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). This survey also explores the relationship between these activation functions and the two weight initialization methods - Xavier and He normal initialization.
在神经网络中，权值的正确初始化对网络的收敛性至关重要。目前对权重初始化的认识主要来自线性激活函数。在本文中，我开发了一个理论的重量初始化与非线性激活。首先，我推导出一个一般的权重初始化策略，任何神经网络使用激活函数可微0。接下来，我推导出了整流线性单元（RECTIFITED LINEAR UNIT）的权重初始化策略，并从理论上解释了为什么Xavier初始化对于RECTIFITED LINEAR UNIT激活是一个糟糕的选择。我的分析提供了一个明确的示范作用的非线性确定适当的权重初始化。
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…
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 …
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…
LSQ+: Improving low-bit quantization through learnable offsets and better initialization
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）
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…
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…
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…
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 …
Weight initialization is critical in being able to successfully train artificial neural networks (ANNs), and even more so for recurrent neural networks (RNNs) which can easily suffer from vanishing and exploding gradients. In neuroevolution, where evolutionary algorithms are applied to neural architecture search, weights typically need to be initialized at three different times: when initial genomes (ANN architectures) are created at the beginning of the search, when offspring genomes are generated by crossover, and when new nodes or edges are created during mutation. This work explores the di…
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 …
Efficient anti-symmetrization of a neural network layer by taming the sign problem
University of California, Berkeley（加州大学伯克利分校）；The Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
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 California, Santa Cruz（加州大学圣克鲁兹分校）；University of California, Irvine（加州大学欧文分校）；Columbia University（哥伦比亚大学）
Recurrent Neural Networks (RNNs) frequently exhibit complicated dynamics, and their sensitivity to the initialization process often renders them notoriously hard to train. Recent works have shed light on such phenomena analyzing when exploding or vanishing gradients may occur, either of which is detrimental for training dynamics. In this paper, we point to a formal connection between RNNs and chaotic dynamical systems and prove a qualitatively stronger phenomenon about RNNs than what exploding gradients seem to suggest. Our main result proves that under standard initialization (e.g., He, Xavie…
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…
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…
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 …
Center for Machine Perception（机器感知中心）；Czech Technical University in Prague（捷克布拉格技术大学）
提出了层序贯单位方差（LSUV）初始化-一种用于深度网络学习的权重初始化的简单方法。该方法包括两个步骤。首先，用正交矩阵预初始化每个卷积或内积层的权重。其次，从第一层到最后一层，将每层输出的方差归一化为1。使用不同的激活函数进行试验（maxout，ReLU-家族，tanh）表明，所提出的初始化导致了对非常深的网络的学习，（i）产生测试精度优于或等于标准方法的网络，（ii）至少与专门为非常深的网络提出的复杂方案一样快，如FitNets（Romero et al.（2015））和Highway（Srivastava et al.（2015））。性能在GoogLeNet、CaffeNet、FitNets和残差网络上进行了评估，在MNIST、CIFAR-10/100和ImageNet数据集上实现了最先进或非常接近的性能。
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…
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 …
UCLA（加州大学洛杉矶分校）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）
Under mild conditions on the network initialization we derive a power series expansion for the Neural Tangent Kernel (NTK) of arbitrarily deep feedforward networks in the infinite width limit. We provide expressions for the coefficients of this power series which depend on both the Hermite coefficients of the activation function as well as the depth of the network. We observe faster decay of the Hermite coefficients leads to faster decay in the NTK coefficients and explore the role of depth. Using this series, first we relate the effective rank of the NTK to the effective rank of the input-dat…
ExpertWeaver: Unlocking the Inherent MoE in Dense LLMs with GLU Activation Patterns
Zhejiang University（浙江大学）；Shanghai AI Laboratory（上海人工智能实验室）；Shanghai Innovation Institute（上海创新研究院）；Fudan University（复旦大学）；Chinese University of Hong Kong（中国香港大学）
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）
Batch Normalization is a key component in almost all state-of-the-art image classifiers, but it also introduces practical challenges: it breaks the independence between training examples within a batch, can incur compute and memory overhead, and often results in unexpected bugs. Building on recent theoretical analyses of deep ResNets at initialization, we propose a simple set of analysis tools to characterize signal propagation on the forward pass, and leverage these tools to design highly performant ResNets without activation normalization layers. Crucial to our success is an adapted version …
University of the Witwatersrand（金山大学）；InstaDeep；Google Brain（谷歌大脑）
Training sparse networks to converge to the same performance as dense neural architectures has proven to be elusive. Recent work suggests that initialization is the key. However, while this direction of research has had some success, focusing on initialization alone appears to be inadequate. In this paper, we take a broader view of training sparse networks and consider the role of regularization, optimization, and architecture choices on sparse models. We propose a simple experimental framework, Same Capacity Sparse vs Dense Comparison (SC-SDC), that allows for a fair comparison of sparse and …
