New Theoretical Advances in Online Convex Optimization with Noise-Adaptive Regret Bounds
Researchers have established three new high-probability regret bound results for online convex optimization (OCO) with strongly convex losses, resolving previously open questions about noise adaptivity, feedback structure, and constraint satisfaction. The work introduces an exponential supermartingale technique to handle unbounded sub-Gaussian noise and formally proves a separation in confidence cost between full-information and bandit feedback settings. These results tighten theoretical guarantees for a class of sequential decision-making algorithms widely used in machine learning.
A paper accepted to ECML-PKDD 2026 presents three theoretical contributions to online convex optimization (OCO) under strongly convex losses. In the full-information setting, the authors prove a noise-adaptive high-probability regret bound where the martingale deviation term scales with the actual noise level σ rather than the worst-case gradient bound G, achieving a multiplicative improvement of G/σ over the classical Azuma-Hoeffding baseline. This is enabled by a novel exponential supermartingale argument that avoids the bounded-difference requirement of Freedman's inequality, allowing direct handling of unbounded sub-Gaussian noise without truncation. For the bandit feedback setting, the paper establishes a minimax lower bound showing that high-probability regret scales linearly in log(1/δ), compared to the √log(1/δ) scaling under full information — a formal separation between the two feedback models. Finally, for constrained OCO with stochastic constraints under a Slater condition, the authors provide simultaneous high-probability guarantees of O(√(T log(m/δ))) regret and O(√T/(ζδ) + m√(T log(m/δ))) constraint violation. Synthetic experiments are reported to corroborate all three theoretical results.
What's missing
The paper relies solely on synthetic experiments to validate theoretical predictions; empirical evaluation on real-world datasets or practical machine learning benchmarks is absent, leaving open questions about the practical magnitude of improvements in applied settings. The tightness of the constrained OCO violation bound with respect to the dependence on δ in the denominator is not fully characterized as a lower bound, leaving open whether it is optimal.
What different sources said
- arXiv cs.LGCenter
Noise-Adaptive High-Probability Regret Bounds for Online Convex Optimization
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