February 2023 Sharp global convergence guarantees for iterative nonconvex optimization with random data
Kabir Aladin Chandrasekher, Ashwin Pananjady, Christos Thrampoulidis
Author Affiliations +
Ann. Statist. 51(1): 179-210 (February 2023). DOI: 10.1214/22-AOS2246


We consider a general class of regression models with normally distributed 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 comparison 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 convergence 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.

Funding Statement

KAC was supported in part by a National Science Foundation Graduate Research Fellowship and the Sony Stanford Graduate Fellowship. AP was supported in part by a research fellowship from the Simons Institute and National Science Foundation Grant CCF-2107455. CT was supported in part by the National Science Foundation Grant CCF-2009030, by an NSERC Discovery Grant and by a research grant from KAUST.


We thank the program on Probability, Geometry and Computation in High Dimensions at the Simons Institute for the Theory of Computing for hosting us when part of this work was performed. We also thank the anonymous referees whose input improved the scope, clarity and presentation of this manuscript. In particular, we are grateful to an anonymous reviewer for a suggestion that led to the improved rate of O˜(n1/2) for first-order methods in Theorem 2.


Download Citation

Kabir Aladin Chandrasekher. Ashwin Pananjady. Christos Thrampoulidis. "Sharp global convergence guarantees for iterative nonconvex optimization with random data." Ann. Statist. 51 (1) 179 - 210, February 2023. https://doi.org/10.1214/22-AOS2246


Received: 1 December 2021; Revised: 1 September 2022; Published: February 2023
First available in Project Euclid: 23 March 2023

Digital Object Identifier: 10.1214/22-AOS2246

Primary: 62J02 , 90C06
Secondary: 90C26

Keywords: convergence rate , nonconvex optimization , precise iterate-by-iterate prediction

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 1 • February 2023
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