Plans for making backward pass more configurable. Neural Network program.

Let's start by examining how the backward_pass function currently works, and then we'll discuss ways to make it more flexible and configurable.

Current Implementation:

1. The backward_pass function takes the neural network, input, expected output, and actual output as parameters.
2. It starts with the output layer and calculates the errors:
   • For softmax activation, it uses the difference between actual and expected outputs.
   • For other activations, it multiplies this difference by the derivative of the activation function.
3. It then iterates through the layers backwards:
   • For each layer, it calculates gradients for weights and biases.
   • It updates weights and biases using the chosen optimization method (via the chooser function).
   • For hidden layers, it calculates errors to be backpropagated to the previous layer.
4. The error calculation for hidden layers involves multiplying the current layer's errors by the weights and then by the activation function's derivative.

Now, let's discuss how we can make this more flexible and what options we can add:

1. Loss Functions: Currently, the code implicitly uses mean squared error (MSE) for non-softmax outputs. We can add support for different loss functions:
   • Cross-entropy loss
   • Huber loss
   • Custom loss functions
2. Output Layer Options: We can add support for different output layer types:
   • Sigmoid for binary classification
   • Linear for regression tasks
   • Custom output activations
3. Regularization: Implement various regularization techniques:
   • L1 regularization (Lasso)
   • L2 regularization (Ridge)
   • Elastic Net (combination of L1 and L2)
   • Dropout (during training)
4. Learning Rate Schedules: Implement different learning rate adjustment strategies:
   • Step decay
   • Exponential decay
   • Cosine annealing
5. Gradient Clipping: Add options for gradient clipping to prevent exploding gradients:
   • By value
   • By norm
6. Second-Order Optimization: Implement more advanced optimization techniques:
   • Newton's method
   • Conjugate gradient
   • L-BFGS
7. Layer-specific Learning Rates: Allow different learning rates for different layers or parameter types (weights vs. biases).
8. Custom Initialization: Provide options for different weight initialization methods:
   • Xavier/Glorot initialization
   • He initialization
   • Custom initialization functions
9. Batch Normalization: Implement batch normalization layers and their corresponding backward pass.
10. Skip Connections: Add support for residual connections or dense connections between layers.

To make these options configurable, we could:

1. Extend the NeuralNet struct to include configuration options for these features.
2. Create a separate Configuration struct that holds all these options.
3. Use function pointers for customizable components (e.g., loss functions, regularization methods).
4. Implement a builder pattern for creating and configuring the neural network.