Robustly Learning Monotone Generalized Linear Models via Data Augmentation

TL;DR

This paper introduces a polynomial-time algorithm for learning monotone GLMs under Gaussian noise, achieving constant-factor approximation for a broad class of activations, thus solving a longstanding open problem.

Contribution

It develops the first robust, data-augmentation-based algorithm for monotone GLMs applicable to a wide range of activations, extending previous methods.

Findings

Achieves constant-factor approximation for all monotone Lipschitz activations.

Works under Gaussian distribution with bounded moments.

Introduces a novel noise injection technique for robust learning.

Abstract

We study the task of learning Generalized Linear models (GLMs) in the agnostic model under the Gaussian distribution. We give the first polynomial-time algorithm that achieves a constant-factor approximation for \textit{any} monotone Lipschitz activation. Prior constant-factor GLM learners succeed for a substantially smaller class of activations. Our work resolves a well-known open problem, by developing a robust counterpart to the classical GLMtron algorithm (Kakade et al., 2011). Our robust learner applies more generally, encompassing all monotone activations with bounded $(2+\zeta)$-moments, for any fixed $\zeta>0$ -- a condition that is essentially necessary. To obtain our results, we leverage a novel data augmentation technique with decreasing Gaussian noise injection and prove a number of structural results that may be useful in other settings.