Agnostic Smoothed Online Learning without Knowledge of the Base Measure

TL;DR

This paper introduces an algorithm for smoothed online learning that operates effectively without prior knowledge of the base measure, bridging the gap between i.i.d. and adversarial models.

Contribution

It presents the first algorithm guaranteeing sublinear regret in agnostic smoothed online learning without knowing the base measure.

Findings

Achieves adaptive regret of  O(T/) for classification.

Ensures sublinear oblivious regret for regression with polynomial fat-shattering dimension.

Extends smoothed online learning to more realistic scenarios without prior measure knowledge.

Abstract

Classical results in statistical learning typically consider two extreme data-generating models: i.i.d. instances from an unknown distribution, or fully adversarial instances, often much more challenging statistically. To bridge the gap between these models, recent work introduced the smoothed framework, in which at each iteration an adversary generates instances from a distribution constrained to have density bounded by $\sigma^{-1}$ compared to some fixed base measure $\mu$. This framework interpolates between the i.i.d. and adversarial cases, depending on the value of $\sigma$. For the classical online prediction problem, most prior results in smoothed online learning rely on the arguably strong assumption that the base measure $\mu$ is known to the learner, contrasting with standard settings in the PAC learning or consistency literature. We consider the general agnostic problem in which the base measure is unknown and values are arbitrary. Along this direction, Block et al. showed that empirical risk minimization has sublinear regret under the well-specified assumption. We propose an algorithm R-Cover based on recursive coverings which is the first to guarantee sublinear regret for agnostic smoothed online learning without prior knowledge of $\mu$. For classification, we prove that R-Cover has adaptive regret $\tilde O(\sqrt{dT/\sigma})$ for function classes with VC dimension $d$, which is optimal up to logarithmic factors. For regression, we establish that R-Cover has sublinear oblivious regret for function classes with polynomial fat-shattering dimension growth.