Preconditioning Kernel Matrices
This addresses scalability issues in kernel machines for large datasets, though it is incremental as it builds on existing conjugate gradient methods.
The paper tackles the poor convergence of conjugate gradient methods for kernel machines due to ill-conditioned kernel matrices by proposing a range of preconditioners, showing that this approach outperforms state-of-the-art approximations for a given computational budget.
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.