Learning Multinomial Logits in $O(n \log n)$ time

TL;DR

This paper presents efficient algorithms for learning the weights of a Multinomial Logit model from query access, achieving near-optimal query complexity with both adaptive and non-adaptive methods, and establishes matching lower bounds.

Contribution

It introduces the first algorithms with provable guarantees for learning MNL weights efficiently, including adaptive and non-adaptive approaches, with tight bounds on query complexity.

Findings

Adaptive algorithm uses $O(n/\

Non-adaptive algorithm uses $O(n^2/\

Lower bounds match the upper bounds up to logarithmic factors.

Abstract

A Multinomial Logit (MNL) model is composed of a finite universe of items $[n]=\{1,..., n\}$, each assigned a positive weight. A query specifies an admissible subset -- called a slate -- and the model chooses one item from that slate with probability proportional to its weight. This query model is also known as the Plackett-Luce model or conditional sampling oracle in the literature. Although MNLs have been studied extensively, a basic computational question remains open: given query access to slates, how efficiently can we learn weights so that, for every slate, the induced choice distribution is within total variation distance $\varepsilon$ of the ground truth? This question is central to MNL learning and has direct implications for modern recommender system interfaces. We provide two algorithms for this task, one with adaptive queries and one with non-adaptive queries. Each algorithm outputs an MNL $M'$ that induces, for each slate $S$, a distribution $M'_S$ on $S$ that is within $\varepsilon$ total variation distance of the true distribution. Our adaptive algorithm makes $O\left(\frac{n}{\varepsilon^{3}}\log n\right)$ queries, while our non-adaptive algorithm makes $O\left(\frac{n^{2}}{\varepsilon^{3}}\log n \log\frac{n}{\varepsilon}\right)$ queries. Both algorithms query only slates of size two and run in time proportional to their query complexity. We complement these upper bounds with lower bounds of $\Omega\left(\frac{n}{\varepsilon^{2}}\log n\right)$ for adaptive queries and $\Omega\left(\frac{n^{2}}{\varepsilon^{2}}\log n\right)$ for non-adaptive queries, thus proving that our adaptive algorithm is optimal in its dependence on the support size $n$, while the non-adaptive one is tight within a $\log n$ factor.