So, I was reading this post Implementing a Neural Network from Scratch in Python and I thought I should break the main function being used to generate the neural net, just so I have a clear understanding of whats going on.
here's his code:
# This function learns parameters for the neural network and returns the model.
#  nn_hdim: Number of nodes in the hidden layer
#  num_passes: Number of passes through the training data for gradient descent > number of training iterations
#  print_loss: If True, print the loss every 1000 iterations
def build_model(nn_hdim, num_passes=20000, print_loss=False):
# Initialize the parameters to random values. We need to learn these.
np.random.seed(0)
W1 = np.random.randn(nn_input_dim, nn_hdim) / np.sqrt(nn_input_dim)
b1 = np.zeros((1, nn_hdim))
W2 = np.random.randn(nn_hdim, nn_output_dim) / np.sqrt(nn_hdim)
b2 = np.zeros((1, nn_output_dim))
# This is what we return at the end
model = {}
# Gradient descent. For each batch...
for i in range(0, num_passes):
# Forward propagation
z1 = X.dot(W1) + b1
a1 = np.tanh(z1)
z2 = a1.dot(W2) + b2
exp_scores = np.exp(z2)
probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)
# Backpropagation
delta3 = probs
delta3[range(num_examples), y] = 1
dW2 = (a1.T).dot(delta3)
db2 = np.sum(delta3, axis=0, keepdims=True)
delta2 = delta3.dot(W2.T) * (1  np.power(a1, 2))
dW1 = np.dot(X.T, delta2)
db1 = np.sum(delta2, axis=0)
# Add regularization terms (b1 and b2 don't have regularization terms)
dW2 += reg_lambda * W2
dW1 += reg_lambda * W1
# Gradient descent parameter update
W1 += epsilon * dW1
b1 += epsilon * db1
W2 += epsilon * dW2
b2 += epsilon * db2
# Assign new parameters to the model
model = { 'W1': W1, 'b1': b1, 'W2': W2, 'b2': b2}
# Optionally print the loss.
# This is expensive because it uses the whole dataset, so we don't want to do it too often.
if print_loss and i % 1000 == 0:
print(f"Loss after iteration %i: %f" %(i, calculate_loss(model)))
return model

let's break it down
# Initialize the parameters to random values. We need to learn these. np.random.seed(0) W1 = np.random.randn(nn_input_dim, nn_hdim) / np.sqrt(nn_input_dim) b1 = np.zeros((1, nn_hdim)) W2 = np.random.randn(nn_hdim, nn_output_dim) / np.sqrt(nn_hdiW1 = np.random.randn(nn_input_dim, nn_hdim) / np.sqrt(nn_input_dim)m) b2 = np.zeros((1, nn_output_dim))

the neural network's parameters are initialized randomly.
W1
andW2
are the weight matrices of the first and second layers, respectively. They are initialized with random values drawn from a normal distribution with a standard deviation scaled by the square root of the input and hidden layer dimensions.b1
andb2
are the bias vectors for the hidden and output layers and are initialized as zero arrays. 
This specific weight initialization method is known as "Xavier" or "Glorot" initialization, and it is designed to improve the convergence and training of neural networks.

In a neural network, the magnitude of the inputs to a neuron can vary significantly. If the weights are initialized with large values, it could lead to exploding gradients during training, making it difficult for the network to learn. Conversely, if weights are initialized with very small values, it can result in vanishing gradients, impeding learning.

The Xavier initialization assumes that the activation functions used in the network are symmetric, such as the hyperbolic tangent (tanh) or the sigmoid function. These functions have zero mean and operate around the origin. The Xavier initialization ensures that the initial weights are centered around zero to align with the activation functions' operating ranges.

In the expression
np.random.randn(nn_input_dim, nn_hdim) / np.sqrt(nn_input_dim)
, the numeratornp.random.randn(nn_input_dim, nn_hdim)
generates random values from a standard normal distribution (mean=0, variance=1). Dividing bynp.sqrt(nn_input_dim)
scales the values down by the square root of the input dimension, which is an approximation of the expected variance of the inputs. This scaling helps maintain a balance between the magnitude of inputs and the magnitudes of weights.
By dividing the randomly generated values by the square root of the input dimension, the values are scaled down. This scaling has a couple of effects:

Magnitude Balance: It ensures that the initial weights are not too large, which can lead to large activations that slow down learning (exploding gradients). It also prevents the initial weights from being too small, which can lead to vanishing gradients and slow convergence.

Signal Propagation: By scaling the weights down, the forward and backward propagations of signals in the network are more balanced. This helps maintain a consistent scale for the signal as it passes through the layers, which can aid in training convergence.



# Forward propagation z1 = X.dot(W1) + b1 a1 = np.tanh(z1) z2 = a1.dot(W2) + b2 exp_scores = np.exp(z2) probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)

Input Layer to Hidden Layer (z1 and a1):

z1 = X.dot(W1) + b1
: Here,X
represents the input data (a matrix where each row is a sample and each column is a feature),W1
is the weight matrix connecting the input layer to the hidden layer, andb1
is the bias vector for the hidden layer. This line calculates the preactivation values for the hidden layer. 
X
represents the input data, where each row is a sample and each column is a feature. For example, if you have 100 samples with 10 features each,X
would have dimensions(100, 10)
. 
W1
is the weight matrix that connects the input layer to the hidden layer. It has dimensions(input_dim, hidden_dim)
, whereinput_dim
is the number of input features andhidden_dim
is the number of nodes in the hidden layer. 
b1
is the bias vector for the hidden layer. It has dimensions(1, hidden_dim)
and is added to the result of the multiplication to ensure that each neuron in the hidden layer has a certain level of activation even if all the inputs are zero. 
Multiplying
X
withW1
and addingb1
effectively applies a linear transformation to the input data. This linear transformation helps the neural network learn to extract relevant features from the input data.


a1 = np.tanh(z1)
: The preactivation values fromz1
are passed through the hyperbolic tangent (tanh) activation function. This introduces nonlinearity to the model and produces the activation values (a1
) for the hidden layer. These activation values represent the hidden layer's output and capture the complex relationships between the input data and the learned parameters.
z1
is the result of the linear transformation (multiplication with weights and addition of bias) applied to the input dataX
. It represents the preactivation values of the neurons in the hidden layer. 
np.tanh(z1)
computes the hyperbolic tangent ofz1
for each element in the matrix. The hyperbolic tangent is a nonlinear activation function that squashes its input values between 1 and 1. 
The reason for using the tanh activation is to introduce nonlinearity to the neural network. Without a nonlinear activation function, the composition of multiple linear transformations would still result in a linear transformation. Nonlinearity is crucial for a neural network to capture complex relationships in the data. The tanh activation function has the desirable property of being symmetric around the origin (output values are centered around 0), which makes it suitable for maintaining a balanced signal in the network during training.


Hidden Layer to Output Layer (z2 and exp_scores):

z2 = a1.dot(W2) + b2
: The activation values from the hidden layer (a1
) are multiplied by the weightsW2
connecting the hidden layer to the output layer, and the biasb2
is added. This calculation gives the preactivation values for the output layer. 
exp_scores = np.exp(z2)
: The preactivation values fromz2
are exponentiated elementwise using the exponential function. These exponentiated values represent the "unnormalized" scores for each class in the output layer. In other words, they reflect the network's raw prediction strength for each class. 
a1
represents the output of the hidden layer, which is obtained after applying the tanh activation function to the preactivation valuesz1
(resulting from the input data being transformed by weights and bias). 
W2
is the weight matrix connecting the hidden layer to the output layer. It has dimensions(hidden_dim, output_dim)
, wherehidden_dim
is the number of nodes in the hidden layer, andoutput_dim
is the number of classes in the classification problem. 
b2
is the bias vector for the output layer, with dimensions(1, output_dim)
. 
Multiplying
a1
withW2
and addingb2
performs another linear transformation. This is similar to what was done in the hidden layer, but this time the goal is to map the hidden layer's output to the final output layer's input. This transformation learns how the hidden layer's features are associated with the final class probabilities.


Softmax Function and Class Probabilities (probs):

probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)
: The exponentiated scores are normalized to obtain class probabilities. The softmax function is used for this normalization. Each unnormalized score is divided by the sum of all unnormalized scores for a particular sample (axis=1). This results in a set of probabilities that sum up to 1 for each sample. 
The calculated probabilities (
probs
) represent the model's prediction for the probability of each class given the input data. The class with the highest probability is considered the predicted class for a given input sample. 
z2
represents the preactivation values of the output layer, which are obtained after the linear transformation in the previous step. 
np.exp(z2)
calculates the exponentiation of each value inz2
. Exponentiating the values is a crucial step, as it converts the raw preactivation values into a form that can represent probabilities. 
The reason for exponentiating the preactivation values is to ensure that they are positive. This is important when interpreting them as probabilities.

np.sum(exp_scores, axis=1, keepdims=True)
calculates the sum of the exponentiated scores along the rows (axis 1). This sum is performed to normalize the scores and transform them into probabilities. Thekeepdims=True
argument ensures that the resulting sum retains the same shape as the original matrix, which is necessary for division. 
probs = exp_scores / np.sum(exp_scores, axis=1, keepdims=True)
divides each exponentiated score by the sum of all exponentiated scores for the corresponding sample. This division normalizes the scores to create a set of probabilities for each class. 
The reason for using the softmax function and these operations is to convert the preactivation values into a probability distribution over the different classes. The softmax function ensures that the resulting probabilities are positive, sum to 1, and represent the model's confidence in its predictions for each class. This allows the neural network to make predictions about the most likely class for a given input sample.


The reason for performing these calculations lies in creating a feedforward process that transforms the input data into meaningful predictions. The transformation involves a sequence of linear and nonlinear operations that allow the neural network to capture complex relationships within the data. The use of activation functions like the hyperbolic tangent and the softmax ensures that the network can model both linear and nonlinear patterns in the data, enabling it to learn and generalize from the training examples to make accurate predictions on new, unseen data.


# Backpropagation delta3 = probs delta3[range(num_examples), y] = 1 dW2 = (a1.T).dot(delta3) db2 = np.sum(delta3, axis=0, keepdims=True) delta2 = delta3.dot(W2.T) * (1  np.power(a1, 2)) dW1 = np.dot(X.T, delta2) db1 = np.sum(delta2, axis=0)

Calculating
delta3
 Gradient of Loss with Respect to Output Layer's PreActivation (delta3 = probs):probs
contains the predicted probabilities for each class obtained from the forward propagation step.delta3
is used to represent the gradient of the loss with respect to the output layer's preactivation values. This gradient indicates how changes in the output layer's preactivation values affect the loss function.

Adjusting
delta3
for Prediction Error (delta3[range(num_examples), y] = 1):delta3
is adjusted based on the difference between the predicted probabilities (probs
) and the true labelsy
. This step represents the calculation of the gradient of the loss function with respect to the output layer's preactivation values, taking into account the prediction error. By subtracting 1 from the appropriate positions in
delta3
, the gradient is calculated to reflect the difference between predicted probabilities and the true class for each sample.

Calculating
dW2
anddb2
 Gradients of Loss with Respect to Output Layer's Weights and Bias:
dW2
represents the gradient of the loss with respect to the weightsW2
connecting the hidden layer to the output layer. 
db2
represents the gradient of the loss with respect to the biasb2
of the output layer. 
These gradients are calculated using the chain rule and are derived from
delta3
. The gradients indicate how changes in the weights and bias of the output layer affect the loss function. 
The matrix multiplication between the transposed hidden layer output
a1.T
and thedelta3
gradient results in a matrix whose elements represent how each weight inW2
affects the overall loss. This is essentially a weighted sum of the contributions of each weight to the loss. 
This operation effectively calculates the gradient of the loss with respect to the weights
W2
. Each entry in the resulting matrix tells us how a specific weight connecting the hidden layer to the output layer contributes to the loss. This information is used to update the weights during training through gradient descent.


Calculating
delta2
 Gradient of Loss with Respect to Hidden Layer's PreActivation:
delta2
represents the gradient of the loss with respect to the hidden layer's preactivation values. 
It is calculated by propagating the gradient
delta3
backward through the weightsW2
. This step involves the chain rule and takes into account how changes in the output layer's preactivation values affect the hidden layer's preactivation values. 
delta3
 Gradient of Loss with Respect to Output Layer's PreActivation: 
delta3
represents the gradient of the loss function with respect to the preactivation values of the output layer. It was calculated in the previous steps of backpropagation and indicates how changes in the output layer's preactivation values affect the overall loss. 
W2.T
 Transposed Weights Matrix: 
W2
is the weight matrix that connects the hidden layer to the output layer. Taking the transposeW2.T
is necessary because we want to propagate the gradient backwards from the output layer to the hidden layer. 
Matrix Multiplication
delta3.dot(W2.T)
 Backpropagating the Gradient: 
This matrix multiplication calculates the gradient of the loss with respect to the hidden layer's preactivation values. It tells us how changes in the output layer's preactivation values affect the hidden layer's preactivation values.

(1  np.power(a1, 2))
 Gradient of Tanh Activation Function: 
The derivative of the hyperbolic tangent (tanh) activation function is given by
1  np.power(a1, 2)
. This derivative indicates how a small change in the hidden layer's preactivation value would affect the hidden layer's output (thea1
values). 
Multiplying the backpropagated gradient by this derivative is a crucial step in calculating the gradient of the loss with respect to the hidden layer's preactivation values. It ensures that the gradient accounts for the nonlinearity introduced by the tanh activation function.

delta2
 Gradient of Loss with Respect to Hidden Layer's PreActivation: 
delta2
represents the gradient of the loss function with respect to the hidden layer's preactivation values. It's calculated by combining the information from the output layer's gradient and the effect of the hidden layer's activation function.


Calculating
dW1
anddb1
 Gradients of Loss with Respect to Hidden Layer's Weights and Bias:
dW1
represents the gradient of the loss with respect to the weightsW1
connecting the input layer to the hidden layer. 
db1
represents the gradient of the loss with respect to the biasb1
of the hidden layer. 
These gradients are calculated using the chain rule and are derived from
delta2
. They indicate how changes in the weights and bias of the hidden layer affect the loss function.



# Add regularization terms (b1 and b2 don't have regularization terms) dW2 += reg_lambda * W2 dW1 += reg_lambda * W1
 Regularization is a technique used to prevent overfitting, which occurs when a model learns to fit the training data too closely and doesn't generalize well to new, unseen data.

# Gradient descent parameter update W1 += epsilon * dW1 b1 += epsilon * db1 W2 += epsilon * dW2 b2 += epsilon * db2
 The necessity of this step lies in the goal of training a neural network: to find the set of parameters that result in the lowest possible value of the loss function. Gradient descent is an iterative process where the parameters are adjusted step by step to reach this optimal set of values. By updating the parameters in the direction that reduces the loss (opposite to the gradient), the network gradually improves its performance on the training data.
