Improved Algorithms for Bandit with Graph Feedback via Regret Decomposition

TL;DR

This paper introduces a novel algorithmic framework for bandit problems with graph feedback, decomposing regret to better understand and optimize learning performance across various graph structures.

Contribution

It proposes a new graph partition-based approach that unifies and generalizes existing algorithms, achieving improved regret bounds for diverse graph families.

Findings

Unified framework for strongly and weakly observable graphs

Achieves optimal regret bounds for bounded degree graphs

Generalizes algorithms for multiple graph structures

Abstract

The problem of bandit with graph feedback generalizes both the multi-armed bandit (MAB) problem and the learning with expert advice problem by encoding in a directed graph how the loss vector can be observed in each round of the game. The mini-max regret is closely related to the structure of the feedback graph and their connection is far from being fully understood. We propose a new algorithmic framework for the problem based on a partition of the feedback graph. Our analysis reveals the interplay between various parts of the graph by decomposing the regret to the sum of the regret caused by small parts and the regret caused by their interaction. As a result, our algorithm can be viewed as an interpolation and generalization of the optimal algorithms for MAB and learning with expert advice. Our framework unifies previous algorithms for both strongly observable graphs and weakly observable graphs, resulting in improved and optimal regret bounds on a wide range of graph families including graphs of bounded degree and strongly observable graphs with a few corrupted arms.