SHARP GLOBAL CONVERGENCE GUARANTEES FOR ITERATIVE NONCONVEX OPTIMIZATION WITH RANDOM DATA
成果类型：
Article
署名作者：
Chandrasekher, Kabir Aladin; Pananjady, Ashwin; Thrampoulidis, Christos
署名单位：
Stanford University; University System of Georgia; Georgia Institute of Technology; University System of Georgia; Georgia Institute of Technology; University System of Georgia; Georgia Institute of Technology; University of British Columbia
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/22-AOS2246
发表日期：
2023
页码：
179-210
关键词：
phase retrieval
em algorithm
parameter-estimation
Lasso
regression
mixtures
SPARSE
RISK
摘要：
We consider a general class of regression models with normally dis-tributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms for this task from a random initialization. In particular, provided each iteration can be written as the solution to a convex optimization problem satisfying some natural conditions, we leverage Gaussian compari-son theorems to derive a deterministic sequence that provides sharp upper and lower bounds on the error of the algorithm with sample splitting. Crucially, this deterministic sequence accurately captures both the convergence rate of the algorithm and the eventual error floor in the finite-sample regime, and is distinct from the commonly used ???population??? sequence that results from taking the infinite-sample limit. We apply our general framework to derive several concrete consequences for parameter estimation in popular statistical models including phase retrieval and mixtures of regressions. Provided the sample size scales near linearly in the dimension, we show sharp global con-vergence rates for both higher-order algorithms based on alternating updates and first-order algorithms based on subgradient descent. These corollaries, in turn, reveal multiple nonstandard phenomena that are then corroborated by extensive numerical experiments.