Alternating minimization and alternating descent over nonconvex sets

We analyze the performance of alternating minimization for loss functions optimized over two variables, where each variable may be restricted to lie in some potentially nonconvex constraint set. This type of setting arises naturally in high-dimensional statistics and signal processing, where the var...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2019-02
Main Authors: Ha, Wooseok, Rina Foygel Barber
Format: Article
Language:English
Subjects:
Algorithms
Concavity
Optimization
Regression analysis
Signal processing
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We analyze the performance of alternating minimization for loss functions optimized over two variables, where each variable may be restricted to lie in some potentially nonconvex constraint set. This type of setting arises naturally in high-dimensional statistics and signal processing, where the variables often reflect different structures or components within the signals being considered. Our analysis relies on the notion of local concavity coefficients, which has been proposed in Barber and Ha to measure and quantify the concavity of a general nonconvex set. Our results further reveal important distinctions between alternating and non-alternating methods. Since computing the alternating minimization steps may not be tractable for some problems, we also consider an inexact version of the algorithm and provide a set of sufficient conditions to ensure fast convergence of the inexact algorithms. We demonstrate our framework on several examples, including low rank + sparse decomposition and multitask regression, and provide numerical experiments to validate our theoretical results.
ISSN:2331-8422