A Minimalist Bayesian Framework for Stochastic Optimization
Kaizheng Wang
TL;DR
The paper addresses the challenge of applying Bayesian methods to structured stochastic optimization by proposing a minimalist framework that places priors only on the parameter of interest (such as the optimum) and profiles out nuisance parameters via the profile likelihood. This yields a generalized posterior that can accommodate constraints and structure, enabling the MINTS algorithm to operate effectively in complex settings like continuum-armed Lipschitz bandits and dynamic pricing. The authors provide near-optimal regret guarantees for MINTS in multi-armed bandits, and offer novel probabilistic interpretations of classical convex-optimization methods through the Bayesian lens. Overall, the approach delivers flexible, structure-aware Bayesian decision-making with solid theoretical guarantees and practical performance enhancements.
Abstract
The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the component of interest, such as the location of the optimum. Nuisance parameters are eliminated via profile likelihood, which naturally handles constraints. As a direct instantiation, we develop a MINimalist Thompson Sampling (MINTS) algorithm. Our framework accommodates structured problems, including continuum-armed Lipschitz bandits and dynamic pricing. It also provides a probabilistic lens on classical convex optimization algorithms such as the center of gravity and ellipsoid methods. We further analyze MINTS for multi-armed bandits and establish near-optimal regret guarantees.