Okay, we've compiled our model, defining how it should learn (optimizer), what it should minimize (loss function), and how we'll track progress (metrics). But how does the optimizer actually use the loss value to improve the network's predictions? This is where the concept of backpropagation comes into play.

Imagine your network has just made a prediction for a training example. You compare this prediction to the actual target label using the loss function, which calculates a single number representing the error or "badness" of the prediction. A high loss means the prediction was far off; a low loss means it was close. The goal of training is to minimize this loss across all training examples.

How do we adjust the network to reduce the loss? The network's behavior is determined by its weights and biases. If we could figure out how much a tiny change in each weight would affect the final loss, we could systematically adjust the weights in the direction that decreases the loss the most.

This is precisely what backpropagation enables. It's an algorithm that calculates the gradient of the loss function with respect to each weight and bias in the network. Remember from calculus that a gradient points in the direction of the steepest ascent of a function. Since we want to minimize the loss, we'll want to adjust the weights in the opposite direction of the gradient.

The Backward Pass: Assigning Blame

Think of backpropagation as a way of assigning responsibility for the error backward through the network, layer by layer.

Calculate Output Layer Error: The process starts at the output layer. We directly compare the network's output with the true target and calculate the loss. We can also determine how much the activations of the output neurons contributed to this final error.
Propagate Error to Previous Layer: Using the gradients calculated for the output layer, backpropagation determines how much the activations of the previous hidden layer contributed to the output layer's error. This involves understanding how the weights connecting these layers influenced the result.
Continue Backward: This process repeats, moving backward layer by layer. At each layer, we calculate how its neurons' activations contributed to the error calculated for the layer after it.
Calculate Weight Gradients: As we propagate the error information backward, we simultaneously calculate the gradient of the loss function with respect to each weight in each layer. This tells us the sensitivity of the loss to small changes in that specific weight. $\frac{\partial L}{\partial W_{ij}}$ essentially asks, "If I nudge this particular weight $W_{ij}$ slightly, how much does the total loss $L$ change, and in which direction?"

Mathematically, this backward flow of information relies heavily on the chain rule from calculus, allowing us to compute these gradients efficiently layer by layer, starting from the final loss.

The forward pass takes input data through the network to produce a prediction. The loss is calculated by comparing the prediction to the target. The backward pass then propagates the error gradient backward from the loss, calculating how changes in weights (W1, W2) and activations (H1, Output) affect the loss.

Using the Gradients: The Update Step

Once backpropagation has computed the gradients for all weights and biases, the optimizer (which you chose during the compile step, like Adam or SGD) takes over. It uses these gradients, along with its specific update rules (often incorporating factors like learning rate and momentum), to adjust the network's weights and biases. The goal is to "nudge" the weights in the direction that reduces the loss.

This entire cycle, forward pass, loss calculation, backward pass (backpropagation), and weight update, constitutes one step of the training process. It's repeated many times over batches of data across multiple epochs, gradually guiding the network's parameters towards values that minimize the overall loss function, making the network better at its task.

While Keras handles the implementation details of backpropagation automatically when you call the fit() method, understanding this conceptual flow is significant for diagnosing training issues and making informed decisions about model architecture, loss functions, and optimizers.