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Register a new userNeural Networks with Smooth Adaptive Activation Functions for Regression
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Le Hou
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In Neural Networks (NN), Adaptive Activation Functions (AAF) have parameters that control the shapes of activation functions. These parameters are trained along with other parameters in the NN. AAFs have improved performance of Neural Networks (NN) in multiple classification tasks. In this paper, we propose and apply AAFs on feedforward NNs for regression tasks. We argue that applying AAFs in the regression (second-to-last) layer of a NN can significantly decrease the bias of the regression NN. However, using existing AAFs may lead to overfitting. To address this problem, we propose a Smooth Adaptive Activation Function (SAAF) with piecewise polynomial form which can approximate any continuous function to arbitrary degree of error. NNs with SAAFs can avoid overfitting by simply regularizing the parameters. In particular, an NN with SAAFs is Lipschitz continuous given a bounded magnitude of the NN parameters. We prove an upper-bound for model complexity in terms of fat-shattering dimension for any Lipschitz continuous regression model. Thus, regularizing the parameters in NNs with SAAFs avoids overfitting. We empirically evaluated NNs with SAAFs and achieved state-of-the-art results on multiple regression datasets.
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We have proposed orthogonal-Pade activation functions, which are trainable activation functions and show that they have faster learning capability and improves the accuracy in standard deep learning datasets and models. Based on our experiments, we have found two best candidates out of six orthogonal-Pade activations, which we call safe Hermite-Pade (HP) activation functions, namely HP-1 and HP-2. When compared to ReLU, HP-1 and HP-2 has an increment in top-1 accuracy by 5.06% and 4.63% respectively in PreActResNet-34, by 3.02% and 2.75% respectively in MobileNet V2 model on CIFAR100 dataset while on CIFAR10 dataset top-1 accuracy increases by 2.02% and 1.78% respectively in PreActResNet-34, by 2.24% and 2.06% respectively in LeNet, by 2.15% and 2.03% respectively in Efficientnet B0.
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 signals 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 implicit neural representations and demonstrate that these networks, dubbed sinusoidal representation networks or Sirens, are ideally suited for representing complex natural signals and their derivatives. We analyze Siren activation statistics to propose a principled initialization scheme and demonstrate the representation of images, wavefields, video, sound, and their derivatives. Further, we show how Sirens can be leveraged to solve challenging boundary value problems, such as particular Eikonal equations (yielding signed distance functions), the Poisson equation, and the Helmholtz and wave equations. Lastly, we combine Sirens with hypernetworks to learn priors over the space of Siren functions.
The scope of research in the domain of activation functions remains limited and centered around improving the ease of optimization or generalization quality of neural networks (NNs). However, to develop a deeper understanding of deep learning, it becomes important to look at the non linear component of NNs more carefully. In this paper, we aim to provide a generic form of activation function along with appropriate mathematical grounding so as to allow for insights into the working of NNs in future. We propose Self-Learnable Activation Functions (SLAF), which are learned during training and are capable of approximating most of the existing activation functions. SLAF is given as a weighted sum of pre-defined basis elements which can serve for a good approximation of the optimal activation function. The coefficients for these basis elements allow a search in the entire space of continuous functions (consisting of all the conventional activations). We propose various training routines which can be used to achieve performance with SLAF equipped neural networks (SLNNs). We prove that SLNNs can approximate any neural network with lipschitz continuous activations, to any arbitrary error highlighting their capacity and possible equivalence with standard NNs. Also, SLNNs can be completely represented as a collections of finite degree polynomial upto the very last layer obviating several hyper parameters like width and depth. Since the optimization of SLNNs is still a challenge, we show that using SLAF along with standard activations (like ReLU) can provide performance improvements with only a small increase in number of parameters.
Neural networks are generally built by interleaving (adaptable) linear layers with (fixed) nonlinear activation functions. To increase their flexibility, several authors have proposed methods for adapting the activation functions themselves, endowing them with varying degrees of flexibility. None of these approaches, however, have gained wide acceptance in practice, and research in this topic remains open. In this paper, we introduce a novel family of flexible activation functions that are based on an inexpensive kernel expansion at every neuron. Leveraging over several properties of kernel-based models, we propose multiple variations for designing and initializing these kernel activation functions (KAFs), including a multidimensional scheme allowing to nonlinearly combine information from different paths in the network. The resulting KAFs can approximate any mapping defined over a subset of the real line, either convex or nonconvex. Furthermore, they are smooth over their entire domain, linear in their parameters, and they can be regularized using any known scheme, including the use of $ell_1$ penalties to enforce sparseness. To the best of our knowledge, no other known model satisfies all these properties simultaneously. In addition, we provide a relatively complete overview on alternative techniques for adapting the activation functions, which is currently lacking in the literature. A large set of experiments validates our proposal.
We study the dynamics of optimization and the generalization properties of one-hidden layer neural networks with quadratic activation function in the over-parametrized regime where the layer width $m$ is larger than the input dimension $d$. We consider a teacher-student scenario where the teacher has the same structure as the student with a hidden layer of smaller width $m^*le m$. We describe how the empirical loss landscape is affected by the number $n$ of data samples and the width $m^*$ of the teacher network. In particular we determine how the probability that there be no spurious minima on the empirical loss depends on $n$, $d$, and $m^*$, thereby establishing conditions under which the neural network can in principle recover the teacher. We also show that under the same conditions gradient descent dynamics on the empirical loss converges and leads to small generalization error, i.e. it enables recovery in practice. Finally we characterize the time-convergence rate of gradient descent in the limit of a large number of samples. These results are confirmed by numerical experiments.
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