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Archive-name: ai-faq/neural-nets/part1
Last-modified: 2002-05-17
URL: ftp://ftp.sas.com/pub/neural/FAQ.html
Maintainer: saswss@unx.sas.com (Warren S. Sarle)
Copyright 1997, 1998, 1999, 2000, 2001, 2002 by Warren S. Sarle, Cary, NC,
USA.
---------------------------------------------------------------
Additions, corrections, or improvements are always welcome.
Anybody who is willing to contribute any information,
please email me; if it is relevant, I will incorporate it.
The monthly posting departs around the 28th of every month.
---------------------------------------------------------------
This is the first of seven parts of a monthly posting to the Usenet
newsgroup comp.ai.neural-nets (as well as comp.answers and news.answers,
where it should be findable at any time). Its purpose is to provide basic
information for individuals who are new to the field of neural networks or
who are just beginning to read this group. It will help to avoid lengthy
discussion of questions that often arise for beginners.
SO, PLEASE, SEARCH THIS POSTING FIRST IF YOU HAVE A QUESTION
and
DON'T POST ANSWERS TO FAQs: POINT THE ASKER TO THIS POSTING
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The FAQ posting departs to comp.ai.neural-nets around the 28th of every
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All changes to the FAQ from the previous month are shown in another monthly
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ftp://ftp.sas.com/pub/neural/changes.txt . There is also a weekly post with
the subject "comp.ai.neural-nets FAQ: weekly reminder" that briefly
describes any major changes to the FAQ.
This FAQ is not meant to discuss any topic exhaustively. It is neither a
tutorial nor a textbook, but should be viewed as a supplement to the many
excellent books and online resources described in Part 4: Books, data, etc..
Disclaimer:
This posting is provided 'as is'. No warranty whatsoever is expressed or
implied, in particular, no warranty that the information contained herein
is correct or useful in any way, although both are intended.
To find the answer of question "x", search for the string "Subject: x"
========== Questions ==========
********************************
Part 1: Introduction
What is this newsgroup for? How shall it be used?
Where is comp.ai.neural-nets archived?
What if my question is not answered in the FAQ?
May I copy this FAQ?
What is a neural network (NN)?
Where can I find a simple introduction to NNs?
Are there any online books about NNs?
What can you do with an NN and what not?
Who is concerned with NNs?
How many kinds of NNs exist?
How many kinds of Kohonen networks exist? (And what is k-means?)
VQ: Vector Quantization and k-means
SOM: Self-Organizing Map
LVQ: Learning Vector Quantization
Other Kohonen networks and references
How are layers counted?
What are cases and variables?
What are the population, sample, training set, design set, validation
set, and test set?
How are NNs related to statistical methods?
Part 2: Learning
What are combination, activation, error, and objective functions?
What are batch, incremental, on-line, off-line, deterministic,
stochastic, adaptive, instantaneous, pattern, epoch, constructive, and
sequential learning?
What is backprop?
What learning rate should be used for backprop?
What are conjugate gradients, Levenberg-Marquardt, etc.?
How does ill-conditioning affect NN training?
How should categories be encoded?
Why not code binary inputs as 0 and 1?
Why use a bias/threshold?
Why use activation functions?
How to avoid overflow in the logistic function?
What is a softmax activation function?
What is the curse of dimensionality?
How do MLPs compare with RBFs?
What are OLS and subset/stepwise regression?
Should I normalize/standardize/rescale the data?
Should I nonlinearly transform the data?
How to measure importance of inputs?
What is ART?
What is PNN?
What is GRNN?
What does unsupervised learning learn?
Help! My NN won't learn! What should I do?
Part 3: Generalization
How is generalization possible?
How does noise affect generalization?
What is overfitting and how can I avoid it?
What is jitter? (Training with noise)
What is early stopping?
What is weight decay?
What is Bayesian learning?
How to combine networks?
How many hidden layers should I use?
How many hidden units should I use?
How can generalization error be estimated?
What are cross-validation and bootstrapping?
How to compute prediction and confidence intervals (error bars)?
Part 4: Books, data, etc.
Books and articles about Neural Networks?
Journals and magazines about Neural Networks?
Conferences and Workshops on Neural Networks?
Neural Network Associations?
Mailing lists, BBS, CD-ROM?
How to benchmark learning methods?
Databases for experimentation with NNs?
Part 5: Free software
Source code on the web?
Freeware and shareware packages for NN simulation?
Part 6: Commercial software
Commercial software packages for NN simulation?
Part 7: Hardware and miscellaneous
Neural Network hardware?
What are some applications of NNs?
General
Agriculture
Chemistry
Face recognition
Finance and economics
Games, sports, gambling
Industry
Materials science
Medicine
Music
Robotics
Weather forecasting
Weird
What to do with missing/incomplete data?
How to forecast time series (temporal sequences)?
How to learn an inverse of a function?
How to get invariant recognition of images under translation, rotation,
etc.?
How to recognize handwritten characters?
What about pulsed or spiking NNs?
What about Genetic Algorithms and Evolutionary Computation?
What about Fuzzy Logic?
Unanswered FAQs
Other NN links?
Subject: What is this newsgroup for? How shall it be
====================================================
used?
=====
The newsgroup comp.ai.neural-nets is intended as a forum for people who want
to use or explore the capabilities of Artificial Neural Networks or
Neural-Network-like structures.
Posts should be in plain-text format, not postscript, html, rtf, TEX, MIME,
or any word-processor format.
Do not use vcards or other excessively long signatures.
Please do not post homework or take-home exam questions. Please do not post
a long source-code listing and ask readers to debug it. And note that chain
letters and other get-rich-quick pyramid schemes are illegal in the USA; for
example, see http://www.usps.gov/websites/depart/inspect/chainlet.htm
There should be the following types of articles in this newsgroup:
1. Requests
+++++++++++
Requests are articles of the form "I am looking for X", where X
is something public like a book, an article, a piece of software. The
most important about such a request is to be as specific as possible!
If multiple different answers can be expected, the person making the
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announce to do so with a phrase like "Please reply by email,
I'll summarize to the group" at the end of the posting.
The Subject line of the posting should then be something like
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++++++++++++
As opposed to requests, questions ask for a larger piece of information
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redundant traffic it is important that the poster provides with the
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Students: please do not ask comp.ai.neural-net readers to do your
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Most news-reader software automatically provides a subject line beginning
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4. Summaries
++++++++++++
In all cases of requests or questions the answers for which can be
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question shall summarize the answers he/she received. Such a summary
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summarize"
In such a case, people who answer to a question should NOT post their
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the original posting, its poster should make the summary of answers and
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Some care should be invested into a summary:
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Note that a good summary is pure gold for the rest of the newsgroup
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summaries are more valuable than any moderator ! :-)
5. Announcements
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Some articles never need any public reaction. These are called
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Announcements should be clearly indicated to be such by giving them a
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Sometimes people spontaneously want to report something to the newsgroup.
This might be special experiences with some software, results of own
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Reports should be clearly indicated to be such by giving them a subject
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7. Discussions
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An especially valuable possibility of Usenet is of course that of
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If somebody explicitly wants to start a discussion, he/she can do so by
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It is quite difficult to keep a discussion from drifting into chaos, but,
unfortunately, as many many other newsgroups show there seems to be no
secure way to avoid this. On the other hand, comp.ai.neural-nets has not
had many problems with this effect in the past, so let's just go and
hope...
8. Jobs Ads
+++++++++++
Advertisements for jobs requiring expertise in artificial neural networks
are appropriate in comp.ai.neural-nets. Job ads should be clearly
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country-state-city abbreviations that are conventional in
misc.jobs.offered, such as: "Job: US-NY-NYC Neural network
engineer". If an employer has more than one job opening, all such
openings should be listed in a single post, not multiple posts. Job ads
should not be reposted more than once per month.
Subject: Where is comp.ai.neural-nets archived?
================================================
The following archives are available for comp.ai.neural-nets:
o http://groups.google.com, formerly Deja News. Does not work very well
yet.
o 94-09-14 through 97-08-16
ftp://ftp.cs.cmu.edu/user/ai/pubs/news/
For more information on newsgroup archives, see
http://starbase.neosoft.com/~claird/news.lists/newsgroup_archives.html
or http://www.pitt.edu/~grouprev/Usenet/Archive-List/newsgroup_archives.html
Subject: What if my question is not answered in the FAQ?
========================================================
If your question is not answered in the FAQ, you can try a web search. The
following search engines are especially useful:
http://www.google.com/
http://search.yahoo.com/
http://www.altavista.com/
http://citeseer.nj.nec.com/cs
Another excellent web site on NNs is Donald Tveter's Backpropagator's Review
at http://www.dontveter.com/bpr/bpr.html or
http://gannoo.uce.ac.uk/bpr/bpr.html.
For feedforward NNs, the best reference book is:
Bishop, C.M. (1995), Neural Networks for Pattern Recognition, Oxford:
Oxford University Press.
If the answer isn't in Bishop, then for more theoretical questions try:
Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge:
Cambridge University Press.
For more practical questions about MLP training, try:
Masters, T. (1993). Practical Neural Network Recipes in C++, San Diego:
Academic Press.
Reed, R.D., and Marks, R.J, II (1999), Neural Smithing: Supervised
Learning in Feedforward Artificial Neural Networks, Cambridge, MA: The
MIT Press.
There are many more excellent books and web sites listed in the Neural
Network FAQ, Part 4: Books, data, etc.
Subject: May I copy this FAQ?
=============================
The intent in providing a FAQ is to make the information freely available to
whoever needs it. You may copy all or part of the FAQ, but please be sure to
include a reference to the URL of the master copy,
ftp://ftp.sas.com/pub/neural/FAQ.html, and do not sell copies of the FAQ. If
you want to include information from the FAQ in your own web site, it is
better to include links to the master copy rather than to copy text from the
FAQ to your web pages, because various answers in the FAQ are updated at
unpredictable times. To cite the FAQ in an academic-style bibliography, use
something along the lines of:
Sarle, W.S., ed. (1997), Neural Network FAQ, part 1 of 7: Introduction,
periodic posting to the Usenet newsgroup comp.ai.neural-nets, URL:
ftp://ftp.sas.com/pub/neural/FAQ.html
Subject: What is a neural network (NN)?
=======================================
The question 'What is a neural network?' is ill-posed.
- Pinkus
(1999)
First of all, when we are talking about a neural network, we should more
properly say "artificial neural network" (ANN), because that is what we mean
most of the time in comp.ai.neural-nets. Biological neural networks are much
more complicated than the mathematical models we use for ANNs. But it is
customary to be lazy and drop the "A" or the "artificial".
There is no universally accepted definition of an NN. But perhaps most
people in the field would agree that an NN is a network of many simple
processors ("units"), each possibly having a small amount of local memory.
The units are connected by communication channels ("connections") which
usually carry numeric (as opposed to symbolic) data, encoded by any of
various means. The units operate only on their local data and on the inputs
they receive via the connections. The restriction to local operations is
often relaxed during training.
Some NNs are models of biological neural networks and some are not, but
historically, much of the inspiration for the field of NNs came from the
desire to produce artificial systems capable of sophisticated, perhaps
"intelligent", computations similar to those that the human brain routinely
performs, and thereby possibly to enhance our understanding of the human
brain.
Most NNs have some sort of "training" rule whereby the weights of
connections are adjusted on the basis of data. In other words, NNs "learn"
from examples, as children learn to distinguish dogs from cats based on
examples of dogs and cats. If trained carefully, NNs may exhibit some
capability for generalization beyond the training data, that is, to produce
approximately correct results for new cases that were not used for training.
NNs normally have great potential for parallelism, since the computations of
the components are largely independent of each other. Some people regard
massive parallelism and high connectivity to be defining characteristics of
NNs, but such requirements rule out various simple models, such as simple
linear regression (a minimal feedforward net with only two units plus bias),
which are usefully regarded as special cases of NNs.
Here is a sampling of definitions from the books on the FAQ maintainer's
shelf. None will please everyone. Perhaps for that reason many NN textbooks
do not explicitly define neural networks.
According to the DARPA Neural Network Study (1988, AFCEA International
Press, p. 60):
... a neural network is a system composed of many simple processing
elements operating in parallel whose function is determined by
network structure, connection strengths, and the processing performed
at computing elements or nodes.
According to Haykin (1994), p. 2:
A neural network is a massively parallel distributed processor that
has a natural propensity for storing experiential knowledge and
making it available for use. It resembles the brain in two respects:
1. Knowledge is acquired by the network through a learning process.
2. Interneuron connection strengths known as synaptic weights are
used to store the knowledge.
According to Nigrin (1993), p. 11:
A neural network is a circuit composed of a very large number of
simple processing elements that are neurally based. Each element
operates only on local information. Furthermore each element operates
asynchronously; thus there is no overall system clock.
According to Zurada (1992), p. xv:
Artificial neural systems, or neural networks, are physical cellular
systems which can acquire, store, and utilize experiential knowledge.
References:
Pinkus, A. (1999), "Approximation theory of the MLP model in neural
networks," Acta Numerica, 8, 143-196.
Haykin, S. (1994), Neural Networks: A Comprehensive Foundation, NY:
Macmillan.
Nigrin, A. (1993), Neural Networks for Pattern Recognition, Cambridge,
MA: The MIT Press.
Zurada, J.M. (1992), Introduction To Artificial Neural Systems, Boston:
PWS Publishing Company.
Subject: Where can I find a simple introduction to NNs?
=======================================================
Several excellent introductory books on NNs are listed in part 4 of the FAQ
under "Books and articles about Neural Networks?" If you want a book with
minimal math, see "The best introductory book for business executives."
Dr. Leslie Smith has a brief on-line introduction to NNs with examples and
diagrams at http://www.cs.stir.ac.uk/~lss/NNIntro/InvSlides.html.
If you are a Java enthusiast, see Jochen Fröhlich's diploma at
http://rfhs8012.fh-regensburg.de/~saj39122/jfroehl/diplom/e-index.html
For a more detailed introduction, see Donald Tveter's excellent
Backpropagator's Review at http://www.dontveter.com/bpr/bpr.html or
http://gannoo.uce.ac.uk/bpr/bpr.html, which contains both answers to
additional FAQs and an annotated neural net bibliography emphasizing on-line
articles.
StatSoft Inc. has an on-line Electronic Statistics Textbook, at
http://www.statsoft.com/textbook/stathome.html that includes a chapter on
neural nets as well as many statistical topics relevant to neural nets.
Subject: Are there any online books about NNs?
==============================================
Kevin Gurney has on-line a preliminary draft of his book, An Introduction to
Neural Networks, at
http://www.shef.ac.uk/psychology/gurney/notes/index.html The book is now in
print and is one of the better general-purpose introductory textbooks on
NNs. Here is the table of contents from the on-line version:
1. Computers and Symbols versus Nets and Neurons
2. TLUs and vectors - simple learning rules
3. The delta rule
4. Multilayer nets and backpropagation
5. Associative memories - the Hopfield net
6. Hopfield nets (contd.)
7. Kohonen nets
8. Alternative node types
9. Cubic nodes (contd.) and Reward Penalty training
10. Drawing things together - some perspectives
Another on-line book by Ben Kröse and Patrick van der Smagt, also called An
Introduction to Neural Networks, can be found at
ftp://ftp.wins.uva.nl/pub/computer-systems/aut-sys/reports/neuro-intro/neuro-intro.ps.gz
or http://www.robotic.dlr.de/Smagt/books/neuro-intro.ps.gz. or
http://www.supelec-rennes.fr/acth/net/neuro-intro.ps.gz
Here is the table of contents:
1. Introduction
2. Fundamantals
3. Perceptron and Adaline
4. Back-Propagation
5. Recurrent Networks
6. Self-Organising Networks
7. Reinforcement Learning
8. Robot Control
9. Vision
10. General Purpose Hardware
11. Dedicated Neuro-Hardware
Subject: What can you do with an NN and what not?
=================================================
In principle, NNs can compute any computable function, i.e., they can do
everything a normal digital computer can do (Valiant, 1988; Siegelmann and
Sontag, 1999; Orponen, 2000; Sima and Orponen, 2001), or perhaps even more,
under some assumptions of doubtful practicality (see Siegelmann, 1998, but
also Hadley, 1999).
Practical applications of NNs most often employ supervised learning. For
supervised learning, you must provide training data that includes both the
input and the desired result (the target value). After successful training,
you can present input data alone to the NN (that is, input data without the
desired result), and the NN will compute an output value that approximates
the desired result. However, for training to be successful, you may need
lots of training data and lots of computer time to do the training. In many
applications, such as image and text processing, you will have to do a lot
of work to select appropriate input data and to code the data as numeric
values.
In practice, NNs are especially useful for classification and function
approximation/mapping problems which are tolerant of some imprecision, which
have lots of training data available, but to which hard and fast rules (such
as those that might be used in an expert system) cannot easily be applied.
Almost any finite-dimensional vector function on a compact set can be
approximated to arbitrary precision by feedforward NNs (which are the type
most often used in practical applications) if you have enough data and
enough computing resources.
To be somewhat more precise, feedforward networks with a single hidden layer
and trained by least-squares are statistically consistent estimators of
arbitrary square-integrable regression functions under certain
practically-satisfiable assumptions regarding sampling, target noise, number
of hidden units, size of weights, and form of hidden-unit activation
function (White, 1990). Such networks can also be trained as statistically
consistent estimators of derivatives of regression functions (White and
Gallant, 1992) and quantiles of the conditional noise distribution (White,
1992a). Feedforward networks with a single hidden layer using threshold or
sigmoid activation functions are universally consistent estimators of binary
classifications (Faragó and Lugosi, 1993; Lugosi and Zeger 1995; Devroye,
Györfi, and Lugosi, 1996) under similar assumptions. Note that these results
are stronger than the universal approximation theorems that merely show the
existence of weights for arbitrarily accurate approximations, without
demonstrating that such weights can be obtained by learning.
Unfortunately, the above consistency results depend on one impractical
assumption: that the networks are trained by an error (L_p error or
misclassification rate) minimization technique that comes arbitrarily close
to the global minimum. Such minimization is computationally intractable
except in small or simple problems (Blum and Rivest, 1989; Judd, 1990). In
practice, however, you can usually get good results without doing a
full-blown global optimization; e.g., using multiple (say, 10 to 1000)
random weight initializations is usually sufficient.
One example of a function that a typical neural net cannot learn is Y=1/X
on the open interval (0,1). An open interval is not a compact set. With any
bounded output activation function, the error will get arbitrarily large as
the input approaches zero. Of course, you could make the output activation
function a reciprocal function and easily get a perfect fit, but neural
networks are most often used in situations where you do not have enough
prior knowledge to set the activation function in such a clever way. There
are also many other important problems that are so difficult that a neural
network will be unable to learn them without memorizing the entire training
set, such as:
o Predicting random or pseudo-random numbers.
o Factoring large integers.
o Determing whether a large integer is prime or composite.
o Decrypting anything encrypted by a good algorithm.
And it is important to understand that there are no methods for training NNs
that can magically create information that is not contained in the training
data.
Feedforward NNs are restricted to finite-dimensional input and output
spaces. Recurrent NNs can in theory process arbitrarily long strings of
numbers or symbols. But training recurrent NNs has posed much more serious
practical difficulties than training feedforward networks. NNs are, at least
today, difficult to apply successfully to problems that concern manipulation
of symbols and rules, but much research is being done.
There have been attempts to pack recursive structures into
finite-dimensional real vectors (Blair, 1997; Pollack, 1990; Chalmers, 1990;
Chrisman, 1991; Plate, 1994; Hammerton, 1998). Obviously, finite precision
limits how far the recursion can go (Hadley, 1999). The practicality of such
methods is open to debate.
As for simulating human consciousness and emotion, that's still in the realm
of science fiction. Consciousness is still one of the world's great
mysteries. Artificial NNs may be useful for modeling some aspects of or
prerequisites for consciousness, such as perception and cognition, but ANNs
provide no insight so far into what Chalmers (1996, p. xi) calls the "hard
problem":
Many books and articles on consciousness have appeared in the past
few years, and one might think we are making progress. But on a
closer look, most of this work leaves the hardest problems about
consciousness untouched. Often, such work addresses what might be
called the "easy problems" of consciousness: How does the brain
process environmental stimulation? How does it integrate information?
How do we produce reports on internal states? These are important
questions, but to answer them is not to solve the hard problem: Why
is all this processing accompanied by an experienced inner life?
For more information on consciousness, see the on-line journal Psyche at
http://psyche.cs.monash.edu.au/index.html.
For examples of specific applications of NNs, see What are some applications
of NNs?
References:
Blair, A.D. (1997), "Scaling Up RAAMs," Brandeis University Computer
Science Technical Report CS-97-192,
http://www.demo.cs.brandeis.edu/papers/long.html#sur97
Blum, A., and Rivest, R.L. (1989), "Training a 3-node neural network is
NP-complete," in Touretzky, D.S. (ed.), Advances in Neural Information
Processing Systems 1, San Mateo, CA: Morgan Kaufmann, 494-501.
Chalmers, D.J. (1990), "Syntactic Transformations on Distributed
Representations," Connection Science, 2, 53-62,
http://ling.ucsc.edu/~chalmers/papers/transformations.ps
Chalmers, D.J. (1996), The Conscious Mind: In Search of a Fundamental
Theory, NY: Oxford University Press.
Chrisman, L. (1991), "Learning Recursive Distributed Representations for
Holistic Computation", Connection Science, 3, 345-366,
ftp://reports.adm.cs.cmu.edu/usr/anon/1991/
Collier, R. (1994), "An historical overview of natural language
processing systems that learn," Artificial Intelligence Review, 8(1),
??-??.
Devroye, L., Györfi, L., and Lugosi, G. (1996), A Probabilistic Theory of
Pattern Recognition, NY: Springer.
Faragó, A. and Lugosi, G. (1993), "Strong Universal Consistency of Neural
Network Classifiers," IEEE Transactions on Information Theory, 39,
1146-1151.
Hadley, R.F. (1999), "Cognition and the computational power of
connectionist networks," http://www.cs.sfu.ca/~hadley/online.html
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Subject: Who is concerned with NNs?
===================================
Neural Networks are interesting for quite a lot of very different people:
o Computer scientists want to find out about the properties of non-symbolic
information processing with neural nets and about learning systems in
general.
o Statisticians use neural nets as flexible, nonlinear regression and
classification models.
o Engineers of many kinds exploit the capabilities of neural networks in
many areas, such as signal processing and automatic control.
o Cognitive scientists view neural networks as a possible apparatus to
describe models of thinking and consciousness (High-level brain
function).
o Neuro-physiologists use neural networks to describe and explore
medium-level brain function (e.g. memory, sensory system, motorics).
o Physicists use neural networks to model phenomena in statistical
mechanics and for a lot of other tasks.
o Biologists use Neural Networks to interpret nucleotide sequences.
o Philosophers and some other people may also be interested in Neural
Networks for various reasons.
For world-wide lists of groups doing research on NNs, see the Foundation for
Neural Networks's (SNN) page at
http://www.mbfys.kun.nl/snn/pointers/groups.html and see Neural Networks
Research on the IEEE Neural Network Council's homepage
http://www.ieee.org/nnc.
Subject: How many kinds of NNs exist?
=====================================
There are many many kinds of NNs by now. Nobody knows exactly how many. New
ones (or at least variations of old ones) are invented every week. Below is
a collection of some of the most well known methods, not claiming to be
complete.
The two main kinds of learning algorithms are supervised and unsupervised.
o In supervised learning, the correct results (target values, desired
outputs) are known and are given to the NN during training so that the NN
can adjust its weights to try match its outputs to the target values.
After training, the NN is tested by giving it only input values, not
target values, and seeing how close it comes to outputting the correct
target values.
o In unsupervised learning, the NN is not provided with the correct results
during training. Unsupervised NNs usually perform some kind of data
compression, such as dimensionality reduction or clustering. See "What
does unsupervised learning learn?"
The distinction between supervised and unsupervised methods is not always
clear-cut. An unsupervised method can learn a summary of a probability
distribution, then that summarized distribution can be used to make
predictions. Furthermore, supervised methods come in two subvarieties:
auto-associative and hetero-associative. In auto-associative learning, the
target values are the same as the inputs, whereas in hetero-associative
learning, the targets are generally different from the inputs. Many
unsupervised methods are equivalent to auto-associative supervised methods.
For more details, see "What does unsupervised learning learn?"
Two major kinds of network topology are feedforward and feedback.
o In a feedforward NN, the connections between units do not form cycles.
Feedforward NNs usually produce a response to an input quickly. Most
feedforward NNs can be trained using a wide variety of efficient
conventional numerical methods (e.g. see "What are conjugate gradients,
Levenberg-Marquardt, etc.?") in addition to algorithms invented by NN
reserachers.
o In a feedback or recurrent NN, there are cycles in the connections. In
some feedback NNs, each time an input is presented, the NN must iterate
for a potentially long time before it produces a response. Feedback NNs
are usually more difficult to train than feedforward NNs.
Some kinds of NNs (such as those with winner-take-all units) can be
implemented as either feedforward or feedback networks.
NNs also differ in the kinds of data they accept. Two major kinds of data
are categorical and quantitative.
o Categorical variables take only a finite (technically, countable) number
of possible values, and there are usually several or more cases falling
into each category. Categorical variables may have symbolic values (e.g.,
"male" and "female", or "red", "green" and "blue") that must be encoded
into numbers before being given to the network (see "How should
categories be encoded?") Both supervised learning with categorical target
values and unsupervised learning with categorical outputs are called
"classification."
o Quantitative variables are numerical measurements of some attribute, such
as length in meters. The measurements must be made in such a way that at
least some arithmetic relations among the measurements reflect analogous
relations among the attributes of the objects that are measured. For more
information on measurement theory, see the Measurement Theory FAQ at
ftp://ftp.sas.com/pub/neural/measurement.html. Supervised learning with
quantitative target values is called "regression."
Some variables can be treated as either categorical or quantitative, such as
number of children or any binary variable. Most regression algorithms can
also be used for supervised classification by encoding categorical target
values as 0/1 binary variables and using those binary variables as target
values for the regression algorithm. The outputs of the network are
posterior probabilities when any of the most common training methods are
used.
Here are some well-known kinds of NNs:
1. Supervised
1. Feedforward
o Linear
o Hebbian - Hebb (1949), Fausett (1994)
o Perceptron - Rosenblatt (1958), Minsky and Papert (1969/1988),
Fausett (1994)
o Adaline - Widrow and Hoff (1960), Fausett (1994)
o Higher Order - Bishop (1995)
o Functional Link - Pao (1989)
o MLP: Multilayer perceptron - Bishop (1995), Reed and Marks (1999),
Fausett (1994)
o Backprop - Rumelhart, Hinton, and Williams (1986)
o Cascade Correlation - Fahlman and Lebiere (1990), Fausett (1994)
o Quickprop - Fahlman (1989)
o RPROP - Riedmiller and Braun (1993)
o RBF networks - Bishop (1995), Moody and Darken (1989), Orr (1996)
o OLS: Orthogonal Least Squares - Chen, Cowan and Grant (1991)
o CMAC: Cerebellar Model Articulation Controller - Albus (1975),
Brown and Harris (1994)
o Classification only
o LVQ: Learning Vector Quantization - Kohonen (1988), Fausett
(1994)
o PNN: Probabilistic Neural Network - Specht (1990), Masters
(1993), Hand (1982), Fausett (1994)
o Regression only
o GNN: General Regression Neural Network - Specht (1991), Nadaraya
(1964), Watson (1964)
2. Feedback - Hertz, Krogh, and Palmer (1991), Medsker and Jain (2000)
o BAM: Bidirectional Associative Memory - Kosko (1992), Fausett
(1994)
o Boltzman Machine - Ackley et al. (1985), Fausett (1994)
o Recurrent time series
o Backpropagation through time - Werbos (1990)
o Elman - Elman (1990)
o FIR: Finite Impulse Response - Wan (1990)
o Jordan - Jordan (1986)
o Real-time recurrent network - Williams and Zipser (1989)
o Recurrent backpropagation - Pineda (1989), Fausett (1994)
o TDNN: Time Delay NN - Lang, Waibel and Hinton (1990)
3. Competitive
o ARTMAP - Carpenter, Grossberg and Reynolds (1991)
o Fuzzy ARTMAP - Carpenter, Grossberg, Markuzon, Reynolds and Rosen
(1992), Kasuba (1993)
o Gaussian ARTMAP - Williamson (1995)
o Counterpropagation - Hecht-Nielsen (1987; 1988; 1990), Fausett
(1994)
o Neocognitron - Fukushima, Miyake, and Ito (1983), Fukushima,
(1988), Fausett (1994)
2. Unsupervised - Hertz, Krogh, and Palmer (1991)
1. Competitive
o Vector Quantization
o Grossberg - Grossberg (1976)
o Kohonen - Kohonen (1984)
o Conscience - Desieno (1988)
o Self-Organizing Map
o Kohonen - Kohonen (1995), Fausett (1994)
o GTM: - Bishop, Svensén and Williams (1997)
o Local Linear - Mulier and Cherkassky (1995)
o Adaptive resonance theory
o ART 1 - Carpenter and Grossberg (1987a), Moore (1988), Fausett
(1994)
o ART 2 - Carpenter and Grossberg (1987b), Fausett (1994)
o ART 2-A - Carpenter, Grossberg and Rosen (1991a)
o ART 3 - Carpenter and Grossberg (1990)
o Fuzzy ART - Carpenter, Grossberg and Rosen (1991b)
o DCL: Differential Competitive Learning - Kosko (1992)
2. Dimension Reduction - Diamantaras and Kung (1996)
o Hebbian - Hebb (1949), Fausett (1994)
o Oja - Oja (1989)
o Sanger - Sanger (1989)
o Differential Hebbian - Kosko (1992)
3. Autoassociation
o Linear autoassociator - Anderson et al. (1977), Fausett (1994)
o BSB: Brain State in a Box - Anderson et al. (1977), Fausett (1994)
o Hopfield - Hopfield (1982), Fausett (1994)
3. Nonlearning
1. Hopfield - Hertz, Krogh, and Palmer (1991)
2. various networks for optimization - Cichocki and Unbehauen (1993)
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Subject: How many kinds of Kohonen networks exist?
==================================================
(And what is k-means?)
======================
Teuvo Kohonen is one of the most famous and prolific researchers in
neurocomputing, and he has invented a variety of networks. But many people
refer to "Kohonen networks" without specifying which kind of Kohonen
network, and this lack of precision can lead to confusion. The phrase
"Kohonen network" most often refers to one of the following three types of
networks:
o VQ: Vector Quantization--competitive networks that can be viewed as
unsupervised density estimators or autoassociators (Kohonen, 1995/1997;
Hecht-Nielsen 1990), closely related to k-means cluster analysis
(MacQueen, 1967; Anderberg, 1973). Each competitive unit corresponds to a
cluster, the center of which is called a "codebook vector". Kohonen's
learning law is an on-line algorithm that finds the codebook vector
closest to each training case and moves the "winning" codebook vector
closer to the training case. The codebook vector is moved a certain
proportion of the distance between it and the training case, the
proportion being specified by the learning rate, that is:
new_codebook = old_codebook * (1-learning_rate)
+ data * learning_rate
Numerous similar algorithms have been developed in the neural net and
machine learning literature; see Hecht-Nielsen (1990) for a brief
historical overview, and Kosko (1992) for a more technical overview of
competitive learning.
MacQueen's on-line k-means algorithm is essentially the same as Kohonen's
learning law except that the learning rate is the reciprocal of the
number of cases that have been assigned to the winnning cluster. Suppose
that when processing a given training case, N cases have been previously
assigned to the winning codebook vector. Then the codebook vector is
updated as:
new_codebook = old_codebook * N/(N+1)
+ data * 1/(N+1)
This reduction of the learning rate makes each codebook vector the mean
of all cases assigned to its cluster and guarantees convergence of the
algorithm to an optimum value of the error function (the sum of squared
Euclidean distances between cases and codebook vectors) as the number of
training cases goes to infinity. Kohonen's learning law with a fixed
learning rate does not converge. As is well known from stochastic
approximation theory, convergence requires the sum of the infinite
sequence of learning rates to be infinite, while the sum of squared
learning rates must be finite (Kohonen, 1995, p. 34). These requirements
are satisfied by MacQueen's k-means algorithm.
Kohonen VQ is often used for off-line learning, in which case the
training data are stored and Kohonen's learning law is applied to each
case in turn, cycling over the data set many times (incremental
training). Convergence to a local optimum can be obtained as the training
time goes to infinity if the learning rate is reduced in a suitable
manner as described above. However, there are off-line k-means
algorithms, both batch and incremental, that converge in a finite number
of iterations (Anderberg, 1973; Hartigan, 1975; Hartigan and Wong, 1979).
The batch algorithms such as Forgy's (1965; Anderberg, 1973) have the
advantage for large data sets, since the incremental methods require you
either to store the cluster membership of each case or to do two
nearest-cluster computations as each case is processed. Forgy's algorithm
is a simple alternating least-squares algorithm consisting of the
following steps:
1. Initialize the codebook vectors.
2. Repeat the following two steps until convergence:
A. Read the data, assigning each case to the nearest (using Euclidean
distance) codebook vector.
B. Replace each codebook vector with the mean of the cases that were
assigned to it.
Fastest training is usually obtained if MacQueen's on-line algorithm is
used for the first pass and off-line k-means algorithms are applied on
subsequent passes (Bottou and Bengio, 1995). However, these training
methods do not necessarily converge to a global optimum of the error
function. The chance of finding a global optimum can be improved by using
rational initialization (SAS Institute, 1989, pp. 824-825), multiple
random initializations, or various time-consuming training methods
intended for global optimization (Ismail and Kamel, 1989; Zeger, Vaisy,
and Gersho, 1992).
VQ has been a popular topic in the signal processing literature, which
has been largely separate from the literature on Kohonen networks and
from the cluster analysis literature in statistics and taxonomy. In
signal processing, on-line methods such as Kohonen's and MacQueen's are
called "adaptive vector quantization" (AVQ), while off-line k-means
methods go by the names of "Lloyd" or "Lloyd I" (Lloyd, 1982) and "LBG"
(Linde, Buzo, and Gray, 1980). There is a recent textbook on VQ by Gersho
and Gray (1992) that summarizes these algorithms as information
compression methods.
Kohonen's work emphasized VQ as density estimation and hence the
desirability of equiprobable clusters (Kohonen 1984; Hecht-Nielsen 1990).
However, Kohonen's learning law does not produce equiprobable
clusters--that is, the proportions of training cases assigned to each
cluster are not usually equal. If there are I inputs and the number of
clusters is large, the density of the codebook vectors approximates the
I/(I+2) power of the density of the training data (Kohonen, 1995, p.
35; Ripley, 1996, p. 202; Zador, 1982), so the clusters are approximately
equiprobable only if the data density is uniform or the number of inputs
is large. The most popular method for obtaining equiprobability is
Desieno's (1988) algorithm which adds a "conscience" value to each
distance prior to the competition. The conscience value for each cluster
is adjusted during training so that clusters that win more often have
larger conscience values and are thus handicapped to even out the
probabilities of winning in later iterations.
Kohonen's learning law is an approximation to the k-means model, which is
an approximation to normal mixture estimation by maximum likelihood
assuming that the mixture components (clusters) all have spherical
covariance matrices and equal sampling probabilities. Hence if the
population contains clusters that are not equiprobable, k-means will tend
to produce sample clusters that are more nearly equiprobable than the
population clusters. Corrections for this bias can be obtained by
maximizing the likelihood without the assumption of equal sampling
probabilities Symons (1981). Such corrections are similar to conscience
but have the opposite effect.
In cluster analysis, the purpose is not to compress information but to
recover the true cluster memberships. K-means differs from mixture models
in that, for k-means, the cluster membership for each case is considered
a separate parameter to be estimated, while mixture models estimate a
posterior probability for each case based on the means, covariances, and
sampling probabilities of each cluster. Balakrishnan, Cooper, Jacob, and
Lewis (1994) found that k-means algorithms recovered cluster membership
more accurately than Kohonen VQ.
o SOM: Self-Organizing Map--competitive networks that provide a
"topological" mapping from the input space to the clusters (Kohonen,
1995). The SOM was inspired by the way in which various human sensory
impressions are neurologically mapped into the brain such that spatial or
other relations among stimuli correspond to spatial relations among the
neurons. In a SOM, the neurons (clusters) are organized into a
grid--usually two-dimensional, but sometimes one-dimensional or (rarely)
three- or more-dimensional. The grid exists in a space that is separate
from the input space; any number of inputs may be used as long as the
number of inputs is greater than the dimensionality of the grid space. A
SOM tries to find clusters such that any two clusters that are close to
each other in the grid space have codebook vectors close to each other in
the input space. But the converse does not hold: codebook vectors that
are close to each other in the input space do not necessarily correspond
to clusters that are close to each other in the grid. Another way to look
at this is that a SOM tries to embed the grid in the input space such
every training case is close to some codebook vector, but the grid is
bent or stretched as little as possible. Yet another way to look at it is
that a SOM is a (discretely) smooth mapping between regions in the input
space and points in the grid space. The best way to undestand this is to
look at the pictures in Kohonen (1995) or various other NN textbooks.
The Kohonen algorithm for SOMs is very similar to the Kohonen algorithm
for AVQ. Let the codebook vectors be indexed by a subscript j, and let
the index of the codebook vector nearest to the current training case be
n. The Kohonen SOM algorithm requires a kernel function K(j,n), where
K(j,j)=1 and K(j,n) is usually a non-increasing function of the
distance (in any metric) between codebook vectors j and n in the grid
space (not the input space). Usually, K(j,n) is zero for codebook
vectors that are far apart in the grid space. As each training case is
processed, all the codebook vectors are updated as:
new_codebook = old_codebook * [1 - K(j,n) * learning_rate]
j j
+ data * K(j,n) * learning_rate
The kernel function does not necessarily remain constant during training.
The neighborhood of a given codebook vector is the set of codebook
vectors for which K(j,n)>0. To avoid poor results (akin to local
minima), it is usually advisable to start with a large neighborhood, and
let the neighborhood gradually shrink during training. If K(j,n)=0
for j not equal to n, then the SOM update formula reduces to the formula
for Kohonen vector quantization. In other words, if the neighborhood size
(for example, the radius of the support of the kernel function K(j,n))
is zero, the SOM algorithm degenerates into simple VQ. Hence it is
important not to let the neighborhood size shrink all the way to zero
during training. Indeed, the choice of the final neighborhood size is the
most important tuning parameter for SOM training, as we will see shortly.
A SOM works by smoothing the codebook vectors in a manner similar to
kernel estimation methods, but the smoothing is done in neighborhoods in
the grid space rather than in the input space (Mulier and Cherkassky
1995). This is easier to see in a batch algorithm for SOMs, which is
similar to Forgy's algorithm for batch k-means, but incorporates an extra
smoothing process:
1. Initialize the codebook vectors.
2. Repeat the following two steps until convergence or boredom:
A. Read the data, assigning each case to the nearest (using Euclidean
distance) codebook vector. While you are doing this, keep track of the
mean and the number of cases for each cluster.
B. Do a nonparametric regression using K(j,n) as a kernel function,
with the grid points as inputs, the cluster means as target values,
and the number of cases in each cluster as an case weight. Replace
each codebook vector with the output of the nonparametric regression
function evaluated at its grid point.
If the nonparametric regression method is Nadaraya-Watson kernel
regression (see What is GRNN?), the batch SOM algorithm produces
essentially the same results as Kohonen's algorithm, barring local
minima. The main difference is that the batch algorithm often converges.
Mulier and Cherkassky (1995) note that other nonparametric regression
methods can be used to provide superior SOM algorithms. In particular,
local-linear smoothing eliminates the notorious "border effect", whereby
the codebook vectors near the border of the grid are compressed in the
input space. The border effect is especially problematic when you try to
use a high degree of smoothing in a Kohonen SOM, since all the codebook
vectors will collapse into the center of the input space. The SOM border
effect has the same mathematical cause as the "boundary effect" in kernel
regression, which causes the estimated regression function to flatten out
near the edges of the regression input space. There are various cures for
the edge effect in nonparametric regression, of which local-linear
smoothing is the simplest (Wand and Jones, 1995). Hence, local-linear
smoothing also cures the border effect in SOMs. Furthermore, local-linear
smoothing is much more general and reliable than the heuristic weighting
rule proposed by Kohonen (1995, p. 129).
Since nonparametric regression is used in the batch SOM algorithm,
various properties of nonparametric regression extend to SOMs. In
particular, it is well known that the shape of the kernel function is not
a crucial matter in nonparametric regression, hence it is not crucial in
SOMs. On the other hand, the amount of smoothing used for nonparametric
regression is crucial, hence the choice of the final neighborhood size in
a SOM is crucial. Unfortunately, I am not aware of any systematic studies
of methods for choosing the final neighborhood size.
The batch SOM algorithm is very similar to the principal curve and
surface algorithm proposed by Hastie and Stuetzle (1989), as pointed out
by Ritter, Martinetz, and Schulten (1992) and Mulier and Cherkassky
(1995). A principal curve is a nonlinear generalization of a principal
component. Given the probability distribution of a population, a
principal curve is defined by the following self-consistency condition:
1. If you choose any point on a principal curve,
2. then find the set of all the points in the input space that are closer
to the chosen point than any other point on the curve,
3. and compute the expected value (mean) of that set of points with
respect to the probability distribution, then
4. you end up with the same point on the curve that you chose originally.
See http://www.iro.umontreal.ca/~kegl/research/pcurves/ for more
information about principal curves and surfaces.
In a multivariate normal distribution, the principal curves are the same
as the principal components. A principal surface is the obvious
generalization from a curve to a surface. In a multivariate normal
distribution, the principal surfaces are the subspaces spanned by any two
principal components.
A one-dimensional local-linear batch SOM can be used to estimate a
principal curve by letting the number of codebook vectors approach
infinity while choosing a kernel function of appropriate smoothness. A
two-dimensional local-linear batch SOM can be used to estimate a
principal surface by letting the number of both rows and columns in the
grid approach infinity. This connection between SOMs and principal curves
and surfaces shows that the choice of the number of codebook vectors is
not crucial, provided the number is fairly large.
If the final neighborhood size in a local-linear batch SOM is large, the
SOM approximates a subspace spanned by principal components--usually the
first principal component if the SOM is one-dimensional, the first two
principal components if the SOM is two-dimensional, and so on. This
result does not depend on the data having a multivariate normal
distribution.
Principal component analysis is a method of data compression, not a
statistical model. However, there is a related method called "common
factor analysis" that is often confused with principal component analysis
but is indeed a statistical model. Common factor analysis posits that the
relations among the observed variables can be explained by a smaller
number of unobserved, "latent" variables. Tibshirani (1992) has proposed
a latent-variable variant of principal curves, and latent-variable
modifications of SOMs have been introduced by Utsugi (1996, 1997) and
Bishop, Svensén, and Williams (1997).
The choice of the number of codebook vectors is usually not critical as
long as the number is fairly large. But results can be sensitive to the
shape of the grid, e.g., square or an elongated rectangle. And the
dimensionality of the grid is a crucial choice. It is difficult to guess
the appropriate shape and dimensionality before analyzing the data.
Determining the shape and dimensionality by trial and error can be quite
tedious. Hence, a variety of methods have been tried for growing SOMs and
related kinds of NNs during training. For more information on growing
SOMs, see Bernd Fritzke's home page at
http://pikas.inf.tu-dresden.de/~fritzke/
Using a 1-by-2 SOM is pointless. There is no "topological structure" in a
1-by-2 grid. A 1-by-2 SOM is essentially the same as VQ with two
clusters, except that the SOM clusters will be closer together than the
VQ clusters if the final neighborhood size for the SOM is large.
In a Kohonen SOM, as in VQ, it is necessary to reduce the learning rate
during training to obtain convergence. Greg Heath has commented in this
regard:
I favor separate learning rates for each winning SOM node (or k-means
cluster) in the form 1/(N_0i + N_i + 1), where N_i is the
count of vectors that have caused node i to be a winner and N_0i
is an initializing count that indicates the confidence in the initial
weight vector assignment. The winning node expression is based on
stochastic estimation convergence constraints and pseudo-Bayesian
estimation of mean vectors. Kohonen derived a heuristic recursion
relation for the "optimal" rate. To my surprise, when I solved the
recursion relation I obtained the same above expression that I've
been using for years.
In addition, I have had success using the similar form
(1/n)/(N_0j + N_j + (1/n)) for the n nodes in the
shrinking updating-neighborhood. Before the final "winners-only"
stage when neighbors are no longer updated, the number of updating
neighbors eventually shrinks to n = 6 or 8 for hexagonal or
rectangular neighborhoods, respectively.
Kohonen's neighbor-update formula is more precise replacing my
constant fraction (1/n) with a node-pair specific h_ij (h_ij
< 1). However, as long as the initial neighborhood is sufficiently
large, the shrinking rate is sufficiently slow, and the final
winner-only stage is sufficiently long, the results should be
relatively insensitive to exact form of h_ij.
Another advantage of batch SOMs is that there is no learning rate, so these
complications evaporate.
Kohonen (1995, p. VII) says that SOMs are not intended for pattern
recognition but for clustering, visualization, and abstraction. Kohonen has
used a "supervised SOM" (1995, pp. 160-161) that is similar to
counterpropagation (Hecht-Nielsen 1990), but he seems to prefer LVQ (see
below) for supervised classification. Many people continue to use SOMs for
classification tasks, sometimes with surprisingly (I am tempted to say
"inexplicably") good results (Cho, 1997).
o LVQ: Learning Vector Quantization--competitive networks for supervised
classification (Kohonen, 1988, 1995; Ripley, 1996). Each codebook vector is
assigned to one of the target classes. Each class may have one or more
codebook vectors. A case is classified by finding the nearest codebook
vector and assigning the case to the class corresponding to the codebook
vector. Hence LVQ is a kind of nearest-neighbor rule.
Ordinary VQ methods, such as Kohonen's VQ or k-means, can easily be used for
supervised classification. Simply count the number of training cases from
each class assigned to each cluster, and divide by the total number of cases
in the cluster to get the posterior probability. For a given case, output
the class with the greatest posterior probability--i.e. the class that forms
a majority in the nearest cluster. Such methods can provide universally
consistent classifiers (Devroye et al., 1996) even when the codebook vectors
are obtained by unsupervised algorithms. LVQ tries to improve on this
approach by adapting the codebook vectors in a supervised way. There are
several variants of LVQ--called LVQ1, OLVQ1, LVQ2, and LVQ3--based on
heuristics. However, a smoothed version of LVQ can be trained as a
feedforward network using a NRBFEQ architecture (see "How do MLPs compare
with RBFs?") and optimizing any of the usual error functions; as the width
of the RBFs goes to zero, the NRBFEQ network approaches an optimized LVQ
network.
There are several other kinds of Kohonen networks described in Kohonen
(1995), including:
o DEC--Dynamically Expanding Context
o LSM--Learning Subspace Method
o ASSOM--Adaptive Subspace SOM
o FASSOM--Feedback-controlled Adaptive Subspace SOM
o Supervised SOM
o LVQ-SOM
More information on the error functions (or absence thereof) used by Kohonen
VQ and SOM is provided under "What does unsupervised learning learn?"
For more on-line information on Kohonen networks and other varieties of
SOMs, see:
o The web page of The Neural Networks Research Centre, Helsinki University
of Technology, at http://www.cis.hut.fi/research/
o The SOM of articles from comp.ai.neural-nets at
http://websom.hut.fi/websom/comp.ai.neural-nets-new/html/root.html
o Akio Utsugi's web page on Bayesian SOMs at the National Institute of
Bioscience and Human-Technology, Agency of Industrial Science and
Technology, M.I.T.I., 1-1, Higashi, Tsukuba, Ibaraki, 305 Japan, at
http://www.aist.go.jp/NIBH/~b0616/Lab/index-e.html
o The GTM (generative topographic mapping) home page at the Neural
Computing Research Group, Aston University, Birmingham, UK, at
http://www.ncrg.aston.ac.uk/GTM/
o Nenet SOM software at http://www.mbnet.fi/~phodju/nenet/nenet.html
o Bernd Fritzke's home page at http://pikas.inf.tu-dresden.de/~fritzke/ has
information on growing SOMs and other related types of NNs
References:
Anderberg, M.R. (1973), Cluster Analysis for Applications, New York:
Academic Press, Inc.
Balakrishnan, P.V., Cooper, M.C., Jacob, V.S., and Lewis, P.A. (1994) "A
study of the classification capabilities of neural networks using
unsupervised learning: A comparison with k-means clustering",
Psychometrika, 59, 509-525.
Bishop, C.M., Svensén, M., and Williams, C.K.I (1997), "GTM: A principled
alternative to the self-organizing map," in Mozer, M.C., Jordan, M.I.,
and Petsche, T., (eds.) Advances in Neural Information Processing
Systems 9, Cambridge, MA: The MIT Press, pp. 354-360. Also see
http://www.ncrg.aston.ac.uk/GTM/
Bottou, L., and Bengio, Y. (1995), "Convergence properties of the K-Means
algorithms," in Tesauro, G., Touretzky, D., and Leen, T., (eds.)
Advances in Neural Information Processing Systems 7, Cambridge, MA: The
MIT Press, pp. 585-592.
Cho, S.-B. (1997), "Self-organizing map with dynamical node-splitting:
Application to handwritten digit recognition," Neural Computation, 9,
1345-1355.
Desieno, D. (1988), "Adding a conscience to competitive learning," Proc.
Int. Conf. on Neural Networks, I, 117-124, IEEE Press.
Devroye, L., Györfi, L., and Lugosi, G. (1996), A Probabilistic Theory of
Pattern Recognition, NY: Springer,
Forgy, E.W. (1965), "Cluster analysis of multivariate data: Efficiency
versus interpretability," Biometric Society Meetings, Riverside, CA.
Abstract in Biomatrics, 21, 768.
Gersho, A. and Gray, R.M. (1992), Vector Quantization and Signal
Compression, Boston: Kluwer Academic Publishers.
Hartigan, J.A. (1975), Clustering Algorithms, NY: Wiley.
Hartigan, J.A., and Wong, M.A. (1979), "Algorithm AS136: A k-means
clustering algorithm," Applied Statistics, 28-100-108.
Hastie, T., and Stuetzle, W. (1989), "Principal curves," Journal of the
American Statistical Association, 84, 502-516.
Hecht-Nielsen, R. (1990), Neurocomputing, Reading, MA: Addison-Wesley.
Ismail, M.A., and Kamel, M.S. (1989), "Multidimensional data clustering
utilizing hybrid search strategies," Pattern Recognition, 22, 75-89.
Kohonen, T (1984), Self-Organization and Associative Memory, Berlin:
Springer-Verlag.
Kohonen, T (1988), "Learning Vector Quantization," Neural Networks, 1
(suppl 1), 303.
Kohonen, T. (1995/1997), Self-Organizing Maps, Berlin: Springer-Verlag.
First edition was 1995, second edition 1997. See
http://www.cis.hut.fi/nnrc/new_book.html for information on the second
edition.
Kosko, B.(1992), Neural Networks and Fuzzy Systems, Englewood Cliffs,
N.J.: Prentice-Hall.
Linde, Y., Buzo, A., and Gray, R. (1980), "An algorithm for vector
quantizer design," IEEE Transactions on Communications, 28, 84-95.
Lloyd, S. (1982), "Least squares quantization in PCM," IEEE Transactions
on Information Theory, 28, 129-137.
MacQueen, J.B. (1967), "Some Methods for Classification and Analysis of
Multivariate Observations,"Proceedings of the Fifth Berkeley Symposium on
Mathematical Statistics and Probability, 1, 281-297.
Max, J. (1960), "Quantizing for minimum distortion," IEEE Transactions on
Information Theory, 6, 7-12.
Mulier, F. and Cherkassky, V. (1995), "Self-Organization as an iterative
kernel smoothing process," Neural Computation, 7, 1165-1177.
Ripley, B.D. (1996), Pattern Recognition and Neural Networks,
Cambridge: Cambridge University Press.
Ritter, H., Martinetz, T., and Schulten, K. (1992), Neural Computation
and Self-Organizing Maps: An Introduction, Reading, MA: Addison-Wesley.
SAS Institute (1989), SAS/STAT User's Guide, Version 6, 4th edition,
Cary, NC: SAS Institute.
Symons, M.J. (1981), "Clustering Criteria and Multivariate Normal
Mixtures," Biometrics, 37, 35-43.
Tibshirani, R. (1992), "Principal curves revisited," Statistics and
Computing, 2, 183-190.
Utsugi, A. (1996), "Topology selection for self-organizing maps,"
Network: Computation in Neural Systems, 7, 727-740, available on-line at
http://www.aist.go.jp/NIBH/~b0616/Lab/index-e.html
Utsugi, A. (1997), "Hyperparameter selection for self-organizing maps,"
Neural Computation, 9, 623-635, available on-line at
http://www.aist.go.jp/NIBH/~b0616/Lab/index-e.html
Wand, M.P., and Jones, M.C. (1995), Kernel Smoothing, London: Chapman &
Hall.
Zador, P.L. (1982), "Asymptotic quantization error of continuous signals
and the quantization dimension," IEEE Transactions on Information Theory,
28, 139-149.
Zeger, K., Vaisey, J., and Gersho, A. (1992), "Globally optimal vector
quantizer design by stochastic relaxation," IEEE Transactions on Signal
Procesing, 40, 310-322.
Subject: How are layers counted?
=================================
How to count layers is a matter of considerable dispute.
o Some people count layers of units. But of these people, some count the
input layer and some don't.
o Some people count layers of weights. But I have no idea how they count
skip-layer connections.
To avoid ambiguity, you should speak of a 2-hidden-layer network, not a
4-layer network (as some would call it) or 3-layer network (as others would
call it). And if the connections follow any pattern other than fully
connecting each layer to the next and to no others, you should carefully
specify the connections.
Subject: What are cases and variables?
======================================
A vector of values presented at one time to all the input units of a neural
network is called a "case", "example", "pattern, "sample", etc. The term
"case" will be used in this FAQ because it is widely recognized,
unambiguous, and requires less typing than the other terms. A case may
include not only input values, but also target values and possibly other
information.
A vector of values presented at different times to a single input unit is
often called an "input variable" or "feature". To a statistician, it is a
"predictor", "regressor", "covariate", "independent variable", "explanatory
variable", etc. A vector of target values associated with a given output
unit of the network during training will be called a "target variable" in
this FAQ. To a statistician, it is usually a "response" or "dependent
variable".
A "data set" is a matrix containing one or (usually) more cases. In this
FAQ, it will be assumed that cases are rows of the matrix, while variables
are columns.
Note that the often-used term "input vector" is ambiguous; it can mean
either an input case or an input variable.
Subject: What are the population, sample, training set,
=======================================================
design set, validation set, and test set?
=========================================
It is rarely useful to have a NN simply memorize a set of data, since
memorization can be done much more efficiently by numerous algorithms for
table look-up. Typically, you want the NN to be able to perform accurately
on new data, that is, to generalize.
There seems to be no term in the NN literature for the set of all cases that
you want to be able to generalize to. Statisticians call this set the
"population". Tsypkin (1971) called it the "grand truth distribution," but
this term has never caught on.
Neither is there a consistent term in the NN literature for the set of cases
that are available for training and evaluating an NN. Statisticians call
this set the "sample". The sample is usually a subset of the population.
(Neurobiologists mean something entirely different by "population,"
apparently some collection of neurons, but I have never found out the exact
meaning. I am going to continue to use "population" in the statistical sense
until NN researchers reach a consensus on some other terms for "population"
and "sample"; I suspect this will never happen.)
In NN methodology, the sample is often subdivided into "training",
"validation", and "test" sets. The distinctions among these subsets are
crucial, but the terms "validation" and "test" sets are often confused.
Bishop (1995), an indispensable reference on neural networks, provides the
following explanation (p. 372):
Since our goal is to find the network having the best performance on
new data, the simplest approach to the comparison of different
networks is to evaluate the error function using data which is
independent of that used for training. Various networks are trained
by minimization of an appropriate error function defined with respect
to a training data set. The performance of the networks is then
compared by evaluating the error function using an independent
validation set, and the network having the smallest error with
respect to the validation set is selected. This approach is called
the hold out method. Since this procedure can itself lead to some
overfitting to the validation set, the performance of the selected
network should be confirmed by measuring its performance on a third
independent set of data called a test set.
And there is no book in the NN literature more authoritative than Ripley
(1996), from which the following definitions are taken (p.354):
Training set:
A set of examples used for learning, that is to fit the parameters [i.e.,
weights] of the classifier.
Validation set:
A set of examples used to tune the parameters [i.e., architecture, not
weights] of a classifier, for example to choose the number of hidden
units in a neural network.
Test set:
A set of examples used only to assess the performance [generalization] of
a fully-specified classifier.
The literature on machine learning often reverses the meaning of
"validation" and "test" sets. This is the most blatant example of the
terminological confusion that pervades artificial intelligence research.
The crucial point is that a test set, by the standard definition in the NN
literature, is never used to choose among two or more networks, so that the
error on the test set provides an unbiased estimate of the generalization
error (assuming that the test set is representative of the population,
etc.). Any data set that is used to choose the best of two or more networks
is, by definition, a validation set, and the error of the chosen network on
the validation set is optimistically biased.
There is a problem with the usual distinction between training and
validation sets. Some training approaches, such as early stopping, require a
validation set, so in a sense, the validation set is used for training.
Other approaches, such as maximum likelihood, do not inherently require a
validation set. So the "training" set for maximum likelihood might encompass
both the "training" and "validation" sets for early stopping. Greg Heath has
suggested the term "design" set be used for cases that are used solely to
adjust the weights in a network, while "training" set be used to encompass
both design and validation sets. There is considerable merit to this
suggestion, but it has not yet been widely adopted.
But things can get more complicated. Suppose you want to train nets with 5
,10, and 20 hidden units using maximum likelihood, and you want to train
nets with 20 and 50 hidden units using early stopping. You also want to use
a validation set to choose the best of these various networks. Should you
use the same validation set for early stopping that you use for the final
network choice, or should you use two separate validation sets? That is, you
could divide the sample into 3 subsets, say A, B, C and proceed as follows:
o Do maximum likelihood using A.
o Do early stopping with A to adjust the weights and B to decide when to
stop (this makes B a validation set).
o Choose among all 3 nets trained by maximum likelihood and the 2 nets
trained by early stopping based on the error computed on B (the
validation set).
o Estimate the generalization error of the chosen network using C (the test
set).
Or you could divide the sample into 4 subsets, say A, B, C, and D and
proceed as follows:
o Do maximum likelihood using A and B combined.
o Do early stopping with A to adjust the weights and B to decide when to
stop (this makes B a validation set with respect to early stopping).
o Choose among all 3 nets trained by maximum likelihood and the 2 nets
trained by early stopping based on the error computed on C (this makes C
a second validation set).
o Estimate the generalization error of the chosen network using D (the test
set).
Or, with the same 4 subsets, you could take a third approach:
o Do maximum likelihood using A.
o Choose among the 3 nets trained by maximum likelihood based on the error
computed on B (the first validation set)
o Do early stopping with A to adjust the weights and B (the first
validation set) to decide when to stop.
o Choose among the best net trained by maximum likelihood and the 2 nets
trained by early stopping based on the error computed on C (the second
validation set).
o Estimate the generalization error of the chosen network using D (the test
set).
You could argue that the first approach is biased towards choosing a net
trained by early stopping. Early stopping involves a choice among a
potentially large number of networks, and therefore provides more
opportunity for overfitting the validation set than does the choice among
only 3 networks trained by maximum likelihood. Hence if you make the final
choice of networks using the same validation set (B) that was used for early
stopping, you give an unfair advantage to early stopping. If you are writing
an article to compare various training methods, this bias could be a serious
flaw. But if you are using NNs for some practical application, this bias
might not matter at all, since you obtain an honest estimate of
generalization error using C.
You could also argue that the second and third approaches are too wasteful
in their use of data. This objection could be important if your sample
contains 100 cases, but will probably be of little concern if your sample
contains 100,000,000 cases. For small samples, there are other methods that
make more efficient use of data; see "What are cross-validation and
bootstrapping?"
References:
Bishop, C.M. (1995), Neural Networks for Pattern Recognition, Oxford:
Oxford University Press.
Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge:
Cambridge University Press.
Tsypkin, Y. (1971), Adaptation and Learning in Automatic Systems, NY:
Academic Press.
Subject: How are NNs related to statistical methods?
=====================================================
There is considerable overlap between the fields of neural networks and
statistics. Statistics is concerned with data analysis. In neural network
terminology, statistical inference means learning to generalize from noisy
data. Some neural networks are not concerned with data analysis (e.g., those
intended to model biological systems) and therefore have little to do with
statistics. Some neural networks do not learn (e.g., Hopfield nets) and
therefore have little to do with statistics. Some neural networks can learn
successfully only from noise-free data (e.g., ART or the perceptron rule)
and therefore would not be considered statistical methods. But most neural
networks that can learn to generalize effectively from noisy data are
similar or identical to statistical methods. For example:
o Feedforward nets with no hidden layer (including functional-link neural
nets and higher-order neural nets) are basically generalized linear
models.
o Feedforward nets with one hidden layer are closely related to projection
pursuit regression.
o Probabilistic neural nets are identical to kernel discriminant analysis.
o Kohonen nets for adaptive vector quantization are very similar to k-means
cluster analysis.
o Kohonen self-organizing maps are discrete approximations to principal
curves and surfaces.
o Hebbian learning is closely related to principal component analysis.
Some neural network areas that appear to have no close relatives in the
existing statistical literature are:
o Reinforcement learning (although this is treated in the operations
research literature on Markov decision processes).
o Stopped training (the purpose and effect of stopped training are similar
to shrinkage estimation, but the method is quite different).
Feedforward nets are a subset of the class of nonlinear regression and
discrimination models. Statisticians have studied the properties of this
general class but had not considered the specific case of feedforward neural
nets before such networks were popularized in the neural network field.
Still, many results from the statistical theory of nonlinear models apply
directly to feedforward nets, and the methods that are commonly used for
fitting nonlinear models, such as various Levenberg-Marquardt and conjugate
gradient algorithms, can be used to train feedforward nets. The application
of statistical theory to neural networks is explored in detail by Bishop
(1995) and Ripley (1996). Several summary articles have also been published
relating statistical models to neural networks, including Cheng and
Titterington (1994), Kuan and White (1994), Ripley (1993, 1994), Sarle
(1994), and several articles in Cherkassky, Friedman, and Wechsler (1994).
Among the many statistical concepts important to neural nets is the
bias/variance trade-off in nonparametric estimation, discussed by Geman,
Bienenstock, and Doursat, R. (1992). Some more advanced results of
statistical theory applied to neural networks are given by White (1989a,
1989b, 1990, 1992a) and White and Gallant (1992), reprinted in White
(1992b).
While neural nets are often defined in terms of their algorithms or
implementations, statistical methods are usually defined in terms of their
results. The arithmetic mean, for example, can be computed by a (very
simple) backprop net, by applying the usual formula SUM(x_i)/n, or by
various other methods. What you get is still an arithmetic mean regardless
of how you compute it. So a statistician would consider standard backprop,
Quickprop, and Levenberg-Marquardt as different algorithms for implementing
the same statistical model such as a feedforward net. On the other hand,
different training criteria, such as least squares and cross entropy, are
viewed by statisticians as fundamentally different estimation methods with
different statistical properties.
It is sometimes claimed that neural networks, unlike statistical models,
require no distributional assumptions. In fact, neural networks involve
exactly the same sort of distributional assumptions as statistical models
(Bishop, 1995), but statisticians study the consequences and importance of
these assumptions while many neural networkers ignore them. For example,
least-squares training methods are widely used by statisticians and neural
networkers. Statisticians realize that least-squares training involves
implicit distributional assumptions in that least-squares estimates have
certain optimality properties for noise that is normally distributed with
equal variance for all training cases and that is independent between
different cases. These optimality properties are consequences of the fact
that least-squares estimation is maximum likelihood under those conditions.
Similarly, cross-entropy is maximum likelihood for noise with a Bernoulli
distribution. If you study the distributional assumptions, then you can
recognize and deal with violations of the assumptions. For example, if you
have normally distributed noise but some training cases have greater noise
variance than others, then you may be able to use weighted least squares
instead of ordinary least squares to obtain more efficient estimates.
Hundreds, perhaps thousands of people have run comparisons of neural nets
with "traditional statistics" (whatever that means). Most such studies
involve one or two data sets, and are of little use to anyone else unless
they happen to be analyzing the same kind of data. But there is an
impressive comparative study of supervised classification by Michie,
Spiegelhalter, and Taylor (1994), which not only compares many
classification methods on many data sets, but also provides unusually
extensive analyses of the results. Another useful study on supervised
classification by Lim, Loh, and Shih (1999) is available on-line. There is
an excellent comparison of unsupervised Kohonen networks and k-means
clustering by Balakrishnan, Cooper, Jacob, and Lewis (1994).
There are many methods in the statistical literature that can be used for
flexible nonlinear modeling. These methods include:
o Polynomial regression (Eubank, 1999)
o Fourier series regression (Eubank, 1999; Haerdle, 1990)
o Wavelet smoothing (Donoho and Johnstone, 1995; Donoho, Johnstone,
Kerkyacharian, and Picard, 1995)
o K-nearest neighbor regression and discriminant analysis (Haerdle, 1990;
Hand, 1981, 1997; Ripley, 1996)
o Kernel regression and discriminant analysis (Eubank, 1999; Haerdle, 1990;
Hand, 1981, 1982, 1997; Ripley, 1996)
o Local polynomial smoothing (Eubank, 1999; Wand and Jones, 1995; Fan and
Gijbels, 1995)
o LOESS (Cleveland and Gross, 1991)
o Smoothing splines (such as thin-plate splines) (Eubank, 1999; Wahba,
1990; Green and Silverman, 1994; Haerdle, 1990)
o B-splines (Eubank, 1999)
o Tree-based models (CART, AID, etc.) (Haerdle, 1990; Lim, Loh, and Shih,
1997; Hand, 1997; Ripley, 1996)
o Multivariate adaptive regression splines (MARS) (Friedman, 1991)
o Projection pursuit (Friedman and Stuetzle, 1981; Haerdle, 1990; Ripley,
1996)
o Various Bayesian methods (Dey, 1998)
o GMDH (Farlow, 1984)
Why use neural nets rather than any of the above methods? There are many
answers to that question depending on what kind of neural net you're
interested in. The most popular variety of neural net, the MLP, tends to be
useful in the same situations as projection pursuit regression, i.e.:
o the number of inputs is fairly large,
o many of the inputs are relevant, but
o most of the predictive information lies in a low-dimensional subspace.
The main advantage of MLPs over projection pursuit regression is that
computing predicted values from MLPs is simpler and faster. Also, MLPs are
better at learning moderately pathological functions than are many other
methods with stronger smoothness assumptions (see
ftp://ftp.sas.com/pub/neural/dojo/dojo.html) as long as the number of
pathological features (such as discontinuities) in the function is not too
large. For more discussion of the theoretical benefits of various types of
neural nets, see How do MLPs compare with RBFs?
Communication between statisticians and neural net researchers is often
hindered by the different terminology used in the two fields. There is a
comparison of neural net and statistical jargon in
ftp://ftp.sas.com/pub/neural/jargon
For free statistical software, see the StatLib repository at
http://lib.stat.cmu.edu/ at Carnegie Mellon University.
There are zillions of introductory textbooks on statistics. One of the
better ones is Moore and McCabe (1989). At an intermediate level, the books
on linear regression by Weisberg (1985) and Myers (1986), on logistic
regression by Hosmer and Lemeshow (1989), and on discriminant analysis by
Hand (1981) can be recommended. At a more advanced level, the book on
generalized linear models by McCullagh and Nelder (1989) is an essential
reference, and the book on nonlinear regression by Gallant (1987) has much
material relevant to neural nets.
Several introductory statistics texts are available on the web:
o David Lane, HyperStat, at
http://www.ruf.rice.edu/~lane/hyperstat/contents.html
o Jan de Leeuw (ed.), Statistics: The Study of Stability in Variation , at
http://www.stat.ucla.edu/textbook/
o StatSoft, Inc., Electronic Statistics Textbook, at
http://www.statsoft.com/textbook/stathome.html
o David Stockburger, Introductory Statistics: Concepts, Models, and
Applications, at http://www.psychstat.smsu.edu/sbk00.htm
o University of Newcastle (Australia) Statistics Department, SurfStat
Australia, http://surfstat.newcastle.edu.au/surfstat/
A more advanced book covering many topics that are also relevant to NNs is:
o Frank Harrell, REGRESSION MODELING STRATEGIES With
Applications to Linear Models, Logistic Regression, and Survival Analysis,
at http://hesweb1.med.virginia.edu/biostat/rms/
References:
Balakrishnan, P.V., Cooper, M.C., Jacob, V.S., and Lewis, P.A. (1994) "A
study of the classification capabilities of neural networks using
unsupervised learning: A comparison with k-means clustering",
Psychometrika, 59, 509-525.
Bishop, C.M. (1995), Neural Networks for Pattern Recognition, Oxford:
Oxford University Press.
Cheng, B. and Titterington, D.M. (1994), "Neural Networks: A Review from
a Statistical Perspective", Statistical Science, 9, 2-54.
Cherkassky, V., Friedman, J.H., and Wechsler, H., eds. (1994), From
Statistics to Neural Networks: Theory and Pattern Recognition
Applications, Berlin: Springer-Verlag.
Cleveland and Gross (1991), "Computational Methods for Local Regression,"
Statistics and Computing 1, 47-62.
Dey, D., ed. (1998) Practical Nonparametric and Semiparametric Bayesian
Statistics, Springer Verlag.
Donoho, D.L., and Johnstone, I.M. (1995), "Adapting to unknown smoothness
via wavelet shrinkage," J. of the American Statistical Association, 90,
1200-1224.
Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., and Picard, D. (1995),
"Wavelet shrinkage: asymptopia (with discussion)?" J. of the Royal
Statistical Society, Series B, 57, 301-369.
Eubank, R.L. (1999), Nonparametric Regression and Spline Smoothing, 2nd
ed., Marcel Dekker, ISBN 0-8247-9337-4.
Fan, J., and Gijbels, I. (1995), "Data-driven bandwidth selection in
local polynomial: variable bandwidth and spatial adaptation," J. of the
Royal Statistical Society, Series B, 57, 371-394.
Farlow, S.J. (1984), Self-organizing Methods in Modeling: GMDH Type
Algorithms, NY: Marcel Dekker. (GMDH)
Friedman, J.H. (1991), "Multivariate adaptive regression splines", Annals
of Statistics, 19, 1-141. (MARS)
Friedman, J.H. and Stuetzle, W. (1981) "Projection pursuit regression,"
J. of the American Statistical Association, 76, 817-823.
Gallant, A.R. (1987) Nonlinear Statistical Models, NY: Wiley.
Geman, S., Bienenstock, E. and Doursat, R. (1992), "Neural Networks and
the Bias/Variance Dilemma", Neural Computation, 4, 1-58.
Green, P.J., and Silverman, B.W. (1994), Nonparametric Regression and
Generalized Linear Models: A Roughness Penalty Approach, London:
Chapman & Hall.
Haerdle, W. (1990), Applied Nonparametric Regression, Cambridge Univ.
Press.
Hand, D.J. (1981) Discrimination and Classification, NY: Wiley.
Hand, D.J. (1982) Kernel Discriminant Analysis, Research Studies Press.
Hand, D.J. (1997) Construction and Assessment of Classification Rules,
NY: Wiley.
Hill, T., Marquez, L., O'Connor, M., and Remus, W. (1994), "Artificial
neural network models for forecasting and decision making," International
J. of Forecasting, 10, 5-15.
Kuan, C.-M. and White, H. (1994), "Artificial Neural Networks: An
Econometric Perspective", Econometric Reviews, 13, 1-91.
Kushner, H. & Clark, D. (1978), Stochastic Approximation Methods for
Constrained and Unconstrained Systems, Springer-Verlag.
Lim, T.-S., Loh, W.-Y. and Shih, Y.-S. ( 1999?), "A comparison of
prediction accuracy, complexity, and training time of thirty-three old
and new classification algorithms," Machine Learning, forthcoming,
preprint available at http://www.recursive-partitioning.com/mach1317.pdf,
and appendix containing complete tables of error rates, ranks, and
training times at http://www.recursive-partitioning.com/appendix.pdf
McCullagh, P. and Nelder, J.A. (1989) Generalized Linear Models, 2nd
ed., London: Chapman & Hall.
Michie, D., Spiegelhalter, D.J. and Taylor, C.C., eds. (1994), Machine
Learning, Neural and Statistical Classification, NY: Ellis Horwood; this
book is out of print but available online at
http://www.amsta.leeds.ac.uk/~charles/statlog/
Moore, D.S., and McCabe, G.P. (1989), Introduction to the Practice of
Statistics, NY: W.H. Freeman.
Myers, R.H. (1986), Classical and Modern Regression with Applications,
Boston: Duxbury Press.
Ripley, B.D. (1993), "Statistical Aspects of Neural Networks", in O.E.
Barndorff-Nielsen, J.L. Jensen and W.S. Kendall, eds., Networks and
Chaos: Statistical and Probabilistic Aspects, Chapman & Hall. ISBN 0 412
46530 2.
Ripley, B.D. (1994), "Neural Networks and Related Methods for
Classification," Journal of the Royal Statistical Society, Series B, 56,
409-456.
Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge:
Cambridge University Press.
Sarle, W.S. (1994), "Neural Networks and Statistical Models,"
Proceedings of the Nineteenth Annual SAS Users Group International
Conference, Cary, NC: SAS Institute, pp 1538-1550. (
ftp://ftp.sas.com/pub/neural/neural1.ps)
Wahba, G. (1990), Spline Models for Observational Data, SIAM.
Wand, M.P., and Jones, M.C. (1995), Kernel Smoothing, London: Chapman &
Hall.
Weisberg, S. (1985), Applied Linear Regression, NY: Wiley
White, H. (1989a), "Learning in Artificial Neural Networks: A Statistical
Perspective," Neural Computation, 1, 425-464.
White, H. (1989b), "Some Asymptotic Results for Learning in Single Hidden
Layer Feedforward Network Models", J. of the American Statistical Assoc.,
84, 1008-1013.
White, H. (1990), "Connectionist Nonparametric Regression: Multilayer
Feedforward Networks Can Learn Arbitrary Mappings," Neural Networks, 3,
535-550.
White, H. (1992a), "Nonparametric Estimation of Conditional Quantiles
Using Neural Networks," in Page, C. and Le Page, R. (eds.), Computing
Science and Statistics.
White, H., and Gallant, A.R. (1992), "On Learning the Derivatives of an
Unknown Mapping with Multilayer Feedforward Networks," Neural Networks,
5, 129-138.
White, H. (1992b), Artificial Neural Networks: Approximation and
Learning Theory, Blackwell.
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Next part is part 2 (of 7).
--
Warren S. Sarle SAS Institute Inc. The opinions expressed here
saswss@unx.sas.com SAS Campus Drive are mine and not necessarily
(919) 677-8000 Cary, NC 27513, USA those of SAS Institute.
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