Unimodal Mono-Partite Matching in a Bandit Setting

TL;DR

This paper introduces a new approach for finding optimal monopartite matchings in weighted graphs within a bandit setting, improving regret bounds by leveraging unimodality and better user comparison methods, with experimental validation.

Contribution

It presents a novel algorithm that reduces regret bounds for unimodal monopartite matching in bandit problems by utilizing unimodality and improved user comparison strategies.

Findings

Regret bound improved to $O(L\frac{1}{\Delta}\log(T))$ using unimodality.

Further reduction to $O(L\frac{\Delta}{\tilde{\Delta}^2}\log(T))$ with better user comparison.

Experimental results support the theoretical improvements.

Abstract

We tackle a new emerging problem, which is finding an optimal monopartite matching in a weighted graph. The semi-bandit version, where a full matching is sampled at each iteration, has been addressed by \cite{ADMA}, creating an algorithm with an expected regret matching $O(\frac{L\log(L)}{\Delta}\log(T))$ with $2L$ players, $T$ iterations and a minimum reward gap $\Delta$. We reduce this bound in two steps. First, as in \cite{GRAB} and \cite{UniRank} we use the unimodality property of the expected reward on the appropriate graph to design an algorithm with a regret in $O(L\frac{1}{\Delta}\log(T))$. Secondly, we show that by moving the focus towards the main question `\emph{Is user $i$ better than user $j$?}' this regret becomes $O(L\frac{\Delta}{\tilde{\Delta}^2}\log(T))$, where $\Tilde{\Delta} > \Delta$ derives from a better way of comparing users. Some experimental results finally show these theoretical results are corroborated in practice.