Analytic and Variational Stability in Deep Learning Systems

We propose a unified analytic and variational framework for stability in deep learning systems viewed as coupled representation-parameter dynamics. The central object is the Learning Stability Profile, which measures how infinitesimal perturbations propagate through representations, parameters, and update mechanisms along the learning trajectory. Our main result, the Fundamental Analytic Stability Theorem, shows that uniform boundedness of these sensitivities is equivalent, up to norm equivalence, to the existence of a Lyapunov-type energy dissipating along the learning flow. In smooth regimes, this yields explicit stability exponents linking spectral norms, activation regularity, step sizes, and learning rates to contractive behavior. Classical spectral stability of feedforward networks, CFL-type conditions for residual architectures, and temporal stability laws for stochastic gradient methods follow as direct consequences. The framework extends to non-smooth 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 theory provides a unified dynamical description of stability across architectures and optimization methods, clarifying how design and training choices jointly control robustness and sensitivity to perturbations.

💡 Research Summary

The paper proposes a unified analytic‑variational framework for studying stability in deep learning systems by modeling learning as coupled dynamics of representations, parameters, and update mechanisms. The central construct is the Learning Stability Profile (LSP), which records the maximal norms of the Jacobian (or Clarke generalized Jacobian in nonsmooth settings) of the one‑step transition map with respect to the representation (σₓ), parameters (σ_θ), and update variables (σ_u) across all layers and training times. By normalizing the logarithm of these supremum norms over depth L and horizon T, the authors define three analytic stability exponents αₓ, α_θ, and α_u. Non‑positive exponents indicate that infinitesimal perturbations remain bounded, while strictly negative values imply exponential decay.

The cornerstone result, the Fundamental Analytic Stability Theorem, establishes an equivalence (up to norm equivalence) between: (1) uniform boundedness of the LSP components (i.e., αₓ, α_θ, α_u ≤ 0) and (2) the existence of a coercive, differentiable Lyapunov‑type energy E that decreases by at least a fixed amount γ along every learning trajectory. The proof proceeds by showing that bounded sensitivities render the one‑step map globally Lipschitz, enabling a discrete‑time Lyapunov construction; conversely, a Lyapunov decrease forces the linearized dynamics to be non‑expansive, precluding unbounded growth of the Jacobian norms. Quantitative bounds on the exponents follow from norm equivalence and the dissipation rate γ.

Applying this general theory yields concrete stability laws for popular architectures and optimization schemes:

• Feedforward networks – If each layer’s weight matrix satisfies ‖Wₖ‖₂ ≤ ρ < 1, the overall Jacobian norm is bounded by ρᴸ, giving αₓ = log ρ < 0. This recovers the classic spectral‑norm control principle as a direct consequence of the Lyapunov framework.

• Residual networks – For blocks of the form Xₖ₊₁ = Xₖ + h gₖ(Xₖ;θₖ) with globally Lipschitz, uniformly dissipative gₖ (Lipschitz constant M_g and dissipation m), the one‑step Lipschitz factor is √(1 − 2hm + h²M_g²). The CFL‑type condition 0 < h < 2m/M_g² guarantees this factor is < 1, leading to αₓ ≤ ½ log(1 − 2hm + h²M_g²) < 0. Thus residual step size and dissipativity jointly control depthwise stability.

• Stochastic Gradient Descent (SGD) – Under strong convexity (μ), smoothness (L), and bounded gradient noise (σ₀, σ₁), the mean‑square Lyapunov recursion E