Smoothed Online Classification can be Harder than Batch Classification

TL;DR

This paper investigates the complexity of online classification under smoothed adversaries, revealing that it can be more challenging than batch classification in certain unbounded label scenarios, unlike previous simpler cases.

Contribution

It demonstrates that smoothed online classification can be harder than iid batch classification when labels are unbounded, and provides conditions linking PAC learnability to smoothed online learnability.

Findings

Smoothed online classification can be harder than iid batch classification with unbounded labels.

A hypothesis class learnable in iid PAC setting may not be learnable in smoothed online setting.

A condition is identified that guarantees PAC learnability implies smoothed online learnability.

Abstract

We study online classification under smoothed adversaries. In this setting, at each time point, the adversary draws an example from a distribution that has a bounded density with respect to a fixed base measure, which is known apriori to the learner. For binary classification and scalar-valued regression, previous works \citep{haghtalab2020smoothed, block2022smoothed} have shown that smoothed online learning is as easy as learning in the iid batch setting under PAC model. However, we show that smoothed online classification can be harder than the iid batch classification when the label space is unbounded. In particular, we construct a hypothesis class that is learnable in the iid batch setting under the PAC model but is not learnable under the smoothed online model. Finally, we identify a condition that ensures that the PAC learnability of a hypothesis class is sufficient for its smoothed online learnability.