Kernel methods are a popular class of nonlinear predictive models in machine learning. Scalable algorithms for learning kernel models need to be iterative in nature, but convergence can be slow due to poor conditioning. Spectral preconditioning is an important tool to speed-up the convergence of such iterative algorithms for training kernel models. However computing and storing a spectral preconditioner can be expensive which can lead to large computational and storage overheads, precluding the application of kernel methods to problems with large datasets. A Nystrom approximation of the spectral preconditioner is often cheaper to compute and store, and has demonstrated success in practical applications. In this paper we analyze the trade-offs of using such an approximated preconditioner. Specifically, we show that a sample of logarithmic size (as a function of the size of the dataset) enables the Nystrom-based approximated preconditioner to accelerate gradient descent nearly as well as the exact preconditioner, while also reducing the computational and storage overheads.

翻译：核方法是一类机器学习中流行的非线性预测模型。可扩展的核模型学习算法本质上需要采用迭代方法，但糟糕的条件数会导致收敛缓慢。谱预处理是加速此类训练核模型的迭代算法收敛的重要工具。然而，计算和存储谱预处理器可能代价高昂，导致巨大的计算和存储开销，从而阻碍核方法应用于大规模数据集问题。Nyström近似谱预处理器通常更易于计算和存储，并在实际应用中取得了成功。本文分析了使用这种近似预处理器的权衡。具体而言，我们表明，对数规模（关于数据集大小的函数）的样本使得基于Nyström的近似预处理器能够几乎与精确预处理器一样快地加速梯度下降，同时降低计算和存储开销。