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This arXiv preprint offers a theoretical framework for analyzing how perturbations propagate through learning systems, using Lyapunov-style sufficient conditions for stability. Its value is structural: it unifies known stability results for feedforward and residual networks, stochastic gradient methods, and nonsmooth updates under one sensitivity calculus, while keeping each guarantee's assumptions explicit. Readers should note it is a sufficient criterion rather than a converse theorem, and that the stochastic extensions are still flagged as needing stronger justification.
A Learning Stability Profile for Finite-Dimensional Learning Dynamics
arXiv:2512.21208v4 Announce Type: replace Abstract: We develop a finite-dimensional sensitivity framework for studying stability in learning systems whose states include representations, parameters, and update variables. The central object is the \emph{Learning Stability Profile}, a collection of directional sensitivity operators that records how perturbations in inputs, parameter initialization, and update mechanisms propagate along a specified learning trajectory. The main result is a Lyapunov criterion for controlling this profile. Under explicit regularity, coercivity, and dissipation assumptions, an incremental Lyapunov energy yields uniform or exponentially decaying bounds on the associated linearized transition operators. The result is stated as a sufficient stability criterion, not as an unconditional converse theorem. The framework also distinguishes terminal decay, profile-wise boundedness, and subexponential growth, avoiding the identification of nonpositive growth exponents with uniform boundedness. The profile is then specialized to several standard learning mechanisms. Spectral bounds give forward sensitivity estimates for feedforward networks. Dissipativity and step-size restrictions give stability bounds for residual architectures. Mean-square contraction assumptions yield parameter and update sensitivity bounds for stochastic gradient methods. Locally Lipschitz systems, including piecewise-linear networks, proximal maps, projected updates, and recurrent or state-space recursions, are handled through Clarke generalized Jacobians and variational Lyapunov inequalities. The resulting framework provides a common stability language for architecture, optimization, stochasticity, and nonsmoothness. Its role is structural: it organizes known stability mechanisms within one perturbation calculus while keeping the hypotheses needed for each guarantee explicit.
Abstract:We develop a finite-dimensional sensitivity framework for studying stability in learning systems whose states include representations, parameters, and update variables. The central object is the \emph{Learning Stability Profile}, a collection of directional sensitivity operators that records how perturbations in inputs, parameter initialization, and update mechanisms propagate along a specified learning trajectory. The main result is a Lyapunov criterion for controlling this profile. Under explicit regularity, coercivity, and dissipation assumptions, an incremental Lyapunov energy yields uniform or exponentially decaying bounds on the associated linearized transition operators. The result is stated as a sufficient stability criterion, not as an unconditional converse theorem. The framework also distinguishes terminal decay, profile-wise boundedness, and subexponential growth, avoiding the identification of nonpositive growth exponents with uniform boundedness. The profile is then specialized to several standard learning mechanisms. Spectral bounds give forward sensitivity estimates for feedforward networks. Dissipativity and step-size restrictions give stability bounds for residual architectures. Mean-square contraction assumptions yield parameter and update sensitivity bounds for stochastic gradient methods. Locally Lipschitz systems, including piecewise-linear networks, proximal maps, projected updates, and recurrent or state-space recursions, are handled through Clarke generalized Jacobians and variational Lyapunov inequalities. The resulting framework provides a common stability language for architecture, optimization, stochasticity, and nonsmoothness. Its role is structural: it organizes known stability mechanisms within one perturbation calculus while keeping the hypotheses needed for each guarantee explicit.
The stochastic sensitivity analysis needs stronger justification when moving from fixed-perturbation expectations to expected operator-norm bounds. The Lyapunov framework remains relevant and potentially repairable
Machine Learning (cs.LG); Dynamical Systems (math.DS); Optimization and Control (math.OC)
arXiv:2512.21208 [cs.LG]
(or arXiv:2512.21208v4 [cs.LG] for this version)
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