在 $O(n \log n)$ 时间内学习多项式对数
Learning Multinomial Logits in $O(n \log n)$ time
AI总结：
在 $O(n \log n)$ 时间内，通过自适应和非自适应查询算法高效学习多项式对数模型的权重，确保总变差距离内的准确性。
摘要：
多项式对数（MNL）模型由一个有限的物品集合 $[n] = \{1, ..., n\}$ 组成，每个物品分配一个正权重。一个查询指定一个可接受的子集——称为一个 slate——模型从该子集中选择一个物品，其概率与权重成正比。这种查询模型也被称为 Plackett-Luce 模型或条件采样 oracle。尽管 MNL 已经被广泛研究，但一个基本的计算问题仍然开放：给定对 slate 的查询访问权限，我们如何高效地学习权重，使得对于每个 slate，所诱导的选择分布在总变差距离上与真实分布相差不超过 $\varepsilon$？这个问题对于 MNL 学习至关重要，并对现代推荐系统界面有直接的影响。我们为此任务提供了两种算法，一种使用自适应查询，另一种使用非自适应查询。每个算法输出一个 MNL $M'$，使得对于每个 slate $S$，$M'_S$ 在 $S$ 上的分布在总变差距离上与真实分布相差不超过 $\varepsilon$。我们的自适应算法做出 $O\left(\frac{n}{\varepsilon^{3}}\log n\right)$ 个查询，而我们的非自适应算法做出 $O\left(\frac{n^{2}}{\varepsilon^{3}}\log n \log\frac{n}{\varepsilon}\right)$ 个查询。这两种算法仅查询大小为两的 slate，并且运行时间与查询复杂度成比例。我们还补充了这些上界，给出了自适应查询的下界为 $Ω\left(\frac{n}{\varepsilon^{2}}\log n\right)$ 和非自适应查询的下界为 $Ω\left(\frac{n^{2}}{\varepsilon^{2}}\log n\right)$，从而证明我们的自适应算法在支持大小 $n$ 的依赖上是最优的，而非自适应算法在 $\log n$ 因子内是紧致的。
英文摘要：
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.
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