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

2026-01-07

AI Summary

This work addresses the problem of efficiently learning the weight vector of a multinomial logit model such that, for any subset of alternatives, the predicted choice distribution is within total variation distance ε of the true distribution. The authors propose both adaptive and non-adaptive algorithms that rely solely on queries to subsets of size two, operating in the conditional sampling oracle model to reconstruct the full weight vector. Their main contributions include achieving the first optimal query complexity of \(O(n/\varepsilon^3 \log n)\) in the adaptive setting and a nearly tight upper bound of \(O(n^2/\varepsilon^3 \log n \log(n/\varepsilon))\) in the non-adaptive setting. Furthermore, they establish information-theoretic lower bounds of \(\Omega(n/\varepsilon^2 \log n)\) and \(\Omega(n^2/\varepsilon^2 \log n)\) for the adaptive and non-adaptive cases, respectively, thereby demonstrating the near-optimality of their algorithms.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Social networks and social learning
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Multinomial Logit
• total variation distance
• query complexity
• Plackett-Luce model
• conditional sampling

Innovation

• Multinomial Logit
• adaptive queries
• query complexity
• total variation distance
• Plackett-Luce model