Batch normalization is a ubiquitous deep learning technique that normalizes activations in intermediate layers. It is associated with improved accuracy and faster learning, but despite its enormous success there is little consensus regarding why it works. We aim to rectify this and take an empirical approach to understanding batch normalization. Our primary observation is that the higher learning rates that batch normalization enables have a regularizing effect that dramatically improves generalization of normalized networks, which is both demonstrated empirically and motivated theoretically. We show how activations become large and how the convolutional channels become increasingly ill-behaved for layers deep in unnormalized networks, and how this results in larger input-independent gradients. Beyond just gradient scaling, we demonstrate how the learning rate in unnormalized networks is further limited by the magnitude of activations growing exponentially with network depth for large parameter updates, a problem batch normalization trivially avoids. Motivated by recent results in random matrix theory, we argue that ill-conditioning of the activations is due to fluctuations in random initialization, shedding new light on classical initialization schemes and their consequences.

For a long time, designing neural architectures that exhibit high performance was considered a dark art that required expert hand-tuning. One of the few well-known guidelines for architecture design is the avoidance of exploding gradients, though even this guideline has remained relatively vague and circumstantial. We introduce the nonlinearity coefficient (NLC), a measurement of the complexity of the function computed by a neural network that is based on the magnitude of the gradient. Via an extensive empirical study, we show that the NLC is a powerful predictor of test error and that attaining a right-sized NLC is essential for optimal performance. The NLC exhibits a range of intriguing and important properties. It is closely tied to the amount of information gained from computing a single network gradient. It is tied to the error incurred when replacing the nonlinearity operations in the network with linear operations. It is not susceptible to the confounders of multiplicative scaling, additive bias and layer width. It is stable from layer to layer. Hence, we argue that the NLC is the first robust predictor of overfitting in deep networks.

In our previous work we have shown that resistive cross point devices, so called Resistive Processing Unit (RPU) devices, can provide significant power and speed benefits when training deep fully connected networks as well as convolutional neural networks. In this work, we further extend the RPU concept for training recurrent neural networks (RNNs) namely LSTMs. We show that the mapping of recurrent layers is very similar to the mapping of fully connected layers and therefore the RPU concept can potentially provide large acceleration factors for RNNs as well. In addition, we study the effect of various device imperfections and system parameters on training performance. Symmetry of updates becomes even more crucial for RNNs; already a few percent asymmetry results in an increase in the test error compared to the ideal case trained with floating point numbers. Furthermore, the input signal resolution to device arrays needs to be at least 7 bits for successful training. However, we show that a stochastic rounding scheme can reduce the input signal resolution back to 5 bits. Further, we find that RPU device variations and hardware noise are enough to mitigate overfitting, so that there is less need for using dropout. We note that the models trained here are roughly 1500 times larger than the fully connected network trained on MNIST dataset in terms of the total number of multiplication and summation operations performed per epoch. Thus, here we attempt to study the validity of the RPU approach for large scale networks.

Abstract--Nyquist ghost artifacts in EPI images are originated from phase mismatch between the even and odd echoes. However, conventional correction methods using reference scans often produce erroneous results especially in high-field MRI due to the nonlinear and time-varying local magnetic field changes. Recently, it was shown that the problem of ghost correction can be transformed into k -space data interpolation problem that can be solved using the annihilating filter-based low-rank Hankel structured matrix completion approach (ALOHA). Another recent discovery has shown that the deep convolutional neural network is closely related to the data-driven Hankel matrix decomposition. By synergistically combining these findings, here we propose a k -space deep learning approach that immediately corrects the k -space phase mismatch without a reference scan. Reconstruction results using 7T in vivo data showed that the proposed reference-free k -space deep learning approach for EPI ghost correction significantly improves the image quality compared to the existing methods, and the computing time is several orders of magnitude faster . Echo-planar imaging (EPI) is one of the widely used MR imaging sequences. This sequence is fast because the wholek - space data is acquired through a single RF pulse by alternating the magnetic field direction in the even and odd lines. This short scan provides high temporal resolution and has therefore been used in many imaging modalities such as functional MRI, diffusion-weighted imaging (DWI) and so on.

The vanilla method to train a Deep Neural Network (DNN) is to use the Stochastic Gradient Descent (SGD) method (a first-order local optimization technique). The gradient of the DNN loss function, represented as a directed computational graph, is calculated using the efficient backpropagation algorithm relying on the chain rule of derivatives (a particular case of automatic differentiation). The ordinary derivative calculus can be encompassed into a more general q-calculus [1, 2] by defining the Jackson's q-derivative (and gradient) as follows: D

Theoretical understanding of deep learning is one of the most important tasks facing the statistics and machine learning communities. While deep neural networks (DNNs) originated as engineering methods and models of biological networks in neuroscience and psychology, they have quickly become a centerpiece of the machine learning toolbox. Unfortunately, DNN adoption powered by recent successes combined with the open-source nature of the machine learning community, has outpaced our theoretical understanding. We cannot reliably identify when and why DNNs will make mistakes. In some applications like text translation these mistakes may be comical and provide for fun fodder in research talks, a single error can be very costly in tasks like medical imaging. As we utilize DNNs in increasingly sensitive applications, a better understanding of their properties is thus imperative. Recent advances in DNN theory are numerous and include many different sources of intuition, such as learning theory, sparse signal analysis, physics, chemistry, and psychology. An interesting pattern begins to emerge in the breadth of possible interpretations. The seemingly limitless approaches are mostly constrained by the lens with which the mathematical operations are viewed. Ultimately, the interpretation of DNNs appears to mimic a type of Rorschach test --- a psychological test wherein subjects interpret a series of seemingly ambiguous ink-blots. Validation for DNN theory requires a convergence of the literature. We must distinguish between universal results that are invariant to the analysis perspective and those that are specific to a particular network configuration. Simultaneously we must deal with the fact that many standard statistical tools for quantifying generalization or empirically assessing important network features are difficult to apply to DNNs.

Due to the success of residual networks (resnets) and related architectures, shortcut connections have quickly become standard tools for building convolutional neural networks. The explanations in the literature for the apparent effectiveness of shortcuts are varied and often contradictory. We hypothesize that shortcuts work primarily because they act as linear counterparts to nonlinear layers. We test this hypothesis by using several variations on the standard residual block, with different types of linear connections, to build small image classification networks. Our experiments show that other kinds of linear connections can be even more effective than the identity shortcuts. Our results also suggest that the best type of linear connection for a given application may depend on both network width and depth.

Deep generative models can learn to generate realistic-looking images on several natural image datasets, but many of the most effective methods are adversarial methods, which require careful balancing of training between a generator network and a discriminator network. Maximum mean discrepancy networks (MMD-nets) avoid this issue using the kernel trick, but unfortunately they have not on their own been able to match the performance of adversarial training. We present a new method of training MMD-nets, based on learning a mapping of samples from the data and from the model into a lower dimensional space, in which MMD training can be more effective. We call these networks ratio matching MMD networks (RM-MMDnets). We train the mapping to preserve density ratios between the densities over the low-dimensional space and the original space. This ensures that matching the model distribution to the data in the low-dimensional space will also match the original distributions. We show that RM-MMDnets have better performance and better stability than recent adversarial methods for training MMD-nets.

Deep learning systems have become ubiquitous in many aspects of our lives. Unfortunately, it has been shown that such systems are vulnerable to adversarial attacks, making them prone to potential unlawful and harmful uses. Designing deep neural networks that are robust to adversarial attacks is a fundamental step in making such systems safer and deployable in a broader variety of applications (e.g., autonomous driving), but more importantly is a necessary step to design novel and more advanced architectures built on new computational paradigms rather than marginally modifying existing ones. In this paper we introduce PeerNets, a novel family of convolutional networks alternating classical Euclidean convolutions with graph convolutions to harness information from a graph of peer samples. This results in a form of non-local forward propagation in the model, where latent features are conditioned on the global structure induced by the data graph, that is up to 3 more robust to a variety of white-and black-box adversarial attacks compared to conventional architectures with almost no drop in accuracy.

Machine learning models are vulnerable to simple model stealing attacks if the adversary can obtain output labels for chosen inputs. To protect against these attacks, it has been proposed to limit the information provided to the adversary by omitting probability scores, significantly impacting the utility of the provided service. In this work, we illustrate how a service provider can still provide useful, albeit misleading, class probability information, while significantly limiting the success of the attack. Our defense forces the adversary to discard the class probabilities, requiring significantly more queries before they can train a model with comparable performance. We evaluate several attack strategies, model architectures, and hyperparameters under varying adversarial models, and evaluate the efficacy of our defense against the strongest adversary. Finally, we quantify the amount of noise injected into the class probabilities to mesure the loss in utility, e.g., adding 1.74 nats per query on CIFAR-10 and 3.27 on MNIST. Our extensive evaluation shows our defense can degrade the accuracy of the stolen model at least 20%, or require 4x more queries while keeping the accuracy of the protected model almost intact.