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Scalable Incremental Robustness Analysis of Neural Network Feedback Systems

arXiv 2609.22576 ↗First posted 18 Sep 2026math.OC cs.LG cs.SYcited by 0

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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.

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References · 21

  1. 2019Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networkscited by 293
  2. 2014Analysis and Design of Optimization Algorithms via Integral Quadratic Constraintscited by 105
  3. 2020Training robust neural networks using Lipschitz boundscited by 99
  4. 2023A Unified Algebraic Perspective on Lipschitz Neural Networkscited by 60
  5. 2023Direct Parameterization of Lipschitz-Bounded Deep Networkscited by 51
  6. 2021Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustnesscited by 45
  7. 2020Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllerscited by 44
  8. 2023Efficient Bound of Lipschitz Constant for Convolutional Layers by Gram Iterationcited by 19
  9. 2020Certifying Incremental Quadratic Constraints for Neural Networks via Convex Optimizationcited by 18
  10. 2022A Quantitative Geometric Approach to Neural-Network Smoothnesscited by 16
  11. 2022Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networkscited by 15
  12. 2022Neural network training under semidefinite constraintscited by 11
9 more
  1. 2024ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networkscited by 10
  2. 2024Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activationscited by 7
  3. 2022Chordal Sparsity for SDP-based Neural Network Verificationcited by 4
  4. 2024Synthesizing Neural Network Controllers with Closed-Loop Dissipativity Guaranteescited by 4
  5. 2024Stability and Performance Analysis of Discrete-Time ReLU Recurrent Neural Networkscited by 4
  6. 2024On Robust Reinforcement Learning with Lipschitz-Bounded Policy Networkscited by 4
  7. 2025Improved Scalable Lipschitz Bounds for Deep Neural Networkscited by 2
  8. 2019Robust Contraction Analysis of Nonlinear Systems via Differential IQCcited by 1
  9. 2023Robust Stability of Neural Network Control Systems with Interval Matrix Uncertaintiescited by 1

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