We study the problem of learning a single neuron $\mathbf{x}\mapsto σ(\mathbf{w}^T\mathbf{x})$ with gradient descent (GD). All the existing positive results are limited to the case where $σ$ is monotonic. However, it is recently observed that non-monotonic activation functions outperform the traditional monotonic ones in many applications. To fill this gap, we establish learnability without assuming monotonicity. Specifically, when the input distribution is the standard Gaussian, we show that mild conditions on $σ$ (e.g., $σ$ has a dominating linear part) are sufficient to guarantee the learnability in polynomial time and polynomial samples. Moreover, with a stronger assumption on the activation function, the condition of input distribution can be relaxed to a non-degeneracy of the marginal distribution. We remark that our conditions on $σ$ are satisfied by practical non-monotonic activation functions, such as SiLU/Swish and GELU. We also discuss how our positive results are related to existing negative results on training two-layer neural networks.
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室公司）；Data61
We propose a new generic type of stochastic neurons, called $q$-neurons, that considers activation functions based on Jackson's $q$-derivatives with stochastic parameters $q$. Our generalization of neural network architectures with $q$-neurons is shown to be both scalable and very easy to implement. We demonstrate experimentally consistently improved performances over state-of-the-art standard activation functions, both on training and testing loss functions.
University of Warwick（华威大学）；Pakistan Institute of Engineering and Applied Sciences (PIEAS)（巴基斯坦工程与应用科学研究院）
An artificial neuron is modelled as a weighted summation followed by an activation function which determines its output. A wide variety of activation functions such as rectified linear units (ReLU), leaky-ReLU, Swish, MISH, etc. have been explored in the literature. In this short paper, we explore what happens when the activation function of each neuron in an artificial neural network is learned natively from data alone. This is achieved by modelling the activation function of each neuron as a small neural network whose weights are shared by all neurons in the original network. We list our pri…
Universidade Federal Rural de Pernambuco（伯南布哥联邦农村大学）；Harvard University（哈佛大学）
The activation function plays a fundamental role in the artificial neural network learning process. However, there is no obvious choice or procedure to determine the best activation function, which depends on the problem. This study proposes a new artificial neuron, named global-local neuron, with a trainable activation function composed of two components, a global and a local. The global component term used here is relative to a mathematical function to describe a general feature present in all problem domain. The local component is a function that can represent a localized behavior, like a t…
Technical University Munich（慕尼黑工业大学）；AI Research, Volkswagen Group（大众集团人工智能研究部）
We propose stochastic, non-parametric activation functions that are fully learnable and individual to each neuron. Complexity and the risk of overfitting are controlled by placing a Gaussian process prior over these functions. The result is the Gaussian process neuron, a probabilistic unit that can be used as the basic building block for probabilistic graphical models that resemble the structure of neural networks. The proposed model can intrinsically handle uncertainties in its inputs and self-estimate the confidence of its predictions. Using variational Bayesian inference and the central lim…
Carnegie Mellon University（卡内基梅隆大学）；University of Alberta（阿尔伯塔大学）
We propose a Laplace approximation that creates a stochastic unit from any smooth monotonic activation function, using only Gaussian noise. This paper investigates the application of this stochastic approximation in training a family of Restricted Boltzmann Machines (RBM) that are closely linked to Bregman divergences. This family, that we call exponential family RBM (Exp-RBM), is a subset of the exponential family Harmoniums that expresses family members through a choice of smooth monotonic non-linearity for each neuron. Using contrastive divergence along with our Gaussian approximation, we s…
Amazon Inc.（亚马逊公司）；University of Oregon（俄勒冈大学）；Microsoft Inc.（微软公司）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
Most deep neural networks use simple, fixed activation functions, such as sigmoids or rectified linear units, regardless of domain or network structure. We introduce differential equation units (DEUs), an improvement to modern neural networks, which enables each neuron to learn a particular nonlinear activation function from a family of solutions to an ordinary differential equation. Specifically, each neuron may change its functional form during training based on the behavior of the other parts of the network. We show that using neurons with DEU activation functions results in a more compact …
Hanyang University（汉阳大学）；NYU（纽约大学）；Baylor College of Medicine（贝勒医学院）；Rice University（莱斯大学）
Neurons in the brain are complex machines with distinct functional compartments that interact nonlinearly. In contrast, neurons in artificial neural networks abstract away this complexity, typically down to a scalar activation function of a weighted sum of inputs. Here we emulate more biologically realistic neurons by learning canonical activation functions with two input arguments, analogous to basal and apical dendrites. We use a network-in-network architecture where each neuron is modeled as a multilayer perceptron with two inputs and a single output. This inner perceptron is shared by all …
The choice of activation function can significantly influence the performance of neural networks. The lack of guiding principles for the selection of activation function is lamentable. We try to address this issue by introducing our variational neural networks, where the activation function is represented as a linear combination of possible candidate functions, and an optimal activation is obtained via minimization of a loss function using gradient descent method. The gradient formulae for the loss function with respect to these expansion coefficients are central for the implementation of grad…
Amazon Web Services（亚马逊云科技）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Microsoft Inc.（微软公司）
Most deep neural networks use simple, fixed activation functions, such as sigmoids or rectified linear units, regardless of domain or network structure. We introduce differential equation units (DEUs), an improvement to modern neural networks, which enables each neuron to learn a particular nonlinear activation function from a family of solutions to an ordinary differential equation. Specifically, each neuron may change its functional form during training based on the behavior of the other parts of the network. We show that using neurons with DEU activation functions results in a more compact …
Many neural network architectures rely on the choice of the activation function for each hidden layer. Given the activation function, the neural network is trained over the bias and the weight parameters. The bias catches the center of the activation, and the weights capture the scale. Here we propose to train the network over a shape parameter as well. This view allows each neuron to tune its own activation function and adapt the neuron curvature towards a better prediction. This modification only adds one further equation to the back-propagation for each neuron. Re-formalizing activation fun…
We consider the fundamental problem of learning a single neuron $x \mapstoσ(w^\top x)$ using standard gradient methods. As opposed to previous works, which considered specific (and not always realistic) input distributions and activation functions $σ(\cdot)$, we ask whether a more general result is attainable, under milder assumptions. On the one hand, we show that some assumptions on the distribution and the activation function are necessary. On the other hand, we prove positive guarantees under mild assumptions, which go beyond those studied in the literature so far. We also point out and st…
We propose $\textit{Mish}$, a novel self-regularized non-monotonic activation function which can be mathematically defined as: $f(x)=x\tanh(softplus(x))$. As activation functions play a crucial role in the performance and training dynamics in neural networks, we validated experimentally on several well-known benchmarks against the best combinations of architectures and activation functions. We also observe that data augmentation techniques have a favorable effect on benchmarks like ImageNet-1k and MS-COCO across multiple architectures. For example, Mish outperformed Leaky ReLU on YOLOv4 with a…
University of California - Irvine（加州大学尔湾分校）；Adobe Research（奥多比研究院）
Artificial neural networks typically have a fixed, non-linear activation function at each neuron. We have designed a novel form of piecewise linear activation function that is learned independently for each neuron using gradient descent. With this adaptive activation function, we are able to improve upon deep neural network architectures composed of static rectified linear units, achieving state-of-the-art performance on CIFAR-10 (7.51%), CIFAR-100 (30.83%), and a benchmark from high-energy physics involving Higgs boson decay modes.
IIIT Delhi（德里英德拉普拉斯塔信息技术学院）；Shaheed Bhagat Singh College, University of Delhi（德里大学沙希德·巴格特·辛格学院）；University of Delhi（德里大学）
An activation function is a crucial component of a neural network that introduces non-linearity in the network. The state-of-the-art performance of a neural network depends also on the perfect choice of an activation function. We propose two novel non-monotonic smooth trainable activation functions, called ErfAct and Pserf. Experiments suggest that the proposed functions improve the network performance significantly compared to the widely used activations like ReLU, Swish, and Mish. Replacing ReLU by ErfAct and Pserf, we have 5.68% and 5.42% improvement for top-1 accuracy on Shufflenet V2 (2.0…
Many activation functions have been proposed in the past, but selecting an adequate one requires trial and error. We propose a new methodology of designing activation functions within a neural network at each layer. We call this technique an "activation ensemble" because it allows the use of multiple activation functions at each layer. This is done by introducing additional variables, $α$, at each activation layer of a network to allow for multiple activation functions to be active at each neuron. By design, activations with larger $α$ values at a neuron is equivalent to having the largest mag…
Michigan Technological University（密歇根理工大学）；Symbiosis Centre for Medical Image Analysis（共生医学图像分析中心）；Symbiosis Institute of Technology（共生技术学院）
This work introduces a novel activation unit that can be efficiently employed in deep neural nets (DNNs) and performs significantly better than the traditional Rectified Linear Units (ReLU). The function developed is a two parameter version of the specialized Richard's Curve and we call it Adaptive Richard's Curve weighted Activation (ARiA). This function is non-monotonous, analogous to the newly introduced Swish, however allows a precise control over its non-monotonous convexity by varying the hyper-parameters. We first demonstrate the mathematical significance of the two parameter ARiA follo…
University of Siena（锡耶纳大学）；University of Florence（佛罗伦萨大学）
The effectiveness of deep neural architectures has been widely supported in terms of both experimental and foundational principles. There is also clear evidence that the activation function (e.g. the rectifier and the LSTM units) plays a crucial role in the complexity of learning. Based on this remark, this paper discusses an optimal selection of the neuron non-linearity in a functional framework that is inspired from classic regularization arguments. It is shown that the best activation function is represented by a kernel expansion in the training set, that can be effectively approximated ove…
This work proposes a novel solution to the problem of internal covariate shift and dying neurons using the concept of linked neurons. We define the neuron linkage in terms of two constraints: first, all neuron activations in the linkage must have the same operating point. That is to say, all of them share input weights. Secondly, a set of neurons is linked if and only if there is at least one member of the linkage that has a non-zero gradient in regard to the input of the activation function. This means that for any input in the activation function, there is at least one member of the linkage …
Department of Physics, Cornell University（康奈尔大学物理系）；Institute of Natural Sciences, School of Mathematical Sciences, Ministry of Education Key Laboratory in Scientific and Engineering Computing, and Global College, Shanghai Jiao Tong University（上海交通大学自然科学院、数学科学学院、教育部科学与工程计算重点实验室及全球学院）；School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）
Department of Physics, Cornell University（康奈尔大学物理系）；Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然科学研究院）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Ministry of Education Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University（上海交通大学教育部科学与工程计算重点实验室）；Global College, Shanghai Jiao Tong University（上海交通大学全球学院）；School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）
ReLU$^2$ Wins: Discovering Efficient Activation Functions for Sparse LLMs
Tsinghua University（清华大学）；Shanghai Jiao Tong University（上海交通大学）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；Renmin University of China（中国人民大学）
Rectified Linear Units (ReLU) are the default choice for activation functions in deep neural networks. While they demonstrate excellent empirical performance, ReLU activations can fall victim to the dead neuron problem. In these cases, the weights feeding into a neuron end up being pushed into a state where the neuron outputs zero for all inputs. Consequently, the gradient is also zero for all inputs, which means that the weights which feed into the neuron cannot update. The neuron is not able to recover from direct back propagation and model capacity is reduced as those parameters can no long…
This paper presents a new reachability analysis approach to compute interval over-approximations of the output set of feedforward neural networks with input uncertainty. We adapt to neural networks an existing mixed-monotonicity method for the reachability analysis of dynamical systems and apply it to each partial network within the main network. This ensures that the intersection of the obtained results is the tightest interval over-approximation of the output of each layer that can be obtained using mixed-monotonicity on any partial network decomposition. Unlike other tools in the literature…
Seoul National University（首尔国立大学）；Hanyang University（汉阳大学）；NAVER Corp（NAVER公司）
An activation boundary for a neuron refers to a separating hyperplane that determines whether the neuron is activated or deactivated. It has been long considered in neural networks that the activations of neurons, rather than their exact output values, play the most important role in forming classification friendly partitions of the hidden feature space. However, as far as we know, this aspect of neural networks has not been considered in the literature of knowledge transfer. In this paper, we propose a knowledge transfer method via distillation of activation boundaries formed by hidden neuron…
Princeton University（普林斯顿大学）；Beijing Institute of Big Data Research（北京大数据研究院）
A numerical and phenomenological study of the gradient descent (GD) algorithm for training two-layer neural network models is carried out for different parameter regimes when the target function can be accurately approximated by a relatively small number of neurons. It is found that for Xavier-like initialization, there are two distinctive phases in the dynamic behavior of GD in the under-parametrized regime: An early phase in which the GD dynamics follows closely that of the corresponding random feature model and the neurons are effectively quenched, followed by a late phase in which the neur…
fortiss - Research Institute of the Free State of Bavaria（fortiss——巴伐利亚自由州研究院）；DENSO CORPORATION（电装公司）
For using neural networks in safety critical domains, it is important to know if a decision made by a neural network is supported by prior similarities in training. We propose runtime neuron activation pattern monitoring - after the standard training process, one creates a monitor by feeding the training data to the network again in order to store the neuron activation patterns in abstract form. In operation, a classification decision over an input is further supplemented by examining if a pattern similar (measured by Hamming distance) to the generated pattern is contained in the monitor. If t…
Yale Department of Computer Science（耶鲁大学计算机科学系）；Yale Department of Genetics（耶鲁大学遗传学系）；Université de Montréal, Dept. of Math. & Stat.（蒙特利尔大学数学与统计系）；Mila（米拉研究院）
While neural networks are powerful approximators used to classify or embed data into lower dimensional spaces, they are often regarded as black boxes with uninterpretable features. Here we propose Graph Spectral Regularization for making hidden layers more interpretable without significantly impacting performance on the primary task. Taking inspiration from spatial organization and localization of neuron activations in biological networks, we use a graph Laplacian penalty to structure the activations within a layer. This penalty encourages activations to be smooth either on a predetermined gra…
Otto von Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）
Deep Learning based Automatic Speech Recognition (ASR) models are very successful, but hard to interpret. To gain better understanding of how Artificial Neural Networks (ANNs) accomplish their tasks, introspection methods have been proposed. Adapting such techniques from computer vision to speech recognition is not straight-forward, because speech data is more complex and less interpretable than image data. In this work, we introduce Gradient-adjusted Neuron Activation Profiles (GradNAPs) as means to interpret features and representations in Deep Neural Networks. GradNAPs are characteristic re…
Despite deep neural network (DNN)'s impressive prediction performance in various domains, it is well known now that a set of DNN models trained with the same model specification and the same data can produce very different prediction results. Ensemble method is one state-of-the-art benchmark for prediction uncertainty estimation. However, ensembles are expensive to train and serve for web-scale traffic. In this paper, we seek to advance the understanding of prediction variation estimated by the ensemble method. Through empirical experiments on two widely used benchmark datasets MovieLens and C…
Brown University（布朗大学）；Massachusetts Institute of Technology（麻省理工学院）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
We propose two approaches of locally adaptive activation functions namely, layer-wise and neuron-wise locally adaptive activation functions, which improve the performance of deep and physics-informed neural networks. The local adaptation of activation function is achieved by introducing a scalable parameter in each layer (layer-wise) and for every neuron (neuron-wise) separately, and then optimizing it using a variant of stochastic gradient descent algorithm. In order to further increase the training speed, an activation slope based slope recovery term is added in the loss function, which furt…
