Budget Online Learning Algorithm for Least Squares SVM

Batch-mode least squares support vector machine (LSSVM) is often associated with unbounded number of support vectors (SVs’), making it unsuitable for applications involving large-scale streaming data. Limited-scale LSSVM, which allows efficient updating, seems to be a good solution to tackle this issue. In this paper, to train the limited-scale LSSVM dynamically, we present a budget online LSSVM (BOLSSVM) algorithm. Methodologically, by setting a fixed budget for SVs’, we are able to update the LSSVM model according to the updated SVs’ set dynamically without retraining from scratch. In particular, when a new small chunk of SVs’ substitute for the old ones, the proposed algorithm employs a low rank correction technology and the Sherman–Morrison–Woodbury formula to compute the inverse of saddle point matrix derived from the LSSVM’s Karush-Kuhn-Tucker (KKT) system, which, in turn, updates the LSSVM model efficiently. In this way, the proposed BOLSSVM algorithm is especially useful for online prediction tasks. Another merit of the proposed BOLSSVM is that it can be used for <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-fold cross validation. Specifically, compared with batch-mode learning methods, the computational complexity of the proposed BOLSSVM method is significantly reduced from <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(n^{4})$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(n^{3})$ </tex-math></inline-formula> for leave-one-out cross validation with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> training samples. The experimental results of classification and regression on benchmark data sets and real-world applications show the validity and effectiveness of the proposed BOLSSVM algorithm.