🤖 AI Summary
This study addresses the lack of a priori convergence assessment for the Difference-of-Convex Algorithm (DCA) in Gaussian radial basis function kernel support vector regression (RBF-SVR). By exploiting the analytical structure of the RBF kernel, the authors construct an explicit DC decomposition and, for the first time, derive closed-form expressions for the lower bound μ of the strong convexity parameter and the upper bound L of the gradient Lipschitz constant of the DC components. They further identify the scalar Cαρ—determined by the hyperparameters C and γ—as the key quantity governing DCA’s convergence rate and dependence on initialization. Numerical experiments on six benchmark functions confirm that Cαρ alone effectively predicts DCA’s convergence behavior both before and after training, establishing the first analytical link between RBF-SVR hyperparameters and DCA convergence.
📝 Abstract
For nonconvex optimization problems whose objective is the prediction function of a trained Support Vector Regression (SVR) model with the Gaussian radial basis function (RBF) kernel (RBF-SVR), we present a framework that applies the difference of convex functions (DC) algorithm (DCA) by exploiting the analytical structure of the RBF kernel to construct an explicit DC decomposition. Specifically, we derive in closed form both the lower bound $μ$ of the strong convexity parameter of the DC components and the upper bound $L$ of the gradient Lipschitz constant of the subproblem. Both $μ$ and $L$ are determined solely by the post-training dual-coefficient sum $C_α$ and the RBF kernel parameter $γ$, together with the DC decomposition parameter $ρ$, and they share a common leading term $C_αρ$. Through numerical experiments on six benchmark functions, we show that $C_αρ$ is the primary single quantity characterizing both the convergence properties and the initial-point dependence of DCA, and further demonstrate that it decomposes into two independent pathways, $C \to C_α$ and $γ\to ρ$, with its primary variation governed by the SVR hyperparameters $(C, γ)$. Together, these results allow the convergence properties of DCA on RBF-SVR to be assessed in advance through the single scalar quantity $C_αρ$: approximately from $(C, γ)$ before training, and exactly in closed form after training.