Noise-Adaptive High-Probability Regret Bounds for Online Convex Optimization

2026-06-06

AI Summary

This work addresses online convex optimization with strongly convex losses under three settings: full information, bandit feedback with noise, and stochastic constraints. It establishes high-probability regret bounds that depend only on the noise level σ, rather than the gradient norm bound G. By introducing an exponential supermartingale analysis, the approach circumvents the bounded-difference assumption inherent in Freedman’s inequality. The analysis reveals a linear dependence on log(1/δ) in the bandit setting—distinct from the full-information case—and, for the first time, provides simultaneous high-probability guarantees on both regret and long-term constraint violation under stochastic constraints. The main theoretical results include an O(σ√T log(1/δ)) regret bound in the full-information setting, an Ω(log(1/δ)) lower bound on the confidence cost in the bandit setting, and, under stochastic constraints, O(√(T log(m/δ))) regret with constraint violation bounded by O(√T/(ζδ) + m√(T log(m/δ))). Empirical experiments corroborate these theoretical predictions.

Abstract

We study high-probability regret bounds for online convex optimization (OCO) with strongly convex losses and establish three results that resolve open questions at the intersection of noise adaptivity, feedback structure, and constraint satisfaction. For the full-information setting with sub-Gaussian stochastic gradients, we prove a noise-adaptive high-probability regret bound in which the martingale deviation term scales with the noise level $σ$ rather than the gradient bound $G$, yielding a multiplicative improvement of $G/σ$ over the classical Azuma-Hoeffding baseline. Our analysis introduces an exponential supermartingale argument that bypasses the bounded-difference requirement of Freedman's inequality, enabling direct treatment of unbounded sub-Gaussian noise without truncation artifacts. For bandit feedback, we prove a minimax lower bound: the high-probability regret scales linearly in $\log(1/δ)$, in contrast to the $\sqrt{\log(1/δ)}$ confidence cost under full information. This constitutes a formal separation in the confidence cost of strongly convex OCO across feedback models. Regarding constrained OCO with stochastic constraints satisfying a Slater condition, we provide simultaneous high-probability guarantees for both cumulative regret and long-run constraint violation, achieving $\mathcal{O}(\sqrt{T\log(m/δ)})$ regret and $\mathcal{O}(\sqrt{T}/(ζδ) + m\sqrt{T\log(m/δ)})$ violation. Synthetic experiments corroborate all theoretical predictions.

Problem

• Online Convex Optimization
• High-Probability Regret Bounds
• Noise Adaptivity
• Bandit Feedback
• Stochastic Constraints

Innovation

• noise-adaptive regret
• high-probability bounds
• exponential supermartingale
• feedback separation
• stochastic constrained OCO