# Output Range Analysis for Deep Feedforward Neural Networks

@inproceedings{Dutta2018OutputRA, title={Output Range Analysis for Deep Feedforward Neural Networks}, author={Souradeep Dutta and Susmit Jha and Sriram Sankaranarayanan and Ashish Tiwari}, booktitle={NFM}, year={2018} }

Given a neural network (NN) and a set of possible inputs to the network described by polyhedral constraints, we aim to compute a safe over-approximation of the set of possible output values. This operation is a fundamental primitive enabling the formal analysis of neural networks that are extensively used in a variety of machine learning tasks such as perception and control of autonomous systems. Increasingly, they are deployed in high-assurance applications, leading to a compelling use case…

## 188 Citations

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## References

SHOWING 1-10 OF 36 REFERENCES

Formal Verification of Piece-Wise Linear Feed-Forward Neural Networks

- Computer ScienceATVA
- 2017

An approach for the verification of feed-forward neural networks in which all nodes have a piece-wise linear activation function and infers additional node phases for the non-linear nodes in the network from partial node phase assignments, similar to unit propagation in classical SAT solving.

Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks

- Computer ScienceCAV
- 2017

Results show that the novel, scalable, and efficient technique presented can successfully prove properties of networks that are an order of magnitude larger than the largest networks verified using existing methods.

Output Reachable Set Estimation and Verification for Multilayer Neural Networks

- Computer ScienceIEEE Transactions on Neural Networks and Learning Systems
- 2018

An application to the safety verification for a robotic arm model with two joints is presented to show the effectiveness of the proposed approaches to output reachable set estimation and safety verification problems for multilayer perceptron (MLP) neural networks.

Piecewise Linear Neural Network verification: A comparative study

- Computer ScienceArXiv
- 2017

Motivated by the need of accelerating progress in this very important area, a number of different approaches based on Mixed Integer Programming, Satisfiability Modulo Theory, as well as a novel method based on the Branch-and-Bound framework are investigated.

PLATO: Policy learning using adaptive trajectory optimization

- Computer Science2017 IEEE International Conference on Robotics and Automation (ICRA)
- 2017

PLATO is proposed, a continuous, reset-free reinforcement learning algorithm that trains complex control policies with supervised learning, using model-predictive control (MPC) to generate the supervision, hence never in need of running a partially trained and potentially unsafe policy.

Reachable Set Estimation and Verification for a Class of Piecewise Linear Systems with Neural Network Controllers

- Mathematics, Computer Science
- 2018

A layer-by-layer approach is developed for the output reachable set computation of ReLU neural networks, which is formulated in the form of a set of manipulations for a union of polytopes.

Intriguing properties of neural networks

- Computer ScienceICLR
- 2014

It is found that there is no distinction between individual highlevel units and random linear combinations of high level units, according to various methods of unit analysis, and it is suggested that it is the space, rather than the individual units, that contains of the semantic information in the high layers of neural networks.

Verifying Neural Networks with Mixed Integer Programming

- Computer ScienceArXiv
- 2017

It is demonstrated that, for networks that are piecewise affine (for example, deep networks with ReLU and maxpool units), proving no adversarial example exists can be naturally formulated as solving a mixed integer program.

Reachable Set Estimation and Safety Verification for Piecewise Linear Systems with Neural Network Controllers

- Mathematics, Computer Science2018 Annual American Control Conference (ACC)
- 2018

The estimated output reachable set can be estimated iteratively for a given finite-time interval and the safety verification for piecewise linear systems with neural network controllers can be performed by checking the existence of intersections of unsafe regions and output reach set.

An Abstraction-Refinement Approach to Verification of Artificial Neural Networks

- Computer ScienceCAV
- 2010

A solution to verify their safety using abstractions to Boolean combinations of linear arithmetic constraints, and it is shown that whenever the abstract MLP is declared to be safe, the same holds for the concrete one.