This paper presents a general framework for norm-based capacity control for $L_{p,q}$ weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an $L_{p,q}$ normalization where $q\le p^*$ and $1/p+1/p^{*}=1$, we discuss properties of a width-independent capacity control, which only depends on the depth by a square root term. We further analyze the approximation properties of $L_{p,q}$ weight normalized deep neural networks. In particular, for an $L_{1,\infty}$ weight normalized network, the approximation error can be controlled by the $L_1$ norm of the output layer, and the corresponding generalization error only depends on the architecture by the square root of the depth.

This paper essentially seems to address 2 questions (a) Under certain natural weight constraints what is the Rademacher complexity upperbound for nets when we allow for bias vectors in each layer? There is a section 4.1 in the paper which is about generalization bounds for nets doing regression. Let me first say at the outset that the writing of the paper seems extremely bad and many of the crucial steps in the proofs look unfollowable. As it stands this paper is hardly fit to be made public and needs a thorough rewriting! If that is the entire point then why is this interesting?

Our human brain learns any objective presented to it with unprecedented speed and accuracy. It has left humankind baffled as to how remarkable our brain is and its ability to learn patterns. Recognition is the core feature in our evolution from savages to intelligent beings. Similarly, activation functions allow machines to differ and learn from their output. Although simple to understand, it's an incredible feat to make machines capable of mastering many arduous tasks which are difficult for us; human beings.

Deep learning is used to classify data into several groups based on nonlinear curved surfaces. In this paper, we focus on the theoretical analysis of deep learning using the rectified linear unit (ReLU) activation function. Because layers approximate a nonlinear curved surface, increasing the number of layers improves the approximation accuracy of the curved surface. While neurons perform a layer-by-layer approximation of the most appropriate hyperplanes, increasing their number cannot improve the results obtained via canonical correlation analysis (CCA). These results illustrate the functions of layers and neurons in deep learning with ReLU.

Artificial neural networks are inspired by the biological neurons within the human body which activate under certain circumstances resulting in a related action performed by the body in response. Artificial neural nets consist of various layers of interconnected artificial neurons powered by activation functions which help in switching them ON/OFF. Like traditional machine learning algorithms, here too, there are certain values that neural nets learn in the training phase. Briefly, each neuron receives a multiplied version of inputs and random weights which is then added with static bias value (unique to each neuron layer), this is then passed to an appropriate activation function which decides the final value to be given out of the neuron. There are various activation functions available as per the nature of input values.

In a neural network, the activation function is responsible for transforming the summed weighted input from the node into the activation of the node or output for that input. The rectified linear activation function or ReLU for short is a piecewise linear function that will output the input directly if it is positive, otherwise, it will output zero. It has become the default activation function for many types of neural networks because a model that uses it is easier to train and often achieves better performance. In this tutorial, you will discover the rectified linear activation function for deep learning neural networks. A Gentle Introduction to the Rectified Linear Activation Function for Deep Learning Neural Networks Photo by Bureau of Land Management, some rights reserved. A neural network is comprised of layers of nodes and learns to map examples of inputs to outputs. For a given node, the inputs are multiplied by the weights in a node and summed together.

Sigmoid and tanh were monotonous, differentiable, and previously more popular activation functions. However, these functions suffer saturation over time, and this leads to problems occurring with vanishing gradients. An alternative and the most popular activation function to overcome this issue is the Rectified Linear Unit (ReLU). The above diagram with the blue line is the representation of the Rectified Linear Unit (ReLU) whereas, the green line is a variant of ReLU called Softplus. The other variants of ReLU include Leaky ReLU, ELU, SiLU, etc., which are used for better performance in some tasks.

Artificial neural networks (ANNs) have occupied the center stage in the realm of deep learning in the recent past. ANNs are made up of several hidden layers, and each hidden layer consists of several neurons. At each neuron, an affine linear map is composed with a nonlinear function known as activation function. During the training of an ANN, the linear map is optimized, however an activation function is usually fixed in the beginning along with the architecture of the ANN. There has been an increasing interest in developing a methodical understanding of activation functions, in particular with regards to the construction of novel activation functions and identifying mathematical properties leading to a better learning [1]. An activation function is considered good if it can increase the learning rate and leaning to better convergence which leads to more accurate results. At the early stage of deep learning research, researchers used shallow networks (fewer hidden layers), and tanh or sigmoid, were used as activation functions. As the research progressed and deeper networks (multiple hidden layers) came into fashion to achieve challenging tasks, Rectified Linear Unit (ReLU)([2], [3], [4]) emerged as the most popular activation function. Despite its simplicity, deep neural networks with ReLU have learned many complex and highly nonlinear functions with high accuracy.

This paper presents a general framework for norm-based capacity control for $L_{p,q}$ weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an $L_{p,q}$ normalization where $q\le p *$ and $1/p 1/p {*} 1$, we discuss properties of a width-independent capacity control, which only depends on the depth by a square root term. We further analyze the approximation properties of $L_{p,q}$ weight normalized deep neural networks. In particular, for an $L_{1,\infty}$ weight normalized network, the approximation error can be controlled by the $L_1$ norm of the output layer, and the corresponding generalization error only depends on the architecture by the square root of the depth.

During the calculations of the values for activations in each layer, we use an activation function right before deciding what exactly the activation value should be. From the previous activations, weights and biases in each layer, we calculate a value for every activation in the next layer. But before sending that value to the activations of the next layer, we use an activation function to scale the output. Here, we will explore different activation functions. The prerequisite for this post is my last post about feedfordward and backpropagation in neural networks, you would have seen that I briefly talked about activation functions, but never actually expanded on what they do for us. Much of what I talk about here will only be relevant if you have the prior knowledge, or have read my previous post. This article has several ads, which links to Amazon. ML From Scratch earns a small commission, when a purchase is made, using the links provided. It goes back into the site; that is, any fees associated with keeping this website live and producing content.