Robust Learning of a Group DRO Neuron

TL;DR

This paper introduces a robust method for learning a single neuron under label noise and distributional shifts across groups, using a primal-dual algorithm to optimize worst-case group performance.

Contribution

It proposes a computationally efficient primal-dual algorithm for group distributionally robust optimization of a neuron, with theoretical guarantees under label noise and distributional shifts.

Findings

Algorithm is competitive with the best possible neuron under worst-case group weighting.

Framework handles nonconvex loss with robustness guarantees.

Promising results on large language model pre-training benchmarks.

Abstract

We study the problem of learning a single neuron under standard squared loss in the presence of arbitrary label noise and group-level distributional shifts, for a broad family of covariate distributions. Our goal is to identify a ''best-fit'' neuron parameterized by $\mathbf{w}_*$ that performs well under the most challenging reweighting of the groups. Specifically, we address a Group Distributionally Robust Optimization problem: given sample access to $K$ distinct distributions $\mathcal p_{[1]},\dots,\mathcal p_{[K]}$, we seek to approximate $\mathbf{w}_*$ that minimizes the worst-case objective over convex combinations of group distributions $\boldsymbol{\lambda} \in \Delta_K$, where the objective is $\sum_{i \in [K]}\lambda_{[i]}\,\mathbb E_{(\mathbf x,y)\sim\mathcal p_{[i]}}(\sigma(\mathbf w\cdot\mathbf x)-y)^2 - \nu d_f(\boldsymbol\lambda,\frac{1}{K}\mathbf1)$ and $d_f$ is an $f$-divergence that imposes (optional) penalty on deviations from uniform group weights, scaled by a parameter $\nu \geq 0$. We develop a computationally efficient primal-dual algorithm that outputs a vector $\widehat{\mathbf w}$ that is constant-factor competitive with $\mathbf{w}_*$ under the worst-case group weighting. Our analytical framework directly confronts the inherent nonconvexity of the loss function, providing robust learning guarantees in the face of arbitrary label corruptions and group-specific distributional shifts. The implementation of the dual extrapolation update motivated by our algorithmic framework shows promise on LLM pre-training benchmarks.