PAC-Bayes Meets Online Contextual Optimization

TL;DR

This paper introduces a novel Bayesian online contextual optimization framework based on PAC-Bayes theory, which achieves low regret, handles nondifferentiable problems, and moves beyond traditional frequentist, gradient-dependent methods.

Contribution

It is the first to integrate PAC-Bayes theory into online contextual optimization, providing a gradient-free, Bayesian approach with theoretical regret guarantees.

Findings

Achieves $\\mathcal{O}(\sqrt{T})$ regret for bounded, mixable losses.

Eliminates gradient dependence using sequential Monte Carlo samplers.

Validated through theoretical analysis and numerical experiments.

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.