Abstract

Contemporary machine learning is overwhelmingly framed as an optimization problem. Gradient descent and its variants define learning as a trajectory in parameter space, governed by step sizes, penalty weights, and convergence criteria. In such formulations, the solution is not assumed it is approached. This work challenges that framing in the linear regime. When a linear system satisfies the compatibility condition A W = b, equilibri um is not produced by optimization. It is already encoded in the algebraic structure of the system. Both L1 and L2 objectives attain zero residual at the same structural solution. The minimum does not emerge from descent dynamics; it exists as a consequence of determinacy. Optimization, in this setting, is epistemic- it reveals a solution. Equilibrium is structural it precedes the algorithm. Regularization does not create equilibrium. It modifies the geometry surrounding an equilibrium that may already be present. It stabilizes degeneracy, suppresses ill-conditioning, and reshapes curvature. But when the system is structurally solvable, the equilib rium is not a product of penalty tuning. Principal Component Analysis exemplifies this distinction. Principal directions are obtained through eigen-decomposition of the covariance matrix. No learning rate governs their emergence. No penalty trajectory produces them. The spectral structure determines them outright. The pre vailing narrative in machine learning equates learning with iterative convergence. This work separates the two. Trajectory-based convergence is necessary when structural determinacy is absent or obscured. But when determinacy is present, equilibrium-based inference supersedes optimization. The distinction is not compu tational, it is ontological, optimization describes how we search and determinacy describes what already is. Recognizing this separation clarifies a central issue: iteration is indispensable in underdetermined or nonline ar regimes, but in structurally determined systems it is often a method of access, not a mechanism of existence.

DOI: doi.org/10.63721/26JPAIR0126

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