# A Systems-Theoretic View on the Convergence of Algorithms under Disturbances

arXiv: https://arxiv.org/abs/2512.17598 · HTML: https://mlsanity.com/paper/2512.17598/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 56 references that are in the map.
- First posted on arXiv 19 Dec 2025.

## Builds on (most cited first, 30 of 30)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Sparsified SGD with Memory](https://mlsanity.com/paper/1809.07599/) (2018)
- [FedPAQ: A Communication-Efficient Federated Learning Method with Periodic Averaging and Quantization](https://mlsanity.com/paper/1909.13014/) (2019)
- [Performative Prediction](https://mlsanity.com/paper/2002.06673/) (2020)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Differentially Private Learning with Adaptive Clipping](https://mlsanity.com/paper/1905.03871/) (2019)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Private Stochastic Convex Optimization with Optimal Rates](https://mlsanity.com/paper/1908.09970/) (2019)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds](https://mlsanity.com/paper/1405.7085/) (2014)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Privacy of Noisy Stochastic Gradient Descent: More Iterations without More Privacy Loss](https://mlsanity.com/paper/2205.13710/) (2022)
- [A Dynamical Systems Perspective on Nesterov Acceleration](https://mlsanity.com/paper/1905.07436/) (2019)
- [Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling](https://mlsanity.com/paper/2407.16936/) (2024)
- [A General Analysis of the Convergence of ADMM](https://mlsanity.com/paper/1502.02009/) (2015)
- [Beyond Convexity -- Contraction and Global Convergence of Gradient Descent](https://mlsanity.com/paper/1806.06655/) (2018)
- [Orthogonal Directions Constrained Gradient Method: from non-linear equality constraints to Stiefel manifold](https://mlsanity.com/paper/2303.09261/) (2023)
- [On Constraints in First-Order Optimization: A View from Non-Smooth Dynamical Systems](https://mlsanity.com/paper/2107.08225/) (2021)
- [Event-triggered Learning](https://mlsanity.com/paper/1904.03042/) (2019)
- [Continuous-time Lower Bounds for Gradient-based Algorithms](https://mlsanity.com/paper/2002.03546/) (2020)
- [Towards a Systems Theory of Algorithms](https://mlsanity.com/paper/2401.14029/) (2024)
- [Decision-Dependent Stochastic Optimization: The Role of Distribution Dynamics](https://mlsanity.com/paper/2503.07324/) (2025)
- [Accelerated First-Order Optimization under Nonlinear Constraints](https://mlsanity.com/paper/2302.00316/) (2023)
- [A Dynamical Systems Perspective on Discrete Optimization](https://mlsanity.com/paper/2305.08536/) (2023)
- [Primal Methods for Variational Inequality Problems with Functional Constraints](https://mlsanity.com/paper/2403.12859/) (2024)
- [Distributed Event-Based Learning via ADMM](https://mlsanity.com/paper/2405.10618/) (2024)
- [Controlling Participation in Federated Learning with Feedback](https://mlsanity.com/paper/2411.19242/) (2024)

## Abstract (arXiv)

Algorithms increasingly operate within complex physical, social, and engineering systems where they are exposed to disturbances, noise, and interconnections with other dynamical systems. This article extends known convergence guarantees of an algorithm operating in isolation (i.e., without disturbances) and systematically derives stability bounds and convergence rates in the presence of such disturbances. By leveraging converse Lyapunov theorems, we derive key inequalities that quantify the impact of disturbances. We further demonstrate how our result can be utilized to assess the effects of disturbances on algorithmic performance in a wide variety of applications, including communication constraints in distributed learning, sensitivity in machine learning generalization, and intentional noise injection for privacy. This underpins the role of our result as a unifying tool for algorithm analysis in the presence of noise, disturbances, and interconnections with other dynamical systems.