# Learning Mixed Multinomial Logit Model from Ordinal Data

arXiv: https://arxiv.org/abs/1411.0073 · HTML: https://mlsanity.com/paper/1411.0073/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Nov 2014.

## Built on by (most cited first, 7 of 7)

- [Learning Rich Rankings](https://mlsanity.com/paper/2312.15081/) (2023)
- [On the identifiability of mixtures of ranking models](https://mlsanity.com/paper/2201.13132/) (2022)
- [The Sign Estimator: LLM Alignment in the Face of Choice Heterogeneity](https://mlsanity.com/paper/2510.23965/) (2025)
- [Fast Learning of MNL Model from General Partial Rankings with Application to Network Formation Modeling](https://mlsanity.com/paper/2112.15575/) (2021)
- [Approximating a RUM from Distributions on k-Slates](https://mlsanity.com/paper/2305.13283/) (2023)
- [Distortion of AI Alignment Revisited: RLHF is a Decent Utilitarian Aligner](https://mlsanity.com/paper/2609.12651/) (2026)
- [Generalized Top-k Mallows Model for Ranked Choices](https://mlsanity.com/paper/2510.22040/) (2025)

## Abstract (arXiv)

Motivated by generating personalized recommendations using ordinal (or preference) data, we study the question of learning a mixture of MultiNomial Logit (MNL) model, a parameterized class of distributions over permutations, from partial ordinal or preference data (e.g. pair-wise comparisons). Despite its long standing importance across disciplines including social choice, operations research and revenue management, little is known about this question. In case of single MNL models (no mixture), computationally and statistically tractable learning from pair-wise comparisons is feasible. However, even learning mixture with two MNL components is infeasible in general. Given this state of affairs, we seek conditions under which it is feasible to learn the mixture model in both computationally and statistically efficient manner. We present a sufficient condition as well as an efficient algorithm for learning mixed MNL models from partial preferences/comparisons data. In particular, a mixture of r MNL components over n objects can be learnt using samples whose size scales polynomially in n and r (concretely, r3.5n3(log n)4, with rll n2/7 when the model parameters are sufficiently incoherent). The algorithm has two phases: first, learn the pair-wise marginals for each component using tensor decomposition; second, learn the model parameters for each component using Rank Centrality introduced by Negahban et al. In the process of proving these results, we obtain a generalization of existing analysis for tensor decomposition to a more realistic regime where only partial information about each sample is available.