Deep belief nets

Содержание

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Some things you will learn in this tutorial How to learn

Some things you will learn in this tutorial
How to learn multi-layer

generative models of unlabelled data by learning one layer of features at a time.
How to add Markov Random Fields in each hidden layer.
How to use generative models to make discriminative training methods work much better for classification and regression.
How to extend this approach to Gaussian Processes and how to learn complex, domain-specific kernels for a Gaussian Process.
How to perform non-linear dimensionality reduction on very large datasets
How to learn binary, low-dimensional codes and how to use them for very fast document retrieval.
How to learn multilayer generative models of high-dimensional sequential data.
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A spectrum of machine learning tasks Low-dimensional data (e.g. less than

A spectrum of machine learning tasks

Low-dimensional data (e.g. less than 100

dimensions)
Lots of noise in the data
There is not much structure in the data, and what structure there is, can be represented by a fairly simple model.
The main problem is distinguishing true structure from noise.

High-dimensional data (e.g. more than 100 dimensions)
The noise is not sufficient to obscure the structure in the data if we process it right.
There is a huge amount of structure in the data, but the structure is too complicated to be represented by a simple model.
The main problem is figuring out a way to represent the complicated structure so that it can be learned.

Typical Statistics------------Artificial Intelligence

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Historical background: First generation neural networks Perceptrons (~1960) used a layer

Historical background: First generation neural networks

Perceptrons (~1960) used a layer of hand-coded

features and tried to recognize objects by learning how to weight these features.
There was a neat learning algorithm for adjusting the weights.
But perceptrons are fundamentally limited in what they can learn to do.

non-adaptive
hand-coded
features

output units e.g. class labels

input units e.g. pixels

Sketch of a typical perceptron from the 1960’s

Bomb

Toy

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Second generation neural networks (~1985) input vector hidden layers outputs Back-propagate

Second generation neural networks (~1985)

input vector

hidden layers

outputs

Back-propagate error signal to get

derivatives for learning

Compare outputs with correct answer to get error signal

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A temporary digression Vapnik and his co-workers developed a very clever

A temporary digression

Vapnik and his co-workers developed a very clever type

of perceptron called a Support Vector Machine.
Instead of hand-coding the layer of non-adaptive features, each training example is used to create a new feature using a fixed recipe.
The feature computes how similar a test example is to that training example.
Then a clever optimization technique is used to select the best subset of the features and to decide how to weight each feature when classifying a test case.
But its just a perceptron and has all the same limitations.
In the 1990’s, many researchers abandoned neural networks with multiple adaptive hidden layers because Support Vector Machines worked better.
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What is wrong with back-propagation? It requires labeled training data. Almost

What is wrong with back-propagation?

It requires labeled training data.
Almost all data

is unlabeled.
The learning time does not scale well
It is very slow in networks with multiple hidden layers.
It can get stuck in poor local optima.
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Overcoming the limitations of back-propagation Keep the efficiency and simplicity of

Overcoming the limitations of back-propagation
Keep the efficiency and simplicity of using

a gradient method for adjusting the weights, but use it for modeling the structure of the sensory input.
Adjust the weights to maximize the probability that a generative model would have produced the sensory input.
Learn p(image) not p(label | image)
If you want to do computer vision, first learn computer graphics
What kind of generative model should we learn?
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Belief Nets A belief net is a directed acyclic graph composed

Belief Nets

A belief net is a directed acyclic graph composed

of stochastic variables.
We get to observe some of the variables and we would like to solve two problems:
The inference problem: Infer the states of the unobserved variables.
The learning problem: Adjust the interactions between variables to make the network more likely to generate the observed data.

stochastic
hidden cause

visible
effect

We will use nets composed of layers of stochastic binary variables with weighted connections. Later, we will generalize to other types of variable.

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Stochastic binary units (Bernoulli variables) These have a state of 1

Stochastic binary units (Bernoulli variables)

These have a state of 1 or 0.
The

probability of turning on is determined by the weighted input from other units (plus a bias)

0

0

1

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Learning Deep Belief Nets It is easy to generate an unbiased

Learning Deep Belief Nets

It is easy to generate an unbiased

example at the leaf nodes, so we can see what kinds of data the network believes in.
It is hard to infer the posterior distribution over all possible configurations of hidden causes.
It is hard to even get a sample from the posterior.
So how can we learn deep belief nets that have millions of parameters?

stochastic
hidden cause

visible
effect

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The learning rule for sigmoid belief nets Learning is easy if

The learning rule for sigmoid belief nets

Learning is easy if we

can get an unbiased sample from the posterior distribution over hidden states given the observed data.
For each unit, maximize the log probability that its binary state in the sample from the posterior would be generated by the sampled binary states of its parents.

j

i

learning rate

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Explaining away (Judea Pearl) Even if two hidden causes are independent,

Explaining away (Judea Pearl)

Even if two hidden causes are independent, they

can become dependent when we observe an effect that they can both influence.
If we learn that there was an earthquake it reduces the probability that the house jumped because of a truck.

truck hits house

earthquake

house jumps

20

20

-20

-10

-10


p(1,1)=.0001
p(1,0)=.4999
p(0,1)=.4999
p(0,0)=.0001

posterior

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Why it is usually very hard to learn sigmoid belief nets

Why it is usually very hard to learn sigmoid belief nets

one layer at a time

To learn W, we need the posterior distribution in the first hidden layer.
Problem 1: The posterior is typically complicated because of “explaining away”.
Problem 2: The posterior depends on the prior as well as the likelihood.
So to learn W, we need to know the weights in higher layers, even if we are only approximating the posterior. All the weights interact.
Problem 3: We need to integrate over all possible configurations of the higher variables to get the prior for first hidden layer. Yuk!

data

hidden variables

hidden variables

hidden variables

likelihood

W

prior

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Two types of generative neural network If we connect binary stochastic

Two types of generative neural network

If we connect binary stochastic neurons

in a directed acyclic graph we get a Sigmoid Belief Net (Radford Neal 1992).
If we connect binary stochastic neurons using symmetric connections we get a Boltzmann Machine (Hinton & Sejnowski, 1983).
If we restrict the connectivity in a special way, it is easy to learn a Boltzmann machine.
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Restricted Boltzmann Machines (Smolensky ,1986, called them “harmoniums”) We restrict the

Restricted Boltzmann Machines (Smolensky ,1986, called them “harmoniums”)

We restrict the connectivity to

make learning easier.
Only one layer of hidden units.
We will deal with more layers later
No connections between hidden units.
In an RBM, the hidden units are conditionally independent given the visible states.
So we can quickly get an unbiased sample from the posterior distribution when given a data-vector.
This is a big advantage over directed belief nets

hidden

i

j

visible

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The Energy of a joint configuration (ignoring terms to do with

The Energy of a joint configuration (ignoring terms to do with biases)

weight

between units i and j

Energy with configuration v on the visible units and h on the hidden units

binary state of visible unit i

binary state of hidden unit j

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Weights ? Energies ? Probabilities Each possible joint configuration of the

Weights ? Energies ? Probabilities
Each possible joint configuration of the visible

and hidden units has an energy
The energy is determined by the weights and biases (as in a Hopfield net).
The energy of a joint configuration of the visible and hidden units determines its probability:
The probability of a configuration over the visible units is found by summing the probabilities of all the joint configurations that contain it.
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Using energies to define probabilities The probability of a joint configuration

Using energies to define probabilities

The probability of a joint configuration over

both visible and hidden units depends on the energy of that joint configuration compared with the energy of all other joint configurations.
The probability of a configuration of the visible units is the sum of the probabilities of all the joint configurations that contain it.

partition function

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A picture of the maximum likelihood learning algorithm for an RBM

A picture of the maximum likelihood learning algorithm for an RBM

i

j

i

j

i

j

i

j

t

= 0 t = 1 t = 2 t = infinity

Start with a training vector on the visible units.
Then alternate between updating all the hidden units in parallel and updating all the visible units in parallel.

a fantasy

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A quick way to learn an RBM i j i j

A quick way to learn an RBM

i

j

i

j

t = 0 t =

1

Start with a training vector on the visible units.
Update all the hidden units in parallel
Update the all the visible units in parallel to get a “reconstruction”.
Update the hidden units again.

This is not following the gradient of the log likelihood. But it works well. It is approximately following the gradient of another objective function (Carreira-Perpinan & Hinton, 2005).

reconstruction

data

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How to learn a set of features that are good for

How to learn a set of features that are good for

reconstructing images of the digit 2

50 binary feature neurons

16 x 16 pixel image

50 binary feature neurons

16 x 16 pixel image

Increment weights between an active pixel and an active feature

Decrement weights between an active pixel and an active feature

data (reality)

reconstruction (better than reality)

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The final 50 x 256 weights Each neuron grabs a different feature.

The final 50 x 256 weights

Each neuron grabs a different feature.


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Reconstruction from activated binary features Data Reconstruction from activated binary features

Reconstruction from activated binary features

Data

Reconstruction from activated binary features

Data

How well can

we reconstruct the digit images from the binary feature activations?

New test images from the digit class that the model was trained on

Images from an unfamiliar digit class (the network tries to see every image as a 2)

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Three ways to combine probability density models (an underlying theme of

Three ways to combine probability density models (an underlying theme of

the tutorial)

Mixture: Take a weighted average of the distributions.
It can never be sharper than the individual distributions. It’s a very weak way to combine models.
Product: Multiply the distributions at each point and then renormalize.
Exponentially more powerful than a mixture. The normalization makes maximum likelihood learning difficult, but approximations allow us to learn anyway.
Composition: Use the values of the latent variables of one model as the data for the next model.
Works well for learning multiple layers of representation, but only if the individual models are undirected.

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Training a deep network (the main reason RBM’s are interesting) First

Training a deep network (the main reason RBM’s are interesting)

First train a

layer of features that receive input directly from the pixels.
Then treat the activations of the trained features as if they were pixels and learn features of features in a second hidden layer.
It can be proved that each time we add another layer of features we improve a variational lower bound on the log probability of the training data.
The proof is slightly complicated.
But it is based on a neat equivalence between an RBM and a deep directed model (described later)
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The generative model after learning 3 layers To generate data: Get

The generative model after learning 3 layers

To generate data:
Get an

equilibrium sample from the top-level RBM by performing alternating Gibbs sampling for a long time.
Perform a top-down pass to get states for all the other layers.
So the lower level bottom-up connections are not part of the generative model. They are just used for inference.

h2

data

h1

h3

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Why does greedy learning work? An aside: Averaging factorial distributions If

Why does greedy learning work? An aside: Averaging factorial distributions

If

you average some factorial distributions, you do NOT get a factorial distribution.
In an RBM, the posterior over the hidden units is factorial for each visible vector.
But the aggregated posterior over all training cases is not factorial (even if the data was generated by the RBM itself).
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Why does greedy learning work? Each RBM converts its data distribution

Why does greedy learning work?

Each RBM converts its data distribution into

an aggregated posterior distribution over its hidden units.
This divides the task of modeling its data into two tasks:
Task 1: Learn generative weights that can convert the aggregated posterior distribution over the hidden units back into the data distribution.
Task 2: Learn to model the aggregated posterior distribution over the hidden units.
The RBM does a good job of task 1 and a moderately good job of task 2.
Task 2 is easier (for the next RBM) than modeling the original data because the aggregated posterior distribution is closer to a distribution that an RBM can model perfectly.

data distribution on visible units

aggregated posterior distribution on hidden units

Task 2

Task 1

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Why does greedy learning work? The weights, W, in the bottom

Why does greedy learning work?

The weights, W, in the bottom level

RBM define p(v|h) and they also, indirectly, define p(h).
So we can express the RBM model as

If we leave p(v|h) alone and improve p(h), we will improve p(v).
To improve p(h), we need it to be a better model of the aggregated posterior distribution over hidden vectors produced by applying W to the data.

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Which distributions are factorial in a directed belief net? In a

Which distributions are factorial in a directed belief net?

In a directed

belief net with one hidden layer, the posterior over the hidden units for each visible vector is non-factorial (due to explaining away).
The aggregated posterior is factorial if the data was generated by the directed model.
It’s the opposite way round from an undirected model.
The intuitions that people have from using directed models are very misleading for undirected models.
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Why does greedy learning fail in a directed module? A directed

Why does greedy learning fail in a directed module?

A directed module

also converts its data distribution into an aggregated posterior
Task 1 is now harder because the posterior for each training case is non-factorial.
Task 2 is performed using an independent prior. This is a bad approximation unless the aggregated posterior is close to factorial.
A directed module attempts to make the aggregated posterior factorial in one step.
This is too difficult and leads to a bad compromise. There is no guarantee that the aggregated posterior is easier to model than the data distribution.

data distribution on visible units

Task 2

Task 1

aggregated posterior distribution on hidden units

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A model of digit recognition 2000 top-level neurons 500 neurons 500

A model of digit recognition

2000 top-level neurons

500 neurons

500 neurons

28 x

28 pixel image

10 label neurons

The model learns to generate combinations of labels and images.
To perform recognition we start with a neutral state of the label units and do an up-pass from the image followed by a few iterations of the top-level associative memory.

The top two layers form an associative memory whose energy landscape models the low dimensional manifolds of the digits.
The energy valleys have names

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Fine-tuning with a contrastive version of the “wake-sleep” algorithm After learning

Fine-tuning with a contrastive version of the “wake-sleep” algorithm

After learning

many layers of features, we can fine-tune the features to improve generation.
1. Do a stochastic bottom-up pass
Adjust the top-down weights to be good at reconstructing the feature activities in the layer below.
Do a few iterations of sampling in the top level RBM
-- Adjust the weights in the top-level RBM.
Do a stochastic top-down pass
Adjust the bottom-up weights to be good at reconstructing the feature activities in the layer above.
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Show the movie of the network generating digits (available at www.cs.toronto/~hinton)

Show the movie of the network generating digits (available at www.cs.toronto/~hinton)

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Samples generated by letting the associative memory run with one label

Samples generated by letting the associative memory run with one label

clamped. There are 1000 iterations of alternating Gibbs sampling between samples.
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Examples of correctly recognized handwritten digits that the neural network had

Examples of correctly recognized handwritten digits that the neural network had never

seen before

Its very good

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How well does it discriminate on MNIST test set with no

How well does it discriminate on MNIST test set with no

extra information about geometric distortions?

Generative model based on RBM’s 1.25%
Support Vector Machine (Decoste et. al.) 1.4%
Backprop with 1000 hiddens (Platt) ~1.6%
Backprop with 500 -->300 hiddens ~1.6%
K-Nearest Neighbor ~ 3.3%
See Le Cun et. al. 1998 for more results
Its better than backprop and much more neurally plausible because the neurons only need to send one kind of signal, and the teacher can be another sensory input.

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Unsupervised “pre-training” also helps for models that have more data and

Unsupervised “pre-training” also helps for models that have more data and

better priors

Ranzato et. al. (NIPS 2006) used an additional 600,000 distorted digits.
They also used convolutional multilayer neural networks that have some built-in, local translational invariance.

Back-propagation alone: 0.49%
Unsupervised layer-by-layer
pre-training followed by backprop: 0.39% (record)

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Another view of why layer-by-layer learning works There is an unexpected

Another view of why layer-by-layer learning works

There is an unexpected equivalence

between RBM’s and directed networks with many layers that all use the same weights.
This equivalence also gives insight into why contrastive divergence learning works.
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An infinite sigmoid belief net that is equivalent to an RBM

An infinite sigmoid belief net that is equivalent to an RBM

The

distribution generated by this infinite directed net with replicated weights is the equilibrium distribution for a compatible pair of conditional distributions: p(v|h) and p(h|v) that are both defined by W
A top-down pass of the directed net is exactly equivalent to letting a Restricted Boltzmann Machine settle to equilibrium.
So this infinite directed net defines the same distribution as an RBM.

v1

h1

v0

h0

v2

h2

etc.

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The variables in h0 are conditionally independent given v0. Inference is

The variables in h0 are conditionally independent given v0.
Inference is trivial.

We just multiply v0 by W transpose.
The model above h0 implements a complementary prior.
Multiplying v0 by W transpose gives the product of the likelihood term and the prior term.
Inference in the directed net is exactly equivalent to letting a Restricted Boltzmann Machine settle to equilibrium starting at the data.

Inference in a directed net with replicated weights

v1

h1

v0

h0

v2

h2

etc.

+

+

+

+

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The learning rule for a sigmoid belief net is: With replicated

The learning rule for a sigmoid belief net is:
With replicated weights

this becomes:

v1

h1

v0

h0

v2

h2

etc.

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First learn with all the weights tied This is exactly equivalent

First learn with all the weights tied
This is exactly equivalent to

learning an RBM
Contrastive divergence learning is equivalent to ignoring the small derivatives contributed by the tied weights between deeper layers.

Learning a deep directed network

v1

h1

v0

h0

v2

h2

etc.

v0

h0

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Then freeze the first layer of weights in both directions and

Then freeze the first layer of weights in both directions and

learn the remaining weights (still tied together).
This is equivalent to learning another RBM, using the aggregated posterior distribution of h0 as the data.

v1

h1

v0

h0

v2

h2

etc.

v1

h0

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How many layers should we use and how wide should they

How many layers should we use and how wide should they

be? (I am indebted to Karl Rove for this slide)

How many lines of code should an AI program use and how long should each line be?
This is obviously a silly question.
Deep belief nets give the creator a lot of freedom.
How best to make use of that freedom depends on the task.
With enough narrow layers we can model any distribution over binary vectors (Sutskever & Hinton, 2007)
If freedom scares you, stick to convex optimization of shallow models that are obviously inadequate for doing Artificial Intelligence.

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What happens when the weights in higher layers become different from

What happens when the weights in higher layers become different from

the weights in the first layer?

The higher layers no longer implement a complementary prior.
So performing inference using the frozen weights in the first layer is no longer correct.
Using this incorrect inference procedure gives a variational lower bound on the log probability of the data.
We lose by the slackness of the bound.
The higher layers learn a prior that is closer to the aggregated posterior distribution of the first hidden layer.
This improves the network’s model of the data.
Hinton, Osindero and Teh (2006) prove that this improvement is always bigger than the loss.

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A stack of RBM’s (Yee-Whye Teh’s idea) Each RBM has the

A stack of RBM’s (Yee-Whye Teh’s idea)

Each RBM has the same subscript

as its hidden layer.
Each RBM defines its own distribution over its visible vectors
Each RBM defines its own distribution over its hidden vectors

v

h1

h2

hL

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The variational bound Now we cancel out all of the partition

The variational bound

Now we cancel out all of the partition functions

except the top one and replace log probabilities by goodnesses using the fact that:

This has simple derivatives that give a more justifiable fine-tuning algorithm than contrastive wake-sleep.

Each time we replace the prior over the hidden units by a better prior, we win by the difference in the probability assigned

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Summary so far Restricted Boltzmann Machines provide a simple way to

Summary so far

Restricted Boltzmann Machines provide a simple way to learn

a layer of features without any supervision.
Maximum likelihood learning is computationally expensive because of the normalization term, but contrastive divergence learning is fast and usually works well.
Many layers of representation can be learned by treating the hidden states of one RBM as the visible data for training the next RBM (a composition of experts).
This creates good generative models that can then be fine-tuned.
Contrastive wake-sleep can fine-tune generation.
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Overview of the rest of the tutorial How to fine-tune a

Overview of the rest of the tutorial

How to fine-tune a greedily

trained generative model to be better at discrimination.
How to learn a kernel for a Gaussian process.
How to use deep belief nets for non-linear dimensionality reduction and document retrieval.
How to use deep belief nets for sequential data.
How to learn a generative hierarchy of conditional random fields.
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BREAK

BREAK

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Fine-tuning for discrimination First learn one layer at a time greedily.

Fine-tuning for discrimination

First learn one layer at a time greedily.
Then treat

this as “pre-training” that finds a good initial set of weights which can be fine-tuned by a local search procedure.
Contrastive wake-sleep is one way of fine-tuning the model to be better at generation.
Backpropagation can be used to fine-tune the model for better discrimination.
This overcomes many of the limitations of standard backpropagation.
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Why backpropagation works better after greedy pre-training Greedily learning one layer

Why backpropagation works better after greedy pre-training

Greedily learning one layer at

a time scales well to really big networks, especially if we have locality in each layer.
We do not start backpropagation until we already have sensible weights that already do well at the task.
So the initial gradients are sensible and backprop only needs to perform a local search.
Most of the information in the final weights comes from modeling the distribution of input vectors.
The precious information in the labels is only used for the final fine-tuning. It slightly modifies the features. It does not need to discover features.
This type of backpropagation works well even if most of the training data is unlabeled. The unlabeled data is still very useful for discovering good features.
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First, model the distribution of digit images 2000 units 500 units

First, model the distribution of digit images

2000 units

500 units

500 units


28 x 28 pixel image

The network learns a density model for unlabeled digit images. When we generate from the model we get things that look like real digits of all classes.
But do the hidden features really help with digit discrimination?
Add 10 softmaxed units to the top and do backpropagation.

The top two layers form a restricted Boltzmann machine whose free energy landscape should model the low dimensional manifolds of the digits.

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Results on permutation-invariant MNIST task Very carefully trained backprop net with

Results on permutation-invariant MNIST task

Very carefully trained backprop net with 1.6%

one or two hidden layers (Platt; Hinton)
SVM (Decoste & Schoelkopf, 2002) 1.4%
Generative model of joint density of 1.25% images and labels (+ generative fine-tuning)
Generative model of unlabelled digits 1.15% followed by gentle backpropagation (Hinton & Salakhutdinov, Science 2006)
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Combining deep belief nets with Gaussian processes Deep belief nets can

Combining deep belief nets with Gaussian processes

Deep belief nets can benefit

a lot from unlabeled data when labeled data is scarce.
They just use the labeled data for fine-tuning.
Kernel methods, like Gaussian processes, work well on small labeled training sets but are slow for large training sets.
So when there is a lot of unlabeled data and only a little labeled data, combine the two approaches:
First learn a deep belief net without using the labels.
Then apply Gaussian process models to the deepest layer of features. This works better than using the raw data.
Then use GP’s to get the derivatives that are back-propagated through the deep belief net. This is a further win. It allows GP’s to fine-tune complicated domain-specific kernels.
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Learning to extract the orientation of a face patch (Salakhutdinov & Hinton, NIPS 2007)

Learning to extract the orientation of a face patch (Salakhutdinov &

Hinton, NIPS 2007)
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The training and test sets 11,000 unlabeled cases 100, 500, or

The training and test sets

11,000 unlabeled cases

100, 500, or 1000 labeled

cases

face patches from new people

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The root mean squared error in the orientation when combining GP’s

The root mean squared error in the orientation when combining GP’s

with deep belief nets

22.2 17.9 15.2
17.2 12.7 7.2
16.3 11.2 6.4

GP on the pixels

GP on top-level features

GP on top-level features with fine-tuning

100 labels
500 labels
1000 labels

Conclusion: The deep features are much better than the pixels. Fine-tuning helps a lot.

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Modeling real-valued data For images of digits it is possible to

Modeling real-valued data

For images of digits it is possible to represent

intermediate intensities as if they were probabilities by using “mean-field” logistic units.
We can treat intermediate values as the probability that the pixel is inked.
This will not work for real images.
In a real image, the intensity of a pixel is almost always almost exactly the average of the neighboring pixels.
Mean-field logistic units cannot represent precise intermediate values.
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The free-energy of a mean-field logistic unit In a mean-field logistic

The free-energy of a mean-field logistic unit

In a mean-field logistic unit,

the total input provides a linear energy-gradient and the negative entropy provides a containment function with fixed curvature.
So it is impossible for the value 0.7 to have much lower free energy than both 0.8 and 0.6. This is no good for modeling real-valued data.

0 output-> 1

F?

energy

- entropy

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An RBM with real-valued visible units Using Gaussian visible units we

An RBM with real-valued visible units
Using Gaussian visible units we can

get much sharper predictions and alternating Gibbs sampling is still easy, though learning is slower.

E ?

energy-gradient produced by the total input to a visible unit

parabolic containment function

Welling et. al. (2005) show how to extend RBM’s to the exponential family. See also Bengio et. al. 2007)

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Deep Autoencoders (Hinton & Salakhutdinov, 2006) They always looked like a

Deep Autoencoders (Hinton & Salakhutdinov, 2006)

They always looked like a really nice

way to do non-linear dimensionality reduction:
But it is very difficult to optimize deep autoencoders using backpropagation.
We now have a much better way to optimize them:
First train a stack of 4 RBM’s
Then “unroll” them.
Then fine-tune with backprop.

1000 neurons

500 neurons

500 neurons

250 neurons

250 neurons

30

1000 neurons

28x28

28x28

linear units

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A comparison of methods for compressing digit images to 30 real

A comparison of methods for compressing digit images to 30 real

numbers.

real data
30-D deep auto
30-D logistic PCA
30-D PCA

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Do the 30-D codes found by the deep autoencoder preserve the

Do the 30-D codes found by the deep autoencoder preserve the

class structure of the data?

Take the 30-D activity patterns in the code layer and display them in 2-D using a new form of non-linear multi-dimensional scaling
The method is called UNI-SNE (Cook et. al. 2007).
It keeps similar patterns close together and tries to push dissimilar ones far apart.
Does the learning find the natural classes?

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entirely unsupervised except for the colors

entirely unsupervised except for the colors

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Retrieving documents that are similar to a query document We can

Retrieving documents that are similar to a query document

We can use

an autoencoder to find low-dimensional codes for documents that allow fast and accurate retrieval of similar documents from a large set.
We start by converting each document into a “bag of words”. This a 2000 dimensional vector that contains the counts for each of the 2000 commonest words.
Слайд 69

How to compress the count vector We train the neural network

How to compress the count vector

We train the neural network

to reproduce its input vector as its output
This forces it to compress as much information as possible into the 10 numbers in the central bottleneck.
These 10 numbers are then a good way to compare documents.

2000 reconstructed counts

500 neurons

2000 word counts

500 neurons

250 neurons

250 neurons

10

input vector

output vector

Слайд 70

Performance of the autoencoder at document retrieval Train on bags of

Performance of the autoencoder at document retrieval

Train on bags of 2000

words for 400,000 training cases of business documents.
First train a stack of RBM’s. Then fine-tune with backprop.
Test on a separate 400,000 documents.
Pick one test document as a query. Rank order all the other test documents by using the cosine of the angle between codes.
Repeat this using each of the 400,000 test documents as the query (requires 0.16 trillion comparisons).
Plot the number of retrieved documents against the proportion that are in the same hand-labeled class as the query document.
Слайд 71

Proportion of retrieved documents in same class as query Number of documents retrieved

Proportion of retrieved documents in same class as query

Number of documents

retrieved
Слайд 72

First compress all documents to 2 numbers using a type of

First compress all documents to 2 numbers using a type of

PCA Then use different colors for different document categories
Слайд 73

First compress all documents to 2 numbers. Then use different colors for different document categories

First compress all documents to 2 numbers. Then use different

colors for different document categories
Слайд 74

Finding binary codes for documents Train an auto-encoder using 30 logistic

Finding binary codes for documents
Train an auto-encoder using 30 logistic units

for the code layer.
During the fine-tuning stage, add noise to the inputs to the code units.
The “noise” vector for each training case is fixed. So we still get a deterministic gradient.
The noise forces their activities to become bimodal in order to resist the effects of the noise.
Then we simply round the activities of the 30 code units to 1 or 0.

2000 reconstructed counts

500 neurons

2000 word counts

500 neurons

250 neurons

250 neurons

30

noise

Слайд 75

Semantic hashing: Using a deep autoencoder as a hash-function for finding

Semantic hashing: Using a deep autoencoder as a hash-function for finding

approximate matches (Salakhutdinov & Hinton, 2007)

hash function

“supermarket search”

Слайд 76

How good is a shortlist found this way? We have only

How good is a shortlist found this way?
We have only

implemented it for a million documents with 20-bit codes --- but what could possibly go wrong?
A 20-D hypercube allows us to capture enough of the similarity structure of our document set.
The shortlist found using binary codes actually improves the precision-recall curves of TF-IDF.
Locality sensitive hashing (the fastest other method) is 50 times slower and has worse precision-recall curves.
Слайд 77

Time series models Inference is difficult in directed models of time

Time series models

Inference is difficult in directed models of time series

if we use non-linear distributed representations in the hidden units.
It is hard to fit Dynamic Bayes Nets to high-dimensional sequences (e.g motion capture data).
So people tend to avoid distributed representations and use much weaker methods (e.g. HMM’s).
Слайд 78

Time series models If we really need distributed representations (which we

Time series models

If we really need distributed representations (which we nearly

always do), we can make inference much simpler by using three tricks:
Use an RBM for the interactions between hidden and visible variables. This ensures that the main source of information wants the posterior to be factorial.
Model short-range temporal information by allowing several previous frames to provide input to the hidden units and to the visible units.
This leads to a temporal module that can be stacked
So we can use greedy learning to learn deep models of temporal structure.
Слайд 79

The conditional RBM model (Sutskever & Hinton 2007) Given the data

The conditional RBM model (Sutskever & Hinton 2007)

Given the data and

the previous hidden state, the hidden units at time t are conditionally independent.
So online inference is very easy
Learning can be done by using contrastive divergence.
Reconstruct the data at time t from the inferred states of the hidden units and the earlier states of the visibles.
The temporal connections can be learned as if they were additional biases

t-2 t-1 t

t

Слайд 80

Why the autoregressive connections do not cause problems The autoregressive connections

Why the autoregressive connections do not cause problems

The autoregressive connections do

not mess up contrastive divergence learning because:
We know the initial state of the visible units, so we know the initial effect of the autoregressive connections.
It is not necessary for the reconstructions to be at equilibrium with the hidden units.
The important thing for contrastive divergence is to ensure the hiddens are in equilibrium with the visibles whenever statistics are measured.
Слайд 81

Generating from a learned model The inputs from the earlier states

Generating from a learned model

The inputs from the earlier states of

the visible units create dynamic biases for the hidden and current visible units.
Perform alternating Gibbs sampling for a few iterations between the hidden units and the current visible units.
This picks new hidden and visible states that are compatible with each other and with the recent history.

t-2 t-1 t

t

Слайд 82

Stacking temporal RBM’s Treat the hidden activities of the first level

Stacking temporal RBM’s

Treat the hidden activities of the first level TRBM

as the data for the second-level TRBM.
So when we learn the second level, we get connections across time in the first hidden layer.
After greedy learning, we can generate from the composite model
First, generate from the top-level model by using alternating Gibbs sampling between the current hiddens and visibles of the top-level model, using the dynamic biases created by the previous top-level visibles.
Then do a single top-down pass through the lower layers, but using the autoregressive inputs coming from earlier states of each layer.
Слайд 83

An application to modeling motion capture data (Taylor, Roweis & Hinton,

An application to modeling motion capture data (Taylor, Roweis & Hinton,

2007)

Human motion can be captured by placing reflective markers on the joints and then using lots of infrared cameras to track the 3-D positions of the markers.
Given a skeletal model, the 3-D positions of the markers can be converted into the joint angles plus 6 parameters that describe the 3-D position and the roll, pitch and yaw of the pelvis.
We only represent changes in yaw because physics doesn’t care about its value and we want to avoid circular variables.

Слайд 84

Modeling multiple types of motion We can easily learn to model

Modeling multiple types of motion

We can easily learn to model walking

and running in a single model.
This means we can share a lot of knowledge.
It should also make it much easier to learn nice transitions between walking and running.
In a switching mixture of dynamical systems its hard to get the latent variables to join up nicely when we switch from one system to another.
Because we can do online inference (slightly incorrectly), we can fill in missing markers in real time.
Слайд 85

Show Graham Taylor’s movies available at www.cs.toronto/~hinton

Show Graham Taylor’s movies available at www.cs.toronto/~hinton

Слайд 86

Generating the parts of an object One way to maintain the

Generating the parts of an object

One way to maintain the

constraints between the parts is to generate each part very accurately
But this would require a lot of communication bandwidth.
Sloppy top-down specification of the parts is less demanding
but it messes up relationships between features
so use redundant features and use lateral interactions to clean up the mess.
Each transformed feature helps to locate the others
This allows a noisy channel

sloppy top-down activation of parts

clean-up using known interactions

pose parameters

features with top-down support

“square”

+

Its like soldiers on a parade ground

Слайд 87

Semi-restricted Boltzmann Machines We restrict the connectivity to make learning easier.

Semi-restricted Boltzmann Machines

We restrict the connectivity to make learning easier.
Contrastive divergence

learning requires the hidden units to be in conditional equilibrium with the visibles.
But it does not require the visible units to be in conditional equilibrium with the hiddens.
All we require is that the visible units are closer to equilibrium in the reconstructions than in the data.
So we can allow connections between the visibles.

hidden

i

j

visible

Слайд 88

Learning a semi-restricted Boltzmann Machine i j i j t =

Learning a semi-restricted Boltzmann Machine

i

j

i

j

t = 0 t = 1

1.

Start with a training vector on the visible units.
2. Update all of the hidden units in parallel
3. Repeatedly update all of the visible units in parallel using mean-field updates (with the hiddens fixed) to get a “reconstruction”.
4. Update all of the hidden units again.

reconstruction

data

k

i

i

k

k

k

update for a lateral weight

Слайд 89

Learning in Semi-restricted Boltzmann Machines Method 1: To form a reconstruction,

Learning in Semi-restricted Boltzmann Machines

Method 1: To form a reconstruction,

cycle through the visible units updating each in turn using the top-down input from the hiddens plus the lateral input from the other visibles.
Method 2: Use “mean field” visible units that have real values. Update them all in parallel.
Use damping to prevent oscillations

total input to i

damping

Слайд 90

Results on modeling natural image patches using a stack of RBM’s

Results on modeling natural image patches using a stack of RBM’s

(Osindero and Hinton)
Stack of RBM’s learned one at a time.
400 Gaussian visible units that see whitened image patches
Derived from 100,000 Van Hateren image patches, each 20x20
The hidden units are all binary.
The lateral connections are learned when they are the visible units of their RBM.
Reconstruction involves letting the visible units of each RBM settle using mean-field dynamics.
The already decided states in the level above determine the effective biases during mean-field settling.

Directed Connections

Directed Connections

Undirected Connections

400 Gaussian units

Hidden MRF with 2000 units

Hidden MRF with 500 units

1000 top-level units. No MRF.

Слайд 91

Without lateral connections real data samples from model

Without lateral connections

real data

samples from model

Слайд 92

With lateral connections real data samples from model

With lateral connections

real data

samples from model

Слайд 93

A funny way to use an MRF The lateral connections form

A funny way to use an MRF

The lateral connections form an

MRF.
The MRF is used during learning and generation.
The MRF is not used for inference.
This is a novel idea so vision researchers don’t like it.
The MRF enforces constraints. During inference, constraints do not need to be enforced because the data obeys them.
The constraints only need to be enforced during generation.
Unobserved hidden units cannot enforce constraints.
This requires lateral connections or observed descendants.
Слайд 94

Why do we whiten data? Images typically have strong pair-wise correlations.

Why do we whiten data?

Images typically have strong pair-wise correlations.
Learning higher

order statistics is difficult when there are strong pair-wise correlations.
Small changes in parameter values that improve the modeling of higher order statistics may be rejected because they form a slightly worse model of the much stronger pair-wise statistics.
So we often remove the second-order statistics before trying to learn the higher-order statistics.
Слайд 95

Whitening the learning signal instead of the data Contrastive divergence learning

Whitening the learning signal instead of the data

Contrastive divergence learning can

remove the effects of the second-order statistics on the learning without actually changing the data.
The lateral connections model the second order statistics
If a pixel can be reconstructed correctly using second order statistics, its will be the same in the reconstruction as in the data.
The hidden units can then focus on modeling high-order structure that cannot be predicted by the lateral connections.
For example, a pixel close to an edge, where interpolation from nearby pixels causes incorrect smoothing.
Слайд 96

Towards a more powerful, multi-linear stackable learning module So far, the

Towards a more powerful, multi-linear stackable learning module

So far, the states

of the units in one layer have only been used to determine the effective biases of the units in the layer below.
It would be much more powerful to modulate the pair-wise interactions in the layer below. (A good way to design a hierarchical system is to allow each level to determine the objective function of the level below.)
For example, a vertical edge represents a breakdown in the usual correlational structure of the pixels: Horizontal interpolation does not work, so it needs to be turned off, but interpolation parallel to the edge is OK.
To modulate pair-wise interactions we need higher-order Boltzmann machines.
Слайд 97

Higher order Boltzmann machines (Sejnowski, ~1986) The usual energy function is

Higher order Boltzmann machines (Sejnowski, ~1986)

The usual energy function is quadratic

in the states:

But we could use higher order interactions:

Unit k acts as a switch. When unit k is on, it switches in the pairwise interaction between unit i and unit j.
Units i and j can also be viewed as switches that control the pairwise interactions between j and k or between i and k.

Слайд 98

A picture of a conditional, higher-order Boltzmann machine (Hinton & Lang,1985)

A picture of a conditional, higher-order Boltzmann machine (Hinton & Lang,1985)

retina-based

features

object-based features

viewing transform

We can view it as a Boltzmann machine in which the inputs create interactions between the other variables.
This type of model is now called a conditional random field.
It is hard to learn with two hidden groups.

Слайд 99

Using conditional higher-order Boltzmann machines to model image transformations (Memisevic and

Using conditional higher-order Boltzmann machines to model image transformations (Memisevic and

Hinton, 2007)

A transformation unit specifies which pixel goes to which other pixel.
Conversely, each pair of similar intensity pixels, one in each image, votes for all the compatible transformation units.

image(t)

image(t+1)

image transformation

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