Online Learning of Noisy Data with Kernels

TL;DR

This paper introduces an online learning method robust to adversarial noise in data, using unbiased estimators and kernel methods, enabling learning with noisy data where only multiple queries per instance are feasible.

Contribution

It develops a novel online learning algorithm that handles unknown, changing noise distributions using unbiased estimators within kernel spaces.

Findings

The method can learn in any dot-product or Gaussian kernel space.

Multiple noisy copies per instance are necessary for learning; single copies are insufficient.

The approach achieves high-probability bounds with a constant number of queries.

Abstract

We study online learning when individual instances are corrupted by adversarially chosen random noise. We assume the noise distribution is unknown, and may change over time with no restriction other than having zero mean and bounded variance. Our technique relies on a family of unbiased estimators for non-linear functions, which may be of independent interest. We show that a variant of online gradient descent can learn functions in any dot-product (e.g., polynomial) or Gaussian kernel space with any analytic convex loss function. Our variant uses randomized estimates that need to query a random number of noisy copies of each instance, where with high probability this number is upper bounded by a constant. Allowing such multiple queries cannot be avoided: Indeed, we show that online learning is in general impossible when only one noisy copy of each instance can be accessed.