It can cause a weight update causes the network to never activate on any data point. These are commonly referred to as dead neurons. To combat the issue of dead neurons, leaky ReLU was introduced which contains a small slope. The purpose of this slope is to keep the updates alive and prevent the production of dead neurons. The leaky and generalized rectified linear unit are slight variations on the basic ReLU function.

The leaky ReLU still has a discontinuity at zero, but the function is no longer flat below zero, it merely has a reduced gradient.

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Maxout is simply the maximum of k linear functions — it directly learns the activation function. Currently, the most successful and widely-used activation function is ReLU. However, swish tends to work better than ReLU on deeper models across a number of challenging datasets.

Swish was developed by Google in Swish is essentially the sigmoid function multiplied by x:. One of the main problems with ReLU that gives rise to the vanishing gradient problem is that its derivative is zero for half of the values of the input x. Swish, on the other hand, is a smooth non-monotonic function that does not suffer from this problem of zero derivatives.

## A Beginner's Guide to Neural Networks and Deep Learning

Swish is still seen as a somewhat magical improvement to neural networks, but the results show that it provides a clear improvement for deep networks. To read more about this, I recommend checking out the original paper on arxiv:. In the next section, we will discuss loss functions in more detail. Loss functions also called cost functions are an important aspect of neural networks.

## A Gentle Introduction to the Challenge of Training Deep Learning Neural Network Models

We have already discussed that neural networks are trained using an optimization process that requires a loss function to calculate the model error. There are many functions that could be used to estimate the error of a set of weights in a neural network. However, we prefer a function where the space of candidate solutions maps onto a smooth but high-dimensional landscape that the optimization algorithm can reasonably navigate via iterative updates to the model weights.

Maximum Likelihood provides a framework for choosing a loss function when training neural networks and machine learning models in general. As such, the loss function to use depends on the output data distribution and is closely coupled to the output unit discussed in the next section. Cross-entropy and mean squared error are the two main types of loss functions to use when training neural network models.

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However, the maximum likelihood approach was adopted for several reasons, but primarily because of the results it produces. More specifically, neural networks for classification that use a sigmoid or softmax activation function in the output layer learn faster and more robustly using a cross-entropy loss function than using mean squared error. Cross-entropy between training data and model distribution i. Below is an example of a sigmoid output coupled with a mean squared error loss. Contrast the above with the below example using a sigmoid output and cross-entropy loss.

In the next section, we will tackle output units and discuss the relationship between the loss function and output units more explicitly.

We have already discussed output units in some detail in the section on activation functions, but it is good to make it explicit as this is an important point. It is relatively easy to forget to use the correct output function and spend hours troubleshooting an underperforming network. For multiclass classification, such as a dataset where we are trying to filter images into the categories of dogs, cats, and humans. This uses the multidimensional generalization of the sigmoid function, known as the softmax function.

There are also specific loss functions that should be used in each of these scenarios, which are compatible with the output type. For example, using MSE on binary data makes very little sense, and hence for binary data, we use the binary cross entropy loss function. Life gets a little more complicated when moving into more complex deep learning problems such as generative adversarial networks GANs or autoencoders, and I suggest looking at my articles on these subjects if you are interested in learning about these types of deep neural architectures.

A summary of the data types, distributions, output layers, and cost functions are given in the table below. In the final section, we will discuss how architectures can affect the ability of the network to approximate functions and look at some rules of thumb for developing high-performing neural architectures.

We will assume our neural network is using ReLU activation functions. A neural network with a single hidden layer gives us only one degree of freedom to play with. So we end up with a pretty poor approximation to the function — notice that this is just a ReLU function. Adding a second node in the hidden layer gives us another degree of freedom to play with, so now we have two degrees of freedom. Our approximation is now significantly improved compared to before, but it is still relatively poor.

Now we will try adding another node and see what happens. With a third hidden node, we add another degree of freedom and now our approximation is starting to look reminiscent of the required function. What happens if we add more nodes? Our neural network can approximate the function pretty well now, using just a single hidden layer.

What differences do we see if we use multiple hidden layers? This result looks similar to the situation where we had two nodes in a single hidden layer. However, note that the result is not exactly the same. What occurs if we add more nodes into both our hidden layers?

We see that the number of degrees of freedom has increased again, as we might have expected. However, notice that the number of degrees of freedom is smaller than with the single hidden layer. We will see that this trend continues with larger networks. Our neural network with 3 hidden layers and 3 nodes in each layer give a pretty good approximation of our function. Choosing architectures for neural networks is not an easy task. We want to select a network architecture that is large enough to approximate the function of interest, but not too large that it takes an excessive amount of time to train.

In general, it is good practice to use multiple hidden layers as well as multiple nodes within the hidden layers, as these seem to result in the best performance. It has been shown by Ian Goodfellow the creator of the generative adversarial network that increasing the number of layers of neural networks tends to improve overall test set accuracy. The same paper also showed that large, shallow networks tend to overfit more — which is one stimulus for using deep neural networks as opposed to shallow neural networks. Selecting hidden layers and nodes will be assessed in further detail in upcoming tutorials.

I hope that you now have a deeper knowledge of how neural networks are constructed and now better understand the different activation functions, loss functions, output units, and the influence of neural architecture on network performance. Future articles will look at code examples involving the optimization of deep neural networks, as well as some more advanced topics such as selecting appropriate optimizers, using dropout to prevent overfitting, random restarts, and network ensembles. The third article focusing on neural network optimization is now available:.

Important neural network articles:. Sign in. Get started. Intermediate Topics in Neural Networks. A detailed overview of neural architecture, activation functions, loss functions, output units. Simple Introduction to Neural Networks. A detailed overview of neural networks with a wealth of examples and simple imagery. To read more about this, I recommend checking out the original paper on arxiv: Searching for Activation Functions. The choice of activation functions in deep networks has a significant effect on the training dynamics and task….

Output Units We have already discussed output units in some detail in the section on activation functions, but it is good to make it explicit as this is an important point. Neural Network Optimization. Covering optimizers, momentum, adaptive learning rates, batch normalization, and more. Towards Data Science Sharing concepts, ideas, and codes. Towards Data Science Follow. Sharing concepts, ideas, and codes. See responses 1. Discover Medium.