Defensive Boosting for Online Probabilistic Forecasting
Authors: Georgy Noarov, Aaron Roth arXiv: 2608.13554 Release date: 2026-08-13
Original 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.
Key Points
- Problem setting: Online probabilistic forecasting of binary outcomes against an adaptive adversary, given a weak online learner for hypothesis class H.
- Two prior guarantees, each incomplete:
- Online gradient boosting competes with the best predictor in the span of H in Brier score, but provides no guarantee when the span lacks an accurate predictor.
- Online weak-to-strong boosting drives classification error to zero under a weak-learning condition, but offers little when that condition fails.
- Contribution — Defensive Booster: A single simple defensive forecasting algorithm that achieves both guarantees simultaneously.
- On every adaptive sequence, Brier score matches the online gradient boosting rate against the best span predictor.
- When the transcript satisfies the smooth weak-learning condition, Brier score and randomized classification error match the online classification boosting rate.
- Dual-view mechanism: When randomized classification error is persistently high, mistake weights induce a smooth reweighting under which every weak hypothesis has low edge, producing ex-post hard-core certificates that weak learning fails.
- Strongly adaptive variant: Satisfies both guarantees on every time interval.
- Efficiency: Accesses only one weak learner, whereas prior online boosting methods maintain large weak-learner ensembles.
- Empirical results: On synthetic and real data streams, the method shows strong predictive performance (sometimes substantially improving over all baselines) with orders-of-magnitude faster runtime.