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

• URL: http://arxiv.org/abs/2601.10873v1
• Date: Thu, 15 Jan 2026 21:58:04 GMT
• Title: Unit-Consistent (UC) Adjoint for GSD and Backprop in Deep Learning Applications
• Abstract summary: We formulate the invariance requirement at the level of the backward adjoint/optimization geometry.<n>By replacing the Euclidean-level recipe, we derive a Unit-consistent gauge descent and backprogation.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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