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Register a new userNeural Network Branch-and-Bound for Neural Network Verification
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Added by
Florian Jaeckle
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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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.
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.
Formal verification of neural networks is essential for their deployment in safety-critical areas. Many available formal verification methods have been shown to be instances of a unified Branch and Bound (BaB) formulation. We propose a novel framework for designing an effective branching strategy for BaB. Specifically, we learn a graph neural network (GNN) to imitate the strong branching heuristic behaviour. Our framework differs from previous methods for learning to branch in two main aspects. Firstly, our framework directly treats the neural network we want to verify as a graph input for the GNN. Secondly, we develop an intuitive forward and backward embedding update schedule. Empirically, our framework achieves roughly $50%$ reduction in both the number of branches and the time required for verification on various convolutional networks when compared to the best available hand-designed branching strategy. In addition, we show that our GNN model enjoys both horizontal and vertical transferability. Horizontally, the model trained on easy properties performs well on properties of increased difficulty levels. Vertically, the model trained on small neural networks achieves similar performance on large neural networks.
Existing neural network verifiers compute a proof that each input is handled correctly under a given perturbation by propagating a convex set of reachable values at each layer. This process is repeated independently for each input (e.g., image) and perturbation (e.g., rotation), leading to an expensive overall proof effort when handling an entire dataset. In this work we introduce a new method for reducing this verification cost based on the key insight that convex sets obtained at intermediate layers can overlap across different inputs and perturbations. Leveraging this insight, we introduce the general concept of shared certificates, enabling proof effort reuse across multiple inputs and driving down overall verification costs. We validate our insight via an extensive experimental evaluation and demonstrate the effectiveness of shared certificates on a range of datasets and attack specifications including geometric, patch and $ell_infty$ input perturbations.
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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