# A Learning Stability Profile for Finite-Dimensional Learning Dynamics

arXiv: https://arxiv.org/abs/2512.21208 · HTML: https://mlsanity.com/paper/2512.21208/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 21 references that are in the map.
- First posted on arXiv 24 Dec 2025.

## Builds on (most cited first, 6 of 6)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [How Good is your Explanation? Algorithmic Stability Measures to Assess the Quality of Explanations for Deep Neural Networks](https://mlsanity.com/paper/2009.04521/) (2020)

## Abstract (arXiv)

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 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.