Near-optimal Swap Regret Minimization for Convex Losses

2026-02-09

AI Summary

This work addresses the problem of efficiently minimizing swap regret for adaptively chosen sequences of Lipschitz convex losses. The authors propose a novel algorithm based on multi-scale binning and randomized online prediction, which improves the expected swap regret bound on the unit interval from the previous Õ(T^{2/3}) to the near-optimal Õ(√T). Notably, this approach eliminates the reliance on the Lipschitz continuity of the identification function required by earlier median calibration methods. The algorithm runs in polynomial time in T and, for the first time, achieves an Õ(√T) calibration error in the median calibration task, substantially advancing the state of the art in online learning and calibration theory.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We give a randomized online algorithm that guarantees near-optimal $\widetilde O(\sqrt T)$ expected swap regret against any sequence of $T$ adaptively chosen Lipschitz convex losses on the unit interval. This improves the previous best bound of $\widetilde O(T^{2/3})$ and answers an open question of Fishelson et al. [2025b]. In addition, our algorithm is efficient: it runs in $\mathsf{poly}(T)$ time. A key technical idea we develop to obtain this result is to discretize the unit interval into bins at multiple scales of granularity and simultaneously use all scales to make randomized predictions, which we call multi-scale binning and may be of independent interest. A direct corollary of our result is an efficient online algorithm for minimizing the calibration error for general elicitable properties. This result does not require the Lipschitzness assumption of the identification function needed in prior work, making it applicable to median calibration, for which we achieve the first $\widetilde O(\sqrt T)$ calibration error guarantee.

Problem

• swap regret
• convex losses
• online learning
• calibration error
• elicitable properties

Innovation

• swap regret minimization
• multi-scale binning
• online learning
• calibration error
• convex losses