arXiv subjects

Samuel S. Schoenholz

Publications and source records attributed to Samuel S. Schoenholz.

At least 37 records · Page 2Linked to original sources

arXiv2019

Dynamical Isometry and a Mean Field Theory of LSTMs and GRUs

Training recurrent neural networks (RNNs) on long sequence tasks is plagued with difficulties arising from the exponential explosion or vanishing of signals as they propagate forward or backward through the network. Many techniques have been proposed to ameliorate these issues, including various algorithmic and architectural modifications. Two of the most successful RNN architectures, the LSTM and the GRU, do exhibit modest improvements over vanilla RNN cells, but they still suffer from instabilities when trained on very long sequences. In this work, we develop a mean field theory of signal propagation in LSTMs and GRUs that enables us to calculate the time scales for signal propagation as well as the spectral properties of the state-to-state Jacobians. By optimizing these quantities in terms of the initialization hyperparameters, we derive a novel initialization scheme that eliminates or reduces training instabilities. We demonstrate the efficacy of our initialization scheme on multiple sequence tasks, on which it enables successful training while a standard initialization either fails completely or is orders of magnitude slower. We also observe a beneficial effect on generalization performance using this new initialization.

cs.LG↗

arXiv2019

A Mean Field Theory of Batch Normalization

We develop a mean field theory for batch normalization in fully-connected feedforward neural networks. In so doing, we provide a precise characterization of signal propagation and gradient backpropagation in wide batch-normalized networks at initialization. Our theory shows that gradient signals grow exponentially in depth and that these exploding gradients cannot be eliminated by tuning the initial weight variances or by adjusting the nonlinear activation function. Indeed, batch normalization itself is the cause of gradient explosion. As a result, vanilla batch-normalized networks without skip connections are not trainable at large depths for common initialization schemes, a prediction that we verify with a variety of empirical simulations. While gradient explosion cannot be eliminated, it can be reduced by tuning the network close to the linear regime, which improves the trainability of deep batch-normalized networks without residual connections. Finally, we investigate the learning dynamics of batch-normalized networks and observe that after a single step of optimization the networks achieve a relatively stable equilibrium in which gradients have dramatically smaller dynamic range. Our theory leverages Laplace, Fourier, and Gegenbauer transforms and we derive new identities that may be of independent interest.

cs.NE↗

arXiv2018

Machine learning determination of atomic dynamics at grain boundaries

In polycrystalline materials, grain boundaries are sites of enhanced atomic motion, but the complexity of the atomic structures within a grain boundary network makes it difficult to link the structure and atomic dynamics. Here we use a machine learning technique to establish a connection between local structure and dynamics of these materials. Following previous work on bulk glassy materials, we define a purely structural quantity, softness, that captures the propensity of an atom to rearrange. This approach correctly identifies crystalline regions, stacking faults, and twin boundaries as having low likelihood of atomic rearrangements, while finding a large variability within high-energy grain boundaries. As has been found in glasses [9,19,26], the probability that atoms of a given softness will rearrange is nearly Arrhenius. This indicates a well-defined energy barrier as well as a well-defined prefactor for the Arrhenius form for atoms of a given softness. The decrease in the prefactor for low-softness atoms indicates that variations in entropy exhibit a dominant influence on the atomic dynamics in grain boundaries.

cond-mat.mtrl-sci↗

arXiv2018

Adversarial Spheres

State of the art computer vision models have been shown to be vulnerable to small adversarial perturbations of the input. In other words, most images in the data distribution are both correctly classified by the model and are very close to a visually similar misclassified image. Despite substantial research interest, the cause of the phenomenon is still poorly understood and remains unsolved. We hypothesize that this counter intuitive behavior is a naturally occurring result of the high dimensional geometry of the data manifold. As a first step towards exploring this hypothesis, we study a simple synthetic dataset of classifying between two concentric high dimensional spheres. For this dataset we show a fundamental tradeoff between the amount of test error and the average distance to nearest error. In particular, we prove that any model which misclassifies a small constant fraction of a sphere will be vulnerable to adversarial perturbations of size $O(1/\sqrt{d})$. Surprisingly, when we train several different architectures on this dataset, all of their error sets naturally approach this theoretical bound. As a result of the theory, the vulnerability of neural networks to small adversarial perturbations is a logical consequence of the amount of test error observed. We hope that our theoretical analysis of this very simple case will point the way forward to explore how the geometry of complex real-world data sets leads to adversarial examples.

cs.CV↗

arXiv2018

Peptide-Spectra Matching from Weak Supervision

As in many other scientific domains, we face a fundamental problem when using machine learning to identify proteins from mass spectrometry data: large ground truth datasets mapping inputs to correct outputs are extremely difficult to obtain. Instead, we have access to imperfect hand-coded models crafted by domain experts. In this paper, we apply deep neural networks to an important step of the protein identification problem, the pairing of mass spectra with short sequences of amino acids called peptides. We train our model to differentiate between top scoring results from a state-of-the art classical system and hard-negative second and third place results. Our resulting model is much better at identifying peptides with spectra than the model used to generate its training data. In particular, we achieve a 43% improvement over standard matching methods and a 10% improvement over a combination of the matching method and an industry standard cross-spectra reranking tool. Importantly, in a more difficult experimental regime that reflects current challenges facing biologists, our advantage over the previous state-of-the-art grows to 15% even after reranking. We believe this approach will generalize to other challenging scientific problems.

q-bio.QM↗

arXiv2018

Dynamical Isometry and a Mean Field Theory of RNNs: Gating Enables Signal Propagation in Recurrent Neural Networks

Recurrent neural networks have gained widespread use in modeling sequence data across various domains. While many successful recurrent architectures employ a notion of gating, the exact mechanism that enables such remarkable performance is not well understood. We develop a theory for signal propagation in recurrent networks after random initialization using a combination of mean field theory and random matrix theory. To simplify our discussion, we introduce a new RNN cell with a simple gating mechanism that we call the minimalRNN and compare it with vanilla RNNs. Our theory allows us to define a maximum timescale over which RNNs can remember an input. We show that this theory predicts trainability for both recurrent architectures. We show that gated recurrent networks feature a much broader, more robust, trainable region than vanilla RNNs, which corroborates recent experimental findings. Finally, we develop a closed-form critical initialization scheme that achieves dynamical isometry in both vanilla RNNs and minimalRNNs. We show that this results in significantly improvement in training dynamics. Finally, we demonstrate that the minimalRNN achieves comparable performance to its more complex counterparts, such as LSTMs or GRUs, on a language modeling task.

stat.ML↗

arXiv2018

Dynamical Isometry and a Mean Field Theory of CNNs: How to Train 10,000-Layer Vanilla Convolutional Neural Networks

In recent years, state-of-the-art methods in computer vision have utilized increasingly deep convolutional neural network architectures (CNNs), with some of the most successful models employing hundreds or even thousands of layers. A variety of pathologies such as vanishing/exploding gradients make training such deep networks challenging. While residual connections and batch normalization do enable training at these depths, it has remained unclear whether such specialized architecture designs are truly necessary to train deep CNNs. In this work, we demonstrate that it is possible to train vanilla CNNs with ten thousand layers or more simply by using an appropriate initialization scheme. We derive this initialization scheme theoretically by developing a mean field theory for signal propagation and by characterizing the conditions for dynamical isometry, the equilibration of singular values of the input-output Jacobian matrix. These conditions require that the convolution operator be an orthogonal transformation in the sense that it is norm-preserving. We present an algorithm for generating such random initial orthogonal convolution kernels and demonstrate empirically that they enable efficient training of extremely deep architectures.

stat.ML↗

arXiv2018

Deep Neural Networks as Gaussian Processes

It has long been known that a single-layer fully-connected neural network with an i.i.d. prior over its parameters is equivalent to a Gaussian process (GP), in the limit of infinite network width. This correspondence enables exact Bayesian inference for infinite width neural networks on regression tasks by means of evaluating the corresponding GP. Recently, kernel functions which mimic multi-layer random neural networks have been developed, but only outside of a Bayesian framework. As such, previous work has not identified that these kernels can be used as covariance functions for GPs and allow fully Bayesian prediction with a deep neural network. In this work, we derive the exact equivalence between infinitely wide deep networks and GPs. We further develop a computationally efficient pipeline to compute the covariance function for these GPs. We then use the resulting GPs to perform Bayesian inference for wide deep neural networks on MNIST and CIFAR-10. We observe that trained neural network accuracy approaches that of the corresponding GP with increasing layer width, and that the GP uncertainty is strongly correlated with trained network prediction error. We further find that test performance increases as finite-width trained networks are made wider and more similar to a GP, and thus that GP predictions typically outperform those of finite-width networks. Finally we connect the performance of these GPs to the recent theory of signal propagation in random neural networks.

stat.ML↗

arXiv2018

The Emergence of Spectral Universality in Deep Networks

Recent work has shown that tight concentration of the entire spectrum of singular values of a deep network's input-output Jacobian around one at initialization can speed up learning by orders of magnitude. Therefore, to guide important design choices, it is important to build a full theoretical understanding of the spectra of Jacobians at initialization. To this end, we leverage powerful tools from free probability theory to provide a detailed analytic understanding of how a deep network's Jacobian spectrum depends on various hyperparameters including the nonlinearity, the weight and bias distributions, and the depth. For a variety of nonlinearities, our work reveals the emergence of new universal limiting spectral distributions that remain concentrated around one even as the depth goes to infinity.

stat.ML↗

arXiv2017

Mean Field Residual Networks: On the Edge of Chaos

We study randomly initialized residual networks using mean field theory and the theory of difference equations. Classical feedforward neural networks, such as those with tanh activations, exhibit exponential behavior on the average when propagating inputs forward or gradients backward. The exponential forward dynamics causes rapid collapsing of the input space geometry, while the exponential backward dynamics causes drastic vanishing or exploding gradients. We show, in contrast, that by adding skip connections, the network will, depending on the nonlinearity, adopt subexponential forward and backward dynamics, and in many cases in fact polynomial. The exponents of these polynomials are obtained through analytic methods and proved and verified empirically to be correct. In terms of the "edge of chaos" hypothesis, these subexponential and polynomial laws allow residual networks to "hover over the boundary between stability and chaos," thus preserving the geometry of the input space and the gradient information flow. In our experiments, for each activation function we study here, we initialize residual networks with different hyperparameters and train them on MNIST. Remarkably, our initialization time theory can accurately predict test time performance of these networks, by tracking either the expected amount of gradient explosion or the expected squared distance between the images of two input vectors. Importantly, we show, theoretically as well as empirically, that common initializations such as the Xavier or the He schemes are not optimal for residual networks, because the optimal initialization variances depend on the depth. Finally, we have made mathematical contributions by deriving several new identities for the kernels of powers of ReLU functions by relating them to the zeroth Bessel function of the second kind.

cs.NE↗

arXiv2017

Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice

It is well known that the initialization of weights in deep neural networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network's input-output Jacobian is $O(1)$ is essential for avoiding the exponential vanishing or explosion of gradients. The stronger condition that all singular values of the Jacobian concentrate near $1$ is a property known as dynamical isometry. For deep linear networks, dynamical isometry can be achieved through orthogonal weight initialization and has been shown to dramatically speed up learning; however, it has remained unclear how to extend these results to the nonlinear setting. We address this question by employing powerful tools from free probability theory to compute analytically the entire singular value distribution of a deep network's input-output Jacobian. We explore the dependence of the singular value distribution on the depth of the network, the weight initialization, and the choice of nonlinearity. Intriguingly, we find that ReLU networks are incapable of dynamical isometry. On the other hand, sigmoidal networks can achieve isometry, but only with orthogonal weight initialization. Moreover, we demonstrate empirically that deep nonlinear networks achieving dynamical isometry learn orders of magnitude faster than networks that do not. Indeed, we show that properly-initialized deep sigmoidal networks consistently outperform deep ReLU networks. Overall, our analysis reveals that controlling the entire distribution of Jacobian singular values is an important design consideration in deep learning.

cs.LG↗

arXiv2017

Intriguing Properties of Adversarial Examples

It is becoming increasingly clear that many machine learning classifiers are vulnerable to adversarial examples. In attempting to explain the origin of adversarial examples, previous studies have typically focused on the fact that neural networks operate on high dimensional data, they overfit, or they are too linear. Here we argue that the origin of adversarial examples is primarily due to an inherent uncertainty that neural networks have about their predictions. We show that the functional form of this uncertainty is independent of architecture, dataset, and training protocol; and depends only on the statistics of the logit differences of the network, which do not change significantly during training. This leads to adversarial error having a universal scaling, as a power-law, with respect to the size of the adversarial perturbation. We show that this universality holds for a broad range of datasets (MNIST, CIFAR10, ImageNet, and random data), models (including state-of-the-art deep networks, linear models, adversarially trained networks, and networks trained on randomly shuffled labels), and attacks (FGSM, step l.l., PGD). Motivated by these results, we study the effects of reducing prediction entropy on adversarial robustness. Finally, we study the effect of network architectures on adversarial sensitivity. To do this, we use neural architecture search with reinforcement learning to find adversarially robust architectures on CIFAR10. Our resulting architecture is more robust to white \emph{and} black box attacks compared to previous attempts.

stat.ML↗

arXiv2017

A Correspondence Between Random Neural Networks and Statistical Field Theory

A number of recent papers have provided evidence that practical design questions about neural networks may be tackled theoretically by studying the behavior of random networks. However, until now the tools available for analyzing random neural networks have been relatively ad-hoc. In this work, we show that the distribution of pre-activations in random neural networks can be exactly mapped onto lattice models in statistical physics. We argue that several previous investigations of stochastic networks actually studied a particular factorial approximation to the full lattice model. For random linear networks and random rectified linear networks we show that the corresponding lattice models in the wide network limit may be systematically approximated by a Gaussian distribution with covariance between the layers of the network. In each case, the approximate distribution can be diagonalized by Fourier transformation. We show that this approximation accurately describes the results of numerical simulations of wide random neural networks. Finally, we demonstrate that in each case the large scale behavior of the random networks can be approximated by an effective field theory.

stat.ML↗

arXiv2017

Combining Machine Learning and Physics to Understand Glassy Systems

Our understanding of supercooled liquids and glasses has lagged significantly behind that of simple liquids and crystalline solids. This is in part due to the many possibly relevant degrees of freedom that are present due to the disorder inherent to these systems and in part to non-equilibrium effects which are difficult to treat in the standard context of statistical physics. Together these issues have resulted in a field whose theories are under-constrained by experiment and where fundamental questions are still unresolved. Mean field results have been successful in infinite dimensions but it is unclear to what extent they apply to realistic systems and assume uniform local structure. At odds with this are theories premised on the existence of structural defects. However, until recently it has been impossible to find structural signatures that are predictive of dynamics. Here we summarize and recast the results from several recent papers offering a data driven approach to building a phenomenological theory of disordered materials by combining machine learning with physical intuition.

stat.ML↗

arXiv2017

Neural Message Passing for Quantum Chemistry

Supervised learning on molecules has incredible potential to be useful in chemistry, drug discovery, and materials science. Luckily, several promising and closely related neural network models invariant to molecular symmetries have already been described in the literature. These models learn a message passing algorithm and aggregation procedure to compute a function of their entire input graph. At this point, the next step is to find a particularly effective variant of this general approach and apply it to chemical prediction benchmarks until we either solve them or reach the limits of the approach. In this paper, we reformulate existing models into a single common framework we call Message Passing Neural Networks (MPNNs) and explore additional novel variations within this framework. Using MPNNs we demonstrate state of the art results on an important molecular property prediction benchmark; these results are strong enough that we believe future work should focus on datasets with larger molecules or more accurate ground truth labels.

cs.LG↗

arXiv2017

Machine learning prediction errors better than DFT accuracy

We investigate the impact of choosing regressors and molecular representations for the construction of fast machine learning (ML) models of thirteen electronic ground-state properties of organic molecules. The performance of each regressor/representation/property combination is assessed using learning curves which report out-of-sample errors as a function of training set size with up to $\sim$117k distinct molecules. Molecular structures and properties at hybrid density functional theory (DFT) level of theory used for training and testing come from the QM9 database [Ramakrishnan et al, {\em Scientific Data} {\bf 1} 140022 (2014)] and include dipole moment, polarizability, HOMO/LUMO energies and gap, electronic spatial extent, zero point vibrational energy, enthalpies and free energies of atomization, heat capacity and the highest fundamental vibrational frequency. Various representations from the literature have been studied (Coulomb matrix, bag of bonds, BAML and ECFP4, molecular graphs (MG)), as well as newly developed distribution based variants including histograms of distances (HD), and angles (HDA/MARAD), and dihedrals (HDAD). Regressors include linear models (Bayesian ridge regression (BR) and linear regression with elastic net regularization (EN)), random forest (RF), kernel ridge regression (KRR) and two types of neural net works, graph convolutions (GC) and gated graph networks (GG). We present numerical evidence that ML model predictions deviate from DFT less than DFT deviates from experiment for all properties. Furthermore, our out-of-sample prediction errors with respect to hybrid DFT reference are on par with, or close to, chemical accuracy. Our findings suggest that ML models could be more accurate than hybrid DFT if explicitly electron correlated quantum (or experimental) data was available.

physics.chem-ph↗

arXiv2017

Deep Information Propagation

We study the behavior of untrained neural networks whose weights and biases are randomly distributed using mean field theory. We show the existence of depth scales that naturally limit the maximum depth of signal propagation through these random networks. Our main practical result is to show that random networks may be trained precisely when information can travel through them. Thus, the depth scales that we identify provide bounds on how deep a network may be trained for a specific choice of hyperparameters. As a corollary to this, we argue that in networks at the edge of chaos, one of these depth scales diverges. Thus arbitrarily deep networks may be trained only sufficiently close to criticality. We show that the presence of dropout destroys the order-to-chaos critical point and therefore strongly limits the maximum trainable depth for random networks. Finally, we develop a mean field theory for backpropagation and we show that the ordered and chaotic phases correspond to regions of vanishing and exploding gradient respectively.

stat.ML↗

arXiv2016

Disconnecting structure and dynamics in glassy thin films

Nanometrically thin glassy films depart strikingly from the behavior of their bulk counterparts. We investigate whether the dynamical differences between bulk and thin film glasses can be understood by differences in local microscopic structure. We employ machine-learning methods that have previously identified strong correlations between local structure and particle rearrangement dynamics in bulk systems. We show that these methods completely fail to detect key aspects of thin-film glassy dynamics. Furthermore, we show that no combination of local structural features drawn from a very general set of two- and multi-point functions is able to distinguish between particles at the center of film and those in intermediate layers where the dynamics are strongly perturbed.

cond-mat.soft↗