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Implicit neural networks are a general class of learning models that replace the layers in traditional feedforward models with implicit algebraic equations. Compared to traditional learning models, implicit networks offer competitive performance and reduced memory consumption. However, they can remain brittle with respect to input adversarial perturbations. This paper proposes a theoretical and computational framework for robustness verification of implicit neural networks; our framework blends together mixed monotone systems theory and contraction theory. First, given an implicit neural network, we introduce a related embedded network and show that, given an infinity-norm box constraint on the input, the embedded network provides an infinity-norm box overapproximation for the output of the original network. Second, using infinity-matrix measures, we propose sufficient conditions for well-posedness of both the original and embedded system and design an iterative algorithm to compute the infinity-norm box robustness margins for reachability and classification problems. Third, of independent value, we show that employing a suitable relative classifier variable in our analysis will lead to tighter bounds on the certified adversarial robustness in classification problems. Finally, we perform numerical simulations on a Non-Euclidean Monotone Operator Network (NEMON) trained on the MNIST dataset. In these simulations, we compare the accuracy and run time of our mixed monotone contractive approach with the existing robustness verification approaches in the literature for estimating the certified adversarial robustness.
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Robustness Analysis for Convolutional Neural Networks with Uncertainty Quantification
This paper presents a novel framework for training convolutional neural networks (CNNs) to quantify the impact of gradual and abrupt uncertainties in the form of adversarial attacks. Uncertainty quantification is achieved by combining the CNN with a Gaussian process (GP) classifier algorithm. The variance of the GP quantifies the impact on the uncertainties and especially their effect on the object classification tasks. Learning from uncertainty provides the proposed CNN-GP framework with flexibility, reliability and robustness to adversarial attacks. The proposed approach includes training the network under noisy conditions. This is accomplished by comparing predictions with classification labels via the Kullback-Leibler divergence, Wasserstein distance and maximum correntropy. The network performance is tested on the classical MNIST, Fashion-MNIST, CIFAR10 and CIFAR 100 datasets. Further tests on robustness to both black-box and white-box attacks are also carried out for MNIST. The results show that the testing accuracy improves for networks that backpropogate uncertainty as compared to methods that do not quantify the impact of uncertainties. A comparison with a state-of-art Monte Carlo dropout method is also presented and the outperformance of the CNN-GP framework with respect to reliability and computational efficiency is demonstrated.
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- Award ID(s):
- 1903466
- PAR ID:
- 10259931
- Date Published:
- Journal Name:
- Proc. of the International Forum on Signal Processing
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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