A Hierarchy of Learning Problems: Computational Efficiency Mappings for Optimization Algorithms

We establish a rigorous mathematical hierarchy for optimization problems that serves as both theoretical classification and practitioner decision system. Problems are exhaustively categorized into three classes: (1) Non-learnable problems where no alpha-averaged operator exists, precluding iterative convergence; (2) Learnable problems that admit averaged operators enabling batch methods; (3) Streamable problems that are learnable with uniform low-rank residual approximation, enabling rank-K streaming algorithms with explicit bias floors. This framework unifies gradient descent, proximal methods, Frank-Wolfe, PCA, and attention mechanisms under operator-theoretic foundations. We provide explicit computational mappings showing memory reductions from quadratic to linear scaling, rigorous Koopman operator connections, and an algorithmic ULRA-test for streamability verification. For typical problems with dimension one million and streamable rank one hundred, streaming achieves ten-thousand-fold memory reduction with identical convergence rates.