# Learning from Comparisons and Choices

arXiv: https://arxiv.org/abs/1704.07228 · HTML: https://mlsanity.com/paper/1704.07228/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Apr 2017.

## Built on by (most cited first, 12 of 12)

- [Principled Reinforcement Learning with Human Feedback from Pairwise or K-wise Comparisons](https://mlsanity.com/paper/2301.11270/) (2023)
- [Human-in-the-loop: Provably Efficient Preference-based Reinforcement Learning with General Function Approximation](https://mlsanity.com/paper/2205.11140/) (2022)
- [Lagrangian Inference for Ranking Problems](https://mlsanity.com/paper/2110.00151/) (2021)
- [Poisoning Attack against Estimating from Pairwise Comparisons](https://mlsanity.com/paper/2107.01854/) (2021)
- [Learning Rich Rankings](https://mlsanity.com/paper/2312.15081/) (2023)
- [Learning Linear Utility Functions From Pairwise Comparison Queries](https://mlsanity.com/paper/2405.02612/) (2024)
- [Minimax Hypothesis Testing for the Bradley-Terry-Luce Model](https://mlsanity.com/paper/2410.08360/) (2024)
- [Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms](https://mlsanity.com/paper/2201.06438/) (2022)
- [RecruitView: A Multimodal Dataset for Predicting Personality and Interview Performance for Human Resources Applications](https://mlsanity.com/paper/2512.00450/) (2025)
- [Approximating a RUM from Distributions on k-Slates](https://mlsanity.com/paper/2305.13283/) (2023)
- [Recommendations with Sparse Comparison Data: Provably Fast Convergence for Nonconvex Matrix Factorization](https://mlsanity.com/paper/2502.20033/) (2025)
- [Population-Level Generative Modeling for Ranking Data](https://mlsanity.com/paper/2608.08422/) (2026)

## Abstract (arXiv)

When tracking user-specific online activities, each user's preference is revealed in the form of choices and comparisons. For example, a user's purchase history is a record of her choices, i.e. which item was chosen among a subset of offerings. A user's preferences can be observed either explicitly as in movie ratings or implicitly as in viewing times of news articles. Given such individualized ordinal data in the form of comparisons and choices, we address the problem of collaboratively learning representations of the users and the items. The learned features can be used to predict a user's preference of an unseen item to be used in recommendation systems. This also allows one to compute similarities among users and items to be used for categorization and search. Motivated by the empirical successes of the MultiNomial Logit (MNL) model in marketing and transportation, and also more recent successes in word embedding and crowdsourced image embedding, we pose this problem as learning the MNL model parameters that best explain the data. We propose a convex relaxation for learning the MNL model, and show that it is minimax optimal up to a logarithmic factor by comparing its performance to a fundamental lower bound. This characterizes the minimax sample complexity of the problem, and proves that the proposed estimator cannot be improved upon other than by a logarithmic factor. Further, the analysis identifies how the accuracy depends on the topology of sampling via the spectrum of the sampling graph. This provides a guideline for designing surveys when one can choose which items are to be compared. This is accompanied by numerical simulations on synthetic and real data sets, confirming our theoretical predictions.