Convergence rates of Newton's method for strongly self-concordant minimization

TL;DR

This paper investigates the local convergence properties of Newton's method for strongly self-concordant functions, revealing faster convergence rates and larger convergence regions compared to general self-concordant functions.

Contribution

It provides the first detailed analysis of Newton's method specifically for strongly self-concordant functions, showing improved convergence rates and regions.

Findings

Quadratic convergence rate is faster for strongly self-concordant functions.

Larger local convergence region for strongly self-concordant functions.

Closes the theoretical gap in understanding Newton's method for this subclass.

Abstract

Newton's method has been thoroughly studied for the class of self-concordant functions. However, a local analysis specific to strongly self-concordant functions (a subclass of the former) is missing from the literature. The local quadratic rate of strongly self-concordant functions follows, of course, from the known results for self-concordant functions. However, it is not known whether strongly self-concordant functions enjoy better theoretical properties. In this paper, we study the local convergence of Newton's method for this subclass. We show that its quadratic convergence rate differs from that of general self-concordant functions. In particular, it is provably faster for a wide range of objective functions and benefits from a larger region of local convergence. Thus, the results of this paper close the gap in the theoretical understanding of Newton's method applied to strongly self-concordant functions.