# Unit-Consistent (UC) Adjoint for GSD and Backprop in Deep Learning Applications

arXiv: https://arxiv.org/abs/2601.10873 · HTML: https://mlsanity.com/paper/2601.10873/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 15 references that are in the map.
- First posted on arXiv 15 Jan 2026.

## Builds on (most cited first, 6 of 6)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [High-Performance Large-Scale Image Recognition Without Normalization](https://mlsanity.com/paper/2102.06171/) (2021)
- [Path-SGD: Path-Normalized Optimization in Deep Neural Networks](https://mlsanity.com/paper/1506.02617/) (2015)

## Abstract (arXiv)

Deep neural networks constructed from linear maps and positively homogeneous nonlinearities (e.g., ReLU) possess a fundamental gauge symmetry: the network function is invariant to node-wise diagonal rescalings. However, standard gradient descent is not equivariant to this symmetry, causing optimization trajectories to depend heavily on arbitrary parameterizations. Prior work has proposed rescaling-invariant optimization schemes for positively homogeneous networks (e.g., path-based or path-space updates). Our contribution is complementary: we formulate the invariance requirement at the level of the backward adjoint/optimization geometry, which provides a simple, operator-level recipe that can be applied uniformly across network components and optimizer state. By replacing the Euclidean transpose with a Unit-Consistent (UC) adjoint, we derive UC gauge-consistent steepest descent and backprogation.