Globally Convergent Newton Methods for Ill-conditioned Generalized Self-concordant Losses

TL;DR

This paper introduces globally convergent Newton methods for ill-conditioned generalized self-concordant losses, achieving optimal complexity and generalization bounds in large-scale convex optimization, especially for logistic and softmax regressions.

Contribution

It proposes a new Newton-based scheme with proven global convergence and improved behavior in ill-conditioned problems, extending to non-parametric settings with theoretical guarantees.

Findings

Algorithm achieves linear convergence with logarithmic dependence on condition number.

Provides an explicit non-parametric algorithm with optimal complexity and no dependence on condition number.

First large-scale method with theoretical guarantees for logistic and softmax regression in ill-conditioned settings.

Abstract

In this paper, we study large-scale convex optimization algorithms based on the Newton method applied to regularized generalized self-concordant losses, which include logistic regression and softmax regression. We first prove that our new simple scheme based on a sequence of problems with decreasing regularization parameters is provably globally convergent, that this convergence is linear with a constant factor which scales only logarithmically with the condition number. In the parametric setting, we obtain an algorithm with the same scaling than regular first-order methods but with an improved behavior, in particular in ill-conditioned problems. Second, in the non parametric machine learning setting, we provide an explicit algorithm combining the previous scheme with Nystr{\"o}m projection techniques, and prove that it achieves optimal generalization bounds with a time complexity of order O(ndf $\lambda$), a memory complexity of order O(df 2 $\lambda$) and no dependence on the condition number, generalizing the results known for least-squares regression. Here n is the number of observations and df $\lambda$ is the associated degrees of freedom. In particular, this is the first large-scale algorithm to solve logistic and softmax regressions in the non-parametric setting with large condition numbers and theoretical guarantees.