We explore computational aspects of maximum likelihood estimation of the mixture proportions of a nonparametric finite mixture model -- a convex optimization problem with old roots in statistics and a key member of the modern data analysis toolkit. Motivated by problems in shape constrained inference, we consider structured variants of this problem with additional convex polyhedral constraints. We propose a new cubic regularized Newton method for this problem and present novel worst-case and local computational guarantees for our algorithm. We extend earlier work by Nesterov and Polyak to the case of a self-concordant objective with polyhedral constraints, such as the ones considered herein. We propose a Frank-Wolfe method to solve the cubic regularized Newton subproblem; and derive efficient solutions for the linear optimization oracles that may be of independent interest. In the particular case of Gaussian mixtures without shape constraints, we derive bounds on how well the finite mixture problem approximates the infinite-dimensional Kiefer-Wolfowitz maximum likelihood estimator. Experiments on synthetic and real datasets suggest that our proposed algorithms exhibit improved runtimes and scalability features over existing benchmarks.

翻译：本文探讨了非参数有限混合模型中混合比例极大似然估计的计算问题——这是一个在统计学中具有悠久历史的凸优化问题，也是现代数据分析工具包中的核心组成部分。受形状约束推断问题的驱动，我们考虑了该问题在附加凸多面体约束下的结构变体。针对该问题，我们提出了一种新的三次正则化牛顿方法，并给出了算法在最坏情形和局部条件下的计算保证。我们将Nesterov与Polyak的早期工作推广至具有自和谐目标函数与多面体约束的情形（如本文所考虑的约束类型）。进一步地，我们提出了一种Frank-Wolfe方法求解三次正则化牛顿子问题，并推导了线性优化预言机的高效求解方案，这些求解方案可能具有独立的研究价值。在无形状约束的高斯混合特例中，我们给出了有限混合问题对无限维Kiefer-Wolfowitz极大似然估计逼近程度的上界。在合成数据集和真实数据集上的实验表明，与现有基准方法相比，本文所提算法在运行时间和可扩展性方面均表现出更优的性能。