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Analytic and Variational Stability of Deep Learning Systems

Ronald Katende

TL;DR

This work develops a unified analytic and variational framework for stability in deep learning by introducing the Learning Stability Profile (LSP), which quantifies how infinitesimal perturbations propagate through coupled representation, parameter, and update dynamics. The core result, the Fundamental Analytic Stability Theorem, shows that bounded LSP signatures are equivalent (up to norm changes) to the existence of a Lyapunov-type energy that dissipates along learning trajectories, enabling explicit stability laws in both smooth and non-smooth regimes. The authors derive forward, parametric, and temporal stability laws for feedforward and residual networks, as well as SGD under constant and decreasing steps, with extensions to non-smooth settings via Clarke derivatives and variational energies. This framework provides design rules linking spectral norms, step sizes, and dissipation rates to robust, contractive learning dynamics and offers a principled lens to analyze robustness, generalisation, and explanations across architectures and optimization methods.

Abstract

We propose a unified analytic and variational framework for studying stability in deep learning systems viewed as coupled representation-parameter dynamics. The central object is the Learning Stability Profile, which tracks the infinitesimal response of representations, parameters, and update mechanisms to perturbations along the learning trajectory. We prove a Fundamental Analytic Stability Theorem showing that uniform boundedness of these stability signatures is equivalent, up to norm equivalence, to the existence of a Lyapunov-type energy that dissipates along the learning flow. In smooth regimes, the framework yields explicit stability exponents linking spectral norms, activation regularity, step sizes, and learning rates to contractivity of the learning dynamics. Classical spectral stability results for feedforward networks, a discrete CFL-type condition for residual architectures, and parametric and temporal stability laws for stochastic gradient methods arise as direct consequences. The theory extends to non-smooth learning systems, including ReLU networks, proximal and projected updates, and stochastic subgradient flows, by replacing classical derivatives with Clarke generalized derivatives and smooth energies with variational Lyapunov functionals. The resulting framework provides a unified dynamical description of stability across architectures and optimization methods, clarifying how architectural and algorithmic choices jointly govern robustness and sensitivity to perturbations. It also provides a foundation for further extensions to continuous-time limits and geometric formulations of learning dynamics.

Analytic and Variational Stability of Deep Learning Systems

TL;DR

This work develops a unified analytic and variational framework for stability in deep learning by introducing the Learning Stability Profile (LSP), which quantifies how infinitesimal perturbations propagate through coupled representation, parameter, and update dynamics. The core result, the Fundamental Analytic Stability Theorem, shows that bounded LSP signatures are equivalent (up to norm changes) to the existence of a Lyapunov-type energy that dissipates along learning trajectories, enabling explicit stability laws in both smooth and non-smooth regimes. The authors derive forward, parametric, and temporal stability laws for feedforward and residual networks, as well as SGD under constant and decreasing steps, with extensions to non-smooth settings via Clarke derivatives and variational energies. This framework provides design rules linking spectral norms, step sizes, and dissipation rates to robust, contractive learning dynamics and offers a principled lens to analyze robustness, generalisation, and explanations across architectures and optimization methods.

Abstract

We propose a unified analytic and variational framework for studying stability in deep learning systems viewed as coupled representation-parameter dynamics. The central object is the Learning Stability Profile, which tracks the infinitesimal response of representations, parameters, and update mechanisms to perturbations along the learning trajectory. We prove a Fundamental Analytic Stability Theorem showing that uniform boundedness of these stability signatures is equivalent, up to norm equivalence, to the existence of a Lyapunov-type energy that dissipates along the learning flow. In smooth regimes, the framework yields explicit stability exponents linking spectral norms, activation regularity, step sizes, and learning rates to contractivity of the learning dynamics. Classical spectral stability results for feedforward networks, a discrete CFL-type condition for residual architectures, and parametric and temporal stability laws for stochastic gradient methods arise as direct consequences. The theory extends to non-smooth learning systems, including ReLU networks, proximal and projected updates, and stochastic subgradient flows, by replacing classical derivatives with Clarke generalized derivatives and smooth energies with variational Lyapunov functionals. The resulting framework provides a unified dynamical description of stability across architectures and optimization methods, clarifying how architectural and algorithmic choices jointly govern robustness and sensitivity to perturbations. It also provides a foundation for further extensions to continuous-time limits and geometric formulations of learning dynamics.
Paper Structure (34 sections, 18 theorems, 174 equations)

This paper contains 34 sections, 18 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Learning Stability Profile
  4. Directional sensitivities of the learning flow
  5. Analytic stability exponents
  6. Analytic Stability Theorem
  7. Residual and Feedforward Stability Laws
  8. Feedforward networks with spectral control
  9. Residual networks and CFL type stability
  10. Temporal stability of stochastic gradient methods
  11. Mean square Lyapunov recursion
  12. Temporal analytic stability for constant and decreasing step sizes
  13. Variational Extension to Non-smooth Learning Systems
  14. Locally Lipschitz learning flows and Clarke derivatives
  15. Generalized Learning Stability Profile and exponents
  16. ...and 19 more sections

Key Result

Proposition 3.4

Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{M}$ inducing operator norms $\|\cdot\|_{1,\mathrm{op}}$ and $\|\cdot\|_{2,\mathrm{op}}$, associated stability signatures $\sigma^{x,1}_{k,t}$, $\sigma^{x,2}_{k,t}$, and exponents $\alpha_x^{(1)}$, $\alpha_x^{(2)}$. Then there exist consta and the exponents coincide, $\alpha_x^{(1)}=\alpha_x^{(2)}$. The same conclusion holds for $\alpha_

Theorems & Definitions (54)

  • Definition 2.1: Learning System
  • Definition 3.1: Analytic stability exponents
  • Definition 3.2: Analytic stability and stability indices
  • Definition 3.3: Analytic learning energy
  • Proposition 3.4: Independence of norm
  • proof
  • Proposition 3.5: Finite horizon bound and exponent sign
  • proof
  • Theorem 4.1: Fundamental Analytic Stability Theorem
  • proof : Sketch of proof
  • ...and 44 more