Section A.1 presents the lemmas used to prove the main results. Section A.2 presents the main results The first two inequalities are owing to the triangle inequality, and the third inequality is due to the definition of L-divergence Eq.(5). We complete the proof by applying Lemma A.1 to bound F ollowing the conditions of Theorem 4.1, the upper bound of null V arnull null D Based on the conditions of Theorem 4.1, we assume We complete the proof by applying Lemma A.3 and Lemma A.4 to bound the Rademacher Following the proof of Theorem 4.1, we have |D F ollowing the conditions of Proposition 4.3, as N, we have, null D Based on the result on Proposition 4.3, for any δ (0, 1), we know that 4LB ( 2 D ln 2 + 1)null We complete the proof by applying the triangle inequality. III: Samples from p and q are labeled with 0 and 1, respectively. All values are averaged over five trials.

Instead their main role isto provide inertia during training. Weinterpret current methods in terms of inertia and provide novel insights into the optimization of BNNs.

Concerned with the reliability of neural networks, researchers have developed verification techniques to prove their robustness. Most verifiers work with real-valued networks. Unfortunately, the exact (complete and sound) verifiers face scalability challenges and provide no correctness guarantees due to floating point errors. We argue that Binarized Neural Networks (BNNs) provide comparable robustness and allow exact and significantly more efficient verification. We present a new system, EEV, for efficient and exact verification of BNNs. EEV consists of two parts: (i) a novel SAT solver that speeds up BNN verification by natively handling the reified cardinality constraints arising in BNN encodings; and (ii) strategies to train solver-friendly robust BNNs by inducing balanced layer-wise sparsity and low cardinality bounds, and adaptively cancelling the gradients. We demonstrate the effectiveness of EEV by presenting the first exact verification results for L-inf-bounded adversarial robustness of nontrivial convolutional BNNs on the MNIST and CIFAR10 datasets. Compared to exact verification of real-valued networks of the same architectures on the same tasks, EEV verifies BNNs hundreds to thousands of times faster, while delivering comparable verifiable accuracy in most cases.

Instead their main role is to provide inertia during training. We interpret current methods in terms of inertia and provide novel insights into the optimization of BNNs.

Concerned with the reliability of neural networks, researchers have developed verification techniques to prove their robustness. Most verifiers work with real-valued networks. Unfortunately, the exact (complete and sound) verifiers face scalability challenges and provide no correctness guarantees due to floating point errors. We argue that Binarized Neural Networks (BNNs) provide comparable robustness and allow exact and significantly more efficient verification. We present a new system, EEV, for efficient and exact verification of BNNs.

Complex-valued neural networks have attracted increasing attention in recent years, while it remains open on the advantages of complex-valued neural networks in comparison with real-valued networks. This work takes one step on this direction by introducing the \emph{complex-reaction network} with fully-connected feed-forward architecture. We prove the universal approximation property for complex-reaction networks, and show that a class of radial functions can be approximated by a complex-reaction network using the polynomial number of parameters, whereas real-valued networks need at least exponential parameters to reach the same approximation level. For empirical risk minimization, our theoretical result shows that the critical point set of complex-reaction networks is a proper subset of that of real-valued networks, which may show some insights on finding the optimal solutions more easily for complex-reaction networks.

Previous studies dominantly target at self-supervised learning on real-valued networks and have achieved many promising results. However, on the more challenging binary neural networks (BNNs), this task has not yet been fully explored in the community. In this paper, we focus on this more difficult scenario: learning networks where both weights and activations are binary, meanwhile, without any human annotated labels. We observe that the commonly used contrastive objective is not satisfying on BNNs for competitive accuracy, since the backbone network contains relatively limited capacity and representation ability. Hence instead of directly applying existing self-supervised methods, which cause a severe decline in performance, we present a novel guided learning paradigm from real-valued to distill binary networks on the final prediction distribution, to minimize the loss and obtain desirable accuracy. Our proposed method can boost the simple contrastive learning baseline by an absolute gain of 5.5~15% on BNNs. We further reveal that it is difficult for BNNs to recover the similar predictive distributions as real-valued models when training without labels. Thus, how to calibrate them is key to address the degradation in performance. Extensive experiments are conducted on the large-scale ImageNet and downstream datasets. Our method achieves substantial improvement over the simple contrastive learning baseline, and is even comparable to many mainstream supervised BNN methods. Code will be made available.

Optimization of Binarized Neural Networks (BNNs) currently relies on real-valued latent weights to accumulate small update steps. In this paper, we argue that these latent weights cannot be treated analogously to weights in real-valued networks. Instead their main role is to provide inertia during training. We interpret current methods in terms of inertia and provide novel insights into the optimization of BNNs. We subsequently introduce the first optimizer specifically designed for BNNs, Binary Optimizer (Bop), and demonstrate its performance on CIFAR-10 and ImageNet. Together, the redefinition of latent weights as inertia and the introduction of Bop enable a better understanding of BNN optimization and open up the way for further improvements in training methodologies for BNNs.

Deep learning has become an area of interest in most scientific areas, including physical sciences. Modern networks apply real-valued transformations on the data. Particularly, convolutions in convolutional neural networks discard phase information entirely. Many deterministic signals, such as seismic data or electrical signals, contain significant information in the phase of the signal. We explore complex-valued deep convolutional networks to leverage non-linear feature maps. Seismic data commonly has a lowcut filter applied, to attenuate noise from ocean waves and similar long wavelength contributions. Discarding the phase information leads to low-frequency aliasing analogous to the Nyquist-Shannon theorem for high frequencies. In non-stationary data, the phase content can stabilize training and improve the generalizability of neural networks. While it has been shown that phase content can be restored in deep neural networks, we show how including phase information in feature maps improves both training and inference from deterministic physical data. Furthermore, we show that the reduction of parameters in a complex network results in training on a smaller dataset without overfitting, in comparison to a real-valued network with the same performance.