Internal Regret with Partial Monitoring. Calibration-Based Optimal Algorithms

TL;DR

This paper introduces new algorithms for sequential decision-making under partial feedback that achieve optimal internal and external regret bounds using a generalized calibration approach based on Laguerre diagrams.

Contribution

It presents the first algorithms with optimal regret bounds in partial monitoring using a novel calibration method based on Laguerre diagrams.

Findings

Expected average internal and external regret bounded by O(n^{-1/3})

Algorithms are consistent and do not depend on outcome observations

First to achieve optimal regret bounds in this framework

Abstract

We provide consistent random algorithms for sequential decision under partial monitoring, i.e. when the decision maker does not observe the outcomes but receives instead random feedback signals. Those algorithms have no internal regret in the sense that, on the set of stages where the decision maker chose his action according to a given law, the average payoff could not have been improved in average by using any other fixed law. They are based on a generalization of calibration, no longer defined in terms of a Voronoi diagram but instead of a Laguerre diagram (a more general concept). This allows us to bound, for the first time in this general framework, the expected average internal -- as well as the usual external -- regret at stage $n$ by $O(n^{-1/3})$, which is known to be optimal.