In the previous article, we explored activation functions and visualized them using Python.

Now, let’s see what gradients are.

Neural networks use activation functions to transform inputs inside them.

But if a neural network gives a wrong output, how does it know what to fix?

This is where gradients come in.

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What is a gradient?

Imagine you are walking on a hill. If the ground is steep, you can feel which direction goes up or down.

If the ground is almost flat, it is hard to tell where to go.

A gradient can be simply thought of as a number that tells us how steep a curve is at a point.

How does this apply in the case of neural networks? Let’s see.

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Why gradients matter in neural networks

In neural networks:

• Gradients tell us how much a parameter should change
• The bigger the gradient, the bigger the update
• If the gradient is 0, then learning stops

When training a neural network:

1. We make a prediction
2. We calculate how wrong it is
3. We update weights to reduce the loss
4. This update depends entirely on gradients

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Gradients of activation functions

Each activation function has:

• A curve
• A gradient curve

Let’s check this in Python, starting first with ReLU.

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Gradient of ReLU

We can define the ReLU gradient as:

def relu_grad(x):
return np.where(x > 0, 1, 0)

This means:

• Gradient = 0 when input ≤ 0
• Gradient = 1 when input > 0

Let’s plot it:

plt.figure()
plt.plot(x, relu_grad(x), label="ReLU Gradient")
plt.title("Gradient of ReLU")
plt.xlabel("Input")
plt.ylabel("Gradient")
plt.grid(True)
plt.legend()
plt.show()

This is our gradient. From the above, you can observe the following:

• The entire negative side has zero gradient
• The positive side has a constant gradient of 1

From this, we can further understand that:

• ReLU learns very fast when active
• ReLU neurons can die if they always receive negative inputs

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Gradient of Softplus

def softplus_grad(x):
return 1 / (1 + np.exp(-x))

Let’s plot it:

plt.figure()
plt.plot(x, softplus_grad(x), label="Softplus Gradient")
plt.title("Gradient of Softplus")
plt.xlabel("Input")
plt.ylabel("Gradient")
plt.grid(True)
plt.legend()
plt.show()

You can observe that it is the same as the sigmoid activation function.

You can also observe that:

• There is a smooth transition
• Learning always continues
• This avoids dying neurons and adds stability, but it is slower than ReLU

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Gradient of Sigmoid

The sigmoid gradient looks like this:

def sigmoid_grad(x):
s = sigmoid(x)
return s * (1 - s)

Let’s plot it:

plt.figure()
plt.plot(x, sigmoid_grad(x), label="Sigmoid Gradient")
plt.title("Gradient of Sigmoid")
plt.xlabel("Input")
plt.ylabel("Gradient")
plt.grid(True)
plt.legend()
plt.show()

From the above, you can observe the following:

• The gradient is very small at both extremes
• It is strong only around the middle
• It is almost zero for large positive or negative values

This leads to a famous problem called the vanishing gradient problem. We will explore this more in the next article.

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You can try the examples out via the Colab notebook.

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