Control Allocation in Neural Network Optimization: Joint Affine Control of Weight and Bias Updates

Optimization algorithms determine not only the magnitude of a neural-network update but also how that update is distributed across parameter channels. We study whether this distribution can be treated as a controllable quantity independently of global training progress. We define operational update allocation through normalized channel energies and analyze two scalar controls: a coordinate-preconditioning exponent and an affine spectral exponent that scales the bias column of an augmented weight--bias matrix. At a frozen state, a common nonzero step-size multiplier leaves normalized allocation unchanged; the coordinate exponent yields affine pairwise log-odds with an explicit inverse; and the affine exponent induces a rank-one positive-semidefinite Gram perturbation and a logistic raw-participation law. We further separate raw affine participation, spectral gain, and the decoded physical bias update, and show that finite polynomial spectral iterations preserve singular subspaces. Same-state replay verifies the exact control laws. On a five-seed controlled benchmark, intermediate controls improve held-out and worst-group metrics, whereas excessive affine control causes underfitting. A four-task single-seed transfer study provides descriptive corroboration. These results establish instantaneous allocation control and a bounded empirical operating regime, but do not imply a task-independent generalization ordering.