The first step of backpropagation in a regularized neural network is to calculate the gradients of the overall objective function $$J = L + s$$ with respect to its individual components: the single-example loss term $$L$$ and the regularization term $$s$$. Since the objective function is a simple sum of these two scalar values, its partial derivative with respect to each term is exactly $$1$$:
$$ \frac{\partial J}{\partial L} = 1 \; \textrm{and} \; \frac{\partial J}{\partial s} = 1 $$

Gradient of Objective Function with Respect to Loss and Regularization Term

The ultimate target for optimization during neural network training is the objective function, denoted as $$J$$. This function combines the network's predictive error with a complexity penalty. For a given data example, it is the sum of the unregularized loss term $$L$$ and the $$\ell_2$$ regularization term $$s$$:

$$J = L + s$$

Minimizing this regularized objective function balances fitting the training data accurately while maintaining small parameter weights to improve overall generalization.

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To evaluate the performance of a neural network on an individual data point, a loss function $$l$$ is applied to its predictions. Given the network's final output layer variable $$\mathbf{o}$$ and the true target label $$y$$, the unregularized loss term $$L$$ for a single example is formulated as:

$$L = l(\mathbf{o}, y)$$

This scalar value quantifies the discrepancy between the model's output and the actual ground truth for that specific data example before any complexity penalties are added.

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