Max-affine regression via first-order methods

TL;DR

This paper analyzes the convergence of gradient-based methods for max-affine regression, demonstrating linear convergence under certain conditions and showing SGD's efficiency in noisy and low-sample scenarios.

Contribution

It provides the first non-asymptotic convergence analysis of GD and SGD for max-affine regression with theoretical guarantees.

Findings

GD and SGD converge linearly to a neighborhood of the true model

SGD outperforms alternating minimization and GD in noisy, low-sample settings

Numerical results support the theoretical convergence guarantees

Abstract

We consider regression of a max-affine model that produces a piecewise linear model by combining affine models via the max function. The max-affine model ubiquitously arises in applications in signal processing and statistics including multiclass classification, auction problems, and convex regression. It also generalizes phase retrieval and learning rectifier linear unit activation functions. We present a non-asymptotic convergence analysis of gradient descent (GD) and mini-batch stochastic gradient descent (SGD) for max-affine regression when the model is observed at random locations following the sub-Gaussianity and an anti-concentration with additive sub-Gaussian noise. Under these assumptions, a suitably initialized GD and SGD converge linearly to a neighborhood of the ground truth specified by the corresponding error bound. We provide numerical results that corroborate the theoretical finding. Importantly, SGD not only converges faster in run time with fewer observations than alternating minimization and GD in the noiseless scenario but also outperforms them in low-sample scenarios with noise.