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：我们探讨对非参数有限混合物模型混合比例的最大可能性估计的计算方面 -- -- 一个与统计中的旧根和现代数据分析工具包关键成员之间的混凝土优化问题。受形状限制的推断问题的影响,我们考虑这一问题的结构变体,加上额外的convex 多元性制约。我们建议为这一问题采用新的立方正规牛顿法,并为我们的算法提供新颖的最坏情况和地方计算保证。我们将Nesterov和Polyak早先的工作扩大到具有多种限制的自我比较目标的情况,例如这里所考虑的那些限制。我们提出了用Frank-Wolfe 方法解决立方牛顿常规立方子问题;为可能具有独立兴趣的线性优化或触角找到有效的解决办法。在高斯混合物没有形状限制的特殊情况下,我们从有限的混合物问题与无限的Kiefer-Wolfowitz的最大可能性相近。对合成和真实数据设置的实验表明,我们提议的算法展示了更好的时间和缩度基准。