# Learning Neural Networks under Input-Output Specifications

arXiv: https://arxiv.org/abs/2202.11246 · HTML: https://mlsanity.com/paper/2202.11246/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 32 references that are in the map.
- First posted on arXiv 23 Feb 2022.

## Builds on (most cited first, 19 of 19)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [A Survey on Bias and Fairness in Machine Learning](https://mlsanity.com/paper/1908.09635/) (2019)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [A Lyapunov-based Approach to Safe Reinforcement Learning](https://mlsanity.com/paper/1805.07708/) (2018)
- [A Survey on Neural Network Interpretability](https://mlsanity.com/paper/2012.14261/) (2020)
- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Scaling provable adversarial defenses](https://mlsanity.com/paper/1805.12514/) (2018)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Training robust neural networks using Lipschitz bounds](https://mlsanity.com/paper/2005.02929/) (2020)
- [Opportunities and Challenges in Deep Learning Adversarial Robustness: A Survey](https://mlsanity.com/paper/2007.00753/) (2020)
- [When Machine Learning Meets Privacy: A Survey and Outlook](https://mlsanity.com/paper/2011.11819/) (2020)
- [Training verified learners with learned verifiers](https://mlsanity.com/paper/1805.10265/) (2018)
- [Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming](https://mlsanity.com/paper/2004.07876/) (2020)
- [Imitation Learning with Stability and Safety Guarantees](https://mlsanity.com/paper/2012.09293/) (2020)
- [Stability-certified reinforcement learning: A control-theoretic perspective](https://mlsanity.com/paper/1810.11505/) (2018)
- [A Berkeley View of Systems Challenges for AI](https://mlsanity.com/paper/1712.05855/) (2017)
- [Verification of Non-Linear Specifications for Neural Networks](https://mlsanity.com/paper/1902.09592/) (2019)

## Built on by (most cited first, 3 of 3)

- [Fast and Reliable N-k Contingency Screening with Input-Convex Neural Networks](https://mlsanity.com/paper/2410.00796/) (2024)
- [Self-supervised Equality Embedded Deep Lagrange Dual for Approximate Constrained Optimization](https://mlsanity.com/paper/2306.06674/) (2023)
- [A Scalable Approach for Safe and Robust Learning via Lipschitz-Constrained Networks](https://mlsanity.com/paper/2506.23977/) (2025)

## Abstract (arXiv)

In this paper, we examine an important problem of learning neural networks that certifiably meet certain specifications on input-output behaviors. Our strategy is to find an inner approximation of the set of admissible policy parameters, which is convex in a transformed space. To this end, we address the key technical challenge of convexifying the verification condition for neural networks, which is derived by abstracting the nonlinear specifications and activation functions with quadratic constraints. In particular, we propose a reparametrization scheme of the original neural network based on loop transformation, which leads to a convex condition that can be enforced during learning. This theoretical construction is validated in an experiment that specifies reachable sets for different regions of inputs.