[MUSIC] So let's return to our problem
of estimating the gradient of the objective with respect
to the parameters phi. In the previous video, we discussed that
if we use something called [INAUDIBLE]. We can build a stochastic
approximation of this gradient. But the variance of this stochastic
approximation will be really high. Therefore, will be really inefficient
to use this approximation to train the [INAUDIBLE]. So let's see,
let's look at a really nice and simple and brilliant idea on how to
make this think much better. Make this approximation much better. So, let's make a change,
let's first of all recall that ti is a sample of distribution
from q of ti, given x sin phi. Let's make a change of labels. So let's say, instead of sampling ti,
we'll sample some new variable x and y from the standard variable and
then we'll make ti from this central line. By multiplying kit for
element y in this way by some standard deviation, si, and
by aiding the [INAUDIBLE]. So this way,
the distribution of this expression of this epsilon i times si
plus mi is the same as, it's just q,
it's the same as [INAUDIBLE] ti. So instead of sampling ti
from this distribution q, we can state sample epsilon and
then apply this deterministic function g. With this multiplying ys and adding m to get the sample from
the actual distribution of ti. So we're doing a change of variables. Instead of sampling from ti,
we're sampling from epsilon i and then converting it to a sample from ti. And now we can change our objective,
we can look at the objective and instead of completing the integral
with respect to the distribution q. So expect the distribution q, we can now complete the expected value
with the distribution epsilon i. And then instead of ti,
use this function of epsilon i everywhere. And this is an exact expression,
we didn't lose anything, we just changed the variables. So instead of considering distribution
on ti, we're considering distribution on epsilon i and then converting these
epsilon i samples to samples from ti. And now this g,
this function that converts epsilon i to tis,
it depends on xi and on phi. And to convert your epsilon i,
it passes your image xi through a convolutional neural
network with parameters phi. And this si and mi, and then multiplies
epsilon i by si and [INAUDIBLE] mi. This is [INAUDIBLE] function,
licensing of one. And now we can push the gradient
sign inside the expected value, so past the probability of epsilon i
because [INAUDIBLE] doesn't depend on phi. It doesn't depend on the parameters,
we are differentiation with respect to. And this means, that now we have
an expected value of some expression. Without ever introducing some artificial distributions like in the previous video. We'll like obtain
the expected value naturally. And now these expected values with respect
to the to the distribution epsilon i, which is just standard normal
without any parameters. And now we can approximate this thing
with a sample from standard normal. And so ultimately we have
re-written our objective, so the gradient of our
objective with respect to phi. Is sum with respect to objects
of expected value with respect to standard normal of
the gradient of some function. Which is just standard gradient
of your whole neural network, which defines you the whole
operation [INAUDIBLE]. Andnow you can redraw
this pictures as follows. You have an input image x. You pass it through a convolutional
neural network with parameters phi. You compute the regional parameters m and
s, then you sample one vector from
standard normal distribution epsilon. And then you use all these free values,
m, n, s and epsilon to deterministically compute ti. And then you put this ti inside this
second convolutional neural network. So when you define your model like this,
you have only one place where you
have stochastic units. This epsilon i from standard normal distribution. And this way, you can differentiate
your whole neural structure with respect to phi and w without trouble. So you're going to just
use tender flow and it will find you gradients with
respect to all the parameters. Because you don't have now some different
shapes with through dissembling, dissembling is kind of outside procedure,
it's just yes or nos to determine each functions. And this is basically implementation of
theory we have just discussed within this urban mutualization. And now we're going to approximate our
gradients by just assembling one point, and then using these gradient
of law of this complex function. And this complex function like
log of p of xi given g and w is just this full neural network
with both encoder and decoder. So to summarize,
we have just get the model that allows you to fit probability distribution,
like p of x, into a complicated structure of data,
for example, into images. And it uses a model of
infinite mixture of Gaussians. But to define the parameters of these
Gaussians, it uses a variational neural network with parameters that
are trained with variational inference. And for learning, we can't use the usual
expectation maximization because we have to approximate. And we can't also use variational
expectation maximization because it also [INAUDIBLE]. So we draft kind of stochastic
version of variational inference. That is applicable to,
first of all, large data sets, because we can use mini batches. And second of all, it's applicable
to the small, so you couldn't have used the usual variational
inference for this complicated. Because it has neural networks inside and
every integral is intractable. And the model with that is
called variational autoencoder, it's like the plain usual autoencoder but
It has noise inside and uses [INAUDIBLE] regularization
to make sure that noise stays. That the [INAUDIBLE] chooses
the right amount of noise to use. And can be used to for example, generate
nice images or to handle missing data or to find [INAUDIBLE] in the data and
stuff like that. [MUSIC]