# Stochastic Sample Approximations of (Local) Moduli of Continuity

arXiv: https://arxiv.org/abs/2509.15368 · HTML: https://mlsanity.com/paper/2509.15368/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 76 references that are in the map.
- First posted on arXiv 18 Sep 2025.

## Builds on (most cited first, 30 of 32)

- [Evaluating the Robustness of Neural Networks: An Extreme Value Theory Approach](https://mlsanity.com/paper/1801.10578/) (2018)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Strong mixed-integer programming formulations for trained neural networks](https://mlsanity.com/paper/1811.08359/) (2018)
- [General Cutting Planes for Bound-Propagation-Based Neural Network Verification](https://mlsanity.com/paper/2208.05740/) (2022)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Which Neural Net Architectures Give Rise To Exploding and Vanishing Gradients?](https://mlsanity.com/paper/1801.03744/) (2018)
- [Measuring Neural Net Robustness with Constraints](https://mlsanity.com/paper/1605.07262/) (2016)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [Limitations of the Lipschitz constant as a defense against adversarial examples](https://mlsanity.com/paper/1807.09705/) (2018)
- [A mathematical model for automatic differentiation in machine learning](https://mlsanity.com/paper/2006.02080/) (2020)
- [Instead of Rewriting Foreign Code for Machine Learning, Automatically Synthesize Fast Gradients](https://mlsanity.com/paper/2010.01709/) (2020)
- [Bandits with many optimal arms](https://mlsanity.com/paper/2103.12452/) (2021)
- [Semialgebraic Representation of Monotone Deep Equilibrium Models and Applications to Certification](https://mlsanity.com/paper/2106.01453/) (2021)
- [Pure Exploration in Infinitely-Armed Bandit Models with Fixed-Confidence](https://mlsanity.com/paper/1803.04665/) (2018)
- [Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks](https://mlsanity.com/paper/2204.00846/) (2022)
- [A Closer Look at the Adversarial Robustness of Deep Equilibrium Models](https://mlsanity.com/paper/2306.01429/) (2023)
- [Analytical bounds on the local Lipschitz constants of ReLU networks](https://mlsanity.com/paper/2104.14672/) (2021)
- [Stability Verification of Neural Network Controllers using Mixed-Integer Programming](https://mlsanity.com/paper/2206.13374/) (2022)
- [Stability and feasibility of neural network-based controllers via output range analysis](https://mlsanity.com/paper/2004.00521/) (2020)
- [ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks](https://mlsanity.com/paper/2404.04375/) (2024)
- [Gradient Masking Causes CLEVER to Overestimate Adversarial Perturbation Size](https://mlsanity.com/paper/1804.07870/) (2018)
- [Robustness Certificates for Implicit Neural Networks: A Mixed Monotone Contractive Approach](https://mlsanity.com/paper/2112.05310/) (2021)
- [Lipschitz constant estimation for general neural network architectures using control tools](https://mlsanity.com/paper/2405.01125/) (2024)
- [On Extensions of CLEVER: A Neural Network Robustness Evaluation Algorithm](https://mlsanity.com/paper/1810.08640/) (2018)
- [Predictability and Fairness in Load Aggregation and Operations of Virtual Power Plants](https://mlsanity.com/paper/2110.03001/) (2021)
- [Sound and Complete Verification of Polynomial Networks](https://mlsanity.com/paper/2209.07235/) (2022)
- [Closed-Loop View of the Regulation of AI: Equal Impact across Repeated Interactions](https://mlsanity.com/paper/2209.01410/) (2022)
- [Improved Scalable Lipschitz Bounds for Deep Neural Networks](https://mlsanity.com/paper/2503.14297/) (2025)

## Built on by (most cited first, 1 of 1)

- [Beyond Lipschitz: Data-Driven Robustness via Discrete Modulus of Continuity](https://mlsanity.com/paper/2605.28729/) (2026)

## Abstract (arXiv)

Modulus of local continuity is used to evaluate the robustness of neural networks and fairness of their repeated uses in closed-loop models. Here, we revisit a connection between generalized derivatives and moduli of local continuity, and present a non-uniform stochastic sample approximation for moduli of local continuity. This is of importance in studying robustness of neural networks and fairness of their repeated uses.