Constrained Optimization via Exact Augmented Lagrangian and Randomized Iterative Sketching

Abstract

We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to optimal control, to PDE-constrained optimization. We develop an adaptive inexact Newton method for this problem class. In each iteration, we solve the Lagrangian Newton system inexactly via a randomized iterative sketching solver, and select a suitable stepsize by performing line search on an exact augmented Lagrangian merit function. The randomized solvers have advantages over deterministic linear system solvers by significantly reducing per-iteration flops complexity and storage cost, when equipped with suitable sketching matrices. Our method adaptively controls the accuracy of the randomized solver and the penalty parameters of the exact augmented Lagrangian, to ensure that the inexact Newton direction is a descent direction of the exact augmented Lagrangian. This allows us to establish a global almost sure convergence. We also show that a unit stepsize is admissible locally, so that our method exhibits a local linear convergence. Furthermore, we prove that the linear convergence can be strengthened to superlinear convergence if we gradually sharpen the adaptive accuracy condition on the randomized solver. We demonstrate the superior performance of our method on benchmark nonlinear problems in CUTEst test set, constrained logistic regression with data from LIBSVM, and a PDE-constrained problem.

Ilgee Hong

MS in Statistics (2021-2023)

Ilgee Hong is a master student in the Statistics department at UChicago, working with Mladen Kolar and Sen Na on randomized numerical methods, with an emphasis on machine learning applications.

Sen Na

Assistant Professor in ISyE

Sen Na is an Assistant Professor in the School of Industrial and Systems Engineering at Georgia Tech. Prior to joining ISyE, he was a postdoctoral researcher in the statistics department and ICSI at UC Berkeley. His research interests broadly lie in the mathematical foundations of data science, with topics including high-dimensional statistics, graphical models, semiparametric models, optimal control, and large-scale and stochastic nonlinear optimization. He is also interested in applying machine learning methods to biology, neuroscience, and engineering.

Michael W. Mahoney

Professor in Statistics and ICSI, Amazon Scholar

Michael Mahoney is a Professor in the Statistics department and ICSI at UC Berkeley. He is also the director of the NSF/TRIPODS-funded Foundations of Data Analysis (FODA) Institute at UC Berkeley. He works on the algorithmic and statistical aspects of modern large-scale data analysis. Much of his recent research has focused on large-scale machine learning, including randomized matrix algorithms and randomized numerical linear algebra.

Mladen Kolar

Associate Professor of Econometrics and Statistics

Mladen Kolar is an Associate Professor of Econometrics and Statistics at the University of Chicago Booth School of Business. His research is focused on high-dimensional statistical methods, graphical models, varying-coefficient models and data mining, driven by the need to uncover interesting and scientifically meaningful structures from observational data.