Automated machine learning (AutoML) methods improve upon existing models by optimizing various aspects of their design. While present methods focus on hyperparameters and neural network topologies, other aspects of neural network design can be optimized as well. To further the state of the art in AutoML, this dissertation introduces techniques for discovering more powerful activation functions and establishing more robust weight initialization for neural networks. These contributions improve performance, but also provide new perspectives on neural network optimization. First, the dissertation demonstrates that discovering solutions specialized to specific architectures and tasks gives better performance than reusing general approaches. Second, it shows that jointly optimizing different components of neural networks is synergistic, and results in better performance than optimizing individual components alone. Third, it demonstrates that learned representations are easier to optimize than hard-coded ones, creating further opportunities for AutoML. The dissertation thus makes concrete progress towards fully automatic machine learning in the future.
A proper initialization of the weights in a neural network is critical to its convergence. Current insights into weight initialization come primarily from linear activation functions. In this paper, I develop a theory for weight initializations with non-linear activations. First, I derive a general weight initialization strategy for any neural network using activation functions differentiable at 0. Next, I derive the weight initialization strategy for the Rectified Linear Unit (RELU), and provide theoretical insights into why the Xavier initialization is a poor choice with RELU activations. My…
In artificial neural network, the activation function and the weight initialization method play important roles in training and performance of a neural network. The question arises is what properties of a function are important/necessary for being a well-performing activation function. Also, the most widely used weight initialization methods - Xavier and He normal initialization have fundamental connection with activation function. This survey discusses the important/necessary properties of activation function and the most widely used activation functions (sigmoid, tanh, ReLU, LReLU and PReLU)…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Cognizant AI Labs（高知特人工智能实验室）
Neural networks require careful weight initialization to prevent signals from exploding or vanishing. Existing initialization schemes solve this problem in specific cases by assuming that the network has a certain activation function or topology. It is difficult to derive such weight initialization strategies, and modern architectures therefore often use these same initialization schemes even though their assumptions do not hold. This paper introduces AutoInit, a weight initialization algorithm that automatically adapts to different neural network architectures. By analytically tracking the me…
The weight initialization and the activation function of deep neural networks have a crucial impact on the performance of the training procedure. An inappropriate selection can lead to the loss of information of the input during forward propagation and the exponential vanishing/exploding of gradients during back-propagation. Understanding the theoretical properties of untrained random networks is key to identifying which deep networks may be trained successfully as recently demonstrated by Schoenholz et al. (2017) who showed that for deep feedforward neural networks only a specific choice of h…
Carnegie Mellon University（卡内基梅隆大学）；Bosch Center for AI（博世人工智能中心）
Neural networks with sinusoidal activations have been proposed as an alternative to networks with traditional activation functions. Despite their promise, particularly for learning implicit models, their training behavior is not yet fully understood, leading to a number of empirical design choices that are not well justified. In this work, we first propose a simplified version of such sinusoidal neural networks, which allows both for easier practical implementation and simpler theoretical analysis. We then analyze the behavior of these networks from the neural tangent kernel perspective and de…
Researchers have proposed various activation functions. These activation functions help the deep network to learn non-linear behavior with a significant effect on training dynamics and task performance. The performance of these activations also depends on the initial state of the weight parameters, i.e., different initial state leads to a difference in the performance of a network. In this paper, we have proposed a cooperative initialization for training the deep network using ReLU activation function to improve the network performance. Our approach uses multiple activation functions in the in…
Australian Institute for Machine Learning（澳大利亚机器学习研究所）；University of Adelaide（阿德莱德大学）
Coordinate-MLPs are emerging as an effective tool for modeling multidimensional continuous signals, overcoming many drawbacks associated with discrete grid-based approximations. However, coordinate-MLPs with ReLU activations, in their rudimentary form, demonstrate poor performance in representing signals with high fidelity, promoting the need for positional embedding layers. Recently, Sitzmann et al. proposed a sinusoidal activation function that has the capacity to omit positional embedding from coordinate-MLPs while still preserving high signal fidelity. Despite its potential, ReLUs are stil…
Department of EECS, Oregon State University（俄勒冈州立大学电子工程与计算机科学系）
Training a neural network (NN) depends on multiple factors, including but not limited to the initial weights. In this paper, we focus on initializing deep NN parameters such that it performs better, comparing to random or zero initialization. We do this by reducing the process of initialization into an SMT solver. Previous works consider certain activation functions on small NNs, however the studied NN is a deep network with different activation functions. Our experiments show that the proposed approach for parameter initialization achieves better performance comparing to randomly initialized …
We demonstrate that in residual neural networks (ResNets) dynamical isometry is achievable irrespectively of the activation function used. We do that by deriving, with the help of Free Probability and Random Matrix Theories, a universal formula for the spectral density of the input-output Jacobian at initialization, in the large network width and depth limit. The resulting singular value spectrum depends on a single parameter, which we calculate for a variety of popular activation functions, by analyzing the signal propagation in the artificial neural network. We corroborate our results with n…
Qualcomm AI Research, Qualcomm Technologies, Inc.（高通AI研究，高通技术公司）；Qualcomm AI Research, Qualcomm Technologies Netherlands B.V.（高通AI研究，高通技术荷兰有限公司）；Seoul National University（首尔大学）
Unlike ReLU, newer activation functions (like Swish, H-swish, Mish) that are frequently employed in popular efficient architectures can also result in negative activation values, with skewed positive and negative ranges. Typical learnable quantization schemes [PACT, LSQ] assume unsigned quantization for activations and quantize all negative activations to zero which leads to significant loss in performance. Naively using signed quantization to accommodate these negative values requires an extra sign bit which is expensive for low-bit (2-, 3-, 4-bit) quantization. To solve this problem, we prop…
Pennsylvania State University（宾夕法尼亚州立大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Weight initialization plays an important role in training neural networks and also affects tremendous deep learning applications. Various weight initialization strategies have already been developed for different activation functions with different neural networks. These initialization algorithms are based on minimizing the variance of the parameters between layers and might still fail when neural networks are deep, e.g., dying ReLU. To address this challenge, we study neural networks from a nonlinear computation point of view and propose a novel weight initialization strategy that is based on…
Indian Institute Of Technology–Roorkee（印度理工学院鲁尔基分校）
Activation functions have come up as one of the essential components of neural networks. The choice of adequate activation function can impact the accuracy of these methods. In this study, we experiment for finding an optimal activation function by defining it as a weighted sum of existing activation functions and then further optimizing these weights while training the network. The study uses three activation functions, ReLU, tanh, and sin, over three popular image datasets, MNIST, FashionMNIST, and KMNIST. We observe that the ReLU activation function can easily overlook other activation func…
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of California, San Diego（加州大学圣地亚哥分校）
We consider the problem of optimization of deep learning models with smooth activation functions. While there exist influential results on the problem from the ``near initialization'' perspective, we shed considerable new light on the problem. In particular, we make two key technical contributions for such models with $L$ layers, $m$ width, and $σ_0^2$ initialization variance. First, for suitable $σ_0^2$, we establish a $O(\frac{\text{poly}(L)}{\sqrt{m}})$ upper bound on the spectral norm of the Hessian of such models, considerably sharpening prior results. Second, we introduce a new analysis …
Reducing the precision of weights and activation functions in neural network training, with minimal impact on performance, is essential for the deployment of these models in resource-constrained environments. We apply mean-field techniques to networks with quantized activations in order to evaluate the degree to which quantization degrades signal propagation at initialization. We derive initialization schemes which maximize signal propagation in such networks and suggest why this is helpful for generalization. Building on these results, we obtain a closed form implicit equation for $L_{\max}$,…
The weight initialization and the activation function of deep neural networks have a crucial impact on the performance of the training procedure. An inappropriate selection can lead to the loss of information of the input during forward propagation and the exponential vanishing/exploding of gradients during back-propagation. Understanding the theoretical properties of untrained random networks is key to identifying which deep networks may be trained successfully as recently demonstrated by Samuel et al (2017) who showed that for deep feedforward neural networks only a specific choice of hyperp…
CISPA Helmholtz Center for Information Security（CISPA亥姆霍兹信息安全中心）
The training success, training speed and generalization ability of neural networks rely crucially on the choice of random parameter initialization. It has been shown for multiple architectures that initial dynamical isometry is particularly advantageous. Known initialization schemes for residual blocks, however, miss this property and suffer from degrading separability of different inputs for increasing depth and instability without Batch Normalization or lack feature diversity. We propose a random initialization scheme, RISOTTO, that achieves perfect dynamical isometry for residual networks w…
Georgia Institute of Technology（佐治亚理工学院）；JD.COM American Technologies Corporation（京东美国技术公司）
Many neural networks use the tanh activation function, however when given a probability distribution as input, the problem of computing the output distribution in neural networks with tanh activation has not yet been addressed. One important example is the initialization of the echo state network in reservoir computing, where random initialization of the reservoir requires time to wash out the initial conditions, thereby wasting precious data and computational resources. Motivated by this problem, we propose a novel solution utilizing a moment based approach to propagate uncertainty through an…
Google Research（谷歌研究院）；Poznan University of Technology（波兹南理工大学）
We present the surprising result that randomly initialized neural networks are good feature extractors in expectation. These random features correspond to finite-sample realizations of what we call Neural Network Prior Kernel (NNPK), which is inherently infinite-dimensional. We conduct ablations across multiple architectures of varying sizes as well as initializations and activation functions. Our analysis suggests that certain structures that manifest in a trained model are already present at initialization. Therefore, NNPK may provide further insight into why neural networks are so effective…
Data Science Institute（数据科学研究所）；Imperial College London（帝国理工学院）
Recent seminal work at the intersection of deep neural networks practice and random matrix theory has linked the convergence speed and robustness of these networks with the combination of random weight initialization and nonlinear activation function in use. Building on those principles, we introduce a process to transform an existing activation function into another one with better properties. We term such transform \emph{static activation normalization}. More specifically we focus on this normalization applied to the ReLU unit, and show empirically that it significantly promotes convergence …
We propose a simple extension to the ReLU-family of activation functions that allows them to shift the mean activation across a layer towards zero. Combined with proper weight initialization, this alleviates the need for normalization layers. We explore the training of deep vanilla recurrent neural networks (RNNs) with up to 144 layers, and show that bipolar activation functions help learning in this setting. On the Penn Treebank and Text8 language modeling tasks we obtain competitive results, improving on the best reported results for non-gated networks. In experiments with convolutional neur…
Using an extended and formalized version of the Q/C map analysis of Poole et al. (2016), along with Neural Tangent Kernel theory, we identify the main pathologies present in deep networks that prevent them from training fast and generalizing to unseen data, and show how these can be avoided by carefully controlling the "shape" of the network's initialization-time kernel function. We then develop a method called Deep Kernel Shaping (DKS), which accomplishes this using a combination of precise parameter initialization, activation function transformations, and small architectural tweaks, all of w…
Shanghai Jiao Tong University（上海交通大学）；CMA-Shanghai（上海数学中心）；Shanghai Center for Brain Science and Brain-Inspired Technology（上海脑科学与类脑研究中心）
Empirical works show that for ReLU neural networks (NNs) with small initialization, input weights of hidden neurons (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) condense on isolated orientations. The condensation dynamics implies that the training implicitly regularizes a NN towards one with a much smaller effective size. In this work, we illustrate the formation of the condensation in multi-layer fully connected NNs and show that the maximal number of condensed orientations in the initial training stage is twice the m…
University of Torino（都灵大学）；Collegio Carlo Alberto（卡洛·阿尔贝托学院）；Bocconi University（博科尼大学）；Cogent Labs（Cogent实验室）
There is a recent and growing literature on large-width asymptotic properties of Gaussian neural networks (NNs), namely NNs whose weights are initialized as Gaussian distributions. Two popular problems are: i) the study of the large-width distributions of NNs, which characterizes the infinitely wide limit of a rescaled NN in terms of a Gaussian stochastic process; ii) the study of the large-width training dynamics of NNs, which characterizes the infinitely wide dynamics in terms of a deterministic kernel, referred to as the neural tangent kernel (NTK), and shows that, for a sufficiently large …
Center for Machine Perception（机器感知中心）；Czech Technical University in Prague（捷克布拉格技术大学）
Layer-sequential unit-variance (LSUV) initialization - a simple method for weight initialization for deep net learning - is proposed. The method consists of the two steps. First, pre-initialize weights of each convolution or inner-product layer with orthonormal matrices. Second, proceed from the first to the final layer, normalizing the variance of the output of each layer to be equal to one. Experiment with different activation functions (maxout, ReLU-family, tanh) show that the proposed initialization leads to learning of very deep nets that (i) produces networks with test accuracy better or…
University of Technology Sydney（悉尼科技大学）；Northwestern University（西北大学）
The prevailing thinking is that orthogonal weights are crucial to enforcing dynamical isometry and speeding up training. The increase in learning speed that results from orthogonal initialization in linear networks has been well-proven. However, while the same is believed to also hold for nonlinear networks when the dynamical isometry condition is satisfied, the training dynamics behind this contention have not been thoroughly explored. In this work, we study the dynamics of ultra-wide networks across a range of architectures, including Fully Connected Networks (FCNs) and Convolutional Neural …
In the current research of neural networks, the activation function is manually specified by human and not able to change themselves during training. This paper focus on how to make the activation function trainable for deep neural networks. We use series and linear combination of different activation functions make activation functions continuously variable. Also, we test the performance of CNNs with Fourier series simulated activation(Fourier-CNN) and CNNs with linear combined activation function (LC-CNN) on Cifar-10 dataset. The result shows our trainable activation function reveals better …
Renmin University of China（中国人民大学）；Massachusetts Institute of Technology（麻省理工学院）
In practice, the problems encountered in Neural Architecture Search (NAS) training are not simple problems, but often a series of difficult combinations (wrong compensation estimation, curse of dimension, overfitting, high complexity, etc.). In this paper, we propose a framework to decouple network structure from operator search space, and use two BOHBs to search alternatively. Considering that activation function and initialization are also important parts of neural network, the generalization ability of the model will be affected. We introduce an activation function and an initialization met…
This work explains in detail the theory behind Complex-Valued Neural Network (CVNN), including Wirtinger calculus, complex backpropagation, and basic modules such as complex layers, complex activation functions, or complex weight initialization. We also show the impact of not adapting the weight initialization correctly to the complex domain. This work presents a strong focus on the implementation of such modules on Python using cvnn toolbox. We also perform simulations on real-valued data, casting to the complex domain by means of the Hilbert Transform, and verifying the potential interest of…
We prove that a single step of gradient decent over depth two network, with $q$ hidden neurons, starting from orthogonal initialization, can memorize $Ω\left(\frac{dq}{\log^4(d)}\right)$ independent and randomly labeled Gaussians in $\mathbb{R}^d$. The result is valid for a large class of activation functions, which includes the absolute value.
Implicitly defined, continuous, differentiable signal representations parameterized by neural networks have emerged as a powerful paradigm, offering many possible benefits over conventional representations. However, current network architectures for such implicit neural representations are incapable of modeling signals with fine detail, and fail to represent a signal's spatial and temporal derivatives, despite the fact that these are essential to many physical signals defined implicitly as the solution to partial differential equations. We propose to leverage periodic activation functions for …
Microsoft Research India（微软印度研究院）；Cornell University（康奈尔大学）
It is well-known that overparametrized neural networks trained using gradient-based methods quickly achieve small training error with appropriate hyperparameter settings. Recent papers have proved this statement theoretically for highly overparametrized networks under reasonable assumptions. These results either assume that the activation function is ReLU or they crucially depend on the minimum eigenvalue of a certain Gram matrix depending on the data, random initialization and the activation function. In the later case, existing works only prove that this minimum eigenvalue is non-zero and do…
It Takes a Good Model to Train a Good Model: Generalized Gaussian Priors for Optimized LLMs
Shenzhen International Graduate School, Tsinghua University（清华大学深圳国际研究生院）；Electrical and Computer Engineering, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校电子与计算机工程系）；Computer Science, New York University Shanghai（纽约大学上海分校计算机科学系）
