A New Computational Framework for Log-Concave Density Estimation

• Activity Number: 386 - Nonparametric Modeling II
• Type: Contributed
• Date/Time: Thursday, August 12, 2021 : 12:00 PM to 1:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Wenyu Chen* and Rahul Mazumder and Richard J. Samworth
• Companies: Massachusetts Institute of Technology and Massachusetts Institute of Technology and University of Cambridge
• Keywords: log-concave density estimation; first-order methods; large-scale convex optimization; stochastic optimization; s-concave density estimation

Abstract

We propose a new computational framework for log-concave density maximum likelihood estimation (MLE). The MLE can be formulated into a convex minimization problem. The objective is nonsmooth and includes an integral term, which makes the problem hard to scale. We propose two different smoothing techniques to approximate the integral, and use techniques from first-order stochastic optimization to solve the problem at scale. We also provide computational guarantees for our proposed algorithms, both in expectation and with high probability. In practice, the algorithm is much faster than the existing solvers for the convex problem. Finally, we show that this computational framework also applies to the s-concave density MLE and quasi-concave density estimation.