[MUSIC] Hi, I am Dmitry Vetrov, research professor
from Higher School of Economics and head of Bayesian Method Research Group and
scientific advisor of Alex and Daniel. Under this lecture,
I would like to tell you about one successful example of how [INAUDIBLE]
can be combined with Bayesian influence. Under this lecture, we will briefly review how budget
methods can be scaled to big data. So, suppose we are given a set of
machine problem containing data X,Y, where X contains observed variables and
Y is hidden variables to be predicted. And we have a probabilistic classifier,
which gives us the probabilities of a hidden components, given observed ones,
which is parameterized by y|x, W. Since we are Bayesians,
we also establish reasonable prior, p(W). And from the Bayesian point of
view at the training stage, we need to compute two posterior
distribution, p(W|X, Y). This posterior distribution contains
all available information about W we could extract from our training data. And this is the result
of Bayesian training. At the test stage, we need to perform [INAUDIBLE] with
respect to this posterior distribution. So we're not applying
just single classifier, we're applying the sample and the weights
of each classifier are given by our posterior distribution p(W) given X,
Y. So this is how it should work in theory,
but in practice of course that is not so, and the problem is in these two integrals. So they're usually intractable, they're
usually in huge dimensional spaces. For example, for
the case of deep learning, the dimensionality of W can be
about tens of millions of per mix. And since the integrals are intractable, we cannot even very
roughly approximate them. This was the reason why,
until very recently, Bayesian methods were
considered to be not scalable. The situation has changed
with the development of so called stochastic variational inference. And sort of trying to solve
basic inference problem exactly, that is to find true
[INAUDIBLE] distribution. P(w) given X, Y [INAUDIBLE] distribution
from some parametric family, so the distribution is q of w given five. [INAUDIBLE] Approximated by minimizing
some kind of distance measure between the two distributions between our rational
approximation and the two posterior. There can be different distance measures,
but one of the most popular one is so-called
KL divergence between q and p. As was mentioned in previous lectures,
in this case, the optimization problem is exactly equivalent to maximizing
so-called ELBO, or evidence lower bound. So ELBO itself is an integral,
again, in a huge dimensional space. And the integral itself
is still untaxable. But we do not need to
compute the integral exactly. All we need to do is to optimize
the integral with the [INAUDIBLE]. And surprisingly, it appears that this is possible using
stochastic optimization framework. So our ELBO actually has
several very nice purposes. One of them is the obtaining of
likelihood, p of Y given X, W. It's inside a logarithm. This means that it can be split into the
sum of individual likelihoods of obtaining objects. And this means that additionalization
of our optimization, we do not need to compute
the whole training of likelihood. We can compute only its unbiased estimate
given by a tiny mini batch of data. So this means that ELBO
supports mini batching. Another good property is that ELBO
still expectation with respect to our relational approximation. And this means that, we need to
obtain unbiased stochastic gradient. We could remove this integral with
its unbiased Monte Carlo estimate. For this purpose, we also need to perform
reparameterization trick in order to reduce the variance of
stochastic gradient. So then we may simply sample from the
distribution, which is now parameter-free. And use Monte Carlo estimate
to compute gradients. Another good property is that
the richer is variational family, the better we approximate
the true posterior distribution, so we do not have a risk of overfitting. The more operational parameters we have, the closer we are to the true
posterior distribution. And finally, we can split ELBO into two parts by
splitting our arithmetic side of integral. So then there will be two integrals,
two terms. The first term is called data term, and it
simply shows the expectation with respect to our relational approximation
of training a likelihood. And the second term is negative KL
diversions between our relational approximation and the prior distribution. Even now if I get the second term, we optimize just the first term with
respect all possible distributions, we'll end up with a delta
function at maximal group point. So there'll be a delta function at WML. And the second term, the regularizer prevents us from
collapsing to delta function. It penalizes too much deviations
from prior distribution. It optimize both terms with respect
to all possible distributions, we'll end up with a true
posterior distribution. But since the true
posterior is untraceable, we'll limit the set of possible
relational approximations. And then they'll end up with a variational
distribution, which is the closest in terms of KL divergence to our
true posterior distribution. So to conclude,
this is stochastic relational inference. This is a highly scalable technique
that provides us with the person with Bayesian inference. The usage of stochastic optimization and reparameterization trick makes SVI
very applicable to large data sets. And in the next section,
I will tell you about dropout and how it can be interpreted
from Bayesian point of view. [MUSIC]