Implementations of artificial neural networks can be quite complex, and it is always a good idea to manually check that we have implemented backpropagation correctly. In this section, we will talk about a simple procedure called gradient checking, which is essentially a comparison between our analytical gradients in the network and numerical gradients. Gradient checking is not specific to feedforward neural networks but can be applied to any other neural network architecture that uses gradient-based optimization. Even if you are planning to implement more trivial algorithms using gradient-based optimization, such as linear regression, logistic regression, and support vector machines, it is generally not a bad idea to check if the gradients are computed correctly.
In the previous sections, we defined a cost function 
where 
is the matrix of the weight coefficients of an artificial network. Note that is—roughly speaking—a "stacked" matrix consisting...