# Scalable Incremental Robustness Analysis of Neural Network Feedback Systems

arXiv: https://arxiv.org/abs/2609.22576 · HTML: https://mlsanity.com/paper/2609.22576/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 37 references that are in the map.
- First posted on arXiv 18 Sep 2026.

## Builds on (most cited first, 21 of 21)

- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [A Unified Algebraic Perspective on Lipschitz Neural Networks](https://mlsanity.com/paper/2303.03169/) (2023)
- [Direct Parameterization of Lipschitz-Bounded Deep Networks](https://mlsanity.com/paper/2301.11526/) (2023)
- [Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness](https://mlsanity.com/paper/2104.05942/) (2021)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [Efficient Bound of Lipschitz Constant for Convolutional Layers by Gram Iteration](https://mlsanity.com/paper/2305.16173/) (2023)
- [Certifying Incremental Quadratic Constraints for Neural Networks via Convex Optimization](https://mlsanity.com/paper/2012.05981/) (2020)
- [A Quantitative Geometric Approach to Neural-Network Smoothness](https://mlsanity.com/paper/2203.01212/) (2022)
- [Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks](https://mlsanity.com/paper/2204.00846/) (2022)
- [Neural network training under semidefinite constraints](https://mlsanity.com/paper/2201.00632/) (2022)
- [ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks](https://mlsanity.com/paper/2404.04375/) (2024)
- [Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activations](https://mlsanity.com/paper/2401.14033/) (2024)
- [Chordal Sparsity for SDP-based Neural Network Verification](https://mlsanity.com/paper/2206.03482/) (2022)
- [Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guarantees](https://mlsanity.com/paper/2404.07373/) (2024)
- [Stability and Performance Analysis of Discrete-Time ReLU Recurrent Neural Networks](https://mlsanity.com/paper/2405.05236/) (2024)
- [On Robust Reinforcement Learning with Lipschitz-Bounded Policy Networks](https://mlsanity.com/paper/2405.11432/) (2024)
- [Improved Scalable Lipschitz Bounds for Deep Neural Networks](https://mlsanity.com/paper/2503.14297/) (2025)
- [Robust Contraction Analysis of Nonlinear Systems via Differential IQC](https://mlsanity.com/paper/1903.08806/) (2019)
- [Robust Stability of Neural Network Control Systems with Interval Matrix Uncertainties](https://mlsanity.com/paper/2311.15109/) (2023)

## Abstract (arXiv)

Semidefinite programming (SDP) certificates for feedback systems containing deep neural networks (NNs) typically scale with the total number of neurons, whereas small-gain tests are scalable but can be highly conservative. This paper develops a unified and scalable framework for incremental robust stability and performance analysis of feedback interconnections involving high-dimensional NNs and unmodeled dynamics. By combining a structured decomposition of the full-order SDP condition with scalable Lipschitz constant estimation algorithms, we derive reduced verification conditions that certify incremental convergence and incremental ℓ2-gain bounds. The dimensions of the resulting control-analysis linear matrix inequalities (LMIs) depend only on the widths of the last two network layers and are textitindependent of network depth. The framework preserves the coupling between the plant and the NN, with the incremental small-gain condition recovered as a special case. To further reduce conservatism, we develop a multi-round alternating update scheme that iteratively refines the coupling variables while preserving scalability. Numerical experiments show that the proposed framework achieves state-of-the-art incremental ℓ2-gain bounds for large-scale NN feedback systems.