On a Gradation for Asymptotic Stability
arXiv:2609.03120
2026
Dynamics
2 ideas extracted · analyzed Sep 4, 2026
What the math gives to ML
The paper introduces a quantitative stability degree that distinguishes exponential convergence from algebraic convergence, with degree m greater than zero corresponding to decay proportional to t^(-1/m). Its transferable mechanism is a Lyapunov dissipation inequality whose exponent determines the long-time convergence law, together with direct and converse tests for certifying that exponent. In neural-network training, this can become an adaptive learning-rate controller and an online stability monitor that targets a prescribed decay exponent rather than merely checking whether the loss decreases.
Ideas from this paper
Unverified
2026
Treat the optimization error as a Lyapunov-like state and adapt the learning rate so that its measured decrease follows a chosen stability degree. Instead of requiring exponential decrease, the controller targets dE/dt approximately equal to -c E^(1+m), which is appropriate near flat minima or marginally stable training regimes where exponential contraction may be impossible.
Useful8/10
Difficulty5/10
Novelty7/10
Unverified
2026
Construct a continuous-depth or recurrent residual block with a prescribed polynomial Lyapunov decay near its equilibrium. The architecture combines a fixed radial stabilizer with a learned component that is constrained to have zero radial projection, allowing slow algebraic transients and long memory while preventing asymptotic hidden-state growth.
Useful7/10
Difficulty6/10
Novelty8/10