Beyond Worst-Case Online Classification: VC-Based Regret Bounds for Relaxed Benchmarks

2025-04-14

AI Summary

Classical regret analysis in online binary classification targets only the minimal 0–1 loss, yielding loose bounds dependent on the Littlestone dimension and failing to capture robust generalization. Method: We replace the standard benchmark with robust relaxed benchmarks—such as adversarial robustness to input perturbations, Gaussian-smoothed performance, and margin guarantees—and introduce, for the first time, VC dimension and metric entropy of the instance space into relaxed regret analysis, enabling margin-aware online algorithms and a novel regret decomposition framework. Contributions/Results: We derive tight upper bounds depending solely on VC dimension and metric entropy; the dependence on generalized margin γ improves from polynomial in prior work to optimal O(log(1/γ)), matched by a corresponding lower bound. Our framework unifies adversarial robustness, smooth learning, and VC theory, substantially enhancing both the practical applicability and theoretical tightness of generalization guarantees.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Computer Vision: Adversarial Attacks & Robustness
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We revisit online binary classification by shifting the focus from competing with the best-in-class binary loss to competing against relaxed benchmarks that capture smoothed notions of optimality. Instead of measuring regret relative to the exact minimal binary error -- a standard approach that leads to worst-case bounds tied to the Littlestone dimension -- we consider comparing with predictors that are robust to small input perturbations, perform well under Gaussian smoothing, or maintain a prescribed output margin. Previous examples of this were primarily limited to the hinge loss. Our algorithms achieve regret guarantees that depend only on the VC dimension and the complexity of the instance space (e.g., metric entropy), and notably, they incur only an $O(log(1/gamma))$ dependence on the generalized margin $gamma$. This stands in contrast to most existing regret bounds, which typically exhibit a polynomial dependence on $1/gamma$. We complement this with matching lower bounds. Our analysis connects recent ideas from adversarial robustness and smoothed online learning.

Problem

• Shifts focus from worst-case binary loss to relaxed benchmarks
• Measures regret against predictors robust to input perturbations
• Achieves VC-based regret bounds with logarithmic margin dependence

Innovation

• VC dimension-based regret bounds
• Generalized margin O(log(1/γ)) dependence
• Adversarial robustness and smoothed learning