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

TL;DR

This work tackles the problem of learning a Multinomial Logit (MNL) model on a universe of $[n]$ items from slate-conditioned samples, aiming to approximate the distribution on every nonempty slate within $\varepsilon$ in $d_1$ distance. It introduces an estimation-forest framework that encodes weight ratios along short paths in a forest built via adaptive clustering of similarly-weighted items, enabling efficient reconstruction of MNL weights from pairwise queries. The authors present two algorithms: an adaptive one with $O\left(\frac{n}{\varepsilon^{3}}\log n\right)$ queries and a non-adaptive one with $O\left(\frac{n^{2}}{\varepsilon^{3}}\log n \log\frac{n}{\varepsilon}\right)$ queries, both querying only slate size-two sets; they complement these with lower bounds $\Omega\left(\frac{n}{\varepsilon^{2}}\log n\right)$ (adaptive) and $\Omega\left(\frac{n^{2}}{\varepsilon^{2}}\log n\right)$ (non-adaptive). The results establish near-optimal query complexity in $n$ and show the adaptive approach is optimal up to constant factors, while the non-adaptive method matches within a $\log n$ factor. The techniques advance conditional distribution learning for complex slate-conditional models and have direct implications for scalable, slate-aware recommender systems. Overall, the paper provides a quasi-linear-time framework to recover MNL weights from restricted queries, enabling accurate slate-wise decision distributions in practical interfaces.

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 $Ω\left(\frac{n}{\varepsilon^{2}}\log n\right)$ for adaptive queries and $Ω\left(\frac{n^{2}}{\varepsilon^{2}}\log n\right)$ 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.

Figures (5)

• Figure 1: The structure of an $(A_1,A_2,\varepsilon)$-cluster graph. The vertices of the graph are the items $[n]$ of the MNL, the cluster centers are depicted as white-filled squares, while the other items are represented by black circles. Items in the same cluster have similar weight (within a factor of $A_1$ of each other). Clusters further to the right contain items of higher weights. Associated with each edge $\{u,v\}$, and each direction (say, $u \to v$), is an estimate $r(u,v)$ of the ratio $w_u/w_v$.
• Figure 2: The structure of an estimation-forest constructed by \ref{['alg:nlogn-build-estimation-forest']} in \ref{['sec:adaptive-algorithm']}. White squares represent cluster centers, while black circles represent the other items of $[n]$. A new level in the forest is created when two nodes are compared and the estimate of their ratio is "$\infty$". If this happens twice consecutively (for the parent node and the children with smallest estimated weight), then a new tree is created. In the figure, we have $i < T-1$ and $j < i-1$.
• Figure 3: The structure of our algorithm to learn MNLs adaptively. The non-adaptive algorithm follows the same overall structure, but the first two steps are replaced by ${\normalfont{\texttt{QuicksortClustering}}}$ (described in \ref{['prop:quicksort-cluster-graph']}) and by \ref{['alg:build-forest-non-adaptive']} respectively.
• Figure 4: Highlight of some vertices of the estimation-forest computed by \ref{['alg:nlogn-build-estimation-forest']} that are used in the proofs of \ref{['lem:nlogn-distant-vertices-real']} and \ref{['lem:nlogn-distant-vertices-estimates']}. White squares are cluster centers and black dots are items that are not cluster centers. Curly edges represent paths of length $\geq 0$. The dashed edge is not present in the tree but indicates that \ref{['alg:nlogn-build-estimation-forest']} observed $r({c_z}, {c_k})=\infty$. Vertex $c_x$ is referenced only in the proof of \ref{['lem:nlogn-distant-vertices-estimates']}. Note that in the figure we assume that $u$ and $v$ are not centers, but it might also be $u=c_{\gamma(u)}$ or $v=c_{\gamma(v)}$. Moreover, we assume that $c_{\gamma(u)}$ and $c_x$ are siblings but it might also be $c_{\gamma(u)}=c_x$ (similarly for $c_k$ and $c_{\gamma(v)}$). In particular, it holds that $R\geq \gamma(u) \geq x \geq z > k \geq \gamma(v)$, and also $\gamma(u) > z$.
• Figure 5: The instance $\overline{M}(n,\vec{p},\pi)$ used for the lower bounds. Among two items in the same pair $\pi(2i-1)$ and $\pi(2i)$ the latter wins with probability $\vec{p}_i$. The winner is always an item of the right-most pair.

Theorems & Definitions (96)

• Definition 1: MNL Learning Problem
• Theorem 2: Informal
• Theorem 3: Informal
• Definition 4: $(t,\varepsilon)$-Estimation-Forest
• Definition 5: Cluster Graph
• Theorem 6
• Lemma 7
• proof
• Theorem 8
• Corollary 9