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This has the desired effect:. The only difference between the case of a single gate and multiple interacting gates that compute arbitrarily complex expressions is this additional multipy operation that now happens in each gate. Lets look again at our example circuit with the numbers filled in. The first circuit shows the raw values, and the second circuit shows the gradients that flow back to the inputs as discussed. This is the default pull on the circuit to have its value increased. After a while you start to notice patterns in how the gradients flow backward in the circuits.
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Similar intuitions apply to, for example, a max x,y gate. Numerical Gradient Check. Before we finish with this section, lets just make sure that the analytic gradient we computed by backprop above is correct as a sanity check. Remember that we can do this simply by computing the numerical gradient and making sure that we get [-4, -4, 3] for x,y,z.
In the previous section you hopefully got the basic intuition behind backpropagation. Lets now look at an even more complicated and borderline practical example. We will consider a 2-dimensional neuron that computes the following function:. Sigmoid function is defined as:.
The gradient with respect to its single input, as you can check on Wikipedia or derive yourself if you know some calculus is given by this expression:. Another thing to note is that technically, the sigmoid function is made up of an entire series of gates in a line that compute more atomic functions: an exponentiation gate, an addition gate and a division gate.
Chapter 2: Machine Learning
Treating it so would work perfectly fine but for this example I chose to collapse all of these gates into a single gate that just computes sigmoid in one shot, because the gradient expression turns out to be simple. Lets take this opportunity to carefully structure the associated code in a nice and modular way. Lets create a simple Unit structure that will store these two values on every wire. Our gates will now operate over Unit s: they will take them as inputs and create them as outputs. Lets start out by implementing a multiply gate.
Just think about these as class methods. Also keep in mind that the way we will use these eventually is that we will first forward all the gates one by one, and then backward all the gates in reverse order. Here is the implementation:. The multiply gate takes two units that each hold a value and creates a unit that stores its output. The gradient is initialized to zero.
Hacker's guide to Neural Networks
Then notice that in the backward function call we get the gradient from the output unit we produced during the forward pass which will by now hopefully have its gradient filled in and multiply it with the local gradient for this gate chain rule! This gate computes multiplication u0. This will allow us to possibly use the output of one gate multiple times think of it as a wire branching out , since it turns out that the gradients from these different branches just add up when computing the final gradient with respect to the circuit output.
The other two gates are defined analogously:.
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Note that, again, the backward function in all cases just computes the local derivative with respect to its input and then multiplies on the gradient from the unit above i. To fully specify everything lets finally write out the forward and backward flow for our 2-dimensional neuron with some example values:. And now lets compute the gradient: Simply iterate in reverse order and call the backward function!
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Remember that we stored the pointers to the units when we did the forward pass, so every gate has access to its inputs and also the output unit it previously produced. Note that the first line sets the gradient at the output very last unit to be 1. In other words, we are pulling on the entire circuit to induce the forces that will increase the output value. If we did not set this to 1, all gradients would be computed as zero due to the multiplications in the chain rule.
Finally, lets make the inputs respond to the computed gradients and check that the function increased:. Finally, lets verify that we implemented the backpropagation correctly by checking the numerical gradient:. Indeed, these all give the same values as the backpropagated gradients [ I hope it is clear that even though we only looked at an example of a single neuron, the code I gave above generalizes in a very straight-forward way to compute gradients of arbitrary expressions including very deep expressions foreshadowing. All you have to do is write small gates that compute local, simple derivatives w.
Over time you will become much more efficient in writing the backward pass, even for complicated circuits and all at once. Lets practice backprop a bit with a few examples. In what follows, lets not worry about Unit, Circuit classes because they obfuscate things a bit, and lets just use variables such as a,b,c,x , and refer to their gradients as da,db,dc,dx respectively.
It remembers what its inputs were, and the gradients on each one will be the value of the other during the forward pass.
And then of course we have to multiply with the gradient from above, which is the chain rule. Where 1. What about adding three numbers?
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So we can do it much faster:. And here is our neuron, lets do it in two steps:. This is actually simple because the backward flow of gradients always adds up.
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In other words nothing changes:. When more complex cases like this come up in practice, I like to split the expression into manageable chunks which are almost always composed of simpler expressions and then I chain them together with chain rule:. Here are a few more useful functions and their local gradients that are useful in practice:. Hopefully you see that we are breaking down expressions, doing the forward pass, and then for every variable such as a we derive its gradient da as we go backwards, one by one, applying the simple local gradients and chaining them with gradients from above.
Okay this is making a very simple thing hard to read. The max function passes on the value of the input that was largest and ignores the other ones. In the backward pass then, the max gate will simply take the gradient on top and route it to the input that actually flowed through it during the forward pass. The gate acts as a simple switch based on which input had the highest value during forward pass. The other inputs will have zero gradient. It is used in Neural Networks in place of the sigmoid function.
It is simply thresholding at zero:. In the backward pass, the gate will pass on the gradient from the top if it was activated during the forawrd pass, or if the original input was below zero, it will stop the gradient flow. I will stop at this point. I hope you got some intuition about how you can compute entire expressions which are made up of many gates along the way and how you can compute backprop for every one of them.
Maybe this is not immediately obvious, but this machinery is a powerful hammer for Machine Learning. In the last chapter we were concerned with real-valued circuits that computed possibly complex expressions of their inputs the forward pass , and also we could compute the gradients of these expressions on the original inputs backward pass.
In this chapter we will see how useful this extremely simple mechanism is in Machine Learning. As we did before, lets start out simple. The simplest, common and yet very practical problem in Machine Learning is binary classification. A lot of very interesting and important problems can be reduced to it. For example, in two dimensions our dataset could look as simple as:. Our goal in binary classification is to learn a function that takes a 2-dimensional vector and predicts the label.
This function is usually parameterized by a certain set of parameters, and we will want to tune the parameters of the function so that its outputs are consistent with the labeling in the provided dataset.
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In the end we can discard the dataset and use the learned parameters to predict labels for previously unseen vectors. We will eventually build up to entire neural networks and complex expressions, but lets start out simple and train a linear classifier very similar to the single neuron we saw at the end of Chapter 1. Anyway, lets use a simple linear function:.