Chapter 20
Deep Generative Models
In this chapter, we present several of the specific kinds of generative models that
can be built and trained using the techniques presented in chapters 13, 18 and
19. All of these models represent probability distributions over multiple variables
in some way. Some allow the probability distribution function to be evaluated
explicitly. Others do not allow the evaluation of the probability distribution
function, but support operations that implicitly require knowledge of it, such as
sampling. Some of these models are structured probabilistic models described in
terms of graphs and factors, as described in chapter 13. Others can not easily be
described in terms of factors, but represent probability distributions nonetheless.
20.1 Boltzmann Machines
Boltzmann machines were originally introduced in Ackley et al. (1985) as a gen-
eral “connectionist” approach to learning arbitrary probability distributions over
binary vectors. Boltzmann Machines form the basis of a large number of popular
variants. Indeed, these variants have long ago surpassed the popularity of the
original and most general incarnation of the Boltzmann machine. In this section
we briefly introduce the general Boltzmann machine and discuss the issues that
come up when trying to train and perform inference in the model.
We define our Boltzmann machine over a d-dimensional binary random vector
x ∈ {0, 1}
d
. The Boltzmann machine is an energy-based model
1
, meaning we
define the joint probability distribution over the model variable using an energy
function.
P (x) =
exp (−E(x))
Z
. (20.1)
Where E(x) is the energy function and Z is the partition function, that ensures
1
For a general discussion of energy based models see Sec. 13.2.4
553
CHAPTER 20. DEEP GENERATIVE MODELS
h
1
h
2
h
3
v
1
v
2
v
3
h
4
h
1
(1)
h
2
(1)
h
3
(1)
v
1
v
2
v
3
h
1
(2)
h
2
(2)
h
3
(2)
h
4
(1)
h
1
(1)
h
2
(1)
h
3
(1)
v
1
v
2
v
3
h
1
(2)
h
2
(2)
h
3
(2)
h
4
(1)
Figure 20.1: Examples of models that may be built with restricted Boltzmann machines.
a) The restricted Boltzmann machine itself is an undirected graphical model based on a
bipartite graph. There are no connections among the visible units, nor any connections
among the hidden units. Typically every visible unit is connected to every hidden unit
but it is possible to construct sparsely connected RBMs such as convolutional RBMs. b)
A deep belief network is a hybrid graphical model involving both directed and undirected
connections. Like an RBM, it has no intra-layer connections. However, a DBN has
multiple hidden layers, and thus there are connections between hidden units that are in
separate layers. All of the local conditional probability distributions needed by the deep
belief network are copied directly from the local conditional probability distributions of
its constituent RBMs. Note that we could also represent the deep belief network with
a completely undirected graph, but it would need intra-layer connections to capture the
dependencies between parents. c) A deep Boltzmann machine is an undirected graphical
model with several layers of latent variables. Like RBMs and DBNs, DBMs lack intra-
layer connections. DBMs are less closely tied to RBMs than DBNs are. When initializing
a DBM from a stack of RBMs, it is necessary to modify the RBM parameters slightly.
Some kinds of DBMs may be trained without first training a set of RBMs.
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CHAPTER 20. DEEP GENERATIVE MODELS
that the
P
x
P (x) = 1. The energy function of the Boltzmann machine is given
by:
E(x) = −x
>
Ux − b
>
x, (20.2)
where U is the “weight” matrix of model parameters and b are the offsets for
each x.
In the general setting of the Boltzmann machine, we could consider that the
goal is that we are given a set of observations, each of which are d-dimensional and
that we are to use the joint probability distribution given in Eq. 20.1 describes
the joint probability distribution over the observed variables (also called visible
units). While this scenario is certainly viable, it does limit the kinds of inter-
actions between the observed variables to those described by the weight matrix.
Specifically it limits the model to 2nd-order interactions.
In the spirit of the “connectionist” approach to density modeling that origi-
nally inspired the Boltzmann machine, it is interesting to consider the case where
not all the variables are observed. In this case, the non-observed variables, or
latent variables can act similarly to hidden units in a multi-layer perceptron and
model higher-order interactions among the visible units.
Formally, we decompose the units into two subsets: the visible units x
v
and
the latent (or hidden) units x
h
. Without loss of generality, we can re-express the
energy function decomposing x into subsets x
v
and x
h
:
E(x
v
, x
h
) = −x
>
v
Rx
v
− x
>
v
W x
h
− x
>
h
Sx
h
− b
>
x
v
− c
>
x
h
, (20.3)
Boltzmann Machine Learning As a probabilistic model, it is natural to con-
sider maximum likelihood as the learning paradigm for Boltzmann machines. Ac-
cording to the ML paradigm, we are interested in choosing the parameters that
(locally) maximize the probability of the visible units over a dataset.
Consider a dataset of n examples X
v
= [x
(1)
v
, . . . , x
(t)
v
, . . . , x
n
v
]. Our goal is to
maximize the likelihood of this dataset under the Boltzmann machine. Assuming
the data is i.i.d, this amounts to maximizing the following:
`(θ) = log P (X
v
) =
n
X
t=1
log P (x
(t)
v
). (20.4)
Of course, our Boltzmann machine does not explicitly parametrize a distribution
over the visible units as P (x
(t)
v
), instead, as given in Eq. 20.3, it is parametrized
via an energy function to joint probability distribution over x
v
and x
h
, the hidden
units. In order to recover P (x
(t)
v
), we need to marginalize out the influence of x
h
.
P (x
(t)
v
) =
X
x
h
P (x
(t)
v
, x
(t)
h
) =
X
x
h
1
Z
exp
n
−E(x
(t)
v
, x
(t)
h
)
o
. (20.5)
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CHAPTER 20. DEEP GENERATIVE MODELS
Combining Eqs. 20.4 and 20.5 gives us our objective function we wish to maxi-
mize. Unfortunately, because Z is a function of the model parameters, maximizing
likelihood function is not amenable to analytically solution. Instead we will do as
we do for the vast majority of deep learning models, we will follow the gradient of
our objective function. The Boltzmann machine likelihood gradient is given by:
∂
∂θ
`(θ) =
∂
∂θ
n
X
t=1
"
1
Z
log
X
x
h
exp
n
−E(x
(t)
v
, x
(t)
h
)
o
#!
(20.6)
=
n
X
t=1
∂
∂θ
"
log
X
x
h
exp
n
−E(x
(t)
v
, x
(t)
h
)
o
#
−
∂Z
∂θ
(20.7)
=
n
X
t=1
X
x
h
exp
n
−E(x
(t)
v
, x
(t)
h
)
o
P
x
h
exp
n
−E(x
(t)
v
, x
(t)
h
)
o
∂
∂θ
E(x
(t)
v
, x
(t)
h
)
−
∂Z
∂θ
(20.8)
20.2 Restricted Boltzmann Machines
Restricted Boltzmann machines are some of the most common building blocks
of deep probabilistic models. They are undirected probabilistic graphical models
containing a layer of observable variables and a single layer of latent variables.
RBMs may be stacked (one on top of the other) to form deeper models. See
Fig. 20.1 for some examples. In particular, Fig. 20.1a shows the graph structure
of an RBM itself. It is a bipartite graph: with no connections permitted between
any variables in the observed layer or between any units in the latent layer.
TODO– review and pointers to other sections of the book This should be the
main place where they are described in detail, earlier they are just an example of
undirected models or an example of a feature learning algorithm.
TODO: please use lower-case letter names for scalars, none of this D and N
stuff. do we even use these variable names anywhere, or do we just define them
and never refer back to them? if they are never used, delete them, don’t make
the reader hold variables in their head for no payoff
We begin with the binary version of the restricted Boltzmann machine, but
as we see later there are extensions to other types of visible and hidden units.
More formally, we will consider the observed layer to consist of a set of D
binary random variables which we refer to collectively with the vector v, where
the ith element, i.e. v
i
is a binary random variable. We will refer to the latent or
hidden layer of N binary random variables collectively as h, with the jth random
elements as h
j
.
Like the general Boltzmann machine, the restricted Boltzmann machine is an
energy-based model with the joint probability distribution specified by its energy
556
CHAPTER 20. DEEP GENERATIVE MODELS
function:
P (v = v, h = h) =
1
Z
exp {−E(v, h)}.
Where E(v, h) is the energy function that parametrizes the relationship between
the visible and hidden variables:
E(v, h) = −b
>
v − c
>
h −v
>
W h, (20.9)
and the Z is the normalizing constant known as the partition function:
Z =
X
v
X
h
exp {−E(v, h)}.
For many undirected models, it is apparent from the definition of the partition
function Z that the naive method of computing Z (exhaustively summing over
all states) would be computationally intractable. However, it is still possible that
a more cleverly designed algorithm could exploit regularities in the probability
distribution to compute Z faster than the naive algorithm, so the exponential cost
of the naive algorithm is not a guarantee of the partiion function’s intractability.
In the case of restricted Boltzmann machines, there is actually a hardness result,
proven by Long and Servedio (2010).
20.2.1 Conditional Distributions
The intractable partition function Z, implies that the joint probability distribu-
tion is also intractable (in the sense that the normalized probability of a given
joint configuration of [v, h] is generally not available). However, due the bipartite
graph structure, the restricted Boltzmann machine has the very special property
that its conditional distributions P(h | v) and P(v | h) are factorial and relatively
simple to compute and sample from. Indeed, it is this property that has made
the RBM a relatively popular model for a wide range of applications including
image modeling (TODO CITE), speech processing (TODO CITE) and natural
language processing (TODO CITE).
Deriving the conditional distributions from the joint distribution is straight-
557
CHAPTER 20. DEEP GENERATIVE MODELS
forward.
p(h | v) =
p(h, v)
p(v)
=
p(h, v)
p(v)
=
1
p(v)
1
Z
exp
n
b
>
v + c
>
h + v
>
W h
o
=
1
Z
0
exp
n
c
>
h + v
>
W h
o
=
1
Z
0
exp
n
X
j=1
c
j
h
j
+
n
X
j=1
v
>
W
:,j
h
j
=
1
Z
0
n
Y
j=1
exp
n
c
j
h
j
+ v
>
W
:,j
h
j
o
Since we are conditioning on the visible units v, we can treat these as constants
w.r.t. the distribution p(h | v). The factorial nature of the conditional p(h | v)
follows immediately from our ability to wright the joint probability over the vector
h as the product of (unnormalized) distributions over the individual elements,
h
j
. It is now a simple matter of normalizing the distributions over the individual
binary h
j
.
P (h
j
= 1 | v) =
˜
P (h
j
= 1 | v)
˜
P (h
j
= 0 | v) +
˜
P (h
j
= 1 | v)
=
exp
c
j
+ v
>
W
:,j
exp {0} + exp {c
j
+ v
>
W
:,j
}
= sigmoid
c
j
+ v
>
W
:,j
. (20.10)
We can now express the full conditional over the hidden layer as the factorial
distribution:
P (h | v) =
n
Y
j=1
sigmoid
c
j
+ v
>
W
:,j
. (20.11)
A similar derivation will show that the other condition of interest to us, P (v |
h), is also a factorial distribution:
P (v | h) =
d
Y
i=1
sigmoid (b
i
+ W
i,:
h) . (20.12)
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CHAPTER 20. DEEP GENERATIVE MODELS
20.2.2 RBM Gibbs Sampling
The factorial nature of these conditions is a very useful property of the RBM,
and allows us to efficiently draw samples from the joint distribution via a block
Gibbs sampling strategy (see section 14.1 for a more complete discussion of Gibbs
sampling methods).
Block Gibbs sampling simply refers to the situation where in each step of Gibbs
sampling, multiple variables (or a “block” of variables) are sampled jointly. In
the case of the RBM, each iteration of block Gibbs sampling consists of two steps.
Step 1: Sample h
(l)
∼ P (h | v
(l)
). Due to the factorial nature of the conditionals,
we can simultaneously and independently sample from all the elements of h
(l)
given v
(l)
. Step 2: Sample v
(l+1)
∼ P (v | h
(l)
). Again, the factorial nature
of the conditional P(v | h
(l)
) allows us can simultaneously and independently
sample from all the elements of v
(l+1)
given h
(l)
.
20.3 Training Restricted Boltzmann Machines
Despite the simplicity of the RBM conditionals, training these models is not with-
out its complications. As a probabilistic model, a sensible inductive principle for
estimating the model parameters is maximum likelihood – though other possi-
bilities are certainly possible Marlin et al. (2010) and will be discussed later in
Sec. 20.3.3. In the following we derive the maximum likelihood gradient with
respect to the model parameters.
Let us consider that we have a batch (or minibatch) of n examples taken from
an i.i.d dataset (independently and identically distributed examples) {v
(1)
, . . . , v
(t)
, . . . , v
(n)
}.
The log-likelihood under the RBM with parameters b (visible unit biases), c (hid-
den unit biases) and W (interaction weights) is given by:
`(W , b, c) =
n
X
t=1
log P (v
(t)
)
=
n
X
t=1
log
X
h
P (v
(t)
n,:
, h)
=
n
X
t=1
log
X
h
exp
n
−E(v
(t)
, h)
o
!
− n log Z
=
n
X
t=1
log
X
h
exp
n
−E(v
(t)
, h)
o
!
− n log
X
v,h
exp {−E(v, h)}
(20.13)
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CHAPTER 20. DEEP GENERATIVE MODELS
In the last line of the equation above, we have used the definition of the partition
function.
To maximize the likelihood of the data under the restricted Boltzmann ma-
chine, we consider the gradient of the likelihood with respect to the model pa-
rameters, which we will refer to collectively as θ = {b, c, W }:
∇
θ
`(θ) = ∇
θ
n
X
t=1
log
X
h
exp
n
−E(v
(t)
, h)
o
!
− n
∂
∂θ
log
X
v,h
exp {−E(v, h)}
=
n
X
t=1
P
h
exp
−E(v
(t)
, h)
∇
θ
− E(v
(t)
, h)
P
h
exp
−E(v
(t)
, h)
− n
P
v,h
exp {−E(v, h)}∇
θ
− E(v, h)
P
v,h
exp {−E(v, h)}
=
n
X
t=1
E
P (h|v
(t)
)
h
∇
θ
− E(v
(t)
, h)
i
− nE
P (v,h)
[∇
θ
− E(v, h)] (20.14)
As we can see from Eq. 20.14, the gradient of the log likelihood is specified as the
difference between two expectations of the gradient of the energy function, The
first expectation (the data term) is with respect to the product of the empirical
distribution over the data, P (v) = 1/n
P
n
t=1
δ(x − v
(t)
)
2
and the conditional
distribution P (h | v
(t)
). The second expectation (the model term) is with respect
to the joint model distribution P (v, h).
This difference between a data-driven term and a model-driven term is not
unique to RBMs, as discussed in some detail in Sec. 18.2, this is a general feature
of the maximum likelihood gradient for all undirected models.
We can complete the derivation of log-likelihood gradient by expanding the
term: ∇
θ
− E(v, h). We will consider first the gradient of the negative energy
function of W .
(20.15)
nabla
W
− E(v, h) =
∂
∂W
b
>
v + c
>
h + v
>
W h
= hv
>
(20.16)
The gradients with respect to b and c are similarly derived:
∇
b
− E(v, h) = v, ∇
c
− E(v, h) = h (20.17)
2
As discussed in Sec. 3.10.5, we use the term empirical distribution to refer to a mixture over
delta functions placed on training examples
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CHAPTER 20. DEEP GENERATIVE MODELS
Putting it all together we can the following equations for the gradients with
respect to the RBM parameters and given n training examples:
∇
W
`(W , b, c) =
n
X
t=1
ˆ
h
(t)
v
(t) >
− NE
P (v,h)
h
hv
>
i
∇
b
`(W , b, c) =
n
X
t=1
v
(t)
− nE
P (v,h)
[v]
∇
c
`(W , b, c) =
n
X
t=1
ˆ
h
(t)
− nE
P (v,h)
[h]
where we have defined
ˆ
h
(t)
as
ˆ
h
(t)
= E
P (h|v
(t)
)
[h] = sigmoid
c + v
(t)
W
. (20.18)
While we are able to write down these expressions for the log-likelihood gra-
dient, unfortunately, in most situations of interest, we are not able to use them
directly to calculate gradients. The problem is the expectations over the joint
model distribution P(v, h). While we have conditional distributions P (v | h)
and P(h | v) that are easy to work with, the RBM joint distribution is not
amenable to analytic evaluation of the expectation E
P (v,h)
[f(v, h)].
This is bad news—it implies that in most cases it is impractical to compute
the exact log-likelihood gradient. Fortunately, as discussed in Sec. 18.2, there
are two widely used approximation strategies that have been applied to the train-
ing of RBM with some degree of success: contrastive divergence and stochastic
maximum likelihood.
In the following sections we discuss two different strategies to approximate this
gradient that have been applied to training the RBM. However, before getting into
the actual training algorithms, it is worth considering what general approaches
are available to us in approximating the log-likelihood gradient. As we mentioned,
our problem stems from the expectation over the joint distribution P (v, h), but we
know that we have access to factorial conditionals and that we can use these as the
basis of a Gibbs sampling procedure to recover samples form the joint distribution
(as discussed in Sec. 20.2.2). Thus, we can imagine using, for example, T MCMC
samples from P (v, h) to form a Monte Carlo estimate of the expectations over
the joint distribution:
E
P (v,h)
[f(v, h)] ≈
1
T
T
X
t=1
f(v
(t)
, h
(t)
). (20.19)
There is a problem with this strategy that has to do with the initialization of the
MCMC chain. MCMC chains typically require a burn-in period, where the chain
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CHAPTER 20. DEEP GENERATIVE MODELS
20.3.1 Contrastive Divergence Training of the RBM
As discussed in a more general context in Sec. 18.2, Contrastive divergence (CD)
seeks to approximate the expectation over the joint distribution with samples
drawn from short Gibbs sampling chains. CD deals with the typical requirement
for an extended burn-in sample sequence by initializing these chains at the data
points used in the data-dependent, conditional term. The result is a biased ap-
proximation of the log-likelihood gradient (Carreira-Perpi˜nan and Hinton, 2005;
Bengio and Delalleau, 2009; Fischer and Igel, 2011), that never-the-less has been
empirically shown to be effective. The constrastive divergence algorithm, as ap-
plied to RBMs, is given in Algorithm 20.1.
Algorithm 20.1 The contrastive divergence algorithm, using gradient ascent as
the optimization procedure.
Set , the step size, to a small positive number
Set k, the number of Gibbs steps, high enough to allow a Markov chain of
p(v; θ) to mix when initializedfrom p
data
. Perhaps 1-20 to train an RBM on a
small image patch.
while Not converged do
Sample a minibatch of m examples from the training set {v
(1)
, . . . , v
(m)
}.
∆
W
←
1
m
P
m
t=1
v
(t)
ˆ
h
(t) >
∆
b
←
1
m
P
m
t=1
v
(t)
∆
c
←
1
m
P
m
t=1
ˆ
h
(t)
for t = 1 to m do
˜
v
(t)
← v
(t)
end for
for l = 1 to k do
for t = 1 to m do
˜
h
(t)
sampled from
Q
n
j=1
sigmoid
c
j
+
˜
v
(t) >
W
:,j
.
˜
v
(t)
sampled from
Q
d
i=1
sigmoid
b
i
+ W
i,:
˜
h
(t)
.
end for
end for
¯
h
(t)
← sigmoid
c +
˜
v
(t) >
W
∆
W
← ∆
W
−
1
m
P
m
t=1
˜
v
(t)
¯
h
(t) >
∆
b
← ∆
b
−
1
m
P
m
t=1
˜
v
(t)
∆
c
← ∆
b
−
1
m
P
m
t=1
¯
h
(t)
W ← W + ∆
W
b ← b + ∆
b
c ← c + ∆
c
end while
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CHAPTER 20. DEEP GENERATIVE MODELS
20.3.2 Stochastic Maximum Likelihood (Persistent Contrastive
Divergence) for the RBM
While contrastive divergence has been the most popular method of training
RBMs, the stochastic maximum likelihood (SML) algorithm (Younes, 1998; Tiele-
man, 2008) is known to be a competitive alternative – especially if we are inter-
ested in recovering the best possible generative model (i.e. achieving the highest
possible test set likelihood). As with CD, the general SML algorithm is described
in Sec. 18.2. Here we are concerned with how to apply the algorithm to training
an RBM.
In comparison to the CD algorithm, SML uses an alternative solution to the
problem of how to approximate the partition function’s contribution to the log-
likelihood gradient. Instead of initializing the k-step MCMC chain with the cur-
rent example from the training set, in SML we initialize the MCMC chain for
training iteration s with the last state of the MCMC chain from the last training
iteration (s−1). Assuming that the gradient updates to the model parameters do
not significantly change the model, the MCMC state of the last iteration should
be close to the equilibrium distribution at iteration s – minimizing the number of
“burn-in” MCMC steps needed to reach equilibrium at the current iteration. As
with CD, in practice we often use just one Gibbs step between learning iterations.
Algorithm 20.2 describes the SML algorithm as applied to RBMs.
TODO: include experimental examples, i.e. an RBM trained with CD on
MNIST
20.3.3 Other Inductive Principles
TODO:Other inductive principles have been used to train RBMs. In this section
we briefly discuss these.
20.4 Deep Belief Networks
Deep belief networks (DBNs) were one of the first successful non-convolutional
architectures. The introduction of deep belief networks in 2006 began the cur-
rent deep learning renaissance. Prior to the introduction of deep belief networks,
deep models were considered too difficult to optimize, due to the vanishing and
exploding gradient problems and the existence of plateaus, negative curvature,
and suboptimal local minima that can arise in neural network objective func-
tions. Kernel machines with convex objective functions dominated the research
landscape. Deep belief networks demonstrated that deep architectures can be
successful, by outperforming kernelized support vector machines on the MNIST
dataset (Hinton et al., 2006). Today, deep belief networks have mostly fallen out
563
CHAPTER 20. DEEP GENERATIVE MODELS
Algorithm 20.2 The stochastic maximum likelihood / persistent contrastive
divergence algorithm for training an RBM.
Set , the step size, to a small positive number
Set k, the number of Gibbs steps, high enough to allow a Markov chain of
p(v, h; θ +∆
θ
) toburn in, starting from samples from p(v, h; θ). Perhaps 1 for
RBM on a small image patch.
Initialize a set of m samples {
˜
v
(1)
, . . . ,
˜
v
(m)
} to random values (e.g., from a uni-
form or normal distribution, or possibly a distribution with marginals matched
to the model’s marginals)
while Not converged do
Sample a minibatch of m examples {v
(1)
, . . . , v
(m)
} from the training set.
∆
W
←
1
m
P
m
t=1
ˆ
h
(t)
v
(t) >
∆
b
←
1
m
P
m
t=1
v
(t)
∆
c
←
1
m
P
m
t=1
ˆ
h
(t)
for l = 1 to k do
for t = 1 to m do
˜
h
(t)
sampled from
Q
n
j=1
sigmoid
c
j
+
˜
v
(t) >
W
:,j
.
˜
v
(t)
sampled from
Q
d
i=1
sigmoid
b
i
+ W
i,:
˜
h
(t)
.
end for
end for
∆
W
← ∆
W
−
1
m
P
m
t=1
˜
v
(t)
˜
h
(t) >
∆
b
← ∆
b
−
1
m
P
m
t=1
˜
v
(t)
∆
c
← ∆
b
−
1
m
P
m
t=1
˜
h
(t)
W ← W + ∆
W
b ← b + ∆
b
c ← c + ∆
c
end while
of favor and are rarely used, even compared to other unsupervised or generative
learning algorithms, but they are still deservedly recognized for their important
role in deep learning history.
Deep belief networks are generative models with several layers of latent vari-
ables. The latent variables are typically binary, and the visible units may be
binary or real. There are no intra-layer connections. Usually, every unit in each
layer is connected to every unit in each neighboring layer, though it is possible to
construct more sparsely connected DBNs. The connections between the top two
layers are undirected. The connections between all other layers are directed, with
the arrows pointed toward the layer that is closest to the data. See Fig. 20.1b for
an example.
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CHAPTER 20. DEEP GENERATIVE MODELS
A DBN with L hidden layers contains L weight matrices: W
(1)
, . . . , W
(L)
. It
also contains L + 1 bias vectors: b
(0)
, . . . , b
(L)
with b
(0)
providing the biases for
the visible layer. The probability distribution represented by the DBN is given
by
p(h
(L)
, h
(L−1)
) ∝ exp
b
(L)
>
h
(L)
+ b
(L−1)>
h
(L−1)
+ h
(L−1)>
W
(L)
h
(L)
,
p(h
(l)
i
= 1 | h
(l+1)
) = σ
b
(l)
i
+ W
(l+1)>
:,i
h
(l+1)
∀i, ∀l ∈ 1, . . . , L − 2,
p(v
i
= 1 | h
(1)
) = σ
b
(0)
i
+ W
(1)>
:,i
h
(1)
∀i.
In the cause of real-valued visible units, substitute
v ∼ N
v | b
(0)
+ W
(1)>
h
(1)
, β
−1
with β diagonal for tractability. Generalizations to other exponential family vis-
ible units are straightforward, at least in theory. Note that a DBN with only one
hidden layer is just an RBM.
To generate a sample from a DBN, we first run several steps of Gibbs sampling
on the top two hidden layers. This stage is essentially drawing a sample from the
RBM defined by the top two hidden layers. We can then use a single pass of
ancestral sampling through the rest of the model to draw a sample from the
visible units.
Inference in a deep belief network is intractable due to the explaining away
effect within each directed layer, and due to the interaction between the two final
hidden layers. Evaluating or maximizing the standard evidence lower bound on
the log-likelihood is also intractable, because the evidence lower bound takes the
expectation of cliques whose size is equal to the network width.
Evaluating or maximizing the log-likelihood requires not just confronting the
problem of intractable inference to marginalize out the latent variables, but also
the problem of an intractable partition function within the undirected model of
the last two layers.
As a hybrid of directed and undirected models, deep belief networks encounter
many of the difficulties associated with both families of models. Because deep be-
lief networks are partially undirected, they require Markov chains for sampling
and have an intractable partition function. Because they are directed and gener-
ally consist of binary random variables, their evidence lower bound is intractable.
TODO–training procedure TODO–discriminative fine-tuning TODO–view of
MLP as variational inference with very loose bound comment on how this does
not capture intra-layer explaining away interactions comment on how this does
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CHAPTER 20. DEEP GENERATIVE MODELS
not capture inter-layer feedback interactions TODO–quantitative analysis with
AIS TODO–wake sleep?
The term “deep belief network” is commonly used incorrectly to refer to any
kind of deep neural network, even networks without latent variable semantics. The
term “deep belief network” should refer specifically to models with undirected
connections in the deepest layer and directed connections pointing downward
between all other pairs of sequential layers.
The term “deep belief network” may also cause some confusion because the
term “belief network” is sometimes used to refer to purely directed models, while
deep belief networks contain an undirected layer. Deep belief networks also share
the acronym DBN with dynamic Bayesian networks, which are Bayesian networks
for representing Markov chains.
20.5 Deep Boltzmann Machines
A deep Boltzmann machine (DBM) is another kind of deep, generative model
(Salakhutdinov and Hinton, 2009a). Unlike the deep belief network (DBN), it
is an entirely undirected model. Unlike the RBM, the DBM has several layers
of latent variables (RBMs have just one). But like the RBM, within each layer,
each of the variables are mutually independent, conditioned on the variables in
the neighboring layers. See Fig. 20.2 for the graph structure.
Like RBMs and DBNs, DBMs typically contain only binary units – as we
assume in our development of the model – but it may sometimes contain real-
valued visible units.
A DBM is an energy-based model, meaning that the the joint probability
distribution over the model variables is parametrized by an energy function E. In
the case of a deep Boltzmann machine with one visible layer, v, and three hidden
layers, h
(1)
, h
(2)
andh
(3)
, the joint probability is given by:
P
v, h
(1)
, h
(2)
, h
(3)
=
1
Z(θ)
exp
−E(v, h
(1)
, h
(2)
, h
(3)
; θ)
. (20.20)
The DBM energy function is:
E(v, h
(1)
, h
(2)
, h
(3)
; θ) = −v
>
W
(1)
h
(1)
− h
(1)>
W
(2)
h
(2)
− h
(2)>
W
(3)
h
(3)
.
(20.21)
In comparison to the RBM energy function (Eq. 20.9), the DBM energy
function includes connections between the hidden units (latent variables) in the
form of the weight matrices (W
(2)
and W
(3)
). As we will see, these connections
have significant consequences for both the model behavior as well as how we go
about performing inference in the model.
566
CHAPTER 20. DEEP GENERATIVE MODELS
connections
(weights)
W
(1)
W
(2)
W
(3)
hidden layers
(binary units)
visible layer
(binary units)
h
(2)
h
(1)
h
(3)
v
Figure 20.2: The deep Boltzmann machine (offsets on all units are present but suppressed
to simplify notation).
567
CHAPTER 20. DEEP GENERATIVE MODELS
In comparison to fully connected Boltzmann machines (with every unit con-
nected to every other unit), the DBM offers some similar advantages as offered by
the RBM. Specifically, as illustrated in Fig. (TODO: include figure), the DBM
layers can be organized into a bipartiite graph, with odd layers on one side and
even layers on the other. This immediately implies that when we condition on the
variables in the even layer, the variables in the odd layers become conditionally
independent. Of course, when we condition on the variables in the odd layers,
the variables in the even layers also become conditionally independent.
We show this explicitly for the conditional distribution P (h
(1)
= 1 | v, h
(2)
),
in the case of a DBM with two hidden layers (of course, this result generalizes to
a DBM with any number of layers).
P (h
(1)
| v, h
(2)
) =
P (h
(1)
, v, h
(2)
)
P (v, h
(2)
)
=
exp
v
>
W
(1)
h
(1)
+ h
(1)>
W
(2)
h
(2)
P
1
h
(1)
1
=0
···
P
1
h
(1)
n
=0
exp
v
>
W
(1)
h
(1)
+ h
(1)>
W
(2)
h
(2)
=
exp
v
>
W
(1)
h
(1)
+ h
(1)>
W
(2)
h
(2)
P
1
h
(1)
1
=0
···
P
1
h
(1)
n
=0
exp
v
>
W
(1)
h
(1)
+ h
(1)>
W
(2)
h
(2)
=
exp
P
n
j=1
v
>
W
(1)
:,j
h
(1)
j
+ h
(1)>
j
W
(2)
j,:
h
(2)
P
1
h
(1)
1
=0
···
P
1
h
(1)
n
=0
exp
P
n
j
0
=1
v
>
W
(1)
:,j
0
h
(1)
j
0
+ h
(1)>
j
0
W
(2)
j
0
,:
h
(2)
=
Q
j
exp
v
>
W
(1)
:,j
h
(1)
j
+ h
(1)>
j
W
(2)
j,:
h
(2)
P
1
h
(1)
1
=0
···
P
1
h
(1)
n
=0
Q
j
0
exp
v
>
W
(1)
:,j
0
h
(1)
j
0
+ h
(1)>
j
0
W
(2)
j
0
,:
h
(2)
=
Y
j
exp
v
>
W
(1)
:,j
h
(1)
j
+ h
(1)>
j
W
(2)
j,:
h
(2)
P
1
h
(1)
j
=0
exp
v
>
W
(1)
:,j
h
(1)
j
+ h
(1)>
j
W
(2)
j,:
h
(2)
=
Y
j
exp
v
>
W
(1)
:,j
h
(1)
j
+ h
(1)>
j
W
(2)
j,:
h
(2)
1 + exp
v
>
W
(1)
:,j
+ W
(2)
j,:
h
(2)
=
Y
j
P (h
(1)
j
| v, h
(2)
). (20.22)
From the above we can conclude that the conditional distribution for any
layer of the DBM conditioned on the neighboring layers, is fractorial (i.e. all
variables in the layer are conditionally independent). Further, we’ve shown that
568
CHAPTER 20. DEEP GENERATIVE MODELS
this conditional distribution is given by a logistic sigmoid function:
P (h
(1)
j
= 1 | v, h
(2)
) =
exp
v
>
W
(1)
:,j
+ W
(2)
j,:
h
(2)
1 + exp
v
>
W
(1)
:,j
+ W
(2)
j,:
h
(2)
=
1
1 + exp
−v
>
W
(1)
:,j
− W
(2)
j,:
h
(2)
= sigmoid
v
>
W
(1)
:,j
+ W
(2)
j,:
h
(2)
. (20.23)
For the two layer DBM, the conditional distributions of the remaining two
layers (v, h
(2)
) also factorize. That is P (v | h
(1)
) =
Q
d
i=1
P (v
i
| h
(1)
), where
P (v
i
= 1 | h
(1)
) = sigmoid
W
(1)
i,:
h
(1)
. (20.24)
Also, P (h
(2)
| h
(1)
) =
Q
m
k=1
P (h
(2)
k
| h
(1)
), where
P (h
(2)
k
= 1 | h
(1)
) = sigmoid
h
(1)>
W
(2)
:,k
. (20.25)
20.5.1 Interesting Properties
TODO: comparison to DBNs TODO: comparison to neuroscience (local learn-
ing) “most biologically plausible” TODO: description of easy mean field TODO:
description of sampling, comparison to general Boltzmann machines,DBNs
20.5.2 DBM Mean Field Inference
For the two hidden layer DBM, the conditional distributions, P (v | h
(1)
), P (h
(1)
|
v, h
(2)
), and P (h
(2)
| h
(1)
) are factorial, however the posterior distribution over all
the hidden units given the visible unit, i.e. P(h
(1)
, h
(2)
| v), can be complicated.
This is, of course, due to the interaction weights W
(2)
between h
(1)
and h
(2)
which render these variables mutually dependent, given an observed v.
So, like the DBN we are left to seek out methods to approximate the DBM
posterior distribution. However, unlike the DBN, the DBM posterior distribution
over their hidden units – while complicated – is easy to approximate with a varia-
tional approximation (as discussed in Sec. 19.1), specifically a mean field approx-
imation. The mean field approximation is a simple form of variational inference,
where we restrict the approximating distribution to fully factorial distributions.
In the context of DBMs, the mean field equations capture the bidirectional in-
teractions between layers. In this section we derive the iterative approximate
inference procedure originally introduced in Salakhutdinov and Hinton (2009a)
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CHAPTER 20. DEEP GENERATIVE MODELS
In variational approximations to inference, we approach the task of approx-
imating a particular target distribution – in our case, the posterior distribution
over the hidden units given the visible units – by some reasonably simple family
of distributions. In the case of the mean field approximation, the approximat-
ing family is the set of distributions where the hidden units are conditionally
independent.
Let Q(h
(1)
, h
(2)
| v) be the approximation of P (h
(1)
, h
(2)
| v). The mean field
assumption implies that
Q(h
(1)
, h
(2)
| v) =
n
Y
j=1
Q(h
(1)
j
| v)
m
Y
k=1
Q(h
(2)
k
| v). (20.26)
The mean field approximation attempts to find for every observation a mem-
ber of this family of distributions that “best fits” the true posterior P (h
(1)
, h
(2)
|
v). By best fit, we specifically mean that we wish to find the approximation Q
that minimizes the KL-divergence with P , i.e. KL(QkP ) where:
KL(QkP ) =
X
h
Q(h
(1)
, h
(2)
| v) log
Q(h
(1)
, h
(2)
| v)
P (h
(1)
, h
(2)
| v)
!
(20.27)
In general, we do not have to provide a parametric form of the approximating
distribution beyond enforcing the independence assumptions. The variational
approximation procedure is generally able to recover a functional form of the
approximate distribution. However, in the case of a mean field assumption on
binary hidden units (the case we are considering here) there is no loss of generality
by fixing a parametrization of the model in advance.
We parametrize Q as a product of Bernoulli distributions, that is we consider
the probability of each element of h
(1)
to be associated with a parameter. Specif-
ically, for each j ∈ {1, . . . , n},
ˆ
h
(1)
j
= P (h
(1)
j
= 1), where
ˆ
h
(1)
j
∈ [0, 1] and for
each k ∈ {1, . . . , m},
ˆ
h
(2)
k
= P (h
(2)
k
= 1), where
ˆ
h
(2)
k
∈ [0, 1]. Thus we have the
following approximation to the posterior:
Q(h
(1)
, h
(2)
| v) =
n
Y
j=1
Q(h
(1)
j
| v)
m
Y
k=1
Q(h
(2)
k
| v)
=
n
Y
j=1
(
ˆ
h
(1)
j
)
h
(1)
j
(1 −
ˆ
h
(1)
j
)
(1−h
(1)
j
)
×
m
Y
k=1
(
ˆ
h
(2)
k
)
h
(2)
k
(1 −
ˆ
h
(2)
k
)
(1−h
(2)
k
)
(20.28)
Of course, for DBMs with more layers the approximate posterior parametrization
can be extended in the obvious way.
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CHAPTER 20. DEEP GENERATIVE MODELS
Now that we have specified our family of approximating distributions Q. It
remains to specify a procedure for choosing the member of this family that best
fits P . One way to do this is to explicitly minimize KL(QkP ) with respect to the
variational parameters of Q. We will approach the selection of Q from a slightly
different, but entirely equivalent, path. Rather than minimize KL(QkP ), we will
maximize the variational lower bound (or evidence lower bound: see Sec. 19.1),
which in the context of the 2-hidden-layer deep Boltzmann machine is given by:
L(Q) =
X
h
(1)
,h
(2)
Q(h
(1)
, h
(2)
| v) log
P (v, h
(1)
, h
(2)
; θ)
q(h
(1)
, h
(2)
| v)
!
= −
X
h
(1)
,h
(2)
Q(h
(1)
, h
(2)
| v)E(v, h
(1)
, h
(2)
; θ) − log Z(θ) + H(Q), (20.29)
where Z(θ) is the DBM partition function and H(Q) is the entropy of the mean
field distribution.
We wish to maximize the variational lower bound in Eq. 20.29 with respect
to the mean field parameters of Q(h
(1)
, h
(2)
| v). Substituting Eq. 20.28 for
Q(h
(1)
, h
(2)
| v) in the variational lower bound, we get:
L(q) =
X
i
X
j
0
v
i
W
(1)
ij
0
ˆ
h
(1)
j
0
+
X
j
0
X
k
0
ˆ
h
(1)
j
0
W
(2)
j
0
k
0
ˆ
h
(2)
k
0
− ln Z(θ) + H(q). (20.30)
We maximize the above expression (Eq. 20.30) by taking deriatives with
respect to the variational parameters and solving for the system of fixed point
equations:
∂
∂
ˆ
h
(1)
j
L(q) = 0 ∀j ∈ {1, . . . , n},
∂
∂
ˆ
h
(2)
k
L(q) = 0 ∀k ∈ {1, . . . , m}
The gradient with respect to, for example,
ˆ
h
(1)
j
is reasonable straightforward
571
CHAPTER 20. DEEP GENERATIVE MODELS
to evaluate:
∂
∂
ˆ
h
(1)
j
L(q) =
∂
∂
ˆ
h
(1)
j
X
i
X
j
0
v
i
W
(1)
ij
0
ˆ
h
(1)
j
0
+
X
j
0
X
k
0
ˆ
h
(1)
j
0
W
(2)
j
0
k
0
ˆ
h
(2)
k
0
− ln Z(θ) + H(q)
=
∂
∂
ˆ
h
(1)
j
X
i
X
j
0
v
i
W
(1)
ij
0
ˆ
h
(1)
j
0
+
X
j
0
X
k
0
ˆ
h
(1)
j
0
W
(2)
j
0
k
0
ˆ
h
(2)
k
0
− ln Z(θ)
−
X
j
0
ˆ
h
(1)
j
0
ln
ˆ
h
(1)
j
0
+ (1 −
ˆ
h
(1)
j
0
) ln(1 −
ˆ
h
(1)
j
0
)
−
X
k
0
ˆ
h
(2)
k
0
ln
ˆ
h
(2)
k
0
+ (1 −
ˆ
h
(2)
k
0
) ln(1 −
ˆ
h
(2)
k
0
)
#
=
X
i
v
i
W
(1)
ij
+
X
k
0
W
(2)
jk
0
ˆ
h
(2)
k
0
− ln
ˆ
h
(1)
j
0
1 −
ˆ
h
(1)
j
0
,
where in the second line, we have just expanded the terms involved in the entropy
H(q). Setting this derivative to zero and solving for
ˆ
h
(1)
j
, we have
∂
∂
ˆ
h
(1)
j
L(q) = 0 =
X
i
v
i
W
(1)
ij
+
X
k
0
W
(2)
jk
0
ˆ
h
(2)
k
0
− ln
ˆ
h
(1)
j
1 −
ˆ
h
(1)
j
!
ˆ
h
(1)
j
= sigmoid
X
i
v
i
W
(1)
ij
+
X
k
0
W
(2)
jk
0
ˆ
h
(2)
k
0
!
A similar derivation leads to the other set of equations for the second hidden
layer variational parameters. Putting these together, we have the following system
of equations:
ˆ
h
(1)
j
= sigmoid
X
i
v
i
W
(1)
ij
+
X
k
0
W
(2)
jk
0
ˆ
h
(2)
k
0
!
, ∀j (20.31)
ˆ
h
(2)
k
= sigmoid
X
j
0
W
(2)
j
0
k
ˆ
h
(1)
j
0
, ∀k (20.32)
At a fixed point of this system of equations, we have a local maximum of our
variational lower bound L(q). Thus they define a iterative algorithm where we
intersperse updates of
ˆ
h
(1)
j
(using Eq. 20.31) and updates of
ˆ
h
(2)
k
(using Eq.
20.32). So variational inference in the two hidden layer deep Boltzmann machine
amounts to iterating these update equations for
ˆ
h
(1)
j
and
ˆ
h
(2)
k
until convergence. In
practice, ≈ 10 iterations is usually sufficient. Extending approximate variational
inference to deeper DBMs is straightforward.
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CHAPTER 20. DEEP GENERATIVE MODELS
20.5.3 DBM Parameter Learning
Because a deep Boltzmann machine contains restricted Boltzmann machines as
components, the hardness results for computing the partition function and sam-
pling that apply to restricted Boltzmann machines also apply to deep Boltzmann
machines. This means that evaluating the probability mass function of a Boltz-
mann machine requires approximate methods such as annealed importance sam-
pling. Likewise, training the model requires approximations to the gradient of the
log partition function. See chapter 18 for a general description of these methods.
The posterior distribution over the hidden units in a deep Boltzmann ma-
chine is intractable, due to the interactions between different hidden layers. This
means that we must use approximate inference during learning. The standard
approach is to use stochastic gradient ascent on the mean field lower bound, as
described in chapter 19. Mean field is incompatible with most of the methods for
approximating the gradients of the log partition function described in chapter 18.
Moreover, it has been observed that for contrastive divergence to work well, it is
important that the samples from the posterior (e.g., for the 2 hidden layer DBM:
P (h
(1)
, h
(2)
| v)) be exact (Salakhutdinov and Hinton, 2009b). In the case of
the DBM, the intractability of the posterior means that we would have to run
a Gibbs sampler until the samples converged to samples from the true posterior
(i.e. until they “burned in”). Thus for the DBM, CD offers no speedup relative
to naive MCMC methods. Instead, DBMs are usually trained using a variant of
stochastic maximum likelihood. The negative phase samples can be generated
simply by running a Gibbs sampling chain that alternates between sampling the
odd-numbered layers and sampling the even-numbered layers.
Learning in the DBM can equivalently be considered as performing a varia-
tional form of the Expectation Maximization (EM) algorithm. Specifically, con-
sider the variational lower bound for the two-layer DBM (making the dependency
on the model parameters explicit):
L(Q, θ) =
X
i
X
j
0
v
i
W
(1)
ij
0
ˆ
h
(1)
j
0
+
X
j
0
X
k
0
ˆ
h
(1)
j
0
W
(2)
j
0
k
0
ˆ
h
(2)
k
0
− ln Z(θ) + H(Q).
This expression lower bounds the likelihood P(v | θ). So by maximizing this
bound we hope to improve the likelihood. Thus we can think of L(Q, θ) as a
surrogate objective function for the DBM. From this perspective it is natural
to consider a 2-step optimization procedure. In the first step (the E-step or
expectation step), we optimize L(Q, θ) with respect to the variational parameters.
In the case of the two-layer DBM this amounts to solving for
ˆ
h
(1)
and
ˆ
h
(2)
via
the iterative scheme introduced above. Then in the second step (the M-step or
maximization step), we optimize L(Q, θ) with respect to the model parameters
θ.
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CHAPTER 20. DEEP GENERATIVE MODELS
Note that maximizing the variational lower bound with respect to the param-
eters does not guarantee that we improve the true likelihood P (v | θ) on every
step.
3
That said, in practice we often find that we are able to make progress in
training DBMs by maximizing the lower bound L(Q, θ).
Unlike the standard M-step we typically have as part of the EM algorithm, our
M-step will not actually maximize L(Q, θ) with respect to θ (holding Q fixed).
The presence of the partition function makes it impractical to solve the system of
equations ∇
θ
L(Q, θ) = 0 for θ. Instead we will be content to make incremental
progress toward this maximum by taking a small step in the direction of the
gradient ∇
θ
L(Q, θ). In the case of the 2-hidden layer DBM, this is given by:
∇
θ
L(Q, θ) =
∂
∂θ
X
i
X
j
0
v
i
W
(1)
ij
0
ˆ
h
(1)
j
0
+
X
j
0
X
k
0
ˆ
h
(1)
j
0
W
(2)
j
0
k
0
ˆ
h
(2)
k
0
− ln Z(θ) + H(Q)
=
∂
∂θ
X
i
X
j
0
v
i
W
(1)
ij
0
ˆ
h
(1)
j
0
+
X
j
0
X
k
0
ˆ
h
(1)
j
0
W
(2)
j
0
k
0
ˆ
h
(2)
k
0
−
∂
∂θ
ln Z(θ)
(20.33)
The first term in Eq. 20.33 is straightforward, once the values of
ˆ
h
(1)
and
ˆ
h
(2)
have been computed in the E-step. Our use of variation approximate inference
has rendered learning in the DBM as analogous to training in the RBM where
the likelihood gradient (Eq. 20.14) is also composed of a analytically tractable
term and a term involving the gradient of the partition function:
− nE
P (v,h
(1)
,h
(2)
)
h
∇
θ
− E(v, h
(1)
, h
(2)
)
i
(20.34)
Similar to the RBM case, the partition function’s contribution to the gradient
of the variational lower bound is intractable. We approximate it using a varia-
tional version of stochastic maximum likelihood
4
(VSML) algorithm. The non-
variational version of stochastic maximum likelihood algorithm is discussed in
Sec. 18.2 and is applied to RBMs in Sec. 20.3.2.
Unlike in the RBM, the interaction between the hidden units of the DBM
precludes a direct application of the contrastive divergence training algorithm.
Specifically the issue is that, in the positive phase, in order to get samples from
3
In standard EM we do have just a guarantee. The difference is that in the case of standard
EM we assume the true posterior is tractable and therefore we can set Q(h | v, θ
(t)
) = P (h |
v, θ
(t)
). Under these conditions the lower bound is tight, i.e. L(Q, θ) = P (v | θ)
4
Salakhutdinov and Hinton (2009a) refer to this algorithm as persistent contrastive diver-
gence. We prefer to distinguish the variational version of the algorithm as applied to DBMs
from the original stochastic maximum likelihood algorithm that directly (though stochastically)
maximizes the likelihood rather than a lower bound on the likelihood as we are doing here.
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CHAPTER 20. DEEP GENERATIVE MODELS
the posterior P (h | v), one may have to wait a significant amount of time for the
samples to “burn-in”. The necessity for this burn-in renders CD an impractical
algorithm for training CD. As far as we known, variants of CD that make use of
the variational approximation for the positive phase gradient approximation have
not been unexplored.
Variational stochastic maximum likelihood as applied to the DBM is given in
Algorithm 20.3. Recall that we have included the offset parameters in the weight
matrices W
(1)
and W
(2)
. Note that the Gibbs sampling in the negative phase of
the stochastic maximum likelihood algorithm can be divided into two blocks of
updates, one including all odd layers (including the visible layer) and the other
including all even layers. Due to the DBM connection pattern, given the even
layers, the distribution over the odd layers is factorial and thus can be sampled
simultaneously and independently as a block. Likewise given the odd layers, the
even layers can be sampled simultaneously and independently as a block.
20.5.4 Practical Training Strategies
Unfortunately, training a DBM using stochastic maximum likelihood (as described
above) from a random initialization usually results in failure. In some cases, the
model fails to learn to represent the distribution adequately. In other cases, the
DBM may represent the distribution well, but with no higher likelihood than
could be obtained with just an RBM. Note that a DBM with very small weights
in all but the first layer represents approximately the same distribution as an
RBM.
It is not clear exactly why this happens. When DBMs are initialized from a
pretrained configuration, training usually succeeds. See section 20.5.4 for details.
One possibility is that it is difficult to coordinate the learning rate of the stochastic
gradient algorithm with the number of Gibbs steps used in the negative phase
of stochastic maximum likelihood. SML relies on the learning rate being small
enough relative to the number of Gibbs steps that the Gibbs chain can mix again
after each update to the model parameters. The distribution represented by
the model can change very rapidly during the earlier parts of training, and this
may make it difficult for the negative chains employed by SML to fully mix.
As described in section 20.5.5, multi-prediction deep Boltzmann machines avoid
the potential inaccuracy of SML by training with a different objective function
that is less principled but easier to compute. Another possible explanation for
the failure of joint training with mean field and SML is that the Hessian matrix
could be poorly conditioned. This perspective motivates centered deep Boltzmann
machines, presented in section 20.5.6, which modify the model family in order to
obtain a better conditioned Hessian matrix.
575
CHAPTER 20. DEEP GENERATIVE MODELS
c)
d)
b)
a)
Figure 20.3: The deep Boltzmann machine training procedure used to obtain the state of
the art classification accuracy on the MNIST dataset (Srivastava et al., 2014; Salakhutdi-
nov and Hinton, 2009a). TODO: this is not state of the art anymore, just best DBM result
a) Train an RBM by using CD to approximately maximize logP (v). b) Train a second
RBM that models h
(1)
and y by using CD-k to approximately maximize log P (h
(1)
, y)
where h
(1)
is drawn from the first RBM’s posterior conditioned on the data. Increase
k from 1 to 20 during learning. c) Combine the two RBMs into a DBM. Train it to
approximately maximize logP(v, y) using stochastic maximum likelihood with k = 5. d)
Delete y from the model. Define a new set of features h
(1)
and h
(
2) that are obtained
by running mean field inference in the model lacking y. Use these features as input to an
MLP whose structure is the same as an additional pass of mean field, with an additional
output layer for the estimate of y. Initialize the MLP’s weights to be the same as the
DBM’s weights. Train the MLP to approximately maximize log P (y | v) using stochastic
gradient descent and dropout. Figure reprinted from (Goodfellow et al., 2013b).
Layerwise Pretraining
The original and most popular method for overcoming the joint training problem
of DBMs is greedy layerwise pretraining. In this method, each layer of the DBM is
trained in isolation as an RBM. The first layer is trained to model the input data.
Each subsequent RBM is trained to model samples from the previous RBM’s
posterior distribution. After all of the RBMs have been trained in this way, they
can be combined to form a DBM. The DBM may then be trained with PCD.
Typically PCD training will only make a small change in the model’s parameters
and its performance as measured by the log likelihood it assigns to the data, or
its ability to classify inputs.
Note that this greedy layerwise training procedure is not just coordinate as-
cent. It bears some passing resemblance to coordinate ascent because we optimize
one subset of the parameters at each step. However, in the case of the greedy
layerwise training procedure, we actually use a different objective function at each
step.
TODO: details of combining stacked RBMs into a DBM TODO: partial mean
field negative phase
576
CHAPTER 20. DEEP GENERATIVE MODELS
20.5.5 Multi-Prediction Deep Boltzmann Machines
TODO– cite stoyanov TODO
20.5.6 Centered Deep Boltzmann Machines
TODO
This chapter has described the tools needed to fit a very broad class of proba-
bilistic models. Which tool to use depends on which aspects of the log-likelihood
are problematic.
For the simplest distributions p, the log-likelihood is tractable, and the model
can be fit with a straightforward application of maximum likelihood estimation
and gradient ascent as described in chapter
In this chapter, I’ve shown what to do in two different difficult cases. If Z is in-
tractable, then one may still use maximum likelihood estimation via the sampling
approximation techniques described in section 18.2. If p(h | v) is intractable, one
may still train the model using the negative variational free energy rather than
the likelihood, as described in 19.4.
It is also possible that both of these difficulties will arise. An example of this
occurs with the deep Boltzmann machine (Salakhutdinov and Hinton, 2009b),
which is essential a sequence of RBMs composed together. The model is depicted
graphically in Fig. 20.1c.
This model still has the same problem with computing the partition function
as the simpler RBM does. It has also discarded the restricted structure that
made P (h | v) easy to represent in the RBM. The typical way to train the DBM
is to minimize the variational free energy rather than maximize the likelihood.
Of course, the variational free energy still depends on the partition function, so
it is necessary to use sampling techniques to approximate its gradient.
TODO: k-NADE
20.6 Boltzmann Machines for Real-Valued Data
While Boltzmann machines were originally developed for use with binary data,
many applications such as image and audio modeling seem to require the ability
to represent probability distributions over real values. In some cases, it is possible
to treat real-valued data in the interval [0, 1] as representing the expectation of a
binary variable (TODO cite some examples). However, this is not a particularly
theoretically satisfying approach.
577
CHAPTER 20. DEEP GENERATIVE MODELS
Figure 20.4: TODO caption and label, reference from text Figure reprinted from (Good-
fellow et al., 2013b).
578
CHAPTER 20. DEEP GENERATIVE MODELS
20.6.1 Gaussian-Bernoulli RBMs
TODO– cite exponential family harmoniums? TODO– multiple ways of parametriz-
ing them (citations?)
20.6.2 mcRBMs
TODO–mcRBMs
5
TODO–HMC
20.6.3 mPoT Model
TODO–mPoT
20.6.4 Spike and Slab Restricted Boltzmann Machines
Spike and slab restricted Boltzmann machines (Courville et al., 2011) or ssRBMs
provide another means of modeling the covariance structure of real-valued data.
Compared to mcRBMs, ssRBMs have the advantage of requiring neither matrix
inversion nor Hamiltonian Monte Carlo methods.
The spike and slab RBM has two sets of hidden units: the spike units h which
are binary, and the slab units s which are real-valued. The mean of the visible
units conditioned on the hidden units is given by (h s)W
>
. In other words,
each column W
:,i
defines a component that can be appear in the input. The cor-
responding spike variable h
i
determines whether that component is present at all.
The corresponding slab variable s
i
determines the brightness of that component,
if it is present. When a spike variable is active, the corresponding slab variable
adds variance to the input along the axis defined by W
:,i
. This allows us to model
the covariance of the inputs. Fortunately, contrastive divergence and persistent
contrastive divergence with Gibbs sampling are still applicable. There is no need
to invert any matrix.
Gating by the spike variables means that the true marginal distribution over
h s is sparse. This is different from sparse coding, where samples from the
model “almost never” (in the measure theoretic sense) contain zeros in the code,
and MAP inference is required to impose sparsity.
The primary disadvantage of the spike and slab restricted Boltzmann machine
is that some settings of the parameters can correspond to a covariance matrix
that is not positive definite. Such a covariance matrix places more unnormalized
probability on values that are farther from the mean, causing the integral over
all possible outcomes to diverge. Generally this issue can be avoided with simple
heuristic tricks. There is not yet any theoretically satisfying solution. Using
5
The term “mcRBM” is pronounced by saying the name of the letters M-C-R-B-M; the ”mc”
is not pronounced like the “Mc” in “McDonald’s.”
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CHAPTER 20. DEEP GENERATIVE MODELS
constrained optimization to explicitly avoid the regions where the probability is
undefined is difficult to do without being overly conservative and also preventing
the model from accessing high-performing regions of parameter space.
Qualitatively, convolutional variants of the ssRBM produce excellent samples
of natural images. Some examples are shown in Fig. 13.1.
The ssRBM allows for several extensions. Including higher-order interactions
and average-pooling of the slab variables (Courville et al., 2014) enables the model
to learn excellent features for a classifier when labeled data is scarce. Adding a
term to the energy function that prevents the partition function from becoming
undefined results in a sparse coding model, spike and slab sparse coding (Good-
fellow et al., 2013c), also known as S3C.
20.7 Convolutional Boltzmann Machines
As seen in chapter 9, extremely high dimensional inputs such as images place
great strain on the computation, memory, and statistical requirements of machine
learning models. Replacing matrix multiplication by discrete convolution with a
small kernel is the standard way of solving these problems for inputs that have
translation invariant spatial or temporal structure. Desjardins and Bengio (2008)
showed that this approach works well when applied to RBMs.
Deep convolutional networks usually require a pooling operation so that the
spatial size of each successive layer decreases. Feedforward convolutional networks
often use a pooling function such as the maximum of the elements to be pooled. It
is unclear how to generalize this to the setting of energy-based models. We could
introduce a binary pooling unit p over n binary detector units d and enforce
p = max
i
d
i
by setting the energy function to be ∞ whenever that constraint is
violated. This does not scale well though, as it requires evaluating 2
n
different
energy configurations to compute the normalization constant. For a small 3 × 3
pooling region this requires 2
9
= 512 energy function evaluations per pooling unit!
Lee et al. (2009) developed a solution to this problem called probabilistic max
pooling (not to be confused with “stochastic pooling,” which is a technique for
implicitly constructing ensembles of convolutional feedforward networks). The
strategy behind probabilistic max pooling is to constrain the detector units so
at most one may be active at a time. This means there are only n + 1 total
states (one state for each of the n detector units being on, and an additional state
corresponding to all of the detector units being off). The pooling unit is on if
and only if one of the detector units is on. The state with all units off is assigned
energy zero. We can think of this as describing a model with a single variable that
has n + 1 states, or equivalently as model that has n + 1 variables that assigns
energy ∞ to all but n + 1 joint assignments of variables.
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CHAPTER 20. DEEP GENERATIVE MODELS
While efficient, probabilistic max pooling does force the detector units to be
mutually exclusive, which may be a useful regularizing constraint in some contexts
or a harmful limit on model capacity in other contexts. It also does not support
overlapping pooling regions. Overlapping pool regions are usually required to
obtain the best performance from feedforward convolutional networks, so this
constraint probably greatly reduces the performance of convolutional Boltzmann
machines.
Lee et al. (2009) demonstrated that probabilistic max pooling could be used
to build convolutional deep Boltzmann machines
6
. This model is able to perform
operations such as filling in missing portions of its input. However, it has not
proven especially useful as a pretraining strategy for supervised learning, per-
forming similarly to shallow baseline models introduced by Pinto et al. (2008).
TODO: comment on partition function changing when you change the image
size, boundary issues
20.8 Other Boltzmann Machines
TODO–Conditional Boltzmann machine TODO-RNN-RBM TODO–discriminative
Boltzmann machine TODO–Heng’s class relevant and irrelevant Boltzmann ma-
chines TODO– Honglak’s recent work
20.9 Directed Generative Nets
So far in this chapter we have focused on undirected generative models, in the
deep learning context these are almost always parametrized via an energy function
E and possess an intractable partition function Z. The exception being the deep
belief net which can be characterized as a hybrid directed / undirected model.
As discussed in Chapter 13, directed graphical models make up a second
prominent class of graphical models. While directed graphical models have been
the very popular within the greater Machine Learning community, within the
smaller Deep Learning community they have until recently been overshadowed
by undirected models such as the RBM.
In this section we will consider some of the standard directed graphical models
that have traditionally associated with the deep learning community
7
.
6
The publication describes the model as a ”deep belief network” but because it can be de-
scribed as a purely undirected model with tractable layer-wise mean field fixed point updates, it
best fits the definition of a deep Boltzmann machine.
7
The list of directed graphical models that we cover here is inevitably going to be incomplete.
The choice of models we include has more to do their prominence within the Deep Learning
581
CHAPTER 20. DEEP GENERATIVE MODELS
TODO: sigmoid belief nets TODO: refer back to DBN TODO: sparse coding
(maybe drop this from the list, covered elsewhere) TODO: deconvolutional nets?
(AC votes for dropping this) TODO: refer back to S3C and BSC (binary sparse
coding) TODO: NADE will be in RNN chapter, refer back to it here make sure
k-NADE and multi-NADE are mentioned somewhere
TODO: refer to DARN and NVIL?
TODO: Stochastic Feedforward nets
20.9.1 Sigmoid Belief Nets
Sigmoid Belief Nets were originally conceived in response to the Neal (1992)
is one of the first
20.9.2 Differentiable Generator Nets
TODO describe how VAEs and GANs both use the same kind of generator net
cite the generating chairs paper to show how this generator net can be trained
with a procedure that isn’t explicitly unsupervised cite both Kevin and Zoubin’s
version of training a generator net with MMD
20.9.3 Variational Autoencoders
The variational autoencoder is model
L (20.35)
TODO
20.9.4 Variational Interpretation of PSD
TODO, develop the explanation of Sec. 9.1 of Bengio et al. (2013c).
20.9.5 Generative Adversarial Networks
TODO: do we want to still use the capital value function here? Should we say
it’s a functional? Note that the G is OK because it’s a distribution
Generative adversarial networks (TODO cite) are another kind of generative
model based on differentiable mappings from input noise to samples that resemble
the data. In this sense, they closely resemble variational autoencoders. However,
the training procedure is different, and generative model is not necessarily coupled
with an inference network. It is theoretically possible to train an inference network
community and the perceived impact that they have had on the community.
582
CHAPTER 20. DEEP GENERATIVE MODELS
using a strategy similar to the wake-sleep algorithm, but there is no need to infer
posterior variables during training.
Generative adversarial networks are based on game theory. A generator net-
work is trained to map input noise z to samples x. This function g(z) defines
the generative model. The distribution p(z) is not learned; it is simply fixed to
some distribution at the start of training (usually a very unstructured distribu-
tion such as a normal or uniform distribution). We can think of g(z) as defining
a conditional distribution
p(x | z) = N(x | g
z),
1
β
I
,
but in all learning rules we take limit as β → ∞ so we can treat g(z) itself as a
sample and ignore the parametrization of the output distribution.
The generator g is pitted against an adversary: a discriminator network d.
The discriminator network receives data or samples x as input and outputs its
estimate of the probability that x was sampled from the data rather than the
model. During training, d tries to maximize and g tries to minimize a value
function measuring the log probability of d being correct:
g
∗
= arg min
g
max
d
V (g, d)
where
v(g, d) = E
x∼p
data
log d(x) + E
x
P
p
model
log (1 − d(x)) .
The optimization of g can be done simply by backpropagating through d then
g, so the learning process requires neither approximate inference nor approxima-
tion of a partition function gradient. In the case where max
d
v(g, d) is convex
(such as the case where optimization is performed directly in the space of prob-
ability density functions) then the procedure is guaranteed to converge and is
asymptotically consistent. In practice, the procedure can be difficult to make
work, because it can be difficult to keep d optimized well enough to provide a
good estimate of how to update g at all times.
20.9.6 Convolutional Generative Networks
TODO– discuss convolutional generator nets (was GANs paper the first?) and
be sure to cover “unpooling” include the unpooling technique from generatoring
chairs paper, and include others if there are relevant others
20.10 A Generative View of Autoencoders
Many kinds of autoencoders can be viewed as probabilistic models. Different
autoencoders can be interpreted as probabilistic models in different ways.
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CHAPTER 20. DEEP GENERATIVE MODELS
One of the first probabilistic interpretations of autoencoders was the view
denoising autoencoders as energy-based models trained using regularized score
matching. See Sections 15.9.1 and 18.5 for details. Since the early work (Vin-
cent, 2011a) made the connection with Gaussian RBMs, this gave denoising auto-
encoders with a particular parametrization a generative interpretation (they could
be sampled from using the MCMC sampling techniques for Gaussian RBMs) .
The next milestone in connecting auto-encoders with a generative interpreta-
tion came with the work of Rifai et al. (2012). It relied on the view of contractive
auto-encoders as estimators of the tangent of the manifold near which probability
concentrates, discussed in Section 15.10 (see also Figures 15.9, 17.3). In this con-
text, Rifai et al. (2012) demonstrated experimentally that good samples could
be obtained from a trained contractive auto-encoder by alternating encoding,
decoding, and adding noise in a particular way.
As discussed in Section 15.9.1, the application of the encoder/decoder pair
moves the input configuration towards a more probable one. This can be exploited
to actually sample from the estimated distribution. If you consider most Monte-
Carlo Markov Chain (MCMC) algorithms, they have two elements:
1. move from lower probability configurations towards higher probability con-
figurations, and
2. inject randomness so that the chain moves around (and does not stay stuck
at some peak of probability, or mode) and has a chance to visit every config-
uration in the whole space, with a relative frequency equal to its probability
under the underlying model.
So conceptually all one needs to do is to perform encode-decode operations (go
towards more probable configurations) as well as inject noise (to move around the
probable configurations), as hinted at in (Mesnil et al., 2012; Rifai et al., 2012).
20.10.1 Markov Chain Associated with any Denoising Auto-Encoder
The above discussion left open the question of what noise to inject and where, in
order to obtain a Markov chain that would generate from the distribution esti-
mated by the auto-encoder. Bengio et al. (2013b) showed how to construct such
a Markov chain for generalized denoising autoencoders. Generalized denoising au-
toencoders are specified by a denoising distribution for sampling an estimate of
the clean input given the corrupted input.
584
CHAPTER 20. DEEP GENERATIVE MODELS
x
t
˜x
t
ω
t
= g(h
t
)
h
t
= f(˜x
t
)
f
g
P ( X|ω)
C(
˜
X|X)
x
t+1
Figure 20.5: Each step of the Markov chain associated with a trained denoising auto-
encoder, that generates the samples from the probabilistic model implicitly trained by
the denoising reconstruction criterion. Each step consists in (a) injecting corruption
C in state x, yielding
˜
x, (b) encoding it with f, yielding h = f(
˜
x), (c) decoding the
result with g, yielding parameters ω for the reconstruction distribution, and (d) given ω,
sampling a new state from the reconstruction distribution P (x | ω = g(f (
˜
x))). In the
typical squared reconstruction error case, g(h) =
ˆ
x, which estimates E[x |
˜
x], corruption
consists in adding Gaussian noise and sampling from P (x | ω) consists in adding another
Gaussian noise to the reconstruction
ˆ
x. The latter noise level should correspond to the
mean squared error of reconstructions, whereas the injected noise is a hyperparameter
that controls the mixing speed as well as the extent to which the estimator smoothes the
empirical distribution (Vincent, 2011b). In the figure, only the C and P conditionals
are stochastic steps (f and g are deterministic computations), although noise can also
be injected inside the auto-encoder, as in generative stochastic networks (Bengio et al.,
2014b)
Each step of the Markov chain that generates from the estimated distribution
consists of the following sub-steps, illustrated in Figure 20.5:
1. starting from the previous state x, inject corruption noise, sampling
˜
x from
C(
˜
x | x).
2. Encode
˜
x into h = f(
˜
x).
3. Decode h to obtain the parameters ω = g(h) of P (x | ω = g(h)) = P(x |
˜
x).
4. Sample the next state x from P (x | ω = g(h)) = P (x |
˜
x).
The theorem states that if the auto-encoder P (x |
˜
x) forms a consistent estimator
of corresponding true conditional distribution, then the stationary distribution of
585
CHAPTER 20. DEEP GENERATIVE MODELS
the above Markov chain forms a consistent estimator (albeit an implicit one) of
the data generating distribution of x.
P (
˜
x|x)
1
P (
˜
x|x)
P (
˜
x|x)
P (x|
˜
x)
C
Figure 20.6: Illustration of one step of the sampling Markov chain associated with a
denoising auto-encoder (see also Figure 20.5). In the figure, the data (black circles) are
sitting near a low-dimensional manifold (a spiral, here), and the two stochastic steps of
the Markov chain are first to corrupt x (clean image of dog, green circle) into
˜
x (noisy
image of dog, blue circle) via C(
˜
x | x) (here an isotropic Gaussian noise in green), and
then to sample a new x via the estimated denoising P(x |
˜
x). Note how there are many
possible x which could have given rise to
˜
x, and these all lie on the manifold in the
neighborhood of
˜
x, hence the flattened shape of P (x |
˜
x) (in blue). Modified from a
figure first created and graciously authorized by Jason Yosinski.
Figure 20.6 illustrates the sampling process of the DAE in a way that com-
plements Figure 20.5, with a specific imagined example. For a more elaborate
discussion of the probabilistic nature of denoising auto-encoders, and their gen-
eralization (Bengio et al., 2014b), Generative Stochastic Networks (GSNs), see
Section 20.11 below. In particular, the noise does not have to be injected only in
the input, and it could be injected anywhere along the chain. GSNs also gener-
alize DAEs by allowing the state of the Markov chain to be extended beyond the
visible variable x, to include also some latent variable h. Finally, Section 20.11
discusses training strategies for DAEs that are aimed at making it a better gen-
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CHAPTER 20. DEEP GENERATIVE MODELS
erative model and not just a good feature learner.
Figure 20.7: Illustration of the effect of the walk-back training procedure, used for denois-
ing auto-encoders or GSNs in general. The objective is to remove spurious modes faster
by letting the Markov chain go towards them (along the red path, starting on the purple
data manifold and following the arrows plus noise), and then punishing the Markov chain
for this behavior (i.e., walking back to the right place) by telling the chain to return
towards the data manifold (reconstruct the original data).
20.10.2 Clamping and Conditional Sampling
Similarly to Boltzmann machines, denoising auto-encoders and GSNs can be used
to sample from a conditional distribution P (x
f
| x
o
), simply by clamping the
observed units x
f
and only resampling the free units x
o
given x
f
and the sampled
latent variables (if any). This has been introduced by Bengio et al. (2014b).
However, note that Proposition 1 of that paper is missing a condition: the
transition operator (defined by the stochastic mapping going from one state of
the chain to the next) should satisfy detailed balance, described in Section 14.1.1.
An experiment in clamping half of the pixels (the right part of the image) and
running the Markov chain on the other half is shown in Figure 20.8.
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CHAPTER 20. DEEP GENERATIVE MODELS
Figure 20.8: Illustration of clamping the right half of the image and running the Markov
by resampling only the left half at each step. These samples come from a GSN trained
to reconstruct MNIST digits at each time step, i.e., using the walkback procedure.
20.10.3 Walk-Back Training Procedure
The walk-back training procedure was proposed by Bengio et al. (2013b) as a way
to speed-up the convergence of generative training of denoising auto-encoders.
Instead of performing a one-step encode-decode reconstruction, this procedure
consists in alternative multiple stochastic encode-decode steps (as in the genera-
tive Markov chain) initialized at a training example (just like with the contrastive
divergence algorithm, described in Sections 18.2) and 20.3.1) and penalizing the
last probabilistic reconstructions (or all of the reconstructions along the way).
It was shown in that paper that training with k steps is equivalent (in the
sense of achieving the same stationary distribution) as training with one step,
but practically has the advantage that spurious modes farther from the data can
be removed more efficiently, as illustrated in Figure 20.7.
Figure 20.9 illustrates the application of the walk-back procedure in a gener-
ative stochastic network, which is described in the next section.
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CHAPTER 20. DEEP GENERATIVE MODELS
X
0
H
1
X
1
H
2
H
3
X
2
H
0
W
1
W
1
W
1
W
2
W
2
W
2
W
3
W
3
X
1
H
1
X
2
X
3
H
2
H
3
W
3
X
0
data target target target
Figure 20.9: Left: graphical model of the generative Markov chain associated with a gen-
erative stochastic network (GSN). Right specific case where the latent variable is formed
of several layers, each connected to the one above and the one below, making the genera-
tive process very similar to Gibbs sampling in a deep Boltzmann machine (Salakhutdinov
and Hinton, 2009b). The walk-back training procedure is used, i.e., at every step the re-
construction probability distribution is pushed towards generating the training example
(which also initializes the chain).
20.11 Generative Stochastic Networks
Generative stochastic networks (Bengio et al., 2014b) or GSNs are generalizations
of denoising auto-encoders that include latent variables in the generative Markov
chain, in addition to the visible variables (usually denoted x). The generative
Markov chain looks like the one in Figure 20.10. An example of a GSN structured
like a deep Boltzmann machine and trained by the walk-back procedure is shown
in Figure 20.9.
X
2
X
0
X
1
H
0
H
1
H
2
Figure 20.10: Markov chain of a GSN (Generative Stochastic Network) with latent vari-
ables with H and visible variable X, i.e., an unfolding of the generative process with X
k
and H
k
at step k of the chain. TODO: please use h and x etc. throughout the GSN
section
A GSN is parametrized by two conditional probability distributions which
specify one step of the Markov chain:
1. P (X
k
| H
k
) tells how to generate the next visible variable given the current
latent state. Such a “reconstruction distribution” is also found in denoising
auto-encoders, RBMs, DBNs and DBMs.
2. P (H
k
| H
k−1
, X
k−1
) tells how to update the latent state variable, given the
previous latent state and visible variable.
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CHAPTER 20. DEEP GENERATIVE MODELS
Denoising auto-encoders and GSNs differ from classical probabilistic models
(directed or undirected) in that it parametrizes the generative process itself rather
than the mathematical specification of the joint distribution of visible and latent
variables. Instead, the latter is defined implicitly, if it exists, as the stationary
distribution of the generative Markov chain. The conditions for existence of the
stationary distribution are mild (basically, the chain mixes) but can be violated
by some choices of the transition distributions (for example, if they were deter-
ministic).
One could imagine different training criteria for GSNs. The one proposed and
evaluated by Bengio et al. (2014b) is simply reconstruction log-probability on the
visible units, just like for denoising auto-encoders. This is achieved by clamping
X
0
= x to the observed example and maximizing the probability of generating x
at some subsequent time steps, i.e., maximizing log P (X
k
= x | H
k
), where H
k
is sampled from the chain, given X
0
= x. In order to estimate the gradient of
log P (X
k
= x | H
k
) with respect to the other pieces of the model, Bengio et al.
(2014b) use the reparametrization trick, introduced in Section 13.5.1.
The walk-back training protocol (described in Section 20.10.3 was used (Bengio
et al., 2014b) to improve training convergence of GSNS.
20.11.1 Discriminant GSNs
Whereas the original formulation of GSNs (Bengio et al., 2014b) was meant for
unsupervised learning and implicitly modeling P (x) for observed data x, it is
possible to modify the framework to optimize P (y | x).
For example, Zhou and Troyanskaya (2014) generalize GSNs in this way, by
only back-propagating the reconstruction log-probability over the output vari-
ables, keeping the input variables fixed. They applied this successfully to model
sequences (protein secondary structure) and introduced a (one-dimensional) con-
volutional structure in the transition operator of the Markov chain. Keep in mind
that, for each step of the Markov chain, one generates a new sequence for each
layer, and that sequence is the input for computing other layer values (say the
one below and the one above) at the next time step, as illustrated in Figure 20.11.
Hence the Markov chain is really over the output variable (and associated
higher-level hidden layers), and the input sequence only serves to condition that
chain, with back-propagation allowing to learn how the input sequence can con-
dition the output distribution implicitly represented by the Markov chain. It is
therefore a case of using the GSN in the context of structured outputs, where
P (y | x) does not have a simple parametric form but instead the components of
y are statistically dependent of each other, given x, in complicated ways.
Z¨ohrer and Pernkopf (2014) considered a hybrid model that combines a su-
pervised objective (as in the above work) and an unsupervised objective (as in
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CHAPTER 20. DEEP GENERATIVE MODELS
Figure 20.11: Markov chain arising out of a discriminant GSN, i.e., where a GSN is used
as a structured output model over a variable y, conditioned on an input X. Reproduced
with permission from Zhou and Troyanskaya (2014). The structure is as in a GSN (over
the output) but with computations being conditioned on the input X at each step.
the original GSN work), by simply adding (with a different weight) the super-
vised and unsupervised costs i.e., the reconstruction log-probabilities of y and x
respectively. Such a hybrid criterion had previously been introduced for RBMs
by Larochelle and Bengio (2008a). They show improved classification performance
using this scheme.
20.12 Methodological Notes
Researchers studying generative models often need to compare one generative
model to another, usually in order to demonstrate that a newly invented genera-
tive model is better at capturing some distribution than the pre-existing models.
This can be a difficult and subtle task. In many cases, we can not actually
evaluate the log probability of the data under the model, but only an approxi-
mation. In these cases, it’s important to think and communicate clearly about
exactly what is being measured. For example, suppose we can evaluate a stochas-
tic estimate of the log-likelihood for model A, and a deterministic lower bound
on the log-likelihood for model B. If model A gets a higher score than model B,
which is better? If we care about determining which model has a better internal
representation of the distribution, we actually cannot tell, unless we have some
way of determining how loose the bound for model B is. However, if we care about
how well we can use the model in practice, for example to perform anomaly de-
tection, then it is fair to say that model A is better based on a criterion specific
to the practical task of interest, e.g., based on ranking test examples and ranking
criterian such as precision and recall.
Another subtletly of evaluating generative models is that the evaluation met-
rics are often hard research problems in and of themselves. It can be very difficult
to establish that models are being compared fairly. For example, suppose we use
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CHAPTER 20. DEEP GENERATIVE MODELS
AIS to estimate log Z in order to compute log ˜p(x) − log Z for a new model we
have just invented. A computationally economical implementation of AIS may
fail to find several modes of the model distribution and underestimate Z, which
will result in us overestimating log p(x). It can thus be difficult to tell whether a
good likelihood estimate is due to a good model or a bad AIS implementation.
Other fields of machine learning usually allow for some variation in the pre-
processing of the data. For example, when comparing the accuracy of object
recognition algorithms, it is usually acceptable to preprocess the input images
slightly differently for each algorithm based on what kind of input requirements
it has. Generative modeling is different because changes in preprocessing, even
very small and subtle ones, are completely unacceptable. Any change to the input
data changes the distribution to be captured and fundamentally alters the task.
For example, multiplying the input by 0.1 will artificially increase likelihood by
10.
Issues with preprocessing commonly arise when benchmarking generative mod-
els on the MNIST dataset, one of the more popular generative modeling bench-
marks. MNIST consists of grayscale images. Some models treat MNIST images
as points in a real vector space, while others treat them as binary. Yet others
treat the grayscale values as probabilities for a binary samples. It is essential to
compare real-valued models only to other real-valued models and binary-valued
models only to other binary-valued models. Otherwise the likelihoods measured
are not on the same space. (For the binary-valued models, the log-likelihood can
be at most 0., while for real-valued models it can be arbitrarily high, since it is
the measurement of a density) Among binary models, it is important to com-
pare models using exactly the same kind of binarization. For example, we might
binarize a gray pixel to 0 or 1 by thresholding at 0.5, or by drawing a random
sample whose probability of being 1 is given by the gray pixel intensity. If we use
the random binarization, we might binarize the whole dataset once, or we might
draw a different random example for each step of training and then draw multiple
samples for evaluation. Each of these three schemes yields wildly different like-
lihood numbers, and when comparing different models it is important that both
models use the same binarization scheme for training and for evaluation. In fact,
researchers who apply a single random binarization step share a file containing
the results of the random binarization, so that there is no difference in results
based on different outcomes of the binarization step.
Finally, in some cases the likelihood seems not to measure any attribute of
the model that we really care about. For example, real-valued models of MNIST
can obtain arbitrarily high likelihood by assigning arbitrarily low variance to
background pixels that never change. Models and algorithms that detect these
constant features can reap unlimited rewards, even though this is not a very useful
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CHAPTER 20. DEEP GENERATIVE MODELS
thing to do. This strongly suggests a need for developing other ways of evaluating
generative models.
Although this is still an open question, this might be achieved by converting
the problem into a classification task. For example, we have seen that the NCE
method (Noise Contrastive Estimation, Section 18.6) compares the density of the
training data according to a learned unnormalized model with its density under
a background model. However, generative models do not always provide us with
an energy function (equivalently, an unnormalized density), e.g., deep Boltzmann
machines, generative stochastic networks, most denoising auto-encoders (that are
not guaranteed to correspond to an energy function), deep Belief networks, etc.
Therefore, it would be interesting to consider a classification task in which one
tries to distinguish the training examples from the generated examples. This is
precisely what is achieved by the discriminator network of generative adversarial
networks (Section 20.9.5). However, it would require an expensive operation
(training a discriminator) each time one would have to evaluate performance
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CHAPTER 20. DEEP GENERATIVE MODELS
Algorithm 20.3 The variational stochastic maximum likelihood algorithm for
training a 2 hidden-layer DBM.
Set , the step size, to a small positive number
Set k, the number of Gibbs steps, high enough to allow a Markov chain of
p(v, h
(1)
, h
(2)
; θ +∆
θ
) toburn in, starting from samples from p(v, h
(1)
, h
(2)
; θ).
Initialize three random matrices,
˜
V ,
˜
H
(1)
and
˜
H
(2)
each with m columns set
to random values (e.g., from bernoulli distributions, possibly with marginals
matched to the model’s marginals).
while Not converged (learning loop) do
Sample a minibatch of m examples from the training data and arrange them
as the columns of a matrix V =
v
(1)
, . . . , v
(m)
from the training set.
while Not converged (Mean-field inference loop) do
˜
H
(1)
← sigmoid
V
>
W
(1)
+
˜
H
(2) >
W
(2) >
.
˜
H
(2)
← sigmoid
˜
H
(1)
W
(2)
.
end while
∆
W
(1)
←
1
m
V
ˆ
H
(1) >
∆
W
(2)
←
1
m
ˆ
H
(1)
ˆ
H
(2) >
for l = 1 to k (Gibbs sampling) do
Gibbs block 1:
˜
V sampled from
Q
m
i=1
Q
d
a=1
sigmoid
W
(1)
a,:
˜
H
(2)
:,i
.
˜
H
(2)
sampled from
Q
m
i=1
Q
m
b=1
sigmoid
˜
H
(1) >
:,i
W
(2)
:,b
.
Gibbs block 2:
˜
H
(1)
sampled from
Q
m
i=1
Q
n
j=1
sigmoid
˜
V
>
:,i
W
(1)
:,j
+ W
(2)
j,:
˜
H
(2)
:,i
.
end for
∆
W
(1)
← ∆
W
(1)
−
1
m
V
ˆ
H
(1) >
∆
W
(2)
← ∆
W
(2)
−
1
m
ˆ
H
(1)
ˆ
H
(2) >
W
(1)
← W
(1)
+ ∆
W
(1)
W
(2)
← W
(2)
+ ∆
W
(2)
end while
594
