Analytical Evaluation of DCA Convergence Properties for Minimizing Prediction Functions of Gaussian RBF Support Vector Regression
For practitioners using DCA to optimize RBF-SVR prediction functions, this provides a pre-training estimate of convergence behavior, but the contribution is incremental as it applies known DC decomposition to a specific model class.
The paper derives closed-form bounds for the strong convexity and Lipschitz constants of DC components in DCA applied to RBF-SVR prediction functions, showing that the product C_α*ρ characterizes convergence and initial-point dependence. Numerical experiments on six benchmarks confirm that this single scalar quantity can predict DCA convergence properties from SVR hyperparameters before training.
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.