Understanding Activation Functions in Neural Networks

Activation functions are really important in neural networks. They help decide how well the training goes, especially when using a method called gradient descent. Each activation function has its own strengths and weaknesses that can speed up or slow down how quickly the model learns. Choosing the right activation function is key. It not only affects how fast the training happens but also how well the model learns from the data.

Let’s take a look at some popular activation functions and see what they do for gradient descent.

1. Sigmoid Function

The sigmoid function looks like this:

$$\sigma(x) = \frac{1}{1 + e^{-x}}$$

This function turns any number into a value between 0 and 1. It was one of the first activation functions used, but it's not perfect, especially when it comes to gradient descent.

• Gradient Saturation: For very high or low numbers, the gradients (or changes) get really small. This means during the learning process, the updates to weights (the model's learning parameters) become tiny, causing the training to slow down, especially in deeper networks.

• Vanishing Gradient Problem: This is a big issue for networks with many layers. As the small changes move back through each layer, they can get so tiny that the earlier layers stop learning altogether.

2. Hyperbolic Tangent (tanh)

The hyperbolic tangent function is another commonly used activation function:

$$tanh(x) = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}$$

The $tanh$ function can output values between -1 and 1, which helps with centering the data. But it also has some of the same problems as the sigmoid.

• Gradient Saturation: Just like the sigmoid, $tanh$ can also face the issue of small gradients for extreme values, but to a lesser degree.

• Faster Convergence: Because $tanh$ outputs centered values, it usually helps the training process go faster compared to the sigmoid function.

3. ReLU (Rectified Linear Unit)

ReLU has become very popular and is defined as:

$$f(x) = \max(0, x)$$

It’s simple and quick to calculate, making it a favorite for many deep learning models.

• Sparsity: ReLU often makes the model more efficient by creating lots of zeros in the output (especially for negative inputs), which reduces unnecessary information.

• Preventing Vanishing Gradient: The gradients stay the same for positive inputs, helping earlier layers continue to learn without getting stuck like with sigmoid or $tanh$.

However, ReLU has a problem called the Dying ReLU Problem. Sometimes, neurons can become inactive and stop working if they keep getting negative inputs.

4. Leaky ReLU

Leaky ReLU is a way to fix the dying ReLU issue. It gives a slight slope for negative values:

$$f(x) = \begin{cases} x & \text{if } x > 0 \\ \alpha x & \text{if } x \leq 0 \end{cases}$$

Here, $\alpha$ is a small number (like 0.01). This helps keep the learning going even for negative inputs.

5. Softmax Function

The softmax function is useful when you have multiple classes to classify items. It turns the model's raw scores into probabilities:

$$\sigma(z_i) = \frac{e^{z_i}}{\sum_{j=1}^{K} e^{z_j}}$$

where $K$ is the number of classes. Softmax also makes sure the output is nicely balanced, which helps with training.

Conclusion

Choosing the right activation function is key to how well gradient descent works. ReLU and its variations generally perform better in deep networks because they help reduce the vanishing gradient problem and are easier to compute.

When creating neural networks, it’s important to consider the data type, the model's structure, and how deep it is. This helps in picking the best activation function, which can greatly affect training time and how well the model learns. Trying out various activation functions can lead to better results, which helps make deep learning systems more efficient.