PAC-Bayes Meets Online Contextual Optimization

2025-11-25

AI Summary

Existing online contextual optimization methods are predominantly frequentist, relying on gradient-based updates and deterministic predictors, resulting in high prediction variance and difficulty handling non-differentiable problems. Method: This paper proposes the first PAC-Bayesian framework for online contextual optimization, adopting a predict-then-optimize paradigm. It employs gradient-free stochastic optimization via the Gibbs posterior, supporting non-convex and non-smooth loss functions. The approach integrates Bayesian updating, sequential Monte Carlo sampling, and PAC-Bayes analysis. Contribution/Results: Under bounded mixability of losses, the method achieves an $O(sqrt{T})$ regret bound. Theoretically and empirically, it significantly reduces prediction variance compared to conventional approaches, while enhancing robustness and generalization in dynamic environments.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

The predict-then-optimize paradigm bridges online learning and contextual optimization in dynamic environments. Previous works have investigated the sequential updating of predictors using feedback from downstream decisions to minimize regret in the full-information settings. However, existing approaches are predominantly frequentist, rely heavily on gradient-based strategies, and employ deterministic predictors that could yield high variance in practice despite their asymptotic guarantees. This work introduces, to the best of our knowledge, the first Bayesian online contextual optimization framework. Grounded in PAC-Bayes theory and general Bayesian updating principles, our framework achieves $mathcal{O}(sqrt{T})$ regret for bounded and mixable losses via a Gibbs posterior, eliminates the dependence on gradients through sequential Monte Carlo samplers, and thereby accommodates nondifferentiable problems. Theoretical developments and numerical experiments substantiate our claims.

Problem

• Develops Bayesian online contextual optimization framework using PAC-Bayes theory
• Addresses limitations of deterministic predictors with high variance
• Eliminates gradient dependence through sequential Monte Carlo samplers

Innovation

• Bayesian online contextual optimization framework
• PAC-Bayes theory with Gibbs posterior
• Sequential Monte Carlo samplers replace gradients