On the Mathematical Foundations of Learning, RKHS Regularization, and Smale's 18th Problem

Smale’s 18th Problem (Steve Smale, 2000) asks: What are the limits of intelligence and learning? What are the foundational mathematical principles governing high-dimensional adaptive systems? In 2002, Felipe Cucker and Steve Smale laid the foundational architecture of computational learning theory in their landmark paper (Bulletin of the AMS). By formalizing the learning task as an optimization problem in a Reproducing Kernel Hilbert Space (RKHS) governed by a probability distribution ρ on X ×Y , they established a rigorous bridge between functional analysis, operator theory, and empirical risk minimization.