Beyond Tikhonov: Faster Learning with Self-Concordant Losses via Iterative Regularization

TL;DR

This paper extends spectral filtering theory to generalized self-concordant losses, demonstrating that iterated Tikhonov regularization achieves faster and optimal learning rates beyond classical Tikhonov methods.

Contribution

It introduces the use of iterated Tikhonov regularization for GSC losses, surpassing classical Tikhonov regularization limitations and achieving faster convergence rates.

Findings

Iterated Tikhonov regularization attains faster convergence rates for GSC losses.

The approach overcomes limitations of classical Tikhonov regularization.

Optimal rates are achieved for a broader class of loss functions.

Abstract

The theory of spectral filtering is a remarkable tool to understand the statistical properties of learning with kernels. For least squares, it allows to derive various regularization schemes that yield faster convergence rates of the excess risk than with Tikhonov regularization. This is typically achieved by leveraging classical assumptions called source and capacity conditions, which characterize the difficulty of the learning task. In order to understand estimators derived from other loss functions, Marteau-Ferey et al. have extended the theory of Tikhonov regularization to generalized self concordant loss functions (GSC), which contain, e.g., the logistic loss. In this paper, we go a step further and show that fast and optimal rates can be achieved for GSC by using the iterated Tikhonov regularization scheme, which is intrinsically related to the proximal point method in optimization, and overcomes the limitation of the classical Tikhonov regularization.