Towards Fast Computation of Certified Robustness for ReLU Networks (1804.09699v4)

Abstract: Verifying the robustness property of a general Rectified Linear Unit (ReLU) network is an NP-complete problem [Katz, Barrett, Dill, Julian and Kochenderfer CAV17]. Although finding the exact minimum adversarial distortion is hard, giving a certified lower bound of the minimum distortion is possible. Current available methods of computing such a bound are either time-consuming or delivering low quality bounds that are too loose to be useful. In this paper, we exploit the special structure of ReLU networks and provide two computationally efficient algorithms Fast-Lin and Fast-Lip that are able to certify non-trivial lower bounds of minimum distortions, by bounding the ReLU units with appropriate linear functions Fast-Lin, or by bounding the local Lipschitz constant Fast-Lip. Experiments show that (1) our proposed methods deliver bounds close to (the gap is 2-3X) exact minimum distortion found by Reluplex in small MNIST networks while our algorithms are more than 10,000 times faster; (2) our methods deliver similar quality of bounds (the gap is within 35% and usually around 10%; sometimes our bounds are even better) for larger networks compared to the methods based on solving linear programming problems but our algorithms are 33-14,000 times faster; (3) our method is capable of solving large MNIST and CIFAR networks up to 7 layers with more than 10,000 neurons within tens of seconds on a single CPU core. In addition, we show that, in fact, there is no polynomial time algorithm that can approximately find the minimum $\ell_1$ adversarial distortion of a ReLU network with a $0.99\ln n$ approximation ratio unless $\mathsf{NP}$=$\mathsf{P}$, where $n$ is the number of neurons in the network.

• The paper introduces Fast-Lin and Fast-Lip that efficiently compute certified lower bounds, achieving speedups of up to 14,000x over prior methods.
• Fast-Lin leverages linear approximations and Fast-Lip estimates local Lipschitz constants, providing robust bounds close to exact distortions.
• The work proves that approximating minimum adversarial distortion is NP-hard, emphasizing the importance of efficient robustness verification.

Essay on "Towards Fast Computation of Certified Robustness for ReLU Networks"

The paper "Towards Fast Computation of Certified Robustness for ReLU Networks" addresses the computational challenge associated with verifying the robustness of neural networks, specifically those activated using Rectified Linear Units (ReLUs). The core difficulty presented is NP-completeness of verifying robustness in ReLU networks, rendering the exact computation of minimum adversarial distortion infeasible for non-trivial networks. The authors contribute to overcoming this by introducing two novel algorithms, Fast-Lin and Fast-Lip, which provide computationally efficient means to establish certified lower bounds on adversarial distortions.

Algorithms and Results

1. Fast-Lin and Fast-Lip: These algorithms exploit the structural properties of ReLU networks. Fast-Lin uses linear approximations to compute non-trivial lower bounds efficiently, while Fast-Lip focuses on estimating local Lipschitz constants. Both approaches yield certified bounds significantly faster than traditional methods.
2. Empirical Performance: Experimental results highlight the efficiency and accuracy of the proposed methods. Fast-Lin delivers bounds close to the exact minimum distortions identified by Reluplex, with speed improvements by more than four orders of magnitude on small networks. Fast-Lip also provides robust lower bounds within a narrow gap to quality benchmarks like Linear Programming methods, achieving 33 to 14,000 times faster results.
3. Large-Scale Networks: Unlike methods such as LP-based approaches which fail to scale effectively, Fast-Lin and Fast-Lip maintain computational feasibility and deliver robust bounds for networks comprising up to 7 layers and over 10,000 neurons in a matter of seconds on a single CPU core.
4. Hardness Result: Extending theoretical groundwork, the paper proves the non-existence of polynomial-time algorithms that can approximately find minimum adversarial distortion with a bounded approximation ratio, unless NP equals P. This substantiates the computational difficulty and the importance of efficient approximation strategies proposed in the paper.

Implications and Future Work

The research conveys significant implications for both theoretical validation and practical deployment of neural networks in critical applications, where robustness against adversarial perturbations is crucial. Practically, the algorithms can be flexibly integrated with existing machine learning systems to provide robustness estimates, aiding defenses in adversarial settings.

Theoretical extensions could focus on enhancing these algorithms to handle more complex architectures, such as convolutional layers, and further reducing the computational cost. Future work may involve the adaptation of these methods to sophisticated networks beyond feedforward models, such as ResNets on large dataset challenges like ImageNet.

In summary, "Towards Fast Computation of Certified Robustness for ReLU Networks" offers substantial advancements in efficiently verifying the robustness of neural networks, contributing important tools and insights for the development of reliable AI systems.