Distributionally-Constrained Adversaries in Online Learning

TL;DR

This paper explores a flexible framework for online learning where adversaries choose data distributions within certain constraints, providing new insights into when and how learning is possible across various distributional settings.

Contribution

It characterizes which distribution classes are learnable against different adversaries and shows that learners can adapt to multiple distribution constraints without prior knowledge.

Findings

Characterizes learnability for various distribution classes.

Generalizes known results from smoothed analysis.

Learners can adapt to multiple distribution constraints simultaneously.

Abstract

There has been much recent interest in understanding the continuum from adversarial to stochastic settings in online learning, with various frameworks including smoothed settings proposed to bridge this gap. We consider the more general and flexible framework of distributionally constrained adversaries in which instances are drawn from distributions chosen by an adversary within some constrained distribution class [RST11]. Compared to smoothed analysis, we consider general distributional classes which allows for a fine-grained understanding of learning settings between fully stochastic and fully adversarial for which a learner can achieve non-trivial regret. We give a characterization for which distribution classes are learnable in this context against both oblivious and adaptive adversaries, providing insights into the types of interplay between the function class and distributional constraints on adversaries that enable learnability. In particular, our results recover and generalize learnability for known smoothed settings. Further, we show that for several natural function classes including linear classifiers, learning can be achieved without any prior knowledge of the distribution class -- in other words, a learner can simultaneously compete against any constrained adversary within learnable distribution classes.