Quantifying Depth Sufficiency in Residual Neural Networks: A First-Order Criterion
How can we determine whether a trained neural network is already deep enough? We study this under a fixed function-preserving residual-growth protocol specifying insertion locations, residual families, zero-output initializations, and zero-state first-order updates. We define first-order residual depth saturation as the absence of a strict local decrease from every admissible insertion. We prove residual non-degeneracy is necessary and sufficient: additional depth has first-order value exactly when conditional activation gradients have a nonzero projection onto at least one admissible residual tangent space. This boundary is shared by descent-compatible zero-state updates and invariant un...
| Authors | Zeyu Liu, Jinhao Zhang, Yunquan Zhang et al. |
|---|---|
| Primary category | cs.LG |
| Published | 1 August 2026 |
| Last updated at source | 1 August 2026 |
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