stability analysis

This study addresses the problem of ensuring algorithmic convergence in complex dynamic systems—spanning physics, social sciences, and engineering—subject to external disturbances, stochastic noise, and coupled interactions. To overcome the limitation of existing theories in characterizing disturbance robustness, we systematically introduce the converse Lyapunov theorem into algorithmic convergence analysis for the first time, establishing a unified theoretical framework that jointly guarantees stability and quantifies convergence rates under perturbations. Our method integrates converse Lyapunov theory, nonlinear stability analysis, and quantitative robustness modeling, yielding explicit, computable bounds on convergence via quantitative perturbation inequalities. The framework is successfully applied to three domains: modeling communication constraints in distributed learning, analyzing generalization sensitivity in machine learning, and designing differential privacy mechanisms with calibrated noise injection. The results provide a verifiable, quantifiable theoretical foundation for dynamic algorithm design across disciplines.

Traditional control theory neglects computational uncertainty—such as mathematical object distortion induced by finite-precision arithmetic—leading to reliability gaps between Lyapunov stability analysis and digital controller implementation. Methodologically, this paper introduces the first constructive control framework that explicitly treats computational uncertainty as an independent modeling dimension in controller synthesis and system analysis. Leveraging tools from computability theory, constructive analysis, and measurable selection, we establish a constructive Danskin theorem and provide computable reconstructions of fundamental objects—including control Lyapunov functions (CLFs), Carathéodory trajectories, and eigenvalue problems. Our primary contribution is a computationally feasible paradigm for stability and stabilization proofs: all mathematical constructs are uniformly approximable by finite-precision algorithms while rigorously preserving required properties. This ensures robustness and implementability of digital controllers under realistic computational constraints.

Traditional linearization-based analyses often fail to accurately characterize the stability of optimization algorithms under nonlinear dynamics, leading to potential misjudgments. This work addresses this limitation by explicitly analyzing the nonlinear behavior of gradient descent (GD) and stochastic gradient descent (SGD) near minima. It proposes a multivariate GD stability criterion based on higher-order derivatives that captures stable oscillatory regimes beyond the reach of linear analysis. Furthermore, the study reveals that the overall stability of SGD can be dominated by a single unstable batch rather than governed by averaging effects. Theoretically, it is proven that when all batches are linearly stable, SGD is nonlinearly stable in expectation; however, the presence of even one unstable batch is sufficient to induce global divergence.

Verifying stability of black-box nonlinear control systems is challenging when no prior dynamical model is available. Method: This paper proposes a model-free, data-driven stability verification method that directly learns the Lie derivative of a Lyapunov function—bypassing explicit system dynamics approximation. It integrates region-wise sampling-based validation with a counterexample-guided inductive synthesis (CEGIS) framework, underpinned by Lipschitz-based error bounds to ensure provably terminating synthesis. Contribution/Results: The approach certifies regional stability for 2D and 3D systems using only thousands of samples—requiring fewer than 0.01% of the samples needed by state-of-the-art black-box methods. It guarantees soundness and completeness within bounded regions, enables certified termination, and supports visualization of hard-to-verify stable regions.

This work addresses the stability of gradient descent coupled with second-order dynamics (e.g., momentum or acceleration terms) under explicitly time-varying cost functions—motivated by robust deployment requirements in real-time, safety-critical machine learning systems. Methodologically, we develop a unified stability analysis framework grounded in Lyapunov stability theory and nonlinear control systems analysis, applicable to general time-varying optimization settings and lifting restrictive assumptions of static or slowly varying objectives. Our key contribution is the first derivation of verifiable, explicit stability criteria that impose design constraints on algorithmic parameters—including step size, damping coefficient, and prediction-correction mechanisms—and enable formal safety certification. The framework provides both theoretical foundations and practical tools for reliability verification of optimizers in dynamic environments such as online learning and adaptive control.

This work addresses the challenge of convergence failure in inverse parallel solvers for nonlinear systems of equations, which often arises due to oscillatory or chaotic dynamics. To enhance stability, the authors propose an adaptive stabilization mechanism based on the local maximum Lyapunov exponent (LLE). By estimating the LLE via k-nearest neighbors and integrating it with sliding-window micro-time-series analysis, the method enables real-time detection of unstable phases along the solution trajectory. A Lyapunov-guided parameter control strategy is then developed to dynamically adjust solver parameters, thereby reinforcing numerical stability. Experimental results demonstrate strong agreement between theoretical stability diagrams and empirical Lyapunov profiles, confirming that the proposed approach significantly improves the robustness and convergence performance of solvers under perturbed initial conditions.

This paper addresses the fundamental question of finite-sample learnability for dynamic systems using only output sequences: *When is learning possible?* We introduce the novel paradigm of *dynamic learnability*, formally characterizing learnability as a finite-sample prediction problem governed by intrinsic dynamical properties—such as stability, observability, spectral radius, and Lyapunov exponents—rather than statistical assumptions (e.g., i.i.d. or stationarity). Methodologically, we integrate spectral filtering, stochastic process modeling, and stability analysis to avoid explicit system identification; for linear systems, this yields model-free, time-step-wise uniformly high-accuracy prediction. Our key contribution is establishing quantitative relationships between learnability and structural system parameters, thereby transcending classical PAC and online learning assumptions. This provides a rigorous theoretical foundation for learning non-stationary and latent-state dynamical systems.

This work investigates perturbation stability of deep learning systems under representation-parameter coupling dynamics. Methodologically, it introduces the *Learning Stability Profile*—a novel concept—and establishes a unified variational and Lyapunov functional analysis framework capable of characterizing stability for both smooth and nonsmooth systems, including those with ReLU activations and subgradient flows. Theoretically, it rigorously proves that bounded stability is equivalent to energy dissipation, derives an explicit stability index formula, and reveals connections to spectral stability, the CFL condition in ResNets, and temporal stability laws in stochastic optimization. The approach integrates Clarke’s generalized derivatives, variational inequalities, and continuous-time learning flow modeling. These contributions provide a unifying explanation of robustness mechanisms across diverse architectures and optimizers, and lay foundational groundwork for geometric learning dynamics and continuous-limit theories of deep learning.

This work investigates the fundamental trade-off between stability and accuracy in statistical estimation by formulating stability as a constraint within the framework of statistical decision theory. It systematically analyzes how worst-case and average-case stability requirements affect estimation accuracy, employing minimax analysis and constrained optimization to construct optimal stable estimators for canonical problems such as mean estimation and regression. The key contribution lies in establishing, for the first time within a unified framework, lower bounds on estimation accuracy under both notions of stability, revealing that average-case stability imposes a strictly weaker constraint than worst-case stability, with the gap depending on the specific estimation task. Furthermore, the paper precisely characterizes the optimal stability–accuracy trade-offs in four representative estimation settings, quantifying the statistical cost incurred by different stability mechanisms.

This work addresses the vulnerability of neural network training to rare yet severe unstable updates, which can cause irreversible divergence or subtle performance degradation—issues that existing optimizers fail to detect or mitigate at runtime. The authors model the optimization process as a controlled stochastic process and introduce the first runtime stability framework that operates without modifying the underlying optimizer. By leveraging secondary signals such as validation probes, the framework automatically detects instability and triggers lightweight interventions grounded in control theory. Designed for memory-constrained settings, the approach offers low computational overhead, broad compatibility with standard optimizers, and theoretical guarantees of bounded degradation and recovery, effectively preventing training collapse and performance deterioration.