# The Bayesian Prophet: A Low-Regret Framework for Online Decision Making

## TL;DR

This paper introduces a Bayesian framework for online decision-making that leverages statistical oracles, developing a novel technique called compensated coupling to establish constant regret bounds for a broad class of problems.

## Contribution

The paper presents a new compensated coupling technique and demonstrates that the Bayes Selector policy achieves constant regret in online allocation problems, generalizing existing results.

## Key findings

- Compensated coupling bounds expected regret for online policies.
- Bayes Selector achieves constant regret in online allocation problems.
- Results unify and extend prior online packing and matching analyses.

## Abstract

We develop a new framework for designing online policies given access to an oracle providing statistical information about an offline benchmark. Having access to such prediction oracles enables simple and natural Bayesian selection policies, and raises the question as to how these policies perform in different settings. Our work makes two important contributions towards this question: First, we develop a general technique we call *compensated coupling* which can be used to derive bounds on the expected regret (i.e., additive loss with respect to a benchmark) for any online policy and offline benchmark. Second, using this technique, we show that a natural greedy policy, which we call *the Bayes Selector*, has constant expected regret (i.e., independent of the number of arrivals and resource levels) for a large class of problems we refer to as Online Allocation with finite types, which includes widely-studied Online Packing and Online Matching problems. Our results generalize and simplify several existing results for Online Packing and Online Matching, and suggest a promising pathway for obtaining oracle-driven policies for other online decision-making settings.