Robust supervised learning with coordinate gradient descent

TL;DR

This paper introduces a robust supervised learning method using coordinate gradient descent with robust partial derivative estimators, effectively handling corrupted data with minimal additional computational cost.

Contribution

It proposes a novel combination of coordinate gradient descent and robust estimators for partial derivatives, achieving robustness with nearly the same complexity as standard methods.

Findings

Provides theoretical bounds on generalization error.

Demonstrates competitive performance in numerical experiments.

Offers an efficient Python implementation in linlearn.

Abstract

This paper considers the problem of supervised learning with linear methods when both features and labels can be corrupted, either in the form of heavy tailed data and/or corrupted rows. We introduce a combination of coordinate gradient descent as a learning algorithm together with robust estimators of the partial derivatives. This leads to robust statistical learning methods that have a numerical complexity nearly identical to non-robust ones based on empirical risk minimization. The main idea is simple: while robust learning with gradient descent requires the computational cost of robustly estimating the whole gradient to update all parameters, a parameter can be updated immediately using a robust estimator of a single partial derivative in coordinate gradient descent. We prove upper bounds on the generalization error of the algorithms derived from this idea, that control both the optimization and statistical errors with and without a strong convexity assumption of the risk. Finally, we propose an efficient implementation of this approach in a new python library called linlearn, and demonstrate through extensive numerical experiments that our approach introduces a new interesting compromise between robustness, statistical performance and numerical efficiency for this problem.