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Infinite width limit has shed light on generalization and optimization aspects of deep learning by establishing connections between neural networks and kernel methods. Despite their importance, the utility of these kernel methods was limited in large-scale learning settings due to their (super-)quadratic runtime and memory complexities. Moreover, most prior works on neural kernels have focused on the ReLU activation, mainly due to its popularity but also due to the difficulty of computing such kernels for general activations. In this work, we overcome such difficulties by providing methods to work with general activations. First, we compile and expand the list of activation functions admitting exact dual activation expressions to compute neural kernels. When the exact computation is unknown, we present methods to effectively approximate them. We propose a fast sketching method that approximates any multi-layered Neural Network Gaussian Process (NNGP) kernel and Neural Tangent Kernel (NTK) matrices for a wide range of activation functions, going beyond the commonly analyzed ReLU activation. This is done by showing how to approximate the neural kernels using the truncated Hermite expansion of any desired activation functions. While most prior works require data points on the unit sphere, our methods do not suffer from such limitations and are applicable to any dataset of points in $\mathbb{R}^d$. Furthermore, we provide a subspace embedding for NNGP and NTK matrices with near input-sparsity runtime and near-optimal target dimension which applies to any \emph{homogeneous} dual activation functions with rapidly convergent Taylor expansion. Empirically, with respect to exact convolutional NTK (CNTK) computation, our method achieves $106\times$ speedup for approximate CNTK of a 5-layer Myrtle network on CIFAR-10 dataset.

The Neural Tangent Kernel (NTK), defined as $\Theta_\theta^f(x_1, x_2) = \left[\partial f(\theta, x_1)\big/\partial \theta\right] \left[\partial f(\theta, x_2)\big/\partial \theta\right]^T$ where $\left[\partial f(\theta, \cdot)\big/\partial \theta\right]$ is a neural network (NN) Jacobian, has emerged as a central object of study in deep learning. In the infinite width limit, the NTK can sometimes be computed analytically and is useful for understanding training and generalization of NN architectures. At finite widths, the NTK is also used to better initialize NNs, compare the conditioning across models, perform architecture search, and do meta-learning. Unfortunately, the finite width NTK is notoriously expensive to compute, which severely limits its practical utility. We perform the first in-depth analysis of the compute and memory requirements for NTK computation in finite width networks. Leveraging the structure of neural networks, we further propose two novel algorithms that change the exponent of the compute and memory requirements of the finite width NTK, dramatically improving efficiency. Our algorithms can be applied in a black box fashion to any differentiable function, including those implementing neural networks. We open-source our implementations within the Neural Tangents package (arXiv:1912.02803) at https://github.com/google/neural-tangents., Comment: Published as a conference paper at ICML 2022

Language models demonstrate both quantitative improvement and new qualitative capabilities with increasing scale. Despite their potentially transformative impact, these new capabilities are as yet poorly characterized. In order to inform future research, prepare for disruptive new model capabilities, and ameliorate socially harmful effects, it is vital that we understand the present and near-future capabilities and limitations of language models. To address this challenge, we introduce the Beyond the Imitation Game benchmark (BIG-bench). BIG-bench currently consists of 204 tasks, contributed by 450 authors across 132 institutions. Task topics are diverse, drawing problems from linguistics, childhood development, math, common-sense reasoning, biology, physics, social bias, software development, and beyond. BIG-bench focuses on tasks that are believed to be beyond the capabilities of current language models. We evaluate the behavior of OpenAI's GPT models, Google-internal dense transformer architectures, and Switch-style sparse transformers on BIG-bench, across model sizes spanning millions to hundreds of billions of parameters. In addition, a team of human expert raters performed all tasks in order to provide a strong baseline. Findings include: model performance and calibration both improve with scale, but are poor in absolute terms (and when compared with rater performance); performance is remarkably similar across model classes, though with benefits from sparsity; tasks that improve gradually and predictably commonly involve a large knowledge or memorization component, whereas tasks that exhibit "breakthrough" behavior at a critical scale often involve multiple steps or components, or brittle metrics; social bias typically increases with scale in settings with ambiguous context, but this can be improved with prompting., 27 pages, 17 figures + references and appendices, repo: https://github.com/google/BIG-bench

We introduce repriorisation, a data-dependent reparameterisation which transforms a Bayesian neural network (BNN) posterior to a distribution whose KL divergence to the BNN prior vanishes as layer widths grow. The repriorisation map acts directly on parameters, and its analytic simplicity complements the known neural network Gaussian process (NNGP) behaviour of wide BNNs in function space. Exploiting the repriorisation, we develop a Markov chain Monte Carlo (MCMC) posterior sampling algorithm which mixes faster the wider the BNN. This contrasts with the typically poor performance of MCMC in high dimensions. We observe up to 50x higher effective sample size relative to no reparametrisation for both fully-connected and residual networks. Improvements are achieved at all widths, with the margin between reparametrised and standard BNNs growing with layer width., Comment: ICML 2022

The effectiveness of machine learning algorithms arises from being able to extract useful features from large amounts of data. As model and dataset sizes increase, dataset distillation methods that compress large datasets into significantly smaller yet highly performant ones will become valuable in terms of training efficiency and useful feature extraction. To that end, we apply a novel distributed kernel based meta-learning framework to achieve state-of-the-art results for dataset distillation using infinitely wide convolutional neural networks. For instance, using only 10 datapoints (0.02% of original dataset), we obtain over 65% test accuracy on CIFAR-10 image classification task, a dramatic improvement over the previous best test accuracy of 40%. Our state-of-the-art results extend across many other settings for MNIST, Fashion-MNIST, CIFAR-10, CIFAR-100, and SVHN. Furthermore, we perform some preliminary analyses of our distilled datasets to shed light on how they differ from naturally occurring data., NeurIPS 2021. Code and datasets available at https://github.com/google-research/google-research/tree/master/kip

There is a growing amount of literature on the relationship between wide neural networks (NNs) and Gaussian processes (GPs), identifying an equivalence between the two for a variety of NN architectures. This equivalence enables, for instance, accurate approximation of the behaviour of wide Bayesian NNs without MCMC or variational approximations, or characterisation of the distribution of randomly initialised wide NNs optimised by gradient descent without ever running an optimiser. We provide a rigorous extension of these results to NNs involving attention layers, showing that unlike single-head attention, which induces non-Gaussian behaviour, multi-head attention architectures behave as GPs as the number of heads tends to infinity. We further discuss the effects of positional encodings and layer normalisation, and propose modifications of the attention mechanism which lead to improved results for both finite and infinitely wide NNs. We evaluate attention kernels empirically, leading to a moderate improvement upon the previous state-of-the-art on CIFAR-10 for GPs without trainable kernels and advanced data preprocessing. Finally, we introduce new features to the Neural Tangents library (Novak et al., 2020) allowing applications of NNGP/NTK models, with and without attention, to variable-length sequences, with an example on the IMDb reviews dataset., ICML 2020

There are currently two parameterizations used to derive fixed kernels corresponding to infinite width neural networks, the NTK (Neural Tangent Kernel) parameterization and the naive standard parameterization. However, the extrapolation of both of these parameterizations to infinite width is problematic. The standard parameterization leads to a divergent neural tangent kernel while the NTK parameterization fails to capture crucial aspects of finite width networks such as: the dependence of training dynamics on relative layer widths, the relative training dynamics of weights and biases, and overall learning rate scale. Here we propose an improved extrapolation of the standard parameterization that preserves all of these properties as width is taken to infinity and yields a well-defined neural tangent kernel. We show experimentally that the resulting kernels typically achieve similar accuracy to those resulting from an NTK parameterization, but with better correspondence to the parameterization of typical finite width networks. Additionally, with careful tuning of width parameters, the improved standard parameterization kernels can outperform those stemming from an NTK parameterization. We release code implementing this improved standard parameterization as part of the Neural Tangents library at https://github.com/google/neural-tangents.

We perform a careful, thorough, and large scale empirical study of the correspondence between wide neural networks and kernel methods. By doing so, we resolve a variety of open questions related to the study of infinitely wide neural networks. Our experimental results include: kernel methods outperform fully-connected finite-width networks, but underperform convolutional finite width networks; neural network Gaussian process (NNGP) kernels frequently outperform neural tangent (NT) kernels; centered and ensembled finite networks have reduced posterior variance and behave more similarly to infinite networks; weight decay and the use of a large learning rate break the correspondence between finite and infinite networks; the NTK parameterization outperforms the standard parameterization for finite width networks; diagonal regularization of kernels acts similarly to early stopping; floating point precision limits kernel performance beyond a critical dataset size; regularized ZCA whitening improves accuracy; finite network performance depends non-monotonically on width in ways not captured by double descent phenomena; equivariance of CNNs is only beneficial for narrow networks far from the kernel regime. Our experiments additionally motivate an improved layer-wise scaling for weight decay which improves generalization in finite-width networks. Finally, we develop improved best practices for using NNGP and NT kernels for prediction, including a novel ensembling technique. Using these best practices we achieve state-of-the-art results on CIFAR-10 classification for kernels corresponding to each architecture class we consider., Comment: 17+11 pages; v2 references added, minor improvements

Recent work has shown that the prior over functions induced by a deep Bayesian neural network (BNN) behaves as a Gaussian process (GP) as the width of all layers becomes large. However, many BNN applications are concerned with the BNN function space posterior. While some empirical evidence of the posterior convergence was provided in the original works of Neal (1996) and Matthews et al. (2018), it is limited to small datasets or architectures due to the notorious difficulty of obtaining and verifying exactness of BNN posterior approximations. We provide the missing theoretical proof that the exact BNN posterior converges (weakly) to the one induced by the GP limit of the prior. For empirical validation, we show how to generate exact samples from a finite BNN on a small dataset via rejection sampling.

Neural Tangents is a library designed to enable research into infinite-width neural networks. It provides a high-level API for specifying complex and hierarchical neural network architectures. These networks can then be trained and evaluated either at finite-width as usual or in their infinite-width limit. Infinite-width networks can be trained analytically using exact Bayesian inference or using gradient descent via the Neural Tangent Kernel. Additionally, Neural Tangents provides tools to study gradient descent training dynamics of wide but finite networks in either function space or weight space. The entire library runs out-of-the-box on CPU, GPU, or TPU. All computations can be automatically distributed over multiple accelerators with near-linear scaling in the number of devices. Neural Tangents is available at www.github.com/google/neural-tangents. We also provide an accompanying interactive Colab notebook.

In practice it is often found that large over-parameterized neural networks generalize better than their smaller counterparts, an observation that appears to conflict with classical notions of function complexity, which typically favor smaller models. In this work, we investigate this tension between complexity and generalization through an extensive empirical exploration of two natural metrics of complexity related to sensitivity to input perturbations. Our experiments survey thousands of models with various fully-connected architectures, optimizers, and other hyper-parameters, as well as four different image classification datasets. We find that trained neural networks are more robust to input perturbations in the vicinity of the training data manifold, as measured by the norm of the input-output Jacobian of the network, and that it correlates well with generalization. We further establish that factors associated with poor generalization $-$ such as full-batch training or using random labels $-$ correspond to lower robustness, while factors associated with good generalization $-$ such as data augmentation and ReLU non-linearities $-$ give rise to more robust functions. Finally, we demonstrate how the input-output Jacobian norm can be predictive of generalization at the level of individual test points., Comment: Published as a conference paper at ICLR 2018

There is a previously identified equivalence between wide fully connected neural networks (FCNs) and Gaussian processes (GPs). This equivalence enables, for instance, test set predictions that would have resulted from a fully Bayesian, infinitely wide trained FCN to be computed without ever instantiating the FCN, but by instead evaluating the corresponding GP. In this work, we derive an analogous equivalence for multi-layer convolutional neural networks (CNNs) both with and without pooling layers, and achieve state of the art results on CIFAR10 for GPs without trainable kernels. We also introduce a Monte Carlo method to estimate the GP corresponding to a given neural network architecture, even in cases where the analytic form has too many terms to be computationally feasible. Surprisingly, in the absence of pooling layers, the GPs corresponding to CNNs with and without weight sharing are identical. As a consequence, translation equivariance, beneficial in finite channel CNNs trained with stochastic gradient descent (SGD), is guaranteed to play no role in the Bayesian treatment of the infinite channel limit - a qualitative difference between the two regimes that is not present in the FCN case. We confirm experimentally, that while in some scenarios the performance of SGD-trained finite CNNs approaches that of the corresponding GPs as the channel count increases, with careful tuning SGD-trained CNNs can significantly outperform their corresponding GPs, suggesting advantages from SGD training compared to fully Bayesian parameter estimation., Comment: Published as a conference paper at ICLR 2019

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., Comment: Published version in ICLR 2018. 10 pages + appendix