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

arXiv: https://arxiv.org/abs/2608.02991 · HTML: https://mlsanity.com/paper/2608.02991/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 17 references that are in the map.
- First posted on arXiv 4 Aug 2026.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients](https://mlsanity.com/paper/1705.07774/) (2017)
- [Convergence of Gradient Descent on Separable Data](https://mlsanity.com/paper/1803.01905/) (2018)
- [Closing the Generalization Gap of Adaptive Gradient Methods in Training Deep Neural Networks](https://mlsanity.com/paper/1806.06763/) (2018)
- [Modular Duality in Deep Learning](https://mlsanity.com/paper/2410.21265/) (2024)
- [Can We Remove the Square-Root in Adaptive Gradient Methods? A Second-Order Perspective](https://mlsanity.com/paper/2402.03496/) (2024)

## Abstract (arXiv)

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.