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Forward Propagation in a Deep Network (C1W4L02)
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- in the last video we described what is the deep l-larry neural network and also talked about the notation we use to describe such networks in this video you
- see how you can perform fold propagation in a deep network as usual let's first go over what forward propagation will look like for a single training example
- X and then later on we'll talk about the vectorized version where you want to carry out forward propagation on the entire training set at the same time but
- um given a single training example X here's how you compute the activations of the first layer so for this first layer you compute v1 equals W 1 times X
- plus b1 so W 1 and B 1 that parameters that affect the activations in layer 1 right where this is layer 1 of the neural network and then you compute the
- activations for that layer to be equal to G of Z 1 and the deactivation function G depends on what layer you're at and maybe index AB has the activation
- function from layer 1 so if you do that you've now computed the activations from there 1 how about layer to say that there well
- you would then compute v2 equals W to a 1 plus B 2 and then so the observation of layer 2 is the way matrix times the outputs of layer 1 so that value plus
- the bias vector for layer 2 and then a2 equals the activation function apply to z2 ok so that's it for layer 2 and so on and so forth until you get to the output
- layer that's layer 4 where you would have that V 4 is equal to the parameters for that layer times the activations from the previous layer
- Plus that bias vector and then similarly a four equals G of v4 and so that's how you you know compute your estimated output Y hat so just one thing to notice
- X here is also equal to a zero because the input feature vector X is also the activations of layer 0 so we scratch out X I'm going to cross out X and put a 0
- here then you know all of these equations basically look the same right the general rule is that ZL is equal to WL times a of L minus 1 plus B L 1 there
- and then the activations for that layer is the activation function applied to the values Z so that's the general for propagation equation so we've done all
- this for a single training example how about for doing it in a vectorized way for the whole training set at the same time the equations look quite similar as
- before for the first layer you would have Capital Z 1 equals W 1 times capital X plus B 1 and then a 1 equals G of Z 1 right and bear in mind that X is
- equal to a 0 these are just Neil the training examples stacked in different columns you could take this let me scratch out X we can put a 0 there and
- then for the next layer a little similar Z 2 equals W 2 A 1 plus B 2 and a 2 equals G of Z 2 right we just take these vector Z or a and so on and
- stacking them up so this is V vector for the first training example V vector for the second training example and so on down to the M training example and
- stacking these and columns and calling this capital V alright and similarly for capital A just as capital X all the training examples are column vectors
- snacks left to right and then they then end of this process you end up with y hat which is equal to G of Z 400 this is also equal to a 4 and that's the
- predictions on all the new training examples of stand horizontally so just to summarize our notation I'm going to modify this up here our notation allows
- us to replace lowercase Z and a with the uppercase counterparts it already looks like a capital D and that gives you the vectorized version of forward
- propagation that you carry out on the entire training set at a time where a 0 is X now if you look at this implementation of vectorization it looks
- like that there is going to be a for loop here where it says left for l equals 1 to 4 for l equals 1 through capital l then you have to compute the
- activations for layer 1 and the layer 2 then for layer 3 and they're gentle therefore so since that there is a for loop here and I know that when
- implementing your networks we usually want to get rid of explicit for loops but this is one place where I don't think there's any way to implement this
- other than explicit for loop so we're implementing for propagation it is perfectly okay to have a for loop that compute the activations for layer 1 then
- there are 2 then they are threes and therefore no one knows and I don't think there is this any way to do this without a for loop that goes from 1 to capital L
- from 1 through the total number of layers and in your network so this place is perfectly okay to have an explicit folder so
- that's it for the notation for deep neural networks as well as how to do forward propagation in these networks if the pieces we've seen so far looks a
- little bit familiar to you that's because what we're seeing is taking a piece very similar to what you see in in the neural network with a single hidden
- layer and just repeating that more times now it turns out that we implemented deep neural network one of the ways to increase your odds of having a bug-free
- implementation is to think very systematic and carefully about the matrix dimensions you're working work so when I'm trying to develop my own code
- I'll often pull a piece of paper and just think carefully through so the dimensions of the matrix I'm working with let's see how you could do that in
- the next video