In the previous video, we talked about the backpropagation algorithm. To a lot of people seeing it for the first time, their first impression is often that wow this is a really complicated algorithm, and there are all these different steps, and I'm not sure how they fit together. And it's kinda this black box of all these complicated steps. In case that's how you're feeling about backpropagation, that's actually okay. Backpropagation maybe unfortunately is a less mathematically clean, or less mathematically simple algorithm, compared to linear regression or logistic regression. And I've actually used backpropagation, you know, pretty successfully for many years. And even today I still don't sometimes feel like I have a very good sense of just what it's doing, or intuition about what back propagation is doing. If, for those of you that are doing the programming exercises, that will at least mechanically step you through the different steps of how to implement back prop. So you'll be able to get it to work for yourself. And what I want to do in this video is look a little bit more at the mechanical steps of backpropagation, and try to give you a little more intuition about what the mechanical steps the back prop is doing to hopefully convince you that, you know, it's at least a reasonable algorithm. In case even after this video in case back propagation still seems very black box and kind of like a, too many complicated steps and a little bit magical to you, that's actually okay. And Even though I've used back prop for many years, sometimes this is a difficult algorithm to understand, but hopefully this video will help a little bit. In order to better understand backpropagation, let's take another closer look at what forward propagation is doing. Here's a neural network with two input units that is not counting the bias unit, and two hidden units in this layer, and two hidden units in the next layer. And then, finally, one output unit. Again, these counts two, two, two, are not counting these bias units on top. In order to illustrate forward propagation, I'm going to draw this network a little bit differently. And in particular I'm going to draw this neuro-network with the nodes drawn as these very fat ellipsis, so that I can write text in them. When performing forward propagation, we might have some particular example. Say some example x i comma y i. And it'll be this x i that we feed into the input layer. So this maybe x i 2 and x i 2 are the values we set the input layer to. And when we forward propagated to the first hidden layer here, what we do is compute z (2) 1 and z (2) 2. So these are the weighted sum of inputs of the input units. And then we apply the sigmoid of the logistic function, and the sigmoid activation function applied to the z value. Here's are the activation values. So that gives us a (2) 1 and a (2) 2. And then we forward propagate again to get here z (3) 1. Apply the sigmoid of the logistic function, the activation function to that to get a (3) 1. And similarly, like so until we get z (4) 1. Apply the activation function. This gives us a (4)1, which is the final output value of the neural network. Let's erase this arrow to give myself some more space. And if you look at what this computation really is doing, focusing on this hidden unit, let's say. We have to add this weight. Shown in magenta there is my weight theta (2) 1 0, the indexing is not important. And this way here, which I'm highlighting in red, that is theta (2) 1 1 and this weight here, which I'm drawing in cyan, is theta (2) 1 2. So the way we compute this value, z(3)1 is, z(3)1 is as equal to this magenta weight times this value. So that's theta (2) 10 x 1. And then plus this red weight times this value, so that's theta(2) 11 times a(2)1. And finally this cyan weight times this value, which is therefore plus theta(2)12 times a(2)1. And so that's forward propagation. And it turns out that as we'll see later in this video, what backpropagation is doing is doing a process very similar to this. Except that instead of the computations flowing from the left to the right of this network, the computations since their flow from the right to the left of the network. And using a very similar computation as this. And I'll say in two slides exactly what I mean by that. To better understand what backpropagation is doing, let's look at the cost function. It's just the cost function that we had for when we have only one output unit. If we have more than one output unit, we just have a summation you know over the output units indexed by k there. If you have only one output unit then this is a cost function. And we do forward propagation and backpropagation on one example at a time. So let's just focus on the single example, x (i) y (i) and focus on the case of having one output unit. So y (i) here is just a real number. And let's ignore regularization, so lambda equals 0. And this final term, that regularization term, goes away. Now if you look inside the summation, you find that the cost term associated with the training example, that is the cost associated with the training example x(i), y(i). That's going to be given by this expression. So, the cost to live off examplie i is written as follows. And what this cost function does is it plays a role similar to the squared arrow. So, rather than looking at this complicated expression, if you want you can think of cost of i being approximately the square difference between what the neural network outputs, versus what is the actual value. Just as in logistic repression, we actually prefer to use the slightly more complicated cost function using the log. But for the purpose of intuition, feel free to think of the cost function as being the sort of the squared error cost function. And so this cost(i) measures how well is the network doing on correctly predicting example i. How close is the output to the actual observed label y(i)? Now let's look at what backpropagation is doing. One useful intuition is that backpropagation is computing these delta superscript l subscript j terms. And we can think of these as the quote error of the activation value that we got for unit j in the layer, in the lth layer. More formally, for, and this is maybe only for those of you who are familiar with calculus. More formally, what the delta terms actually are is this, they're the partial derivative with respect to z,l,j, that is this weighted sum of inputs that were confusing these z terms. Partial derivatives with respect to these things of the cost function. So concretely, the cost function is a function of the label y and of the value, this h of x output value neural network. And if we could go inside the neural network and just change those z l j values a little bit, then that will affect these values that the neural network is outputting. And that will end up changing the cost function. And again really, this is only for those of you who are expert in Calculus. If you're comfortable with partial derivatives, what these delta terms are is they turn out to be the partial derivative of the cost function, with respect to these intermediate terms that were confusing. And so they're a measure of how much would we like to change the neural network's weights, in order to affect these intermediate values of the computation. So as to affect the final output of the neural network h(x) and therefore affect the overall cost. In case this lost part of this partial derivative intuition, in case that doesn't make sense. Don't worry about the rest of this, we can do without really talking about partial derivatives. But let's look in more detail about what backpropagation is doing. For the output layer, the first set's this delta term, delta (4) 1, as y (i) if we're doing forward propagation and back propagation on this training example i. That says y(i) minus a(4)1. So this is really the error, right? It's the difference between the actual value of y minus what was the value predicted, and so we're gonna compute delta(4)1 like so. Next we're gonna do, propagate these values backwards. I'll explain this in a second, and end up computing the delta terms for the previous layer. We're gonna end up with delta(3)1. Delta(3)2. And then we're gonna propagate this further backward, and end up computing delta(2)1 and delta(2)2. Now the backpropagation calculation is a lot like running the forward propagation algorithm, but doing it backwards. So here's what I mean. Let's look at how we end up with this value of delta(2)2. So we have delta(2)2. And similar to forward propagation, let me label a couple of the weights. So this weight, which I'm going to draw in cyan. Let's say that weight is theta(2)1 2, and this one down here when we highlight this in red. That is going to be let's say theta(2) of 2 2. So if we look at how delta(2)2, is computed, how it's computed with this note. It turns out that what we're going to do, is gonna take this value and multiply it by this weight, and add it to this value multiplied by that weight. So it's really a weighted sum of these delta values, weighted by the corresponding edge strength. So completely, let me fill this in, this delta(2)2 is going to be equal to, Theta(2)1 2 is that magenta lay times delta(3)1. Plus, and the thing I had in red, that's theta (2)2 times delta (3)2. So it's really literally this red wave times this value, plus this magenta weight times this value. And that's how we wind up with that value of delta. And just as another example, let's look at this value. How do we get that value? Well it's a similar process. If this weight, which I'm gonna highlight in green, if this weight is equal to, say, delta (3) 1 2. Then we have that delta (3) 2 is going to be equal to that green weight, theta (3) 12 times delta (4) 1. And by the way, so far I've been writing the delta values only for the hidden units, but excluding the bias units. Depending on how you define the backpropagation algorithm, or depending on how you implement it, you know, you may end up implementing something that computes delta values for these bias units as well. The bias units always output the value of plus one, and they are just what they are, and there's no way for us to change the value. And so, depending on your implementation of back prop, the way I usually implement it. I do end up computing these delta values, but we just discard them, we don't use them. Because they don't end up being part of the calculation needed to compute a derivative. So hopefully that gives you a little better intuition about what back propegation is doing. In case of all of this still seems sort of magical, sort of black box, in a later video, in the putting it together video, I'll try to get a little bit more intuition about what backpropagation is doing. But unfortunately this is a difficult algorithm to try to visualize and understand what it is really doing. But fortunately I've been, I guess many people have been using very successfully for many years. And if you implement the algorithm you can have a very effective learning algorithm. Even though the inner workings of exactly how it works can be harder to visualize.
