Polynomial-Time Robust Multiclass Linear Classification under Gaussian Marginals

TL;DR

This paper presents a polynomial-time robust learning algorithm for multiclass linear classifiers under Gaussian distributions, overcoming previous exponential complexity barriers for multiple classes.

Contribution

It introduces new structural insights and frameworks for efficient, dimension-independent error guarantees in multiclass linear classification.

Findings

Standard perceptron requires super-polynomial samples for multiclass cases.

Developed a pairwise improper-learning framework with near-optimal error bounds.

Achieved error $O( ext{opt})+ ext{epsilon}$ for 3-class classifiers.

Abstract

We study the task of agnostic learning of multiclass linear classifiers under the Gaussian distribution. Given labeled examples $(x, y)$ from a distribution over $\mathbb{R}^d \times [k]$, with Gaussian $x$-marginal, the goal is to output a hypothesis whose error is comparable to that of the best $k$-class linear classifier. While the binary case $k=2$ has a well-developed algorithmic theory, much less is known for $k \ge 3$. Even for $k=3$, prior robust algorithms incur exponential dependence on the inverse of the desired accuracy in both complexity and representation size. In this work, we develop new structural results for multiclass linear classifiers and use them to design fully polynomial-time robust learners with dimension-independent error guarantees. Our first result shows that the standard multiclass perceptron algorithm requires super-polynomially many samples and updates, even with clean labels and Gaussian marginals, revealing a basic obstruction absent in the binary case. Our main positive result is a pairwise improper-learning framework which yields an efficient learner with error $\widetilde O(k^{3/2}\sqrt{\mathrm{opt}})+\epsilon$ for general $k$. Additionally, we develop a sharper localization-based framework which leads to error $O(\mathrm{opt})+\epsilon$ for $k=3$, and error $\mathrm{poly}(k)\mathrm{opt}+\epsilon$ for geometrically regular $k$-class linear classifiers.