Toward Simultaneously Optimal Regret in U-Calibration

2026-06-16

AI Summary

This work addresses a key limitation of existing U-calibration methods, which struggle to simultaneously achieve an $\widetilde{O}(\sqrt{T})$ regret bound for all bounded proper losses and adaptively attain an $O(\log T)$ regret bound for smooth losses such as the squared loss. To overcome this, the paper proposes a novel Follow-the-Perturbed-Leader algorithm that introduces a perturbation mechanism in the prediction space based on self-concordant functions, coupled with a refined regret analysis tailored to complex noise structures. This approach is the first to unify both guarantees: it achieves an $\widetilde{O}(\sqrt{T})$ regret bound for any bounded proper loss and an $O(\log T)$ regret bound for smooth—and even relatively log-barrier smooth—proper losses, thereby offering significantly improved generality and adaptivity over current techniques.

Abstract

U-calibration studies online forecasting algorithms whose predictions can be consumed by any unknown downstream agent, guaranteeing sublinear regret simultaneously for all proper loss functions. Existing U-calibration algorithms achieve worst-case optimal $O(\sqrt{T})$ regret for every bounded proper loss, but they fail to adapt to easier losses: as we show, even for smooth losses such as squared loss, they incur $Ω(\sqrt{T})$ regret instead of the optimal $O(\log T)$ regret. In this work, we show that this limitation is not inherent. Specifically, we design a single forecast algorithm that simultaneously achieves $\tilde O(\sqrt{T})$ regret for every bounded proper loss and $O(\log T)$ regret for every bounded smooth proper loss. More generally, our algorithm also attains logarithmic regret for losses that are smooth relative to the log-barrier, which include several non-Lipschitz examples. Our approach is based on a novel variant of Follow-the-Perturbed-Leader (FTPL) in which perturbations are applied directly in the prediction space using self-concordant noise. The resulting analysis also departs substantially from prior FTPL analyses due to the complex nature of this noise and may be of independent interest.

Problem

• U-calibration
• regret minimization
• smooth losses
• online forecasting
• proper scoring rules

Innovation

• U-calibration
• adaptive regret
• smooth proper loss
• Follow-the-Perturbed-Leader
• self-concordant noise