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Register a new userBeta-CROWN: Efficient Bound Propagation with Per-neuron Split Constraints for Complete and Incomplete Neural Network Verification
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Added by
Huan Zhang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Recent works in neural network verification show that cheap incomplete verifiers such as CROWN, based upon bound propagations, can effectively be used in Branch-and-Bound (BaB) methods to accelerate complete verification, achieving significant speedups compared to expensive linear programming (LP) based techniques. However, they cannot fully handle the per-neuron split constraints introduced by BaB like LP verifiers do, leading to looser bounds and hurting their verification efficiency. In this work, we develop $beta$-CROWN, a new bound propagation based method that can fully encode per-neuron splits via optimizable parameters $beta$. When the optimizable parameters are jointly optimized in intermediate layers, $beta$-CROWN has the potential of producing better bounds than typical LP verifiers with neuron split constraints, while being efficiently parallelizable on GPUs. Applied to the complete verification setting, $beta$-CROWN is close to three orders of magnitude faster than LP-based BaB methods for robustness verification, and also over twice faster than state-of-the-art GPU-based complete verifiers with similar timeout rates. By terminating BaB early, our method can also be used for incomplete verification. Compared to the state-of-the-art semidefinite-programming (SDP) based verifier, we show a substantial leap forward by greatly reducing the gap between verified accuracy and empirical adversarial attack accuracy, from 35% (SDP) to 12% on an adversarially trained MNIST network ($epsilon=0.3$), while being 47 times faster. Our code is available at https://github.com/KaidiXu/Beta-CROWN
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Many available formal verification methods have been shown to be instances of a unified Branch-and-Bound (BaB) formulation. We propose a novel machine learning framework that can be used for designing an effective branching strategy as well as for computing better lower bounds. Specifically, we learn two graph neural networks (GNN) that both directly treat the network we want to verify as a graph input and perform forward-backward passes through the GNN layers. We use one GNN to simulate the strong branching heuristic behaviour and another to compute a feasible dual solution of the convex relaxation, thereby providing a valid lower bound. We provide a new verification dataset that is more challenging than those used in the literature, thereby providing an effective alternative for testing algorithmic improvements for verification. Whilst using just one of the GNNs leads to a reduction in verification time, we get optimal performance when combining the two GNN approaches. Our combined framework achieves a 50% reduction in both the number of branches and the time required for verification on various convolutional networks when compared to several state-of-the-art verification methods. In addition, we show that our GNN models generalize well to harder properties on larger unseen networks.
Formal verification of neural networks (NNs) is a challenging and important problem. Existing efficient complete solvers typically require the branch-and-bound (BaB) process, which splits the problem domain into sub-domains and solves each sub-domain using faster but weaker incomplete verifiers, such as Linear Programming (LP) on linearly relaxed sub-domains. In this paper, we propose to use the backward mode linear relaxation based perturbation analysis (LiRPA) to replace LP during the BaB process, which can be efficiently implemented on the typical machine learning accelerators such as GPUs and TPUs. However, unlike LP, LiRPA when applied naively can produce much weaker bounds and even cannot check certain conflicts of sub-domains during splitting, making the entire procedure incomplete after BaB. To address these challenges, we apply a fast gradient based bound tightening procedure combined with batch splits and the design of minimal usage of LP bound procedure, enabling us to effectively use LiRPA on the accelerator hardware for the challenging complete NN verification problem and significantly outperform LP-based approaches. On a single GPU, we demonstrate an order of magnitude speedup compared to existing LP-based approaches.
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.
Previous works proved that the combination of the linear neuron network with nonlinear activation functions (e.g. ReLu) can achieve nonlinear function approximation. However, simply widening or deepening the network structure will introduce some training problems. In this work, we are aiming to build a comprehensive second-order CNN implementation framework that includes neuron/network design and system deployment optimization.
We improve the scalability of Branch and Bound (BaB) algorithms for formally proving input-output properties of neural networks. First, we propose novel bounding algorithms based on Lagrangian Decomposition. Previous works have used off-the-shelf solvers to solve relaxations at each node of the BaB tree, or constructed weaker relaxations that can be solved efficiently, but lead to unnecessarily weak bounds. Our formulation restricts the optimization to a subspace of the dual domain that is guaranteed to contain the optimum, resulting in accelerated convergence. Furthermore, it allows for a massively parallel implementation, which is amenable to GPU acceleration via modern deep learning frameworks. Second, we present a novel activation-based branching strategy. By coupling an inexpensive heuristic with fast dual bounding, our branching scheme greatly reduces the size of the BaB tree compared to previous heuristic methods. Moreover, it performs competitively with a recent strategy based on learning algorithms, without its large offline training cost. Finally, we design a BaB framework, named Branch and Dual Network Bound (BaDNB), based on our novel bounding and branching algorithms. We show that BaDNB outperforms previous complete verification systems by a large margin, cutting average verification times by factors up to 50 on adversarial robustness properties.
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