Defensive Boosting for Online Probabilistic Forecasting

We study online probabilistic forecasting of binary outcomes chosen by an adaptive adversary. Given an online learning algorithm for a weak hypothesis class $\mathcal{H}$, 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 $\mathcal{H}$ 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 $\mathcal{H}$ 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 Defensive Booster'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 and provides local hard-core certificates on every time interval. The Defensive Booster 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 strong predictive performance coupled with orders-of-magnitude faster runtime.