Preconditioning Kernel Matrices

TL;DR

This paper introduces preconditioned conjugate gradient methods tailored for kernel machines, enhancing scalability and convergence by developing effective preconditioners and scalable hyperparameter learning techniques.

Contribution

It proposes a novel preconditioning approach for kernel matrices, improving the efficiency and scalability of kernel machine training and hyperparameter optimization.

Findings

Outperforms state-of-the-art approximations within the same computational budget.

Provides a scalable method for solving kernel machines and learning hyperparameters.

Demonstrates exactness in the limit of iterations.

Abstract

The computational and storage complexity of kernel machines presents the primary barrier to their scaling to large, modern, datasets. A common way to tackle the scalability issue is to use the conjugate gradient algorithm, which relieves the constraints on both storage (the kernel matrix need not be stored) and computation (both stochastic gradients and parallelization can be used). Even so, conjugate gradient is not without its own issues: the conditioning of kernel matrices is often such that conjugate gradients will have poor convergence in practice. Preconditioning is a common approach to alleviating this issue. Here we propose preconditioned conjugate gradients for kernel machines, and develop a broad range of preconditioners particularly useful for kernel matrices. We describe a scalable approach to both solving kernel machines and learning their hyperparameters. We show this approach is exact in the limit of iterations and outperforms state-of-the-art approximations for a given computational budget.