Deep neural networks (DNNs) have garnered significant attention in various fields of science and technology in recent years. Activation functions define how neurons in DNNs process incoming signals for them. They are essential for learning non-linear transformations and for performing diverse computations among successive neuron layers. In the last few years, researchers have investigated the approximation ability of DNNs to explain their power and success. In this paper, we explore the approximation ability of DNNs using a different activation function, called SignReLU. Our theoretical results demonstrate that SignReLU networks outperform rational and ReLU networks in terms of approximation performance. Numerical experiments are conducted comparing SignReLU with the existing activations such as ReLU, Leaky ReLU, and ELU, which illustrate the competitive practical performance of SignReLU.

In most control applications, theoretical analysis of the systems is crucial in ensuring stability or convergence, so as to ensure safe and reliable operations and also to gain a better understanding of the systems for further developments. However, most current deep learning methods are black-box approaches that are more focused on empirical studies. Recently, some results have been obtained for convergence analysis of end-to end deep learning based on non-smooth ReLU activation functions, which may result in chattering for control tasks. This paper presents a convergence analysis for end-to-end deep learning of fully connected neural networks (FNN) with smooth activation functions. The proposed method therefore avoids any potential chattering problem, and it also does not easily lead to gradient vanishing problems. The proposed End-to-End algorithm trains multiple two-layer fully connected networks concurrently and collaborative learning can be used to further combine their strengths to improve accuracy. A classification case study based on fully connected networks and MNIST dataset was done to demonstrate the performance of the proposed approach. Then an online kinematics control task of a UR5e robot arm was performed to illustrate the regression approximation and online updating ability of our algorithm.

Generative adversarial networks (GANs) have attracted intense interest in the field of generative models. However, few investigations focusing either on the theoretical analysis or on algorithm design for the approximation ability of the generator of GANs have been reported. This paper will first theoretically analyze GANs' approximation property. Similar to the universal approximation property of the full connected neural networks with one hidden layer, we prove that the generator with the input latent variable in GANs can universally approximate the potential data distribution given the increasing hidden neurons. Furthermore, we propose an approach named stochastic data generation (SDG) to enhance GANs' approximation ability. Our approach is based on the simple idea of imposing randomness through data generation in GANs by a prior distribution on the conditional probability between the layers. Our approach can be easily implemented by using the reparameterization trick. The experimental results on synthetic dataset verify the improved approximation ability obtained by this SDG approach. In the practical dataset, the NSGAN/WGANGP with SDG can also outperform traditional GANs with little change in the model architectures.

There has been a growing interest in expressivity of deep neural networks. But most of existing work about this topic focus only on the specific activation function such as ReLU or sigmoid. In this paper, we investigate the approximation ability of deep neural networks with a quite general class of activation functions. This class of activation functions includes most of frequently used activation functions. We derive the required depth, width and sparsity of a deep neural network to approximate any H\"older smooth function upto a given approximation error for the large class of activation functions. Based on our approximation error analysis, we derive the minimax optimality of the deep neural network estimators with the general activation functions in both regression and classification problems.

This paper considers the power of deep neural networks (deep nets for short) in realizing data features. Based on refined covering number estimates, we find that, to realize some complex data features, deep nets can improve the performances of shallow neural networks (shallow nets for short) without requiring additional capacity costs. This verifies the advantage of deep nets in realizing complex features. On the other hand, to realize some simple data feature like the smoothness, we prove that, up to a logarithmic factor, the approximation rate of deep nets is asymptotically identical to that of shallow nets, provided that the depth is fixed. This exhibits a limitation of deep nets in realizing simple features.

Evolutionary algorithms (EAs) are a large family of heuristic optimization algorithms inspired by natural phenomena, and are often used in practice to obtain satisficing instead of optimal solutions. In this work, we investigate a largely underexplored issue: the approximation performance of EAs in terms of how close the obtained solution is to an optimal solution. We study an EA framework named simple EA with isolated population (SEIP) that can be implemented as a single- or multi-objective EA. We present general approximation results of SEIP, and specifically on the minimum set cover problem, we find that SEIP achieves the currently best-achievable approximation ratio. Moreover, on an instance class of the k -set cover problem, we disclose how SEIP can overcome the difficulty that limits the greedy algorithm.