Sharp Stability Threshold and Certification for Designing Stable Residual Architectures

• URL: http://arxiv.org/abs/2607.14576v1
• Date: Thu, 16 Jul 2026 05:13:44 GMT
• Title: Sharp Stability Threshold and Certification for Designing Stable Residual Architectures
• Abstract summary: We propose a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field.<n>The threshold $q = 1$ is established via two independent arguments.<n>An arithmetic of input-magnitude exponents under the five operations that build residual blocks enables efficient certification of $q_k le 1$ at the level of architectural primitives.
• Score: 4.781989580907985
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose \emph{the sublinear-growth principle} for deep residual architectures -- a sharp stability threshold on the input-magnitude exponent of every residual block's velocity field: $$\|v(x, t)\| \leq c\,\|x\|^q + b, \qquad q \in [0, 1].$$ The threshold $q = 1$ is established via two independent arguments. Classical ODE theory gives a global forward flow on $[0, T]$ at $q \le 1$ and exhibits divergent velocity fields at any $q > 1$. The optimal-control analysis, via the Hamilton-Jacobi-Bellman equation, sharpens this to a selection statement: the training optimum is bang-bang on the boundary of the admissible class, so the optimum at $q > 1$ blows up while the optimum at $q \le 1$ is safe by construction. The exponent criterion $q \le 1$ is thereby a necessary and sufficient condition for stable training. It clarifies architectural placements that ensure the stability of training and inference, explaining, for instance, the stabilizing role of layer normalization. The sublinear-growth velocity fields form \emph{the right function space} on which forward dynamics, adjoint sensitivity, and architectural composition are all well-controlled. An arithmetic of input-magnitude exponents under the five operations that build residual blocks enables efficient certification of $q_k \le 1$ at the level of architectural primitives, in place of ad hoc trial and error in the search for stable neural architectural designs. A parameter-free modification reduces the supercritical Mamba block from $q = 5$ to $q = 1$ without layer normalization, demonstrating this point. Experiments on Mamba and PatchTST confirm that the $q \le 1$ variants train stably: the criterion is the input-magnitude exponent, not the presence of a normalization layer.

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