1. Introduction

In this tutorial, we’ll study the family of ReLU activation functions.

2. Activation Functions

Before we dive into ReLU, let us talk about the activation function and its role in a generic neural network.

2.1. Role in Neural Networks

We define an artificial neural network as a higher-order mathematical model that we compose from interconnected nodes or artificial neurons. It mimics the structure and function of the human brain. For example, we can use it for information processing to solve complex problems such as image classification:

Artificial neural networks

Moving on, our activation function is a mathematical function that a neuron executes. So, it calculates the output of a neuron as per its inputs and the associated weights on individual inputs.

It can be linear, such as unit function f(x)=\alpha x, or non-linear such as the sigmoid function f_ \theta (x) = \frac{\mathrm{1} }{\mathrm{1} + e^- \theta^Tx }. Additionally, we also call these functions as the transfer functions:artificial neuron

We typically use a non-linear activation function to solve complex real-world problems. This is so because most of these problems have complex relationships among their features.

2.2. Common Activation Functions

Next, we move to artificial neural networks’ most common activation functions.

Identity function f(x)=x is the most common linear activation. Its range is (-\infty, \infty):

Linear

Among non-linear functions, ReLU is the most commonly used activation function. ReLU stands for rectified linear unit. Here, f(x) = 0 when x < 0 and f(x) = x when x \ge 0. Further on, its range is (0, \infty):ReLU

The second most used non-linear activation function is the sigmoid function. We also call it the logistic activation function. To define, it takes input x and gives a value between (0, 1). Therefore, we use it for models where we have to predict the probability as an output:Sigmoid

Another widespread non-linear activation is the tanh or hyperbolic tangent activation function. It works in the same way as the sigmoid. Further, the range of the tanh function is from (-1, 1). It offers the advantage of mapping negative inputs to strictly negative output and mapping limiting zero inputs to near zero values in the tanh graph:

tanh

3. Limitations of Sigmoid and Tanh Activation Functions

While training the model, the sigmoid and tanh activation suffer vanishing and exploding gradient problems (especially in hidden layers).

3.1. Gradient Problems

The vanishing gradient problem happens during training, whenever our gradients used to update the network become extremely small (vanish) during backpropagation from the output layers to the earlier layers. Thus, the weights need to be updated more to discover the underlying patterns in the data.

The exploding gradient problem occurs when our gradients become extremely large during the training process. As a result, model parameter weights get a significant update. Thus, this makes our model unstable, and continuing this way, it fails to learn all intractable patterns from our training data.

3.2. Other Problems

Apart from the above problems, we also have a few more. Both these functions are not centered around zero and thus are non-symmetric around the origin. Further, they are computationally expensive and have shallow slopes. Hence, they have low differentiability near the asymptotes, and thus, our training process takes longer.

4. ReLU Activation

In this section, we explore the ReLU function.

4.1. Basic ReLU

The basic version of Relu behaves as a unit function for positive inputs and 0 for negative inputs. Further, we find ReLU faster than sigmoid and tanh functions. Further, it doesn’t suffer from vanishing or exploding backpropagation errors.

However, it suffers from a dying-relu problem. ReLU makes all the negative input values zero immediately in the model graph. This makes the ReLU neuron inactive. We term this condition as the dead state of the ReLU neuron. It is challenging to recover in this state because the gradient 0 is 0. This problem is elevated when most of the training inputs are harmful, or the derivative of the ReLU function is 0.

We use Leaky ReLU to overcome this problem.

4.2. Leaky ReLU

In this version, we use a slight non-negative gradient when the input is negative:Leaky ReLU

It allows small and non-zero gradients when the unit is inactive. This way, we overcame the dying real problem. However, it fails to give consistent predictions for negative inputs. It uses a slope \alpha as a constant parameter throughout the training process. As the training values become more negative, their output becomes more and more linear. This way, we lose non-linearity at the cost of better gradient backpropagation.

4.3. Parametric ReLU

Parametric ReLU or PReLU uses \alpha as a hyperparameter that the model learns during the training process:PReLU

We use the PReLU activation function to overcome the shortcomings of ReLU and LeakyReLU activation functions. PReLU offers an increase in the accuracy of the model. Further, it gives us faster model convergence compared to LReLU and ReLU. However, it has one disadvantage. We must manually modify the parameter \alpha by trial and error. As you can see, this is a very time-consuming exercise, especially when our dataset is diverse.

4.4. GeLU

GeLU or Gaussian Error Linear Unit is a smooth and differentiable approximation of the rectifier function. So, for every input x, it outputs the product of x with its Gaussian cumulative distribution function \Phi(x). Going deeper, we find that the GeLU activation function scales inputs by their percentile, whereas the ReLU family of activation functions scales the inputs by their sign:
GeLU

5. ReLU vs. Leaky ReLU vs. Parametric ReLU

The following table summarizes the key differences between vanilla Relu and its two variants.

Property ReLU LReLU PReLU
Advantage Solves gradient problems Solves gradient problems Solves gradient problems
Disadvantage Dying relu problem Inconsistent output for negative input Fine-tune \alpha
Hyperparameter None None 1
Speed Fastest Faster Fast
Accuracy High Higher Highest
Convergence Slow Fast Fastest

6. Conclusion

In this article, we’ve gone through activation functions in an artificial neural network. After that, we delved into ReLU and its variants. ReLU is a simple yet powerful activation function that allows the neural network to learn intractable dependencies by introducing non-linearity. Moreover, its variants solve the gradient problems and give consistent output for negative input values.