Robustly Learning Monotone Generalized Linear Models via Data Augmentation
This work provides a robust algorithm for learning GLMs, addressing a fundamental challenge in machine learning with broad implications for robust statistical estimation.
The paper tackles the problem of learning Generalized Linear Models (GLMs) under the agnostic model with Gaussian distribution, achieving a constant-factor approximation for any monotone Lipschitz activation, which resolves a well-known open problem.
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 $(2+ζ)$-moments, for any fixed $ζ>0$ -- 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.