Machine learning fundamentals (II): Neural networks

In my previous post I outlined how machine learning works by demonstrating the central role that cost functions and gradient descent play in the learning process. This post builds on these concepts by exploring how neural networks and deep learning work. This post is light on explanation and heavy on code. The reason for this is that I cannot think of any way to elucidate the internal workings of a neural network more clearly that the incredible videos put together by three blue one brown -- see the full playlist here. These videos show how neural networks can be fed raw data -- such as images of digits -- and can output labels for these images with amazing accuracy.