Truncated Kernel Stochastic Gradient Descent with General Losses and Spherical Radial Basis Functions
This work addresses the scalability bottleneck of kernel methods for practitioners dealing with large datasets, though it represents an incremental improvement over existing kernel SGD approaches.
The authors tackled the problem of scaling kernel stochastic gradient descent for large-scale supervised learning by proposing a novel regularization strategy that projects gradients onto finite-dimensional spaces using spherical radial basis functions. Their algorithm achieves minimax-optimal convergence rates while reducing computational complexity from quadratic to linear, enabling efficient processing of streaming data.
In this paper, we propose a novel kernel stochastic gradient descent (SGD) algorithm for large-scale supervised learning with general losses. Compared to traditional kernel SGD, our algorithm improves efficiency and scalability through an innovative regularization strategy. By leveraging the infinite series expansion of spherical radial basis functions, this strategy projects the stochastic gradient onto a finite-dimensional hypothesis space, which is adaptively scaled according to the bias-variance trade-off, thereby enhancing generalization performance. Based on a new estimation of the spectral structure of the kernel-induced covariance operator, we develop an analytical framework that unifies optimization and generalization analyses. We prove that both the last iterate and the suffix average converge at minimax-optimal rates, and we further establish optimal strong convergence in the reproducing kernel Hilbert space. Our framework accommodates a broad class of classical loss functions, including least-squares, Huber, and logistic losses. Moreover, the proposed algorithm significantly reduces computational complexity and achieves optimal storage complexity by incorporating coordinate-wise updates from linear SGD, thereby avoiding the costly pairwise operations typical of kernel SGD and enabling efficient processing of streaming data. Finally, extensive numerical experiments demonstrate the efficiency of our approach.