of pages
- 10
- Language
- English
- Conference
- 38th International Conference on Machine Learning (ICML 2021), 38th (Virtual, 18/Jul/2021 - 24/Jul/2021)
- Grant note
- This paper was partially funded by MIT-IBM Watson AI Lab and Defense Science and Technology Agency. This work was also supported by the QuantERA grant C’MONQSENS!. Alireza Fallah acknowledges support from MathWorks Engineering Fellowship.
- Record Identifier
- 993271020103596
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