Strategic Littlestone Dimension: Improved Bounds on Online Strategic Classification

TL;DR

This paper introduces the Strategic Littlestone Dimension, a new measure that characterizes the complexity of online strategic classification problems involving manipulative agents, leading to improved mistake and regret bounds.

Contribution

It defines the Strategic Littlestone Dimension and demonstrates its role in characterizing optimal mistake bounds and regret in strategic online classification, including scenarios with unknown manipulation graphs.

Findings

Characterizes mistake bounds using the Strategic Littlestone Dimension.

Achieves improved regret bounds in the agnostic setting.

Extends analysis to unknown manipulation graphs with regret guarantees.

Abstract

We study the problem of online binary classification in settings where strategic agents can modify their observable features to receive a positive classification. We model the set of feasible manipulations by a directed graph over the feature space, and assume the learner only observes the manipulated features instead of the original ones. We introduce the Strategic Littlestone Dimension, a new combinatorial measure that captures the joint complexity of the hypothesis class and the manipulation graph. We demonstrate that it characterizes the instance-optimal mistake bounds for deterministic learning algorithms in the realizable setting. We also achieve improved regret in the agnostic setting by a refined agnostic-to-realizable reduction that accounts for the additional challenge of not observing agents' original features. Finally, we relax the assumption that the learner knows the manipulation graph, instead assuming their knowledge is captured by a family of graphs. We derive regret bounds in both the realizable setting where all agents manipulate according to the same graph within the graph family, and the agnostic setting where the manipulation graphs are chosen adversarially and not consistently modeled by a single graph in the family.