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Neural Nets Aren’t Black Boxes
If you think neural nets are black boxes, you’re certainly not alone. While they may not be as interpretable as something like a random forest (at least not yet), we can still understand how they process data to arrive at their predictions. In this post we’ll do just that as we build our own network from scratch, starting with logistic regression.
If you think neural nets are black boxes, you’re certainly not alone. While they may not be as interpretable as something like a random forest (at least not yet), we can still understand how they process data to arrive at their predictions. In this post we’ll do just that as we build our own network from scratch, starting with logistic regression.
This post is very much inspired by this fantastic post by Sylvain Gugger. We won’t pretend to improve upon Sylvain’s post; we just want to explain things in our own way to help us understand things a little better. This post is the first of a series of posts in which we’ll build our own DNN, CNN, and RNN in numpy. You can find all of the source code at tinytorch.
Objective
Our goal is to construct a binary logistic classifier as a neural network. The network will consist of a single linear layer followed by a sigmoid activation with binary cross entropy as the loss. We’ll begin by deriving the back-prop equations for our particular scenario and in doing so we’ll see that what we’ve done generalizes immediately to networks with arbitrary layers and activations. In other words, we’ll have developed a framework that can model any feedforward network — all by starting from ordinary logistic regression.
Actually, this isn’t all that surprising when you think about it. Logistic regression is a linear layer followed by sigmoid and feedforward networks are just a bunch of linear layers stacked together with non-linearities in between.
Back-propagation
Back-propagation is nothing more than the chain rule. We can view our logistic network as the composition of three functions
While the loss function is not usually viewed as a layer of the network, treating it as the final layer makes computing the gradients easier. Let’s denote the output of the i-th layer by xi so that
The first gradient we have to compute is the gradient of BCE with respect to the activations x2.
Next we have to compute the gradient with respect to the linear outputs x1. The chain rule tells us
Last, we’ll need to compute the gradient with respect to the original inputs x.
Notice a pattern? The first gradient we computed — the gradient with respect to the network’s final activations — is used to compute the next gradient — the gradient with respect to the linear outputs — which is in turn used to compute the gradient with respect to the original inputs. To compute the gradients of any network, we simply start at the last layer and successively pass the gradients backwards to the preceding layer until we arrive at the original inputs. That’s the reason it’s called back-propagation. It really is helpful to picture passing the gradients backwards through the network like a baton.
We’ll compute each of these gradients in turn, starting with the last layer and working our way backwards to the original inputs.
#deep-learning #data-science #neural-networks #machine-learning #python #deep learning
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