**Building your Deep Neural Network: Step by Step an Intuitive approach**

**Goal of the Code:** The goal is to set up a neural network that will take the input features and make predictions on the best movies to recommend. The process involves setting up the initial settings (parameters) for the recommendation system, running a series of transformations (forward propagation), evaluating the outcome (computing cost), and refining the settings (backward propagation and parameter updates).

### Breakdown:

**Setting Up the Recommendation System (initialize_parameters)**Imagine we’re setting up our recommendation system with the right parameters. The goal is to initialize the weights and biases (parameters) for the neural network. For example, we decide on the initial importance of each feature like genre preference, director preference, actor preference, average rating of previously watched movies, and movie duration preference, and set them to small random values. We also prepare the biases, which start as zeros.**Forward Propagation (L_model_forward)**This step is like processing user preferences step by step to get a movie recommendation. The goal is to process the input data through each layer of the network, applying transformations (weights and biases) to produce an output. We start with the initial parameters and pass them through a series of steps (layers) where each step refines the input (user preferences) until we get a prediction on which movie to recommend.**Computing Cost (compute_cost)**This step is like checking how good the movie recommendation is. The goal is to measure how good or bad the prediction is compared to the actual result. We compare the predicted movie rating with the actual rating given by the user and see if they match. The cost function gives us a score indicating how close our prediction is to the actual rating.**Backward Propagation (L_model_backward)**This is like adjusting the recommendation parameters based on feedback. The goal is to figure out how to change the weights and biases to improve the predictions. For example, if a certain genre was over-recommended, we reduce its weight for the next batch of recommendations. This involves calculating gradients, which tell us how to tweak each parameter.**Updating Parameters (update_parameters)**This step involves actually making the changes to the recommendation parameters for the next round. The goal is to update the weights and biases based on the gradients calculated in the backward propagation step. We adjust the importance of each feature and the biases according to the feedback we received.

### Putting It All Together:

Think of the neural network as our smart recommendation system that learns and improves its movie recommendation process over time. Each part of the code helps refine its parameters to produce better recommendations. Here’s a simplified flow:

**Initialization:**Start with small random values for weights and zero for biases.**Forward Propagation:**Process user preferences through the current parameters to predict movie ratings.**Compute Cost:**Check the predicted ratings against actual user ratings and score the accuracy.**Backward Propagation:**Determine how to adjust the parameters based on the cost.**Update Parameters:**Make the necessary adjustments to the parameters for the next round.

By repeatedly going through these steps, the recommendation system becomes smarter and more efficient at predicting and recommending the best movies.

### Part 1: Initializing Parameters

**Intuitive Explanation:** Imagine setting up the recommendation system and you need to prepare your initial parameters before starting to make recommendations. In this context:

**Weights (W1, W2):**These are like the importance values you start with for each feature. You don’t want them to be too large initially, so you use small random values to begin with.**Biases (b1, b2):**These are like your adjustments that start at zero.

By setting up these initial parameters, you are ready to start the recommendation process, which involves forward propagation (processing preferences), computing cost (checking accuracy), backward propagation (adjusting parameters), and updating parameters until you get the best recommendations.

**Example:** Let’s say we have:

- 5 input features (e.g., genre preference, director preference, actor preference, average rating of previously watched movies, movie duration preference)
- 2 neurons in the hidden layer
- 1 output (predicting the movie rating)

When we call `initialize_parameters(5, 2, 1)`

, it would create:

**W1**with shape (2, 5)**b1**with shape (2, 1)**W2**with shape (1, 2)**b2**with shape (1, 1)

These parameters are the starting point for our neural network.

### Part 2: Forward Propagation

**Intuitive Explanation:** Imagine you are processing user preferences step-by-step according to a system:

**Linear Forward (Mixing Preferences):**You mix the preferences (input features) with specific weights and add biases. For each step, you combine genre preference, director preference, actor preference, average rating of previously watched movies, and movie duration preference in specific proportions (weights) and add some adjustments (biases).**Linear Activation Forward (Processing Reaction):**After mixing, the preferences go through a reaction (activation function). If you’re processing the preferences (ReLU activation), the system decides which features are more significant. If you’re making a final decision on the movie rating (linear activation), it gives a score.**Model Forward (Complete Process):**Follow the entire process from start to finish, with each step transforming the preferences closer to the final rating. You start with the preferences (input layer), process through the hidden layers, and finally decide the movie rating (output layer).

**Detailed Example:**

**Linear Forward:**

**Input A (previous activations or input data).****Weight W and bias b for the current layer.****Compute Z = W * A + b.**

**Linear Activation Forward:**

- Compute Z using linear_forward.
- Apply activation function (ReLU or linear) to Z to get A.

**Model Forward:**

- Initialize with input features.
- Iterate through layers:
- For each hidden layer, use ReLU activation.
- For the final layer, use a linear activation.

- Store activations and intermediates in caches for backpropagation.

By combining these steps, the neural network processes the input features through each layer, transforming them and making predictions at the output layer.

### Part 3: Computing Cost

**Intuitive Explanation:** Imagine you are recommending different movies and trying to predict the rating a user would give based on some input features (like genre preference, director preference, etc.). You make some predictions and then compare them with the actual ratings given by the user.

**Predictions (AL):**These are like your guesses on which movie rating would be given. For instance, you might predict a 4.5 out of 5 for a particular movie.**Actual Outcomes (Y):**These are the actual ratings the user gave. For example, if the user actually rated the movie 5 out of 5, the label would be 5.**Cost Function (Rating Comparison):**The cost function is like a comparison score that tells you how well your predictions matched the actual ratings. If your guess was 4.5 and the user rated it 5, your score would be close. If your guess was 2 and the user rated it 5, your score would be poor. The cost function aggregates these scores across all your predictions to give you an overall sense of how well you’re doing.

**Detailed Example:** Suppose you have 3 examples (predictions):

**Predictions (AL):**[4.5, 3.0, 4.8]**Actual outcomes (Y):**[5, 3, 5]

Using the mean squared error (MSE) cost formula:

- For the first example: (4.5 - 5)²
- For the second example: (3.0 - 3)²
- For the third example: (4.8 - 5)²

The total cost is the average of these values: cost=13((4.5−5)2+(3.0−3)2+(4.8−5)2)\text{cost} = \frac{1}{3} \left( (4.5 - 5)² + (3.0 - 3)² + (4.8 - 5)² \right)cost=31((4.5−5)2+(3.0−3)2+(4.8−5)2)

This cost value quantifies how good or bad your predictions are. Lower cost values indicate better predictions.

### Part 4: Backward Propagation

**Intuitive Explanation:** Imagine you’re in the recommendation system, and you need to adjust your parameters based on user feedback to improve the recommendation quality.

**Linear Backward (Adjusting Weights and Biases):**You determine how much each weight and bias contributed to the prediction error. For example, if a certain genre was over-recommended, you calculate how much of the genre weight and other features need to be adjusted.**Linear Activation Backward (Adjusting Activations):**After adjusting weights and biases, you evaluate how the reactions (activations) in each layer of the network need to change. If a certain feature’s importance wasn’t calculated well (ReLU), you figure out how the combination of features (linear) needs to be changed to improve it.**Model Backward (Full Process Adjustment):**You go through the entire process, starting from the final rating (output layer) back to the initial preferences (input layer), adjusting each step. For the final rating (linear), you see how the prediction can be improved. For each step before (hidden layers), you adjust based on how they contributed to the final rating.

**Detailed Example:**

**Linear Backward:**

- Given the gradient of the cost with respect to Z (dZ), calculate:
- dW=1m⋅np.dot(dZ,AprevT)dW = \frac{1}{m} \cdot \text{np.dot}(dZ, A_{\text{prev}}^T)dW=m1⋅np.dot(dZ,AprevT)
- db=1m⋅np.sum(dZ,axis=1,keepdims=True)db = \frac{1}{m} \cdot \text{np.sum}(dZ, \text{axis}=1, \text{keepdims=True})db=m1⋅np.sum(dZ,axis=1,keepdims=True)
- dAprev=np.dot(WT,dZ)dA_{\text{prev}} = \text{np.dot}(W^T, dZ)dAprev=np.dot(WT,dZ)

**Linear Activation Backward:**

- Given the gradient of the cost with respect to the activation (dA):
- Calculate dZ using relu_backward or linear_backward.
- Use linear_backward to get dW, db, and dA_prev.

**Model Backward:**

- Initialize the gradient of the cost with respect to the output layer’s activation (dAL).
- For the output layer:
- Calculate gradients using linear activation.

- For each hidden layer (from L-1 to 1):
- Calculate gradients using ReLU activation.

- Store all gradients in the grads dictionary.

By performing these steps, the neural network learns how to adjust its parameters to improve predictions and reduce the cost.

### Part 5: Updating Parameters

**Intuitive Explanation:** Imagine you’re in the recommendation system, and after adjusting the parameters based on user feedback (backward propagation), you now need to make the actual changes to the weights and biases for the next round of recommendations.

**Parameters (Weights and Biases):**These are like the quantities of importance for each feature. For example, you have a certain weight for genre preference and a bias for the overall recommendation system.**Gradients (Feedback Adjustments):**These are like the feedback adjustments you calculated to improve the recommendation quality. For example, you figured out you need a little less importance on genre and more on director preference.**Learning Rate (Adjustment Intensity):**This is like how strongly you apply the feedback adjustments. If your learning rate is high, you make big changes to the weights and biases. If it’s low, you make smaller, more gradual adjustments.**Updating Parameters (Parameter Changes):**You use the feedback to update your parameters for the next round of recommendations. You subtract a bit of importance from the genre weight and add a bit to the director preference based on the learning rate and the feedback.

**Detailed Example:** Let’s say we have the following parameters and gradients for a single layer:

**W = 0.5****b = 0.1****dW = -0.2****db = 0.05**- Learning rate (α) = 0.01

The update rules are:

**W := W - α * dW****b := b - α * db**

Applying the update:

**W := 0.5 - 0.01 * (-0.2) = 0.5 + 0.002 = 0.502****b := 0.1 - 0.01 * 0.05 = 0.1 - 0.0005 = 0.0995**

The parameters are updated slightly, and this process is repeated for each training iteration to gradually improve the network’s performance.

### Full Process Overview:

**Initialize Parameters:**Set up the initial weights and biases.**Forward Propagation:**Calculate predictions based on the current parameters.**Compute Cost:**Evaluate the accuracy of predictions.**Backward Propagation:**Calculate gradients to adjust the parameters.**Update Parameters:**Apply gradients to update the parameters.

By iterating through these steps, the neural network learns to make better predictions and reduces the cost.