Approachability, Regret and Calibration; implications and equivalences

TL;DR

This paper explores the deep connections between approachability, regret minimization, and calibration in sequential decision problems, developing a unified geometric framework and new algorithms based on these links.

Contribution

It generalizes Blackwell's approachability theory and demonstrates how it underpins the construction of consistent and calibrated strategies in decision-making.

Findings

Approachability can be derived from the existence of calibrated strategies.

New algorithms are developed using geometric properties of approachability.

The theory unifies various criteria in sequential decision problems.

Abstract

Blackwell approachability, regret minimization and calibration are three criteria evaluating a strategy (or an algorithm) in different sequential decision problems, or repeated games between a player and Nature. Although they have at first sight nothing in common, links between have been discovered: both consistent and calibrated strategies can be constructed by following, in some auxiliary game, an approachability strategy. We gathered famous or recent results and provide new ones in order to develop and generalize Blackwell's elegant theory. The final goal is to show how it can be used as a basic powerful tool to exhibit a new class of intuitive algorithms, based on simple geometric properties. In order to be complete, we also prove that approachability can be seen as a byproduct of the very existence of consistent or calibrated strategies.