Revisit of Logistic Regression: Efficient Optimization and Kernel Extensions

Logistic regression (LR) is widely applied as a powerful classification method in various fields, and a variety of optimization methods have been developed. To cope with large-scale problems, an efficient optimization method for LR is required in terms of computational cost and memory usage. In this paper, we propose an efficient optimization method using non-linear conjugate gradient (CG) descent. In each CG iteration, the proposed method employs the optimized step size without exhaustive line search, which significantly reduces the number of iterations, making the whole optimization process fast. In addition, on the basis of such CG-based optimization scheme, a novel optimization method for kernel logistic regression (KLR) is proposed. Unlike the ordinary KLR methods, the proposed method optimizes the kernel-based classifier, which is naturally formulated as the linear combination of sample kernel functions, directly in the reproducing kernel Hilbert space (RKHS), not the linear coefficients. Subsequently, we also propose the multiple-kernel logistic regression (MKLR) along with the optimization of KLR. The MKLR effectively combines the multiple types of kernels with optimizing the weights for the kernels in the framework of the logistic regression. These proposed methods are all based on CG-based optimization and matrix-matrix computation which is easily parallelized such as by using multi-thread programming. In the experimental results on multi-class classifications using various datasets, the proposed methods exhibit favorable performances in terms of classification accuracies and computation times.