# Minimax Rate for Learning From Pairwise Comparisons in the BTL Model

ICML, pp. 4193-4202, 2020.

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Abstract:

We consider the problem of learning the qualities w1, . . . , wn of a collection of items by performing noisy comparisons among them. A standard assumption is that there is a fixed “comparison graph” and every neighboring pair of items is compared k times. We will study the popular Bradley-Terry-Luce model, where the probability that item...More

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Introduction

- Estimation of item qualities from user preferences is a common problem across multiple domains in e-commerce, health care, and social science.
- The dominant approach is to rely on raw scores provided by users; for instance, Amazon asks customers for ratings on a scale ranging from 1-5 stars, which are aggregated to produce an average rating for each item.
- Such user-provided scores can be poorly calibrated.
- Many additional examples can be given and the authors refer the reader to (Cattelan, 2012) for an extensive overview of comparison models and their uses

Highlights

- Estimation of item qualities from user preferences is a common problem across multiple domains in e-commerce, health care, and social science
- The outcome of a sports game can be viewed as the result of a noisy comparison of the strengths of the two teams
- The purpose of the present paper is to present a new algorithm, coupled with new upper and lower bounds, which characterize the minimax rate for this problem
- Our main contribution is the determination of the asymptotic minimax rate for inference from pairwise comparisons
- Besides the conjectures discussed in Section 3, the most natural open question raised by our work is to understand how big the number of samples per edge k has to be for the minimax rate derived in this paper to kick in

Conclusion

- The authors' main contribution is the determination of the asymptotic minimax rate for inference from pairwise comparisons.
- Besides the conjectures discussed in Section 3, the most natural open question raised by the work is to understand how big the number of samples per edge k has to be for the minimax rate derived in this paper to kick in.
- The authors would conjecture that tr(L†γ)/||w||22 is, up to constant factors, the minimax rate and the sample complexity of recovering w

Summary

## Introduction:

Estimation of item qualities from user preferences is a common problem across multiple domains in e-commerce, health care, and social science.- The dominant approach is to rely on raw scores provided by users; for instance, Amazon asks customers for ratings on a scale ranging from 1-5 stars, which are aggregated to produce an average rating for each item.
- Such user-provided scores can be poorly calibrated.
- Many additional examples can be given and the authors refer the reader to (Cattelan, 2012) for an extensive overview of comparison models and their uses
## Conclusion:

The authors' main contribution is the determination of the asymptotic minimax rate for inference from pairwise comparisons.- Besides the conjectures discussed in Section 3, the most natural open question raised by the work is to understand how big the number of samples per edge k has to be for the minimax rate derived in this paper to kick in.
- The authors would conjecture that tr(L†γ)/||w||22 is, up to constant factors, the minimax rate and the sample complexity of recovering w

Reference

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