A Fast Algorithm for Maximum Likelihood Estimation of Mixture Proportions Using Sequential Quadratic Programming

Maximum likelihood estimation of mixture proportions has a long history, and continues to play an important role in modern statistics, including in development of nonparametric empirical Bayes methods. Maximum likelihood of mixture proportions has traditionally been solved using the expectation maximization (EM) algorithm, but recent work by Koenker and Mizera shows that modern convex optimization techniques—in particular, interior point methods—are substantially faster and more accurate than EM. Here, we develop a new solution based on sequential quadratic programming (SQP). It is substantially faster than the interior point method, and just as accurate. Our approach combines several ideas: first, it solves a reformulation of the original problem; second, it uses an SQP approach to make the best use of the expensive gradient and Hessian computations; third, the SQP iterations are implemented using an active set method to exploit the sparse nature of the quadratic subproblems; fourth, it uses accurate low-rank approximations for more efficient gradient and Hessian computations. We illustrate the benefits of the SQP approach in experiments on synthetic datasets and a large genetic association dataset. In large datasets (n≈106 observations, m≈103 mixture components), our implementation achieves at least 100-fold reduction in runtime compared with a state-of-the-art interior point solver. Our methods are implemented in Julia and in an R package available on CRAN (https://CRAN.R-project.org/package=mixsqp). Supplementary materials for this article are available online.