Title: Achieving Gradient-Variation Interval Regret in Online Learning

Abstract:

This study explores non-stationary online learning through the lens of interval regret, a performance metric that demands an algorithm to maintain high efficacy across all time intervals. We introduce the inaugural online learning algorithm capable of delivering an interval regret bound that scales with gradient variation. This metric serves as a fundamental indicator of the cumulative shifts in online function gradients, linking directly to stochastic optimization and other related domains, while also correlating with various problem-specific quantities.

Our proposed approach utilizes a straightforward yet efficient two-layer online ensemble framework, which ensures robust theoretical assurances. The method achieves a regret bound that concurrently adapts to diverse problem-dependent factors without sacrificing the minimax-optimal rate under worst-case scenarios. Addressing the difficulties associated with hyperparameter selection, we also present a variant that remains agnostic to Lipschitz and smoothness constants, automatically adjusting to these potentially unknown parameters. This adaptability is largely driven by a novel Lipschitz-adaptive meta algorithm, a contribution that holds significance beyond the scope of this specific problem.

In addition to improving interval regret, our results offer wider implications. The method generates versatile bounds for interval dynamic regret, a more stringent measure that competes with varying comparators across any interval. Furthermore, it establishes the first piecewise characterization for stochastic extended adversarial optimization. We support our theoretical conclusions with experimental validation.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC