Optimization Algorithms & AI Integration

Learning-Augmented Quasi-Gradient Operators for Constrained Optimization: A Contraction-Bias-Variance Decomposition

This paper introduces a rigorous operator-theoretic framework for learning-augmented quasi-gradient methods in constrained optimization. It develops an explicit contraction–bias-variance decomposition of iterative dynamics, revealing how curvature-induced contraction, bias-induced directional distortion, and variance-induced dispersion interact. The analysis establishes convergence guarantees under strong convexity, Polyak-Łojasiewicz condition, and smooth nonconvexity, showing that stability is preserved when learning-induced bias satisfies operator-alignment conditions and variance is bounded. The framework is validated through a reproducible computational study, confirming theoretical predictions and demonstrating compatibility with modern AI-enhanced optimization architectures like online linear models, neural networks, and adaptive momentum schemes.