Branch and Bound for Piecewise Linear Neural Network Verification (1909.06588v4)

Abstract: The success of Deep Learning and its potential use in many safety-critical applications has motivated research on formal verification of Neural Network (NN) models. In this context, verification involves proving or disproving that an NN model satisfies certain input-output properties. Despite the reputation of learned NN models as black boxes, and the theoretical hardness of proving useful properties about them, researchers have been successful in verifying some classes of models by exploiting their piecewise linear structure and taking insights from formal methods such as Satisifiability Modulo Theory. However, these methods are still far from scaling to realistic neural networks. To facilitate progress on this crucial area, we exploit the Mixed Integer Linear Programming (MIP) formulation of verification to propose a family of algorithms based on Branch-and-Bound (BaB). We show that our family contains previous verification methods as special cases. With the help of the BaB framework, we make three key contributions. Firstly, we identify new methods that combine the strengths of multiple existing approaches, accomplishing significant performance improvements over previous state of the art. Secondly, we introduce an effective branching strategy on ReLU non-linearities. This branching strategy allows us to efficiently and successfully deal with high input dimensional problems with convolutional network architecture, on which previous methods fail frequently. Finally, we propose comprehensive test data sets and benchmarks which includes a collection of previously released testcases. We use the data sets to conduct a thorough experimental comparison of existing and new algorithms and to provide an inclusive analysis of the factors impacting the hardness of verification problems.

• The paper introduces a unified Branch-and-Bound framework for PL-NN verification using Mixed Integer Linear Programming techniques.
• It presents a novel ReLU branching strategy that efficiently manages high-dimensional inputs and convolutional architectures.
• Numerical experiments demonstrate up to 100x speedup, enhancing the framework's applicability in safety-critical systems.

A Formal Approach to Verification of Piecewise Linear Neural Networks

The paper "Branch and Bound for Piecewise Linear Neural Network Verification" addresses the challenge of verifying piecewise linear neural networks (PL-NNs), which are fundamentally important for deploying NNs in safety-critical applications. The authors explore the formal verification of neural networks, a crucial task, given the decision-making role that NNs might play in scenarios such as autonomous driving and medical diagnosis.

Key Contributions

The paper outlines several significant advancements in the field of neural network verification:

1. Unified Branch-and-Bound Framework: The authors introduce a comprehensive Branch-and-Bound (BaB) framework for the verification of PL-NNs. Utilizing Mixed Integer Linear Programming (MIP) formulations, this approach encompasses existing verification methods as specific instances. The BaB framework facilitates identifying and rectifying the limitations of previous methods, thereby delivering substantial performance gains.
2. ReLU Branching Strategy: A novel branching strategy for ReLU non-linearities is proposed, enhancing the ability to handle large networks effectively, particularly those involving high-dimensional inputs and convolutional architectures. This strategy is computationally efficient and improves existing BaB-based methods that previously focused merely on input-domain branching.
3. Advanced Benchmarks and Testing: Comprehensive datasets are introduced, including convolutional networks that were previously underrepresented in verification tests. Through rigorous experimental comparisons on these datasets, the authors provide insights into factors influencing the complexity of verification problems.

Numerical Results and Their Implications

The paper demonstrates substantial numerical improvements in verification speed and scalability. Implementations based on the proposed BaB framework show performance improvements ranging up to two orders of magnitudes in specific cases compared to baseline methods. Such numerical results underscore the practical applicability of the framework in real-world scenarios involving complex neural networks.

The implications of these results are manifold:

• Scalability: The BaB framework encourages effective verification of much larger and deeper networks than was previously feasible, pushing forward the boundaries of deployable NNs in critical systems.
• Adaptability: The ability to handle convolutional networks and efficiently manage ReLU nodes provides robustness against adversarial inputs, potentially mitigating security risks in NN applications.

Future Directions

Given the current trajectory, future developments might focus on extending the BaB framework to cover more activation functions beyond ReLU, such as sigmoid and tanh, which, while less frequently used in modern practice, still hold relevance in specific contexts. Moreover, integrating machine learning techniques to optimize branching heuristics dynamically could further enhance verification efficiency and make the BaB approach more adaptive to varying network architectures and complexities.

Additionally, exploring higher levels of relaxations within the Sherali-Adams hierarchy could yield even tighter bounds and further reduce computational costs associated with neural network verification.

In conclusion, the paper represents a significant advancement in the formal verification of neural networks, addressing both theoretical complexities and practical scalability issues. Through the proposed methodologies, verification of PL-NNs is not only more efficient but also more robust, paving the way for safer deployment of neural networks in sensitive applications.