# BOCO: Bayesian Online Contextual Optimization for Decision-Focused Online Learning

arXiv: https://arxiv.org/abs/2511.20413 · HTML: https://mlsanity.com/paper/2511.20413/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 45 references that are in the map.
- First posted on arXiv 25 Nov 2025.

## Builds on (most cited first, 23 of 23)

- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Online Learning: A Comprehensive Survey](https://mlsanity.com/paper/1802.02871/) (2018)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [From Predictive to Prescriptive Analytics](https://mlsanity.com/paper/1402.5481/) (2014)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [PAC-Bayesian Theory Meets Bayesian Inference](https://mlsanity.com/paper/1605.08636/) (2016)
- [Generalization Bounds in the Predict-then-Optimize Framework](https://mlsanity.com/paper/1905.11488/) (2019)
- [Assigning a value to a power likelihood in a general Bayesian model](https://mlsanity.com/paper/1701.08515/) (2017)
- [Function Space Particle Optimization for Bayesian Neural Networks](https://mlsanity.com/paper/1902.09754/) (2019)
- [An invitation to sequential Monte Carlo samplers](https://mlsanity.com/paper/2007.11936/) (2020)
- [Online Non-Convex Learning: Following the Perturbed Leader is Optimal](https://mlsanity.com/paper/1903.08110/) (2019)
- [A comparison of learning rate selection methods in generalized Bayesian inference](https://mlsanity.com/paper/2012.11349/) (2020)
- [Statistical Inference for Online Decision-Making: In a Contextual Bandit Setting](https://mlsanity.com/paper/2010.07283/) (2020)
- [Online Contextual Decision-Making with a Smart Predict-then-Optimize Method](https://mlsanity.com/paper/2206.07316/) (2022)
- [Bayesian inference for misspecified generative models](https://mlsanity.com/paper/2305.08429/) (2023)
- [BPQP: A Differentiable Convex Optimization Framework for Efficient End-to-End Learning](https://mlsanity.com/paper/2411.19285/) (2024)
- [Dissecting the Impact of Model Misspecification in Data-driven Optimization](https://mlsanity.com/paper/2503.00626/) (2025)
- [Joint Online Learning and Decision-making via Dual Mirror Descent](https://mlsanity.com/paper/2104.09750/) (2021)
- [Benchmarking PtO and PnO Methods in the Predictive Combinatorial Optimization Regime](https://mlsanity.com/paper/2311.07633/) (2023)
- [Scenario Predict-then-Optimize for Data-Driven Online Inventory Routing](https://mlsanity.com/paper/2401.17787/) (2024)

## Abstract (arXiv)

Decision-focused learning (DFL) trains predictive models to optimize downstream decisions rather than prediction accuracy alone. While recent studies have extended this paradigm to online settings with streaming data, existing online DFL methods generally maintain a point estimate, while their gradient-based updates require either a differentiable optimization layer or a problem-specific surrogate loss. Consequently, they can be unstable under limited data and difficult to apply across heterogeneous optimization problems. We introduce Bayesian Online Contextual Optimization (textttBOCO), a framework that maintains a decision-focused posterior over model parameters. textttBOCO aggregates the resulting predictions when prescribing decisions, thereby accounting for parameter uncertainty. To track this posterior in evolving environments, we develop two particle-based inference algorithms: a sequential Monte Carlo sampler for nondifferentiable problems and a function-space Stein variational gradient descent algorithm for differentiable problems. Across both real-world tasks, textttBOCO reduces running mean regret and temporal regret variability relative to two frequentist online DFL baselines. At the full horizon, relative to the best frequentist baseline, BOCO reduces running mean regret and temporal regret variability by 6.9% and 6.3% on knapsack and by 49.3% and 38.1% on energy scheduling, respectively. The gains are larger early in the data stream, a pattern consistent with a benefit from accounting for parameter uncertainty when data are limited.