Minimizing Quasi-Self-Concordant Functions by Gradient Regularization of Newton Method

TL;DR

This paper introduces a simple, gradient-regularized Newton method for Quasi-Self-Concordant functions, achieving fast global linear convergence and improving upon previous trust-region schemes in complexity and simplicity.

Contribution

It proposes a gradient-regularized Newton method and an accelerated variant with proven fast convergence for Quasi-Self-Concordant functions, simplifying implementation and improving efficiency.

Findings

Achieves fast global linear convergence rate matching trust-region methods.

Simplifies implementation to basic matrix inversion at each step.

Applies to practical problems like Logistic Regression and Matrix Scaling without strong convexity assumptions.

Abstract

We study the composite convex optimization problems with a Quasi-Self-Concordant smooth component. This problem class naturally interpolates between classic Self-Concordant functions and functions with Lipschitz continuous Hessian. Previously, the best complexity bounds for this problem class were associated with trust-region schemes and implementations of a ball-minimization oracle. In this paper, we show that for minimizing Quasi-Self-Concordant functions we can use instead the basic Newton Method with Gradient Regularization. For unconstrained minimization, it only involves a simple matrix inversion operation (solving a linear system) at each step. We prove a fast global linear rate for this algorithm, matching the complexity bound of the trust-region scheme, while our method remains especially simple to implement. Then, we introduce the Dual Newton Method, and based on it, develop the corresponding Accelerated Newton Scheme for this problem class, which further improves the complexity factor of the basic method. As a direct consequence of our results, we establish fast global linear rates of simple variants of the Newton Method applied to several practical problems, including Logistic Regression, Soft Maximum, and Matrix Scaling, without requiring additional assumptions on strong or uniform convexity for the target objective.