# Model-Agnostic Online Certificate-Driven Calibration for Time Series Forecasting Under Distribution Shift

arXiv: https://arxiv.org/abs/2609.31960 · HTML: https://mlsanity.com/paper/2609.31960/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 30 references that are in the map.
- First posted on arXiv 25 Sep 2026.

## Builds on (most cited first, 15 of 15)

- [Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting](https://mlsanity.com/paper/2012.07436/) (2020)
- [Autoformer: Decomposition Transformers with Auto-Correlation for Long-Term Series Forecasting](https://mlsanity.com/paper/2106.13008/) (2021)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [One Fits All:Power General Time Series Analysis by Pretrained LM](https://mlsanity.com/paper/2302.11939/) (2023)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [OneNet: Enhancing Time Series Forecasting Models under Concept Drift by Online Ensembling](https://mlsanity.com/paper/2309.12659/) (2023)
- [PAC-Bayes with Backprop](https://mlsanity.com/paper/1908.07380/) (2019)
- [Calibration of Time-Series Forecasting: Detecting and Adapting Context-Driven Distribution Shift](https://mlsanity.com/paper/2310.14838/) (2023)
- [Proactive Model Adaptation Against Concept Drift for Online Time Series Forecasting](https://mlsanity.com/paper/2412.08435/) (2024)
- [PAC-Bayes and Domain Adaptation](https://mlsanity.com/paper/1707.05712/) (2017)
- [Out-of-Distribution Generalization in Time Series: A Survey](https://mlsanity.com/paper/2503.13868/) (2025)
- [Understanding the Generalization Ability of Deep Learning Algorithms: A Kernelized Renyi's Entropy Perspective](https://mlsanity.com/paper/2305.01143/) (2023)
- [Benchmarking Physics-Informed Time-Series Models for Operational Global Station Weather Forecasting](https://mlsanity.com/paper/2406.14399/) (2024)
- [Learning via Surrogate PAC-Bayes](https://mlsanity.com/paper/2410.10230/) (2024)

## Abstract (arXiv)

Time series out-of-distribution generalization requires forecasters to remain reliable when deployment dynamics differ from training conditions due to covariate shift, concept shift, and temporal dependence. Probably Approximately Correct Bayesian domain adaptation provides computable certificates by decomposing target risk into a source risk term, a source-to-target mismatch term, and a complexity term, but standard analyses rely on independent sampling and distributional stability, assumptions that are violated in time series by serial dependence and nonstationary shift. We propose a model-agnostic online martingale Probably Approximately Correct Bayesian framework that yields finite-sample certificates under temporal dependence and distribution shift. The certificate replaces independent-sample concentration with martingale concentration that adapts to loss scale and predictable variation. We use the certificate as a surrogate regularizer for online calibration by training a gated residual Bayesian head on top of a fixed forecasting backbone, producing a corrective update that reverts to the backbone prediction when the gate is closed. Online calibration combines a source risk anchor, a posterior-shift penalty, and a time-adaptive mismatch term computed from target windows observed before forecasting. It follows a predict-then-update protocol in which outcomes become available only after forecasting and are used to update subsequent predictions. Experiments across convolutional, attention-based, and large language model-based forecasters show improved stability and accuracy under covariate and concept shift.