The provided code appears to aim at implementing a neural...

August 18, 2025 at 08:39 AM

import numpy as np 

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

x = np.array([[3], [4], [5]])
w1_t = np.array([[11, 0, 0],
                 [12, 22, 0],
                 [0, 23, 33],
                 [0, 0, 34]])
b1 = np.array([[1], [1], [1]])
f_1 = np.dot(w1_t, x) + b1
y_1 = sigmoid(f_1)

w2_t = np.array([[22, 42, 0, 0],
                 [0, 44, 64, 0],
                 [0, 0, 66, 86]])
b2 = np.array([[2], [2], [2]])
f_2 = np.dot(w2_t, y_1) + b2
y_2 = sigmoid(f_2)

w3_t = np.array([[11, 21, 31]])
f_3 = np.dot(w3_t, y_2)
output = sigmoid(f_3)
print(output)

learning_rate = 0.1

def sigmoid_derivative(x):
    return x * (1 - x)

w1 = w1_t.copy()
b1 = b1
w2 = w2_t.copy()
b2 = b2
w3 = w3_t.copy()

num_iterations = 1500

for i in range(num_iterations):
    x = np.array([[3], [4], [5]])
    f_1 = np.dot(w1, x) + b1
    y_1 = sigmoid(f_1)
    f_2 = np.dot(w2, y_1) + b2
    y_2 = sigmoid(f_2)
    f_3 = np.dot(w3, y_2)
    output = sigmoid(f_3)
    
    d_output =np.dot((output - np.array([[1]])),sigmoid_derivative(output))
    
    d_w3 = np.dot(d_output, y_2.T)
    
    d_y_2 = np.dot((y_2 - b2) * sigmoid_derivative(y_2))
    
    d_w2 = np.dot(d_y_2, y_1.T)
    
    d_y_1 = np.dot((y_1 - b1) * sigmoid_derivative(y_1))
    
    d_w2 = np.dot(d_y_1, x)

The provided code appears to aim at implementing a neural network with three layers, including forward propagation and an attempt at backpropagation. However, the code has some issues, such as incorrect calculations and missing steps. Here's a detailed explanation of what each part of the code is intended to do and its current issues:

1. Definitions and Initial Forward Pass

- `import numpy as np`

Imports the NumPy library to handle numerical calculations.

- `def sigmoid(x):`

Defines the sigmoid activation function, which is used to introduce non-linearity into the neural network.
Problem: The implementation is incorrect:

    return 1/1 + np.exp(-x)

This will not work as intended because 1 / 1 + np.exp(-x) adds 1 to np.exp(-x), then divides by 1. The correct formula for the sigmoid function is:

    return 1 / (1 + np.exp(-x))

- Input, Weight Matrices, and Forward Propagation:

x is the input vector: [[3], [4], [5]]. It has 3 elements corresponding to the features or inputs.
w1_t, w2_t, and w3_t are the transposed weight matrices for layers 1, 2, and 3, respectively.
b1 and b2 are the bias terms for layer 1 and layer 2.

The forward propagation equations calculate:

Layer 1 output: (f_1 = w1_t \cdot x + b1), (y_1 = \text{sigmoid}(f_1))
Layer 2 output: (f_2 = w2_t \cdot y_1 + b2), (y_2 = \text{sigmoid}(f_2))
Layer 3 output (final output): (f_3 = w3_t \cdot y_2), (output = \text{sigmoid}(f_3))

The output represents the prediction of this three-layer neural network for the given input x.

Important Notes: The code performs matrix multiplications using np.dot.

2. Training Loop (Neural Network Training)

The code attempts to perform gradient-based optimization for 1500 iterations using the sigmoid derivative. However, there are multiple issues:

Sigmoid Derivative Function:
```
def sigmoid_derivative(x):
    return x * (1 - x)
```
This assumes x is the output of the sigmoid function. However, in backpropagation, we must carefully apply this derivative to intermediate layers.
Backpropagation:
- The code seems to attempt backpropagation by calculating gradients for the weight matrices w1, w2, and w3. However, the observed code has multiple errors:
  - The computation for d_output is incorrect:
```
d_output = np.dot((output - np.array([[1]])), sigmoid_derivative(output))
```
    This should probably be element-wise multiplication (*) with sigmoid_derivative(output), not a dot product.
  - d_y_2, d_y_1, and their gradients (e.g., d_w2, d_w1) are incorrectly calculated. In backpropagation, these gradients should be propagated properly layer by layer using the chain rule, but this is not correctly implemented.
Weights and Biases Update:
- The weight matrices (w1, w2, w3) are not updated at any point in the loop.
- To update weights, the gradients (d_w1, d_w2, d_w3) must be subtracted from the weights, scaled by a learning rate:
```
w3 -= learning_rate * d_w3
# Similar updates for w2 and w1
```
- Without weight updates, the model cannot learn.
Missing Bias Updates:
- Biases b1 and b2 are defined but never updated in the training loop.

3. Overall Function of the Code

Intended Purpose: The code appears to aim at implementing forward and backward propagation for a 3-layer fully connected neural network, using the sigmoid activation function at each layer and training the weights (w1, w2, w3) through gradient descent.
Issues:
1. The sigmoid function implementation has a bug.
2. Backpropagation is incorrectly computed, and weight updates are missing.
3. Biases are defined but never updated.
4. The code will likely not converge to a solution or train the model correctly due to these issues.
Output: Despite errors in the calculations, the forward pass will produce some numeric output based on the initial weights and biases. However, no meaningful learning occurs due to the flaws in the training loop.

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