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

arXiv: https://arxiv.org/abs/2007.03133 · HTML: https://mlsanity.com/paper/2007.03133/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jul 2020.

## Built on by (most cited first, 9 of 9)

- [Near-Optimal Pure Exploration in Matrix Games: A Generalization of Stochastic Bandits & Dueling Bandits](https://mlsanity.com/paper/2310.16252/) (2023)
- [Finding Optimal Arms in Non-stochastic Combinatorial Bandits with Semi-bandit Feedback and Finite Budget](https://mlsanity.com/paper/2202.04487/) (2022)
- [Semi-verified PAC Learning from the Crowd](https://mlsanity.com/paper/2106.07080/) (2021)
- [Dueling Bandits with Team Comparisons](https://mlsanity.com/paper/2107.02738/) (2021)
- [Identifying Copeland Winners in Dueling Bandits with Indifferences](https://mlsanity.com/paper/2310.00750/) (2023)
- [Active Learners as Efficient PRP Rerankers](https://mlsanity.com/paper/2605.14236/) (2026)
- [Tree-Guided Identify-Then-Exploit: A Unified Framework of Best Arm Identification and Regret Minimization for Dueling Bandits](https://mlsanity.com/paper/2606.01799/) (2026)
- [Ranking-and-Selection with Multiple Correct Answers and Non-Answerable Estimates](https://mlsanity.com/paper/2606.21889/) (2026)
- [Optimal Top-k Identification from Pairwise Comparisons](https://mlsanity.com/paper/2607.08979/) (2026)

## Abstract (arXiv)

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.