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

Perez-Lechuga, Gilberto

doi:10.3390/math14071202

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

2026-04-01

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Perez-Lechuga, Gilberto *

C

Coronel Garcia, Marco Antonio

M

Martinez Salazar, Ana Lidia

DOI：10.3390/math14071202

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This paper develops a rigorous operator-theoretic framework for learning-augmented quasi-gradient methods in constrained optimization. We consider the minimization of an objective function over a closed convex feasible set, where feasibility is enforced via projection and directional updates may incorporate data-driven corrections. Such settings arise naturally in modern optimization algorithms that integrate artificial intelligence components under structural constraints. The proposed formulation introduces an explicit contraction-bias-variance decomposition of the iterative dynamics. Curvature induces deterministic contraction, alignment distortion-quantified by a geometric parameter-modifies the effective contraction margin, and stochastic learning components inject controlled dispersion. Explicit error recursions yield convergence guarantees under strong convexity, the Polyak-& Lstrok;ojasiewicz condition, and smooth nonconvexity. The analysis establishes that stability regions and first-order complexity bounds are preserved whenever alignment distortion remains below unity and bounded second-moment conditions hold. A fully reproducible computational study provides quantitative validation: the empirically observed steady-state error closely matches the theoretical prediction proportional to sigma 2/mu(1-eta). Comparative experiments with gradient, stochastic gradient, and momentum methods confirm that the proposed operator retains classical stability margins and conditioning sensitivity while enabling principled integration of learned directional components. The results provide a transparent mathematical bridge between stochastic approximation theory and contemporary AI-enhanced constrained optimization.

Keywords:

learning-augmented optimization

quasi-gradient methods

constrained optimization

operator-theoretic analysis

contraction-bias-variance decomposition

stochastic approximation

alignment geometry

convergence guarantees

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

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