The Sample Complexity of Best-$k$ Items Selection from Pairwise Comparisons

TL;DR

This paper analyzes the number of comparisons needed to identify the top-$k$ items from noisy pairwise comparisons, providing tight bounds and algorithms for both approximate and exact selection under stochastic assumptions.

Contribution

It establishes tight bounds and proposes algorithms for best-$k$ item selection from noisy comparisons, improving understanding of sample complexity in active learning scenarios.

Findings

Sample complexity bounds match lower bounds up to constants.

Algorithms are nearly optimal for both PAC and exact best-$k$ selection.

Provides theoretical guarantees under stochastic transitivity and triangle inequality.

Abstract

This paper studies the sample complexity (aka number of comparisons) bounds for the active best-$k$ items selection from pairwise comparisons. From a given set of items, the learner can make pairwise comparisons on every pair of items, and each comparison returns an independent noisy result about the preferred item. At any time, the learner can adaptively choose a pair of items to compare according to past observations (i.e., active learning). The learner's goal is to find the (approximately) best-$k$ items with a given confidence, while trying to use as few comparisons as possible. In this paper, we study two problems: (i) finding the probably approximately correct (PAC) best-$k$ items and (ii) finding the exact best-$k$ items, both under strong stochastic transitivity and stochastic triangle inequality. For PAC best-$k$ items selection, we first show a lower bound and then propose an algorithm whose sample complexity upper bound matches the lower bound up to a constant factor. For the exact best-$k$ items selection, we first prove a worst-instance lower bound. We then propose two algorithms based on our PAC best items selection algorithms: one works for $k=1$ and is sample complexity optimal up to a loglog factor, and the other works for all values of $k$ and is sample complexity optimal up to a log factor.