In the last video you saw how very deep neural networks can have the problems of vanishing and exploding gradients. It turns out that a partial solution to this, doesn't solve it entirely but helps a lot, is better or more careful choice of the random initialization for your neural network. To understand this, let's start with the example of initializing the ways for a single neuron, and then we're go on to generalize this to a deep network. Let's go through this with an example with just a single neuron, and then we'll talk about the deep net later. So with a single neuron, you might input four features, x1 through x4, and then you have some a=g(z) and then it outputs some y. And later on for a deeper net, you know these inputs will be right, some layer a(l), but for now let's just call this x for now. So z is going to be equal to w1x1 + w2x2 +... + I guess WnXn. And let's set b=0 so, you know, let's just ignore b for now. So in order to make z not blow up and not become too small, you notice that the larger n is, the smaller you want Wi to be, right? Because z is the sum of the WiXi. And so if you're adding up a lot of these terms, you want each of these terms to be smaller. One reasonable thing to do would be to set the variance of W to be equal to 1 over n, where n is the number of input features that's going into a neuron. So in practice, what you can do is set the weight matrix W for a certain layer to be np.random.randn you know, and then whatever the shape of the matrix is for this out here, and then times square root of 1 over the number of features that I fed into each neuron in layer l. So there's going to be n(l-1) because that's the number of units that I'm feeding into each of the units in layer l. It turns out that if you're using a ReLu activation function that, rather than 1 over n it turns out that, set in the variance of 2 over n works a little bit better. So you often see that in initialization, especially if you're using a ReLu activation function. So if gl(z) is ReLu(z), oh and it depends on how familiar you are with random variables. It turns out that something, a Gaussian random variable and then multiplying it by a square root of this, that sets the variance to be quoted this way, to be 2 over n. And the reason I went from n to this n superscript l-1 was, in this example with logistic regression which is at n input features, but the more general case layer l would have n(l-1) inputs each of the units in that layer. So if the input features of activations are roughly mean 0 and standard variance and variance 1 then this would cause z to also take on a similar scale. And this doesn't solve, but it definitely helps reduce the vanishing, exploding gradients problem, because it's trying to set each of the weight matrices w, you know, so that it's not too much bigger than 1 and not too much less than 1 so it doesn't explode or vanish too quickly. I've just mention some other variants. The version we just described is assuming a ReLu activation function and this by a paper by her et al. A few other variants, if you are using a TanH activation function then there's a paper that shows that instead of using the constant 2, it's better use the constant 1 and so 1 over this instead of 2. And so you multiply it by the square root of this. So this square root term will replace this term and you use this if you're using a TanH activation function. This is called Xavier initialization. And another version we're taught by Yoshua Bengio and his colleagues, you might see in some papers, but is to use this formula, which you know has some other theoretical justification, but I would say if you're using a ReLu activation function, which is really the most common activation function, I would use this formula. If you're using TanH you could try this version instead, and some authors will also use this. But in practice I think all of these formulas just give you a starting point. It gives you a default value to use for the variance of the initialization of your weight matrices. If you wish the variance here, this variance parameter could be another thing that you could tune with your hyperparameters. So you could have another parameter that multiplies into this formula and tune that multiplier as part of your hyperparameter surge. Sometimes tuning the hyperparameter has a modest size effect. It's not one of the first hyperparameters I would usually try to tune, but I've also seen some problems where tuning this helps a reasonable amount. But this is usually lower down for me in terms of how important it is relative to the other hyperparameters you can tune. So I hope that gives you some intuition about the problem of vanishing or exploding gradients as well as choosing a reasonable scaling for how you initialize the weights. Hopefully that makes your weights not explode too quickly and not decay to zero too quickly, so you can train a reasonably deep network without the weights or the gradients exploding or vanishing too much. When you train deep networks, this is another trick that will help you make your neural networks trained much more quickly.
