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Learning an arbitrary mixture of two multinomial logits
arXiv - CS - Computational Complexity Pub Date : 2020-07-01 , DOI: arxiv-2007.00204
Wenpin Tang

In this paper, we consider mixtures of multinomial logistic models (MNL), which are known to $\epsilon$-approximate any random utility model. Despite its long history and broad use, rigorous results are only available for learning a uniform mixture of two MNLs. Continuing this line of research, we study the problem of learning an arbitrary mixture of two MNLs. We show that the identifiability of the mixture models may only fail on an algebraic variety of a negligible measure. This is done by reducing the problem of learning a mixture of two MNLs to the problem of solving a system of univariate quartic equations. We also devise an algorithm to learn any mixture of two MNLs using a polynomial number of samples and a linear number of queries, provided that a mixture of two MNLs over some finite universe is identifiable. Several numerical experiments and conjectures are also presented.

中文翻译:

学习两个多项式对数的任意混合

在本文中,我们考虑多项逻辑模型 (MNL) 的混合,这些模型已知 $\epsilon$-近似于任何随机效用模型。尽管其历史悠久且用途广泛,但严格的结果仅适用于学习两种 MNL 的均匀混合。继续这条研究线,我们研究学习两个 MNL 的任意混合的问题。我们表明,混合模型的可识别性可能只会在可忽略不计的代数变体上失败。这是通过将学习两个 MNL 的混合问题简化为求解单变量四次方程组的问题来完成的。我们还设计了一种算法来使用多项式样本数和线性查询数来学习两个 MNL 的任何混合,前提是在某个有限的宇宙中两个 MNL 的混合是可识别的。
更新日期:2020-09-29
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