We propose \emph{Taylorized training} as an initiative towards better understanding neural network training at finite width. Taylorized training involves training the $k$-th order Taylor expansion of the neural network at initialization, and is a principled extension of linearized training---a recently proposed theory for understanding the success of deep learning.
We experiment with Taylorized training on modern neural network architectures, and show that Taylorized training (1) agrees with full neural network training increasingly better as we increase $k$, and (2) can significantly close the performance gap between linearized and full training. Compared with linearized training, higher-order training works in more realistic settings such as standard parameterization and large (initial) learning rate. We complement our experiments with theoretical results showing that the approximation error of $k$-th order Taylorized models decay exponentially over $k$ in wide neural networks.
Perimeter Institute for Theoretical Physics（圆周理论物理研究所）；Blueshift, Alphabet（Alphabet旗下Blueshift）
Empirical studies demonstrate that the performance of neural networks improves with increasing number of parameters. In most of these studies, the number of parameters is increased by increasing the network width. This begs the question: Is the observed improvement due to the larger number of parameters, or is it due to the larger width itself? We compare different ways of increasing model width while keeping the number of parameters constant. We show that for models initialized with a random, static sparsity pattern in the weight tensors, network width is the determining factor for good perfo…
Stanford University（斯坦福大学）；Singapore University of Technology and Design（新加坡科技设计大学）
The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new connection with dynamical systems, using a generalized notion of fixed points of a continuous map $f$, called periodic points. In this work, we strengthen the connection with dynamical systems and we improve the existing width lower bounds along several aspects. Our first main result is period-specific width lower bounds that hold under the stronger notion of $L^1…
The University of Tokyo（东京大学）；Carnegie Mellon University（卡内基梅隆大学）
This work theoretically studies stochastic neural networks, a main type of neural network in use. We prove that as the width of an optimized stochastic neural network tends to infinity, its predictive variance on the training set decreases to zero. Our theory justifies the common intuition that adding stochasticity to the model can help regularize the model by introducing an averaging effect. Two common examples that our theory can be relevant to are neural networks with dropout and Bayesian latent variable models in a special limit. Our result thus helps better understand how stochasticity af…
Department of Neuroscience, Columbia University（哥伦比亚大学神经科学系）；Data Science Institute, Columbia University（哥伦比亚大学数据科学研究所）；Google（谷歌公司）；Mountain View, CA（山景城（加州）（注：此处为地址，非机构））
Transferability of learned features between tasks can massively reduce the cost of training a neural network on a novel task. We investigate the effect of network width on learned features using activation atlases --- a visualization technique that captures features the entire hidden state responds to, as opposed to individual neurons alone. We find that, while individual neurons do not learn interpretable features in wide networks, groups of neurons do. In addition, the hidden state of a wide network contains more information about the inputs than that of a narrow network trained to the same …
Comparing Bayesian neural networks (BNNs) with different widths is challenging because, as the width increases, multiple model properties change simultaneously, and, inference in the finite-width case is intractable. In this work, we empirically compare finite- and infinite-width BNNs, and provide quantitative and qualitative explanations for their performance difference. We find that when the model is mis-specified, increasing width can hurt BNN performance. In these cases, we provide evidence that finite-width BNNs generalize better partially due to the properties of their frequency spectrum…
We provide several new depth-based separation results for feed-forward neural networks, proving that various types of simple and natural functions can be better approximated using deeper networks than shallower ones, even if the shallower networks are much larger. This includes indicators of balls and ellipses; non-linear functions which are radial with respect to the $L_1$ norm; and smooth non-linear functions. We also show that these gaps can be observed experimentally: Increasing the depth indeed allows better learning than increasing width, when training neural networks to learn an indicat…
When narrower is better: the narrow width limit of Bayesian parallel branching neural networks
Harvard University（哈佛大学）；Center for Brain Science, Harvard University（哈佛大学脑科学中心）；Kempner Institute（坎普纳研究所）；Edmond and Lily Safra Center for Brain Sciences, Hebrew University of Jerusalem（耶路撒冷希伯来大学埃德蒙和莉莉·萨夫拉脑科学中心）
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 ove…
Modern deep learning models have achieved great success in predictive accuracy for many data modalities. However, their application to many real-world tasks is restricted by poor uncertainty estimates, such as overconfidence on out-of-distribution (OOD) data and ungraceful failing under distributional shift. Previous benchmarks have found that ensembles of neural networks (NNs) are typically the best calibrated models on OOD data. Inspired by this, we leverage recent theoretical advances that characterize the function-space prior of an ensemble of infinitely-wide NNs as a Gaussian process, ter…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；University of Guelph（圭尔夫大学）
Theoretical results show that neural networks can be approximated by Gaussian processes in the infinite-width limit. However, for fully connected networks, it has been previously shown that for any fixed network width, $n$, the Gaussian approximation gets worse as the network depth, $d$, increases. Given that modern networks are deep, this raises the question of how well modern architectures, like ResNets, are captured by the infinite-width limit. To provide a better approximation, we study ReLU ResNets in the infinite-depth-and-width limit, where both depth and width tend to infinity as their…
Study of neural networks with infinite width is important for better understanding of the neural network in practical application. In this work, we derive the equivalence of the deep, infinite-width maxout network and the Gaussian process (GP) and characterize the maxout kernel with a compositional structure. Moreover, we build up the connection between our deep maxout network kernel and deep neural network kernels. We also give an efficient numerical implementation of our kernel which can be adapted to any maxout rank. Numerical results show that doing Bayesian inference based on the deep max…
Wide neural networks have proven to be a rich class of architectures for both theory and practice. Motivated by the observation that finite width convolutional networks appear to outperform infinite width networks, we study scaling laws for wide CNNs and networks with skip connections. Following the approach of (Dyer & Gur-Ari, 2019), we present a simple diagrammatic recipe to derive the asymptotic width dependence for many quantities of interest. These scaling relationships provide a solvable description for the training dynamics of wide convolutional networks. We test these relations across …
Optimizing the channel counts for different layers of a CNN has shown great promise in improving the efficiency of CNNs at test-time. However, these methods often introduce large computational overhead (e.g., an additional 2x FLOPs of standard training). Minimizing this overhead could therefore significantly speed up training. In this work, we propose width transfer, a technique that harnesses the assumptions that the optimized widths (or channel counts) are regular across sizes and depths. We show that width transfer works well across various width optimization algorithms and networks. Specif…
Oden Institute for Computational Engineering and Sciences（计算工程与科学奥登研究所）；Department of Computer Science（计算机科学系）；University of Texas, Austin（德克萨斯大学奥斯汀分校）
Dense nets are an integral part of any classification and regression problem. Recently, these networks have found a new application as solvers for known representations in various domains. However, one crucial issue with dense nets is it's feature interpretation and lack of reproducibility over multiple training runs. In this work, we identify a basis collapse issue as a primary cause and propose a modified loss function that circumvents this problem. We also provide a few general guidelines relating the choice of activations to loss surface roughness and appropriate scaling for designing low-…
The Neural Tangent Kernel (NTK), defined as $Θ_θ^f(x_1, x_2) = \left[\partial f(θ, x_1)\big/\partial θ\right] \left[\partial f(θ, x_2)\big/\partial θ\right]^T$ where $\left[\partial f(θ, \cdot)\big/\partial θ\right]$ 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…
Recent works on Bayesian neural networks (BNNs) have highlighted the need to better understand the implications of using Gaussian priors in combination with the compositional structure of the network architecture. Similar in spirit to the kind of analysis that has been developed to devise better initialization schemes for neural networks (cf. He- or Xavier initialization), we derive a precise characterization of the prior predictive distribution of finite-width ReLU networks with Gaussian weights. While theoretical results have been obtained for their heavy-tailedness, the full characterizatio…
Deep Linear Network Training Dynamics from Random Initialization: Data, Width, Depth, and Hyperparameter Transfer
John Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·保罗森工程与应用科学学校）；Center for Brain Sciences（脑科学研究中心）；Kempner Institute（凯普纳研究所）
Peking University（北京大学）；Georgia Institute of Technology（佐治亚理工学院）
Deep residual networks (ResNets) have demonstrated better generalization performance than deep feedforward networks (FFNets). However, the theory behind such a phenomenon is still largely unknown. This paper studies this fundamental problem in deep learning from a so-called "neural tangent kernel" perspective. Specifically, we first show that under proper conditions, as the width goes to infinity, training deep ResNets can be viewed as learning reproducing kernel functions with some kernel function. We then compare the kernel of deep ResNets with that of deep FFNets and discover that the class…
University of California, Los Angeles（加州大学洛杉矶分校）；University of California, Irvine（加州大学尔湾分校）
We present LBW-Net, an efficient optimization based method for quantization and training of the low bit-width convolutional neural networks (CNNs). Specifically, we quantize the weights to zero or powers of two by minimizing the Euclidean distance between full-precision weights and quantized weights during backpropagation. We characterize the combinatorial nature of the low bit-width quantization problem. For 2-bit (ternary) CNNs, the quantization of $N$ weights can be done by an exact formula in $O(N\log N)$ complexity. When the bit-width is three and above, we further propose a semi-analytic…
Spectral Evolution and Invariance in Linear-width Neural Networks
University of California, San Diego（加利福尼亚大学圣迭戈分校）；Pacific Northwest National Laboratory（太平洋西北国家实验室）；Rutgers, The State University of New Jersey（罗格斯大学）；University of Washington（华盛顿大学）；University of Texas at El Paso（得克萨斯大学埃尔帕索分校）
Neural network quantization is frequently used to optimize model size, latency and power consumption for on-device deployment of neural networks. In many cases, a target bit-width is set for an entire network, meaning every layer get quantized to the same number of bits. However, for many networks some layers are significantly more robust to quantization noise than others, leaving an important axis of improvement unused. As many hardware solutions provide multiple different bit-width settings, mixed-precision quantization has emerged as a promising solution to find a better performance-efficie…
Convolutional Neural Networks (ConvNets) are commonly developed at a fixed resource budget, and then scaled up for better accuracy if more resources are available. In this paper, we systematically study model scaling and identify that carefully balancing network depth, width, and resolution can lead to better performance. Based on this observation, we propose a new scaling method that uniformly scales all dimensions of depth/width/resolution using a simple yet highly effective compound coefficient. We demonstrate the effectiveness of this method on scaling up MobileNets and ResNet. To go even …
College of Computer Science, Zhejiang University（浙江大学计算机学院）；College of Information Science and Technology, Pennsylvania State University（宾夕法尼亚州立大学信息科学与技术学院）；Department of Computer Science, University of California, Los Angeles（加利福尼亚大学洛杉矶分校计算机科学系）
Adversarial training is a powerful type of defense against adversarial examples. Previous empirical results suggest that adversarial training requires wider networks for better performances. However, it remains elusive how neural network width affects model robustness. In this paper, we carefully examine the relationship between network width and model robustness. Specifically, we show that the model robustness is closely related to the tradeoff between natural accuracy and perturbation stability, which is controlled by the robust regularization parameter $λ$. With the same $λ$, wider networks…
The University of Sydney（悉尼大学）；Beijing University of Posts and Telecommunications（北京邮电大学）；SenseTime Research Centre（商汤科技研究中心）；Tsinghua University（清华大学）
Searching for a more compact network width recently serves as an effective way of channel pruning for the deployment of convolutional neural networks (CNNs) under hardware constraints. To fulfill the searching, a one-shot supernet is usually leveraged to efficiently evaluate the performance \wrt~different network widths. However, current methods mainly follow a \textit{unilaterally augmented} (UA) principle for the evaluation of each width, which induces the training unfairness of channels in supernet. In this paper, we introduce a new supernet called Bilaterally Coupled Network (BCNet) to add…
In this paper we propose a novel network adaption method called Differentiable Network Adaption (DNA), which can adapt an existing network to a specific computation budget by adjusting the width and depth in a differentiable manner. The gradient-based optimization allows DNA to achieve an automatic optimization of width and depth rather than previous heuristic methods that heavily rely on human priors. Moreover, we propose a new elastic search space that can flexibly condense or expand during the optimization process, allowing the network optimization of width and depth in a bi-direction manne…
John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学约翰·A·保尔森工程与应用科学学院）
To better understand the theoretical behavior of large neural networks, several works have analyzed the case where a network's width tends to infinity. In this regime, the effect of random initialization and the process of training a neural network can be formally expressed with analytical tools like Gaussian processes and neural tangent kernels. In this paper, we review methods for quantifying uncertainty in such infinite-width neural networks and compare their relationship to Gaussian processes in the Bayesian inference framework. We make use of several equivalence results along the way to o…
Harvard Medical School（哈佛医学院）；Massachusetts Institute of Technology（麻省理工学院）；University of California San Francisco（加州大学旧金山分校）
A remarkable characteristic of overparameterized deep neural networks (DNNs) is that their accuracy does not degrade when the network's width is increased. Recent evidence suggests that developing compressible representations is key for adjusting the complexity of large networks to the learning task at hand. However, these compressible representations are poorly understood. A promising strand of research inspired from biology is understanding representations at the unit level as it offers a more granular and intuitive interpretation of the neural mechanisms. In order to better understand what …
RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所涌现物质科学中心（CEMS））；The University of Tokyo（东京大学）
It has been recognized that a heavily overparameterized artificial neural network exhibits surprisingly good generalization performance in various machine-learning tasks. Recent theoretical studies have made attempts to unveil the mystery of the overparameterization. In most of those previous works, the overparameterization is achieved by increasing the width of the network, while the effect of increasing the depth has remained less well understood. In this work, we investigate the effect of increasing the depth within an overparameterized regime. To gain an insight into the advantage of depth…
University of Guelph（圭尔夫大学）；Vector Institute for Artificial Intelligence（向量人工智能研究所）；The NSF AI Institute for Artificial Intelligence and Fundamental Interactions（美国国家科学基金会人工智能与基础交互研究所）；Massachusetts Institute of Technology（麻省理工学院）；University of Calgary（卡尔加里大学）
Estimating the Generalization Error (GE) of Deep Neural Networks (DNNs) is an important task that often relies on availability of held-out data. The ability to better predict GE based on a single training set may yield overarching DNN design principles to reduce a reliance on trial-and-error, along with other performance assessment advantages. In search of a quantity relevant to GE, we investigate the Mutual Information (MI) between the input and final layer representations, using the infinite-width DNN limit to bound MI. An existing input compression-based GE bound is used to link MI and GE. …
Samsung Advanced Institute of Technology, Samsung Electronics（三星电子三星先进技术研究院）
Post-training quantization is a representative technique for compressing neural networks, making them smaller and more efficient for deployment on edge devices. However, an inaccessible user dataset often makes it difficult to ensure the quality of the quantized neural network in practice. In addition, existing approaches may use a single uniform bit-width across the network, resulting in significant accuracy degradation at extremely low bit-widths. To utilize multiple bit-width, sensitivity metric plays a key role in balancing accuracy and compression. In this paper, we propose a novel sensit…
New Jersey Institute of Technology（新泽西理工学院）；National University of Singapore（新加坡国立大学）
Obtaining high-quality magnetic and velocity fields through Stokes inversion is crucial in solar physics. In this paper, we present a new deep learning method, named Stacked Deep Neural Networks (SDNN), for inferring line-of-sight (LOS) velocities and Doppler widths from Stokes profiles collected by the Near InfraRed Imaging Spectropolarimeter (NIRIS) on the 1.6 m Goode Solar Telescope (GST) at the Big Bear Solar Observatory (BBSO). The training data of SDNN is prepared by a Milne-Eddington (ME) inversion code used by BBSO. We quantitatively assess SDNN, comparing its inversion results with th…
