Beyond Tikhonov: Faster Learning with Self-Concordant Losses via Iterative Regularization
This work addresses a theoretical bottleneck in machine learning for researchers and practitioners using kernel methods with non-least-squares losses, offering incremental improvements over existing regularization approaches.
The paper tackles the problem of achieving faster convergence rates for learning with generalized self-concordant loss functions, such as logistic loss, by proposing an iterated Tikhonov regularization scheme, which overcomes limitations of classical Tikhonov regularization and yields optimal rates.
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.