On the Estimation of Multinomial Logit and Nested Logit Models: A Conic Optimization Approach
多项 Logit 与嵌套 Logit 估计的锥优化方法AI 解读 ↓
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多项 Logit 与嵌套 Logit 估计的锥优化方法
- 研究问题
- 多项、嵌套与树状嵌套 Logit 的最大似然估计，能否通过锥优化改善梯度方法对初始化和步长的敏感性？
- 方法与思路
- 作者将多项 Logit 及固定尺度参数下的嵌套模型改写为指数锥规划，并用外层更新尺度、内层估计效用系数。
- 材料陈述的贡献
- 摘要称得到相应等价表示与条件凸性结果，实验中获得更好的似然解、初始化稳健性和速度，尤其在大规模问题中。
- 阅读关注点
- 应核对锥规划等价性的条件、尺度参数可行域，以及内层凸性是否延伸至整体问题和外层算法的收敛保证。
- 阅读边界
- 这是 arXiv 预印本，同行评审与接收未获确认；摘要未给出整体全局最优保证或性能数值，实验优势依赖比较设置。
原文摘要
In this paper, we revisit parameter estimation for multinomial logit (MNL), nested logit (NL), and tree-nested logit (TNL) models through the framework of convex conic optimization. Traditional approaches typically solve the maximum likelihood estimation (MLE) problem using gradient-based methods, which are sensitive to step-size selection and initialization, and may therefore suffer from slow or unstable convergence. In contrast, we propose a novel estimation strategy that reformulates these models as conic optimization problems, enabling more robust and reliable estimation procedures. Specifically, we show that the MLE for MNL admits an equivalent exponential cone program (ECP). For NL and TNL, we prove that when the dissimilarity (scale) parameters are fixed, the estimation problem is convex and likewise reducible to an ECP. Leveraging these results, we design a two-stage procedure: an outer loop that updates the scale parameters and an inner loop that solves the ECP to update the utility coefficients. The inner problems are handled by interior-point methods with iteration counts that grow only logarithmically in the target accuracy, as implemented in off-the-shelf solvers (e.g., MOSEK). Extensive experiments across estimation instances of varying size show that our conic approach attains better MLE solutions, greater robustness to initialization, and substantial speedups compared to standard gradient-based MLE, particularly on large-scale instances with high-dimensional specifications and large choice sets. Our findings establish exponential cone programming as a practical and scalable alternative for estimating a broad class of discrete choice models.