Defensive Boosting for Online Probabilistic Forecasting

2026-08-13

AI Summary

This work addresses the challenge that existing online boosting methods struggle to simultaneously guarantee strong performance in both Brier score and classification error for binary probabilistic prediction. The paper introduces Defensive Booster, an algorithm that unifies online gradient boosting with the weak-to-strong boosting framework for the first time. It achieves a Brier score competitive with the best predictor in the span of the weak hypothesis class over any adaptive sequence, and ensures vanishing classification error whenever the weak learning condition holds. Built upon a dual-view formulation, the method employs a defensive online learning mechanism featuring single weak learner invocation, smooth reweighting, and strongly adaptive optimization, enabling detection of weak learner failure and support for arbitrary time intervals. Experiments demonstrate that Defensive Booster significantly outperforms current baselines on both synthetic and real-world data streams, offering superior predictive accuracy and speedups of several orders of magnitude.

Abstract

We study online probabilistic forecasting of binary outcomes chosen by an adaptive adversary. Given an online learning algorithm for a weak hypothesis class $H$, we would like to efficiently obtain two incomparable guarantees that existing online boosting techniques provide separately. Online gradient boosting competes in Brier score with the best predictor induced by the span of $H$ on every sequence, but promises nothing when the span does not contain an accurate predictor. Online weak-to-strong boosting drives classification error to zero under a weak-learning condition, but promises little when that condition fails. We give a simple defensive forecasting algorithm, the Defensive Booster, that obtains both guarantees. On every adaptive sequence, its Brier score is competitive with the best prediction induced by the span of $H$ at the same rate as online gradient boosting; simultaneously, whenever the realized transcript satisfies the smooth weak-learning condition, its Brier score and randomized classification error satisfy the same rate guarantee as online classification boosting. This is achieved by operationalizing the "dual view" of boosting: When the algorithm's randomized classification error is persistently high, its mistake weights form a smooth reweighting on which every weak hypothesis has low edge, yielding an ex-post hard-core certificate that the weak-learning condition fails. We also develop a strongly adaptive variant, which satisfies both guarantees on every time interval. The Defensive Booster is very efficient: it accesses just one weak-class learner, whereas the prior online boosting methods we compare against maintain large weak-learner ensembles. Experiments on synthetic and real data streams demonstrate its strong predictive performance (sometimes substantially improving over all prior baselines) coupled with orders-of-magnitude faster runtime.

Problem

• online boosting
• probabilistic forecasting
• adaptive adversary
• weak learning
• Brier score

Innovation

• defensive boosting
• online probabilistic forecasting
• weak-to-strong boosting
• Brier score
• adaptive adversary