# Defensive Boosting for Online Probabilistic Forecasting

arXiv: https://arxiv.org/abs/2608.13554 · HTML: https://mlsanity.com/paper/2608.13554/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 58 references that are in the map.
- First posted on arXiv 13 Aug 2026.

## Builds on (most cited first, 25 of 25)

- [Multiaccuracy: Black-Box Post-Processing for Fairness in Classification](https://mlsanity.com/paper/1805.12317/) (2018)
- [Strongly Adaptive Online Learning](https://mlsanity.com/paper/1502.07073/) (2015)
- [A Second-order Bound with Excess Losses](https://mlsanity.com/paper/1402.2044/) (2014)
- [Scale-Free Online Learning](https://mlsanity.com/paper/1601.01974/) (2016)
- [Outcome Indistinguishability](https://mlsanity.com/paper/2011.13426/) (2020)
- [Practical Adversarial Multivalid Conformal Prediction](https://mlsanity.com/paper/2206.01067/) (2022)
- [Loss Minimization through the Lens of Outcome Indistinguishability](https://mlsanity.com/paper/2210.08649/) (2022)
- [Challenges in Benchmarking Stream Learning Algorithms with Real-world Data](https://mlsanity.com/paper/2005.00113/) (2020)
- [Online Multivalid Learning: Means, Moments, and Prediction Intervals](https://mlsanity.com/paper/2101.01739/) (2021)
- [High-Dimensional Prediction for Sequential Decision Making](https://mlsanity.com/paper/2310.17651/) (2023)
- [Stronger Calibration Lower Bounds via Sidestepping](https://mlsanity.com/paper/2012.03454/) (2020)
- [Oracle Efficient Online Multicalibration and Omniprediction](https://mlsanity.com/paper/2307.08999/) (2023)
- [Breaking the T2/3 Barrier for Sequential Calibration](https://mlsanity.com/paper/2406.13668/) (2024)
- [Complexity-Theoretic Implications of Multicalibration](https://mlsanity.com/paper/2312.17223/) (2023)
- [Online Gradient Boosting](https://mlsanity.com/paper/1506.04820/) (2015)
- [Optimal and Adaptive Algorithms for Online Boosting](https://mlsanity.com/paper/1502.02651/) (2015)
- [Online Agnostic Boosting via Regret Minimization](https://mlsanity.com/paper/2003.01150/) (2020)
- [Boosting for Online Convex Optimization](https://mlsanity.com/paper/2102.09305/) (2021)
- [How Global Calibration Strengthens Multiaccuracy](https://mlsanity.com/paper/2504.15206/) (2025)
- [Networked Information Aggregation via Machine Learning](https://mlsanity.com/paper/2507.09683/) (2025)
- [Optimal Lower Bounds for Online Multicalibration](https://mlsanity.com/paper/2601.05245/) (2026)
- [In Defense of Defensive Forecasting](https://mlsanity.com/paper/2506.11848/) (2025)
- [Efficient Swap Multicalibration of Elicitable Properties](https://mlsanity.com/paper/2511.04907/) (2025)
- [An Efficient Black-Box Reduction from Online Learning to Multicalibration, and a New Route to Phi-Regret Minimization](https://mlsanity.com/paper/2604.19592/) (2026)
- [Gradient Boosting on Stochastic Data Streams](https://mlsanity.com/paper/1703.00377/) (2017)

## Abstract (arXiv)

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.