Bandits with an Edge

TL;DR

This paper studies a graph-based bandit problem where rewards are inferred through pairwise comparisons, analyzing how graph topology influences the number of queries needed to identify near-optimal nodes with high confidence.

Contribution

It introduces a novel bandit model based on graph comparisons and characterizes how graph structure affects sample complexity for finding optimal nodes.

Findings

Graphs with low diameter reduce sample complexity

Sample complexity depends critically on graph topology

Efficient algorithms can leverage graph structure for better performance

Abstract

We consider a bandit problem over a graph where the rewards are not directly observed. Instead, the decision maker can compare two nodes and receive (stochastic) information pertaining to the difference in their value. The graph structure describes the set of possible comparisons. Consequently, comparing between two nodes that are relatively far requires estimating the difference between every pair of nodes on the path between them. We analyze this problem from the perspective of sample complexity: How many queries are needed to find an approximately optimal node with probability more than $1-\delta$ in the PAC setup? We show that the topology of the graph plays a crucial in defining the sample complexity: graphs with a low diameter have a much better sample complexity.