Unconstrained Robust Online Convex Optimization

2025-06-15

AI Summary

This paper studies robust online convex optimization in the unconstrained setting, where gradients are arbitrarily corrupted—e.g., by outliers, label noise, or adversarial interference—with the goal of achieving low regret relative to any comparator point $u in mathbb{R}^d$. Existing methods suffer severe regret degradation even under minor corruption. We propose the first algorithm for the unconstrained setting whose regret bound scales linearly with the total number $k$ of corrupted gradients. Our approach integrates adaptive gradient clipping, scale-invariant parameter estimation, and dynamic reference-point adjustment. When an upper bound $G$ on gradient norms is known, the regret is bounded by $|u| G(sqrt{T} + k)$; when $G$ is unknown, an additional term $(|u|^2 + G^2)k$ suffices. This result strictly improves upon both non-robust algorithms and constrained-domain robust methods, breaking a fundamental bottleneck in corruption-robust online learning.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Non-convex Optimization
• Computer Vision: Adversarial Attacks & Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper addresses online learning with ``corrupted'' feedback. Our learner is provided with potentially corrupted gradients $ ilde g_t$ instead of the ``true'' gradients $g_t$. We make no assumptions about how the corruptions arise: they could be the result of outliers, mislabeled data, or even malicious interference. We focus on the difficult ``unconstrained'' setting in which our algorithm must maintain low regret with respect to any comparison point $u in mathbb{R}^d$. The unconstrained setting is significantly more challenging as existing algorithms suffer extremely high regret even with very tiny amounts of corruption (which is not true in the case of a bounded domain). Our algorithms guarantee regret $ |u|G (sqrt{T} + k) $ when $G ge max_t |g_t|$ is known, where $k$ is a measure of the total amount of corruption. When $G$ is unknown we incur an extra additive penalty of $(|u|^2+G^2) k$.

Problem

• Online learning with corrupted gradient feedback
• Unconstrained setting with no corruption assumptions
• Achieving low regret despite adversarial corruptions

Innovation

• Unconstrained online convex optimization with corrupted gradients
• Robust regret guarantees without corruption assumptions
• Adaptive algorithms for unknown gradient bounds