From: saswss@unx.sas.com (Warren Sarle)
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Subject: comp.ai.neural-nets FAQ, Part 3 of 7: Generalization
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in part 1 of the FAQ.
This is part 3 (of 7) of a monthly posting to the Usenet newsgroup
comp.ai.neural-nets. See the part 1 of this posting for full information
what it is all about.
========== Questions ==========
********************************
Part 1: Introduction
Part 2: Learning
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.
Part 5: Free software
Part 6: Commercial software
Part 7: Hardware and miscellaneous
------------------------------------------------------------------------
Subject: How is generalization possible?
=========================================
During learning, the outputs of a supervised neural net come to approximate
the target values given the inputs in the training set. This ability may be
useful in itself, but more often the purpose of using a neural net is to
generalize--i.e., to have the outputs of the net approximate target values
given inputs that are not in the training set. Generalizaton is not always
possible, despite the blithe assertions of some authors. For example,
Caudill and Butler, 1990, p. 8, claim that "A neural network is able to
generalize", but they provide no justification for this claim, and they
completely neglect the complex issues involved in getting good
generalization. Anyone who reads comp.ai.neural-nets is well aware from the
numerous posts pleading for help that artificial neural networks do not
automatically generalize.
Generalization requires prior knowledge, as pointed out by Hume (1739/1978),
Russell (1948), and Goodman (1954/1983) and rigorously proved by Wolpert
(1995a, 1996a, 1996b). For any practical application, you have to know what
the relevant inputs are (you can't simply include every imaginable input).
You have to know a restricted class of input-output functions that contains
an adequate approximation to the function you want to learn (you can't use a
learning method that is capable of fitting every imaginable function). And
you have to know that the cases you want to generalize to bear some
resemblance to the training cases. Thus, there are three conditions that are
typically necessary--although not sufficient--for good generalization:
1. The first necessary condition is that the inputs to the network contain
sufficient information pertaining to the target, so that there exists a
mathematical function relating correct outputs to inputs with the desired
degree of accuracy. You can't expect a network to learn a nonexistent
function--neural nets are not clairvoyant! For example, if you want to
forecast the price of a stock, a historical record of the stock's prices
is rarely sufficient input; you need detailed information on the
financial state of the company as well as general economic conditions,
and to avoid nasty surprises, you should also include inputs that can
accurately predict wars in the Middle East and earthquakes in Japan.
Finding good inputs for a net and collecting enough training data often
take far more time and effort than training the network.
2. The second necessary condition is that the function you are trying to
learn (that relates inputs to correct outputs) be, in some sense, smooth.
In other words, a small change in the inputs should, most of the time,
produce a small change in the outputs. For continuous inputs and targets,
smoothness of the function implies continuity and restrictions on the
first derivative over most of the input space. Some neural nets can learn
discontinuities as long as the function consists of a finite number of
continuous pieces. Very nonsmooth functions such as those produced by
pseudo-random number generators and encryption algorithms cannot be
generalized by neural nets. Often a nonlinear transformation of the input
space can increase the smoothness of the function and improve
generalization.
For classification, if you do not need to estimate posterior
probabilities, then smoothness is not theoretically necessary. In
particular, feedforward networks with one hidden layer trained by
minimizing the error rate (a very tedious training method) are
universally consistent classifiers if the number of hidden units grows at
a suitable rate relative to the number of training cases (Devroye,
Györfi, and Lugosi, 1996). However, you are likely to get better
generalization with realistic sample sizes if the classification
boundaries are smoother.
For Boolean functions, the concept of smoothness is more elusive. It
seems intuitively clear that a Boolean network with a small number of
hidden units and small weights will compute a "smoother" input-output
function than a network with many hidden units and large weights. If you
know a good reference characterizing Boolean functions for which good
generalization is possible, please inform the FAQ maintainer
(saswss@unx.sas.com).
3. The third necessary condition for good generalization is that the
training cases be a sufficiently large and representative subset
("sample" in statistical terminology) of the set of all cases that you
want to generalize to (the "population" in statistical terminology). The
importance of this condition is related to the fact that there are,
loosely speaking, two different types of generalization: interpolation
and extrapolation. Interpolation applies to cases that are more or less
surrounded by nearby training cases; everything else is extrapolation. In
particular, cases that are outside the range of the training data require
extrapolation. Cases inside large "holes" in the training data may also
effectively require extrapolation. Interpolation can often be done
reliably, but extrapolation is notoriously unreliable. Hence it is
important to have sufficient training data to avoid the need for
extrapolation. Methods for selecting good training sets are discussed in
numerous statistical textbooks on sample surveys and experimental design.
Thus, for an input-output function that is smooth, if you have a test case
that is close to some training cases, the correct output for the test case
will be close to the correct outputs for those training cases. If you have
an adequate sample for your training set, every case in the population will
be close to a sufficient number of training cases. Hence, under these
conditions and with proper training, a neural net will be able to generalize
reliably to the population.
If you have more information about the function, e.g. that the outputs
should be linearly related to the inputs, you can often take advantage of
this information by placing constraints on the network or by fitting a more
specific model, such as a linear model, to improve generalization.
Extrapolation is much more reliable in linear models than in flexible
nonlinear models, although still not nearly as safe as interpolation. You
can also use such information to choose the training cases more efficiently.
For example, with a linear model, you should choose training cases at the
outer limits of the input space instead of evenly distributing them
throughout the input space.
References:
Caudill, M. and Butler, C. (1990). Naturally Intelligent Systems. MIT
Press: Cambridge, Massachusetts.
Devroye, L., Györfi, L., and Lugosi, G. (1996), A Probabilistic Theory of
Pattern Recognition, NY: Springer.
Goodman, N. (1954/1983), Fact, Fiction, and Forecast, 1st/4th ed.,
Cambridge, MA: Harvard University Press.
Holland, J.H., Holyoak, K.J., Nisbett, R.E., Thagard, P.R. (1986),
Induction: Processes of Inference, Learning, and Discovery, Cambridge, MA:
The MIT Press.
Howson, C. and Urbach, P. (1989), Scientific Reasoning: The Bayesian
Approach, La Salle, IL: Open Court.
Hume, D. (1739/1978), A Treatise of Human Nature, Selby-Bigge, L.A.,
and Nidditch, P.H. (eds.), Oxford: Oxford University Press.
Plotkin, H. (1993), Darwin Machines and the Nature of Knowledge,
Cambridge, MA: Harvard University Press.
Russell, B. (1948), Human Knowledge: Its Scope and Limits, London:
Routledge.
Stone, C.J. (1977), "Consistent nonparametric regression," Annals of
Statistics, 5, 595-645.
Stone, C.J. (1982), "Optimal global rates of convergence for
nonparametric regression," Annals of Statistics, 10, 1040-1053.
Vapnik, V.N. (1995), The Nature of Statistical Learning Theory, NY:
Springer.
Wolpert, D.H. (1995a), "The relationship between PAC, the statistical
physics framework, the Bayesian framework, and the VC framework," in
Wolpert (1995b), 117-214.
Wolpert, D.H. (ed.) (1995b), The Mathematics of Generalization: The
Proceedings of the SFI/CNLS Workshop on Formal Approaches to
Supervised Learning, Santa Fe Institute Studies in the Sciences of
Complexity, Volume XX, Reading, MA: Addison-Wesley.
Wolpert, D.H. (1996a), "The lack of a priori distinctions between
learning algorithms," Neural Computation, 8, 1341-1390.
Wolpert, D.H. (1996b), "The existence of a priori distinctions between
learning algorithms," Neural Computation, 8, 1391-1420.
------------------------------------------------------------------------
Subject: How does noise affect generalization?
===============================================
"Statistical noise" means variation in the target values that is
unpredictable from the inputs of a specific network, regardless of the
architecture or weights. "Physical noise" refers to variation in the target
values that is inherently unpredictable regardless of what inputs are used.
Noise in the inputs usually refers to measurement error, so that if the same
object or example is presented to the network more than once, the input
values differ.
Noise in the actual data is never a good thing, since it limits the accuracy
of generalization that can be achieved no matter how extensive the training
set is. On the other hand, injecting artificial noise (jitter) into the
inputs during training is one of several ways to improve generalization for
smooth functions when you have a small training set.
Certain assumptions about noise are necessary for theoretical results.
Usually, the noise distribution is assumed to have zero mean and finite
variance. The noise in different cases is usually assumed to be independent
or to follow some known stochastic model, such as an autoregressive process.
The more you know about the noise distribution, the more effectively you can
train the network (e.g., McCullagh and Nelder 1989).
If you have noise in the target values, what the network learns depends
mainly on the error function. For example, if the noise is independent with
finite variance for all training cases, a network that is well-trained using
least squares will produce outputs that approximate the conditional mean of
the target values (White, 1990; Bishop, 1995). Note that for a binary 0/1
variable, the mean is equal to the probability of getting a 1. Hence, the
results in White (1990) immediately imply that for a categorical target with
independent noise using 1-of-C coding (see "How should categories be
encoded?"), a network that is well-trained using least squares will produce
outputs that approximate the posterior probabilities of each class (see
Rojas, 1996, if you want a simple explanation of why least-squares estimates
probabilities). Posterior probabilities can also be learned using
cross-entropy and various other error functions (Finke and Müller, 1994;
Bishop, 1995). The conditional median can be learned by least-absolute-value
training (White, 1992a). Conditional modes can be approximated by yet other
error functions (e.g., Rohwer and van der Rest 1996). For noise
distributions that cannot be adequately approximated by a single location
estimate (such as the mean, median, or mode), a network can be trained to
approximate quantiles (White, 1992a) or mixture components (Bishop, 1995;
Husmeier, 1999).
If you have noise in the target values, the mean squared generalization
error can never be less than the variance of the noise, no matter how much
training data you have. But you can estimate the mean of the target values,
conditional on a given set of input values, to any desired degree of
accuracy by obtaining a sufficiently large and representative training set,
assuming that the function you are trying to learn is one that can indeed be
learned by the type of net you are using, and assuming that the complexity
of the network is regulated appropriately (White 1990).
Noise in the target values increases the danger of overfitting (Moody 1992).
Noise in the inputs limits the accuracy of generalization, but in a more
complicated way than does noise in the targets. In a region of the input
space where the function being learned is fairly flat, input noise will have
little effect. In regions where that function is steep, input noise can
degrade generalization severely.
Furthermore, if the target function is Y=f(X), but you observe noisy inputs
X+D, you cannot obtain an arbitrarily accurate estimate of f(X) given X+D no
matter how large a training set you use. The net will not learn f(X), but
will instead learn a convolution of f(X) with the distribution of the noise
D (see "What is jitter?)"
For more details, see one of the statistically-oriented references on neural
nets such as:
Bishop, C.M. (1995), Neural Networks for Pattern Recognition, Oxford:
Oxford University Press, especially section 6.4.
Finke, M., and Müller, K.-R. (1994), "Estimating a-posteriori
probabilities using stochastic network models," in Mozer, Smolensky,
Touretzky, Elman, & Weigend, eds., Proceedings of the 1993 Connectionist
Models Summer School, Hillsdale, NJ: Lawrence Erlbaum Associates, pp.
324-331.
Geman, S., Bienenstock, E. and Doursat, R. (1992), "Neural Networks and
the Bias/Variance Dilemma", Neural Computation, 4, 1-58.
Husmeier, D. (1999), Neural Networks for Conditional Probability
Estimation: Forecasting Beyond Point Predictions, Berlin: Springer
Verlag, ISBN 185233095.
McCullagh, P. and Nelder, J.A. (1989) Generalized Linear Models, 2nd
ed., London: Chapman & Hall.
Moody, J.E. (1992), "The Effective Number of Parameters: An Analysis of
Generalization and Regularization in Nonlinear Learning Systems", in
Moody, J.E., Hanson, S.J., and Lippmann, R.P., Advances in Neural
Information Processing Systems 4, 847-854.
Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge:
Cambridge University Press.
Rohwer, R., and van der Rest, J.C. (1996), "Minimum description length,
regularization, and multimodal data," Neural Computation, 8, 595-609.
Rojas, R. (1996), "A short proof of the posterior probability property of
classifier neural networks," Neural Computation, 8, 41-43.
White, H. (1990), "Connectionist Nonparametric Regression: Multilayer
Feedforward Networks Can Learn Arbitrary Mappings," Neural Networks, 3,
535-550. Reprinted in White (1992).
White, H. (1992a), "Nonparametric Estimation of Conditional Quantiles
Using Neural Networks," in Page, C. and Le Page, R. (eds.), Proceedings
of the 23rd Sympsium on the Interface: Computing Science and Statistics,
Alexandria, VA: American Statistical Association, pp. 190-199. Reprinted
in White (1992b).
White, H. (1992b), Artificial Neural Networks: Approximation and
Learning Theory, Blackwell.
------------------------------------------------------------------------
Subject: What is overfitting and how can I avoid it?
=====================================================
The critical issue in developing a neural network is generalization: how
well will the network make predictions for cases that are not in the
training set? NNs, like other flexible nonlinear estimation methods such as
kernel regression and smoothing splines, can suffer from either underfitting
or overfitting. A network that is not sufficiently complex can fail to
detect fully the signal in a complicated data set, leading to underfitting.
A network that is too complex may fit the noise, not just the signal,
leading to overfitting. Overfitting is especially dangerous because it can
easily lead to predictions that are far beyond the range of the training
data with many of the common types of NNs. Overfitting can also produce wild
predictions in multilayer perceptrons even with noise-free data.
For an elementary discussion of overfitting, see Smith (1996). For a more
rigorous approach, see the article by Geman, Bienenstock, and Doursat (1992)
on the bias/variance trade-off (it's not really a dilemma). We are talking
about statistical bias here: the difference between the average value of an
estimator and the correct value. Underfitting produces excessive bias in the
outputs, whereas overfitting produces excessive variance. There are
graphical examples of overfitting and underfitting in Sarle (1995, 1999).
The best way to avoid overfitting is to use lots of training data. If you
have at least 30 times as many training cases as there are weights in the
network, you are unlikely to suffer from much overfitting, although you may
get some slight overfitting no matter how large the training set is. For
noise-free data, 5 times as many training cases as weights may be
sufficient. But you can't arbitrarily reduce the number of weights for fear
of underfitting.
Given a fixed amount of training data, there are at least six approaches to
avoiding underfitting and overfitting, and hence getting good
generalization:
o Model selection
o Jittering
o Early stopping
o Weight decay
o Bayesian learning
o Combining networks
The first five approaches are based on well-understood theory. Methods for
combining networks do not have such a sound theoretical basis but are the
subject of current research. These six approaches are discussed in more
detail under subsequent questions.
The complexity of a network is related to both the number of weights and the
size of the weights. Model selection is concerned with the number of
weights, and hence the number of hidden units and layers. The more weights
there are, relative to the number of training cases, the more overfitting
amplifies noise in the targets (Moody 1992). The other approaches listed
above are concerned, directly or indirectly, with the size of the weights.
Reducing the size of the weights reduces the "effective" number of
weights--see Moody (1992) regarding weight decay and Weigend (1994)
regarding early stopping. Bartlett (1997) obtained learning-theory results
in which generalization error is related to the L_1 norm of the weights
instead of the VC dimension.
Overfitting is not confined to NNs with hidden units. Overfitting can occur
in generalized linear models (networks with no hidden units) if either or
both of the following conditions hold:
1. The number of input variables (and hence the number of weights) is large
with respect to the number of training cases. Typically you would want at
least 10 times as many training cases as input variables, but with
noise-free targets, twice as many training cases as input variables would
be more than adequate. These requirements are smaller than those stated
above for networks with hidden layers, because hidden layers are prone to
creating ill-conditioning and other pathologies.
2. The input variables are highly correlated with each other. This condition
is called "multicollinearity" in the statistical literature.
Multicollinearity can cause the weights to become extremely large because
of numerical ill-conditioning--see "How does ill-conditioning affect NN
training?"
Methods for dealing with these problems in the statistical literature
include ridge regression (similar to weight decay), partial least squares
(similar to Early stopping), and various methods with even stranger names,
such as the lasso and garotte (van Houwelingen and le Cessie, ????).
References:
Bartlett, P.L. (1997), "For valid generalization, the size of the weights
is more important than the size of the network," in Mozer, M.C., Jordan,
M.I., and Petsche, T., (eds.) Advances in Neural Information Processing
Systems 9, Cambrideg, MA: The MIT Press, pp. 134-140.
Geman, S., Bienenstock, E. and Doursat, R. (1992), "Neural Networks and
the Bias/Variance Dilemma", Neural Computation, 4, 1-58.
Moody, J.E. (1992), "The Effective Number of Parameters: An Analysis of
Generalization and Regularization in Nonlinear Learning Systems", in
Moody, J.E., Hanson, S.J., and Lippmann, R.P., Advances in Neural
Information Processing Systems 4, 847-854.
Sarle, W.S. (1995), "Stopped Training and Other Remedies for
Overfitting," Proceedings of the 27th Symposium on the Interface of
Computing Science and Statistics, 352-360,
ftp://ftp.sas.com/pub/neural/inter95.ps.Z (this is a very large
compressed postscript file, 747K, 10 pages)
Sarle, W.S. (1999), "Donoho-Johnstone Benchmarks: Neural Net Results,"
ftp://ftp.sas.com/pub/neural/dojo/dojo.html
Smith, M. (1996). Neural Networks for Statistical Modeling, Boston:
International Thomson Computer Press, ISBN 1-850-32842-0.
van Houwelingen,H.C., and le Cessie, S. (????), "Shrinkage and penalized
likelihood as methods to improve predictive accuracy,"
http://www.medstat.medfac.leidenuniv.nl/ms/HH/Files/shrinkage.pdf and
http://www.medstat.medfac.leidenuniv.nl/ms/HH/Files/figures.pdf
Weigend, A. (1994), "On overfitting and the effective number of hidden
units," Proceedings of the 1993 Connectionist Models Summer School,
335-342.
------------------------------------------------------------------------
Subject: What is jitter? (Training with noise)
===============================================
Jitter is artificial noise deliberately added to the inputs during training.
Training with jitter is a form of smoothing related to kernel regression
(see "What is GRNN?"). It is also closely related to regularization methods
such as weight decay and ridge regression.
Training with jitter works because the functions that we want NNs to learn
are mostly smooth. NNs can learn functions with discontinuities, but the
functions must be piecewise continuous in a finite number of regions if our
network is restricted to a finite number of hidden units.
In other words, if we have two cases with similar inputs, the desired
outputs will usually be similar. That means we can take any training case
and generate new training cases by adding small amounts of jitter to the
inputs. As long as the amount of jitter is sufficiently small, we can assume
that the desired output will not change enough to be of any consequence, so
we can just use the same target value. The more training cases, the merrier,
so this looks like a convenient way to improve training. But too much jitter
will obviously produce garbage, while too little jitter will have little
effect (Koistinen and Holmström 1992).
Consider any point in the input space, not necessarily one of the original
training cases. That point could possibly arise as a jittered input as a
result of jittering any of several of the original neighboring training
cases. The average target value at the given input point will be a weighted
average of the target values of the original training cases. For an infinite
number of jittered cases, the weights will be proportional to the
probability densities of the jitter distribution, located at the original
training cases and evaluated at the given input point. Thus the average
target values given an infinite number of jittered cases will, by
definition, be the Nadaraya-Watson kernel regression estimator using the
jitter density as the kernel. Hence, training with jitter is an
approximation to training with the kernel regression estimator as target.
Choosing the amount (variance) of jitter is equivalent to choosing the
bandwidth of the kernel regression estimator (Scott 1992).
When studying nonlinear models such as feedforward NNs, it is often helpful
first to consider what happens in linear models, and then to see what
difference the nonlinearity makes. So let's consider training with jitter in
a linear model. Notation:
x_ij is the value of the jth input (j=1, ..., p) for the
ith training case (i=1, ..., n).
X={x_ij} is an n by p matrix.
y_i is the target value for the ith training case.
Y={y_i} is a column vector.
Without jitter, the least-squares weights are B = inv(X'X)X'Y, where
"inv" indicates a matrix inverse and "'" indicates transposition. Note that
if we replicate each training case c times, or equivalently stack c copies
of the X and Y matrices on top of each other, the least-squares weights are
inv(cX'X)cX'Y = (1/c)inv(X'X)cX'Y = B, same as before.
With jitter, x_ij is replaced by c cases x_ij+z_ijk, k=1, ...,
c, where z_ijk is produced by some random number generator, usually with
a normal distribution with mean 0 and standard deviation s, and the
z_ijk's are all independent. In place of the n by p matrix X, this
gives us a big matrix, say Q, with cn rows and p columns. To compute the
least-squares weights, we need Q'Q. Let's consider the jth diagonal
element of Q'Q, which is
2 2 2
sum (x_ij+z_ijk) = sum (x_ij + z_ijk + 2 x_ij z_ijk)
i,k i,k
which is approximately, for c large,
2 2
c(sum x_ij + ns )
i
which is c times the corresponding diagonal element of X'X plus ns^2.
Now consider the u,vth off-diagonal element of Q'Q, which is
sum (x_iu+z_iuk)(x_iv+z_ivk)
i,k
which is approximately, for c large,
c(sum x_iu x_iv)
i
which is just c times the corresponding element of X'X. Thus, Q'Q equals
c(X'X+ns^2I), where I is an identity matrix of appropriate size.
Similar computations show that the crossproduct of Q with the target values
is cX'Y. Hence the least-squares weights with jitter of variance s^2 are
given by
2 2 2
B(ns ) = inv(c(X'X+ns I))cX'Y = inv(X'X+ns I)X'Y
In the statistics literature, B(ns^2) is called a ridge regression
estimator with ridge value ns^2.
If we were to add jitter to the target values Y, the cross-product X'Y
would not be affected for large c for the same reason that the off-diagonal
elements of X'X are not afected by jitter. Hence, adding jitter to the
targets will not change the optimal weights; it will just slow down training
(An 1996).
The ordinary least squares training criterion is (Y-XB)'(Y-XB).
Weight decay uses the training criterion (Y-XB)'(Y-XB)+d^2B'B,
where d is the decay rate. Weight decay can also be implemented by
inventing artificial training cases. Augment the training data with p new
training cases containing the matrix dI for the inputs and a zero vector
for the targets. To put this in a formula, let's use A;B to indicate the
matrix A stacked on top of the matrix B, so (A;B)'(C;D)=A'C+B'D.
Thus the augmented inputs are X;dI and the augmented targets are Y;0,
where 0 indicates the zero vector of the appropriate size. The squared error
for the augmented training data is:
(Y;0-(X;dI)B)'(Y;0-(X;dI)B)
= (Y;0)'(Y;0) - 2(Y;0)'(X;dI)B + B'(X;dI)'(X;dI)B
= Y'Y - 2Y'XB + B'(X'X+d^2I)B
= Y'Y - 2Y'XB + B'X'XB + B'(d^2I)B
= (Y-XB)'(Y-XB)+d^2B'B
which is the weight-decay training criterion. Thus the weight-decay
estimator is:
inv[(X;dI)'(X;dI)](X;dI)'(Y;0) = inv(X'X+d^2I)X'Y
which is the same as the jitter estimator B(d^2), i.e. jitter with
variance d^2/n. The equivalence between the weight-decay estimator and
the jitter estimator does not hold for nonlinear models unless the jitter
variance is small relative to the curvature of the nonlinear function (An
1996). However, the equivalence of the two estimators for linear models
suggests that they will often produce similar results even for nonlinear
models. Details for nonlinear models, including classification problems, are
given in An (1996).
B(0) is obviously the ordinary least-squares estimator. It can be shown
that as s^2 increases, the Euclidean norm of B(ns^2) decreases; in
other words, adding jitter causes the weights to shrink. It can also be
shown that under the usual statistical assumptions, there always exists some
value of ns^2 > 0 such that B(ns^2) provides better expected
generalization than B(0). Unfortunately, there is no way to calculate a
value of ns^2 from the training data that is guaranteed to improve
generalization. There are other types of shrinkage estimators called Stein
estimators that do guarantee better generalization than B(0), but I'm not
aware of a nonlinear generalization of Stein estimators applicable to neural
networks.
The statistics literature describes numerous methods for choosing the ridge
value. The most obvious way is to estimate the generalization error by
cross-validation, generalized cross-validation, or bootstrapping, and to
choose the ridge value that yields the smallest such estimate. There are
also quicker methods based on empirical Bayes estimation, one of which
yields the following formula, useful as a first guess:
2 p(Y-XB(0))'(Y-XB(0))
s = --------------------
1 n(n-p)B(0)'B(0)
You can iterate this a few times:
2 p(Y-XB(0))'(Y-XB(0))
s = --------------------
l+1 2 2
n(n-p)B(s )'B(s )
l l
Note that the more training cases you have, the less noise you need.
References:
An, G. (1996), "The effects of adding noise during backpropagation
training on a generalization performance," Neural Computation, 8,
643-674.
Bishop, C.M. (1995), Neural Networks for Pattern Recognition, Oxford:
Oxford University Press.
Holmström, L. and Koistinen, P. (1992) "Using additive noise in
back-propagation training", IEEE Transaction on Neural Networks, 3,
24-38.
Koistinen, P. and Holmström, L. (1992) "Kernel regression and
backpropagation training with noise," NIPS4, 1033-1039.
Reed, R.D., and Marks, R.J, II (1999), Neural Smithing: Supervised
Learning in Feedforward Artificial Neural Networks, Cambridge, MA: The
MIT Press, ISBN 0-262-18190-8.
Scott, D.W. (1992) Multivariate Density Estimation, Wiley.
Vinod, H.D. and Ullah, A. (1981) Recent Advances in Regression Methods,
NY: Marcel-Dekker.
------------------------------------------------------------------------
Subject: What is early stopping?
=================================
NN practitioners often use nets with many times as many parameters as
training cases. E.g., Nelson and Illingworth (1991, p. 165) discuss training
a network with 16,219 parameters with only 50 training cases! The method
used is called "early stopping" or "stopped training" and proceeds as
follows:
1. Divide the available data into training and validation sets.
2. Use a large number of hidden units.
3. Use very small random initial values.
4. Use a slow learning rate.
5. Compute the validation error rate periodically during training.
6. Stop training when the validation error rate "starts to go up".
It is crucial to realize that the validation error is not a good estimate
of the generalization error. One method for getting an unbiased estimate of
the generalization error is to run the net on a third set of data, the test
set, that is not used at all during the training process. For other methods,
see "How can generalization error be estimated?"
Early stopping has several advantages:
o It is fast.
o It can be applied successfully to networks in which the number of weights
far exceeds the sample size.
o It requires only one major decision by the user: what proportion of
validation cases to use.
But there are several unresolved practical issues in early stopping:
o How many cases do you assign to the training and validation sets? Rules
of thumb abound, but appear to be no more than folklore. The only
systematic results known to the FAQ maintainer are in Sarle (1995), which
deals only with the case of a single input. Amari et al. (1995) attempts
a theoretical approach but contains serious errors that completely
invalidate the results, especially the incorrect assumption that the
direction of approach to the optimum is distributed isotropically.
o Do you split the data into training and validation sets randomly or by
some systematic algorithm?
o How do you tell when the validation error rate "starts to go up"? It may
go up and down numerous times during training. The safest approach is to
train to convergence, then go back and see which iteration had the lowest
validation error. For more elaborate algorithms, see Prechelt (1994,
1998).
Statisticians tend to be skeptical of stopped training because it appears to
be statistically inefficient due to the use of the split-sample technique;
i.e., neither training nor validation makes use of the entire sample, and
because the usual statistical theory does not apply. However, there has been
recent progress addressing both of the above concerns (Wang 1994).
Early stopping is closely related to ridge regression. If the learning rate
is sufficiently small, the sequence of weight vectors on each iteration will
approximate the path of continuous steepest descent down the error surface.
Early stopping chooses a point along this path that optimizes an estimate of
the generalization error computed from the validation set. Ridge regression
also defines a path of weight vectors by varying the ridge value. The ridge
value is often chosen by optimizing an estimate of the generalization error
computed by cross-validation, generalized cross-validation, or bootstrapping
(see "What are cross-validation and bootstrapping?") There always exists a
positive ridge value that will improve the expected generalization error in
a linear model. A similar result has been obtained for early stopping in
linear models (Wang, Venkatesh, and Judd 1994). In linear models, the ridge
path lies close to, but does not coincide with, the path of continuous
steepest descent; in nonlinear models, the two paths can diverge widely. The
relationship is explored in more detail by Sjöberg and Ljung (1992).
References:
S. Amari, N.Murata, K.-R. Muller, M. Finke, H. Yang. Asymptotic
Statistical Theory of Overtraining and Cross-Validation. METR 95-06,
1995, Department of Mathematical Engineering and Information Physics,
University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan.
Finnof, W., Hergert, F., and Zimmermann, H.G. (1993), "Improving model
selection by nonconvergent methods," Neural Networks, 6, 771-783.
Nelson, M.C. and Illingworth, W.T. (1991), A Practical Guide to Neural
Nets, Reading, MA: Addison-Wesley.
Orr, G.B., and Mueller, K.-R., eds. (1998), Neural Networks: Tricks of
the Trade, Berlin: Springer, ISBN 3-540-65311-2.
Prechelt, L. (1998), "Early stopping--But when?" in Orr and Mueller
(1998), 55-69.
Prechelt, L. (1994), "PROBEN1--A set of neural network benchmark problems
and benchmarking rules," Technical Report 21/94, Universitat Karlsruhe,
76128 Karlsruhe, Germany,
ftp://ftp.ira.uka.de/pub/papers/techreports/1994/1994-21.ps.gz.
Sarle, W.S. (1995), "Stopped Training and Other Remedies for
Overfitting," Proceedings of the 27th Symposium on the Interface of
Computing Science and Statistics, 352-360,
ftp://ftp.sas.com/pub/neural/inter95.ps.Z (this is a very large
compressed postscript file, 747K, 10 pages)
Sjöberg, J. and Ljung, L. (1992), "Overtraining, Regularization, and
Searching for Minimum in Neural Networks," Technical Report
LiTH-ISY-I-1297, Department of Electrical Engineering, Linkoping
University, S-581 83 Linkoping, Sweden, http://www.control.isy.liu.se .
Wang, C. (1994), A Theory of Generalisation in Learning Machines with
Neural Network Application, Ph.D. thesis, University of Pennsylvania.
Wang, C., Venkatesh, S.S., and Judd, J.S. (1994), "Optimal Stopping and
Effective Machine Complexity in Learning," NIPS6, 303-310.
Weigend, A. (1994), "On overfitting and the effective number of hidden
units," Proceedings of the 1993 Connectionist Models Summer School,
335-342.
------------------------------------------------------------------------
Subject: What is weight decay?
===============================
Weight decay adds a penalty term to the error function. The usual penalty is
the sum of squared weights times a decay constant. In a linear model, this
form of weight decay is equivalent to ridge regression. See "What is
jitter?" for more explanation of ridge regression.
Weight decay is a subset of regularization methods. The penalty term in
weight decay, by definition, penalizes large weights. Other regularization
methods may involve not only the weights but various derivatives of the
output function (Bishop 1995).
The weight decay penalty term causes the weights to converge to smaller
absolute values than they otherwise would. Large weights can hurt
generalization in two different ways. Excessively large weights leading to
hidden units can cause the output function to be too rough, possibly with
near discontinuities. Excessively large weights leading to output units can
cause wild outputs far beyond the range of the data if the output activation
function is not bounded to the same range as the data. To put it another
way, large weights can cause excessive variance of the output (Geman,
Bienenstock, and Doursat 1992). According to Bartlett (1997), the size (L_1
norm) of the weights is more important than the number of weights in
determining generalization.
Other penalty terms besides the sum of squared weights are sometimes used.
Weight elimination (Weigend, Rumelhart, and Huberman 1991) uses:
(w_i)^2
sum -------------
i (w_i)^2 + c^2
where w_i is the ith weight and c is a user-specified constant. Whereas
decay using the sum of squared weights tends to shrink the large
coefficients more than the small ones, weight elimination tends to shrink
the small coefficients more, and is therefore more useful for suggesting
subset models (pruning).
The generalization ability of the network can depend crucially on the decay
constant, especially with small training sets. One approach to choosing the
decay constant is to train several networks with different amounts of decay
and estimate the generalization error for each; then choose the decay
constant that minimizes the estimated generalization error. Weigend,
Rumelhart, and Huberman (1991) iteratively update the decay constant during
training.
There are other important considerations for getting good results from
weight decay. You must either standardize the inputs and targets, or adjust
the penalty term for the standard deviations of all the inputs and targets.
It is usually a good idea to omit the biases from the penalty term.
A fundamental problem with weight decay is that different types of weights
in the network will usually require different decay constants for good
generalization. At the very least, you need three different decay constants
for input-to-hidden, hidden-to-hidden, and hidden-to-output weights.
Adjusting all these decay constants to produce the best estimated
generalization error often requires vast amounts of computation.
Fortunately, there is a superior alternative to weight decay: hierarchical
Bayesian learning. Bayesian learning makes it possible to estimate
efficiently numerous decay constants.
References:
Bartlett, P.L. (1997), "For valid generalization, the size of the weights
is more important than the size of the network," in Mozer, M.C., Jordan,
M.I., and Petsche, T., (eds.) Advances in Neural Information Processing
Systems 9, Cambrideg, MA: The MIT Press, pp. 134-140.
Bishop, C.M. (1995), Neural Networks for Pattern Recognition, Oxford:
Oxford University Press.
Geman, S., Bienenstock, E. and Doursat, R. (1992), "Neural Networks and
the Bias/Variance Dilemma", Neural Computation, 4, 1-58.
Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge:
Cambridge University Press.
Weigend, A. S., Rumelhart, D. E., & Huberman, B. A. (1991).
Generalization by weight-elimination with application to forecasting. In:
R. P. Lippmann, J. Moody, & D. S. Touretzky (eds.), Advances in Neural
Information Processing Systems 3, San Mateo, CA: Morgan Kaufmann.
------------------------------------------------------------------------
Subject: What is Bayesian Learning?
===================================
By Radford Neal.
Conventional training methods for multilayer perceptrons ("backprop" nets)
can be interpreted in statistical terms as variations on maximum likelihood
estimation. The idea is to find a single set of weights for the network that
maximize the fit to the training data, perhaps modified by some sort of
weight penalty to prevent overfitting.
The Bayesian school of statistics is based on a different view of what it
means to learn from data, in which probability is used to represent
uncertainty about the relationship being learned (a use that is shunned in
conventional--i.e., frequentist--statistics). Before we have seen any data,
our prior opinions about what the true relationship might be can be
expresssed in a probability distribution over the network weights that
define this relationship. After we look at the data (or after our program
looks at the data), our revised opinions are captured by a posterior
distribution over network weights. Network weights that seemed plausible
before, but which don't match the data very well, will now be seen as being
much less likely, while the probability for values of the weights that do
fit the data well will have increased.
Typically, the purpose of training is to make predictions for future cases
in which only the inputs to the network are known. The result of
conventional network training is a single set of weights that can be used to
make such predictions. In contrast, the result of Bayesian training is a
posterior distribution over network weights. If the inputs of the network
are set to the values for some new case, the posterior distribution over
network weights will give rise to a distribution over the outputs of the
network, which is known as the predictive distribution for this new case. If
a single-valued prediction is needed, one might use the mean of the
predictive distribution, but the full predictive distribution also tells you
how uncertain this prediction is.
Why bother with all this? The hope is that Bayesian methods will provide
solutions to such fundamental problems as:
o How to judge the uncertainty of predictions. This can be solved by
looking at the predictive distribution, as described above.
o How to choose an appropriate network architecture (eg, the number hidden
layers, the number of hidden units in each layer).
o How to adapt to the characteristics of the data (eg, the smoothness of
the function, the degree to which different inputs are relevant).
Good solutions to these problems, especially the last two, depend on using
the right prior distribution, one that properly represents the uncertainty
that you probably have about which inputs are relevant, how smooth the
function is, how much noise there is in the observations, etc. Such
carefully vague prior distributions are usually defined in a hierarchical
fashion, using hyperparameters, some of which are analogous to the weight
decay constants of more conventional training procedures. The use of
hyperparameters is discussed by Mackay (1992a, 1992b, 1995) and Neal (1993a,
1996), who in particular use an "Automatic Relevance Determination" scheme
that aims to allow many possibly-relevant inputs to be included without
damaging effects.
Selection of an appropriate network architecture is another place where
prior knowledge plays a role. One approach is to use a very general
architecture, with lots of hidden units, maybe in several layers or groups,
controlled using hyperparameters. This approach is emphasized by Neal
(1996), who argues that there is no statistical need to limit the complexity
of the network architecture when using well-designed Bayesian methods. It is
also possible to choose between architectures in a Bayesian fashion, using
the "evidence" for an architecture, as discussed by Mackay (1992a, 1992b).
Implementing all this is one of the biggest problems with Bayesian methods.
Dealing with a distribution over weights (and perhaps hyperparameters) is
not as simple as finding a single "best" value for the weights. Exact
analytical methods for models as complex as neural networks are out of the
question. Two approaches have been tried:
1. Find the weights/hyperparameters that are most probable, using methods
similar to conventional training (with regularization), and then
approximate the distribution over weights using information available at
this maximum.
2. Use a Monte Carlo method to sample from the distribution over weights.
The most efficient implementations of this use dynamical Monte Carlo
methods whose operation resembles that of backprop with momentum.
The first method comes in two flavours. Buntine and Weigend (1991) describe
a procedure in which the hyperparameters are first integrated out
analytically, and numerical methods are then used to find the most probable
weights. MacKay (1992a, 1992b) instead finds the values for the
hyperparameters that are most likely, integrating over the weights (using an
approximation around the most probable weights, conditional on the
hyperparameter values). There has been some controversy regarding the merits
of these two procedures, with Wolpert (1993) claiming that analytically
integrating over the hyperparameters is preferable because it is "exact".
This criticism has been rebutted by Mackay (1993). It would be inappropriate
to get into the details of this controversy here, but it is important to
realize that the procedures based on analytical integration over the
hyperparameters do not provide exact solutions to any of the problems of
practical interest. The discussion of an analogous situation in a different
statistical context by O'Hagan (1985) may be illuminating.
Monte Carlo methods for Bayesian neural networks have been developed by Neal
(1993a, 1996). In this approach, the posterior distribution is represented
by a sample of perhaps a few dozen sets of network weights. The sample is
obtained by simulating a Markov chain whose equilibrium distribution is the
posterior distribution for weights and hyperparameters. This technique is
known as "Markov chain Monte Carlo (MCMC)"; see Neal (1993b) for a review.
The method is exact in the limit as the size of the sample and the length of
time for which the Markov chain is run increase, but convergence can
sometimes be slow in practice, as for any network training method.
Work on Bayesian neural network learning has so far concentrated on
multilayer perceptron networks, but Bayesian methods can in principal be
applied to other network models, as long as they can be interpreted in
statistical terms. For some models (eg, RBF networks), this should be a
fairly simple matter; for others (eg, Boltzmann Machines), substantial
computational problems would need to be solved.
Software implementing Bayesian neural network models (intended for research
use) is available from the home pages of David MacKay and Radford Neal.
There are many books that discuss the general concepts of Bayesian
inference, though they mostly deal with models that are simpler than neural
networks. Here are some recent ones:
Bernardo, J. M. and Smith, A. F. M. (1994) Bayesian Theory, New York:
John Wiley.
Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (1995) Bayesian
Data Analysis, London: Chapman & Hall, ISBN 0-412-03991-5.
O'Hagan, A. (1994) Bayesian Inference (Volume 2B in Kendall's Advanced
Theory of Statistics), ISBN 0-340-52922-9.
Robert, C. P. (1995) The Bayesian Choice, New York: Springer-Verlag.
The following books and papers have tutorial material on Bayesian learning
as applied to neural network models:
Bishop, C. M. (1995) Neural Networks for Pattern Recognition, Oxford:
Oxford University Press.
Lee, H.K.H (1999), Model Selection and Model Averaging for Neural
Networks, Doctoral dissertation, Carnegie Mellon University,
Pittsburgh, USA, http://lib.stat.cmu.edu/~herbie/thesis.html
MacKay, D. J. C. (1995) "Probable networks and plausible predictions - a
review of practical Bayesian methods for supervised neural networks",
available at ftp://wol.ra.phy.cam.ac.uk/pub/www/mackay/network.ps.gz.
Mueller, P. and Insua, D.R. (1995) "Issues in Bayesian Analysis of Neural
Network Models," Neural Computation, 10, 571-592, (also Institute of
Statistics and Decision Sciences Working Paper 95-31),
ftp://ftp.isds.duke.edu/pub/WorkingPapers/95-31.ps
Neal, R. M. (1996) Bayesian Learning for Neural Networks, New York:
Springer-Verlag, ISBN 0-387-94724-8.
Ripley, B. D. (1996) Pattern Recognition and Neural Networks,
Cambridge: Cambridge University Press.
Thodberg, H. H. (1996) "A review of Bayesian neural networks with an
application to near infrared spectroscopy", IEEE Transactions on Neural
Networks, 7, 56-72.
Some other references:
Bernardo, J.M., DeGroot, M.H., Lindley, D.V. and Smith, A.F.M., eds.,
(1985), Bayesian Statistics 2, Amsterdam: Elsevier Science Publishers B.V.
(North-Holland).
Buntine, W. L. and Weigend, A. S. (1991) "Bayesian back-propagation",
Complex Systems, 5, 603-643.
MacKay, D. J. C. (1992a) "Bayesian interpolation", Neural Computation,
4, 415-447.
MacKay, D. J. C. (1992b) "A practical Bayesian framework for
backpropagation networks," Neural Computation, 4, 448-472.
MacKay, D. J. C. (1993) "Hyperparameters: Optimize or Integrate Out?",
available at ftp://wol.ra.phy.cam.ac.uk/pub/www/mackay/alpha.ps.gz.
Neal, R. M. (1993a) "Bayesian learning via stochastic dynamics", in C. L.
Giles, S. J. Hanson, and J. D. Cowan (editors), Advances in Neural
Information Processing Systems 5, San Mateo, California: Morgan
Kaufmann, 475-482.
Neal, R. M. (1993b) Probabilistic Inference Using Markov Chain Monte
Carlo Methods, available at
ftp://ftp.cs.utoronto.ca/pub/radford/review.ps.Z.
O'Hagan, A. (1985) "Shoulders in hierarchical models", in J. M. Bernardo,
M. H. DeGroot, D. V. Lindley, and A. F. M. Smith (editors), Bayesian
Statistics 2, Amsterdam: Elsevier Science Publishers B.V. (North-Holland),
697-710.
Sarle, W. S. (1995) "Stopped Training and Other Remedies for
Overfitting," Proceedings of the 27th Symposium on the Interface of
Computing Science and Statistics, 352-360,
ftp://ftp.sas.com/pub/neural/inter95.ps.Z (this is a very large
compressed postscript file, 747K, 10 pages)
Wolpert, D. H. (1993) "On the use of evidence in neural networks", in C.
L. Giles, S. J. Hanson, and J. D. Cowan (editors), Advances in Neural
Information Processing Systems 5, San Mateo, California: Morgan
Kaufmann, 539-546.
Finally, David MacKay maintains a FAQ about Bayesian methods for neural
networks, at http://wol.ra.phy.cam.ac.uk/mackay/Bayes_FAQ.html .
Comments on Bayesian learning
+++++++++++++++++++++++++++++
By Warren Sarle.
Bayesian purists may argue over the proper way to do a Bayesian analysis,
but even the crudest Bayesian computation (maximizing over both parameters
and hyperparameters) is shown by Sarle (1995) to generalize better than
early stopping when learning nonlinear functions. This approach requires the
use of slightly informative hyperpriors and at least twice as many training
cases as weights in the network. A full Bayesian analysis by MCMC can be
expected to work even better under even broader conditions. Bayesian
learning works well by frequentist standards--what MacKay calls the
"evidence framework" is used by frequentist statisticians under the name
"empirical Bayes." Although considerable research remains to be done,
Bayesian learning seems to be the most promising approach to training neural
networks.
Bayesian learning should not be confused with the "Bayes classifier." In the
latter, the distribution of the inputs given the target class is assumed to
be known exactly, and the prior probabilities of the classes are assumed
known, so that the posterior probabilities can be computed by a
(theoretically) simple application of Bayes' theorem. The Bayes classifier
involves no learning--you must already know everything that needs to be
known! The Bayes classifier is a gold standard that can almost never be used
in real life but is useful in theoretical work and in simulation studies
that compare classification methods. The term "Bayes rule" is also used to
mean any classification rule that gives results identical to those of a
Bayes classifier.
Bayesian learning also should not be confused with the "naive" or "idiot's"
Bayes classifier (Warner et al. 1961; Ripley, 1996), which assumes that the
inputs are conditionally independent given the target class. The naive Bayes
classifier is usually applied with categorical inputs, and the distribution
of each input is estimated by the proportions in the training set; hence the
naive Bayes classifier is a frequentist method.
The term "Bayesian network" often refers not to a neural network but to a
belief network (also called a causal net, influence diagram, constraint
network, qualitative Markov network, or gallery). Belief networks are more
closely related to expert systems than to neural networks, and do not
necessarily involve learning (Pearl, 1988; Ripley, 1996). Here are some URLs
on Bayesian belief networks:
o http://bayes.stat.washington.edu/almond/belief.html
o http://www.cs.orst.edu/~dambrosi/bayesian/frame.html
o http://www2.sis.pitt.edu/~genie
o http://www.research.microsoft.com/dtg/msbn
References for comments:
Pearl, J. (1988) Probabilistic Reasoning in Intelligent Systems: Networks
of Plausible Inference, San Mateo, CA: Morgan Kaufmann.
Ripley, B. D. (1996) Pattern Recognition and Neural Networks,
Cambridge: Cambridge University Press.
Warner, H.R., Toronto, A.F., Veasy, L.R., and Stephenson, R. (1961), "A
mathematical model for medical diagnosis--application to congenital heart
disease," J. of the American Medical Association, 177, 177-184.
------------------------------------------------------------------------
Subject: How to combine networks?
==================================
Methods for combining networks are a subject of active research. Many
different methods with different purposes have been proposed. The properties
and relationships of these methods are just beginning to be understood. Some
methods, such as boosting, are remedies for underfitting. Other methods,
such as bagging, are mainly remedies for overfitting or instability.
Bayesian learning naturally leads to model averaging (Hoeting et al., 1999).
A good general reference is the book edited by Sharkey (1999), especially
the article by Drucker (1999). Regarding the effects of bagging and weight
decay used together, see Taniguchi and Tresp (1997).
Here is a list of terms used for various methods of combining models, mostly
taken from Christoph M. Friedrich's web page (see below):
o Adaboost
o ADDEMUP
o arcing: adaptive recombination of classifiers
o bagging: bootstrap aggregation
o bag-stacking: bagging plus stacking
o boosting
o cascading
o combination of classifiers
o committees of networks
o consensus theory
o cragging: cross aggregation (like k-fold cross validation)
o dagging: disjoint-sample aggregation
o dag-stacking: dagging plus stacking
o divide and conquer classifiers
o ensembles
o hagging: half-sample aggregation
o mixture of experts
o multiple classifier systems:
o multi-stage and multi-level classifiers
o OLC: optimal linear combination
o pandemonium of reflective agents
o sieving algorithms
o stacking: feeding outputs of several models (and possibly the the
original inputs) into a second-level model
o voting
URLs:
o Christoph M. Friedrich's web page, "Combinations of Classifiers and
Regressors Bibliography and Guide to Internet Resources" at
http://www.tussy.uni-wh.de/~chris/ensemble/ensemble.html
o Tirthankar RayChaudhuri's web page on combining estimators at
http://www-comp.mpce.mq.edu.au/~tirthank/combest.html
o Robert E. Schapire's boosting page at
http://www.research.att.com/~schapire/boost.html
o http://www.boosting.org/
References:
Clemen, Robert T. (1989), "Combining forecasts: A review and annotated
bibliography", International Journal of Forecasting, Vol 5, pp 559-584.
Drucker, H. (1999), "Boosting using neural networks," in Sharkey (1999),
pp. 51-78.
Hoeting, J. A., Madigan, D., Raftery, A.E., and Volinsky, C.T. (1999)
"Bayesian Model Averaging: A Tutorial (with discussion)," Statistical
Science, 14:4, 382-417. Corrected version available at
http://www.stat.washington.edu/www/research/online/hoeting1999.pdf
Sharkey, A.J.C. (1999), Combining Artificial Neural Nets: Ensemble and
Modular Multi-Net Systems, London: Springer.
Taniguchi, M., and Tresp, V. (1997), "Averaging regularized estimators,"
Neural Computation, 9, 1163-1178.
------------------------------------------------------------------------
Subject: How many hidden layers should I use?
==============================================
You may not need any hidden layers at all. Linear and generalized linear
models are useful in a wide variety of applications (McCullagh and Nelder
1989). And even if the function you want to learn is mildly nonlinear, you
may get better generalization with a simple linear model than with a
complicated nonlinear model if there is too little data or too much noise to
estimate the nonlinearities accurately.
In MLPs with step/threshold/Heaviside activation functions, you need two
hidden layers for full generality (Sontag 1992). For further discussion, see
Bishop (1995, 121-126).
In MLPs with any of a wide variety of continuous nonlinear hidden-layer
activation functions, one hidden layer with an arbitrarily large number of
units suffices for the "universal approximation" property (e.g., Hornik,
Stinchcombe and White 1989; Hornik 1993; for more references, see Bishop
1995, 130, and Ripley, 1996, 173-180). But there is no theory yet to tell
you how many hidden units are needed to approximate any given function.
If you have only one input, there seems to be no advantage to using more
than one hidden layer. But things get much more complicated when there are
two or more inputs. To illustrate, examples with two inputs and one output
will be used so that the results can be shown graphically. In each example
there are 441 training cases on a regular 21-by-21 grid. The test sets have
1681 cases on a regular 41-by-41 grid over the same domain as the training
set. If you are reading the HTML version of this document via a web browser,
you can see surface plots based on the test set by clicking on the file
names mentioned in the folowing text. Each plot is a gif file, approximately
9K in size.
Consider a target function of two inputs, consisting of a Gaussian hill in
the middle of a plane (hill.gif). An MLP with an identity output activation
function can easily fit the hill by surrounding it with a few sigmoid
(logistic, tanh, arctan, etc.) hidden units, but there will be spurious
ridges and valleys where the plane should be flat (h_mlp_6.gif). It takes
dozens of hidden units to flatten out the plane accurately (h_mlp_30.gif).
Now suppose you use a logistic output activation function. As the input to a
logistic function goes to negative infinity, the output approaches zero. The
plane in the Gaussian target function also has a value of zero. If the
weights and bias for the output layer yield large negative values outside
the base of the hill, the logistic function will flatten out any spurious
ridges and valleys. So fitting the flat part of the target function is easy
(h_mlpt_3_unsq.gif and h_mlpt_3.gif). But the logistic function also tends
to lower the top of the hill.
If instead of a rounded hill, the target function was a mesa with a large,
flat top with a value of one, the logistic output activation function would
be able to smooth out the top of the mesa just like it smooths out the plane
below. Target functions like this, with large flat areas with values of
either zero or one, are just what you have in many noise-free classificaton
problems. In such cases, a single hidden layer is likely to work well.
When using a logistic output activation function, it is common practice to
scale the target values to a range of .1 to .9. Such scaling is bad in a
noise-free classificaton problem, because it prevents the logistic function
from smoothing out the flat areas (h_mlpt1-9_3.gif).
For the Gaussian target function, [.1,.9] scaling would make it easier to
fit the top of the hill, but would reintroduce undulations in the plane. It
would be better for the Gaussian target function to scale the target values
to a range of 0 to .9. But for a more realistic and complicated target
function, how would you know the best way to scale the target values?
By introducing a second hidden layer with one sigmoid activation function
and returning to an identity output activation function, you can let the net
figure out the best scaling (h_mlp1_3.gif). Actually, the bias and weight
for the output layer scale the output rather than the target values, and you
can use whatever range of target values is convenient.
For more complicated target functions, especially those with several hills
or valleys, it is useful to have several units in the second hidden layer.
Each unit in the second hidden layer enables the net to fit a separate hill
or valley. So an MLP with two hidden layers can often yield an accurate
approximation with fewer weights than an MLP with one hidden layer. (Chester
1990).
To illustrate the use of multiple units in the second hidden layer, the next
example resembles a landscape with a Gaussian hill and a Gaussian valley,
both elliptical (hillanvale.gif). The table below gives the RMSE (root mean
squared error) for the test set with various architectures. If you are
reading the HTML version of this document via a web browser, click on any
number in the table to see a surface plot of the corresponding network
output.
The MLP networks in the table have one or two hidden layers with a tanh
activation function. The output activation function is the identity. Using a
squashing function on the output layer is of no benefit for this function,
since the only flat area in the function has a target value near the middle
of the target range.
Hill and Valley Data: RMSE for the Test Set
(Number of weights in parentheses)
MLP Networks
HUs in HUs in Second Layer
First ----------------------------------------------------------
Layer 0 1 2 3 4
1 0.204( 5) 0.204( 7) 0.189( 10) 0.187( 13) 0.185( 16)
2 0.183( 9) 0.163( 11) 0.147( 15) 0.094( 19) 0.096( 23)
3 0.159( 13) 0.095( 15) 0.054( 20) 0.033( 25) 0.045( 30)
4 0.137( 17) 0.093( 19) 0.009( 25) 0.021( 31) 0.016( 37)
5 0.121( 21) 0.092( 23) 0.010( 37) 0.011( 44)
6 0.100( 25) 0.092( 27) 0.007( 43) 0.005( 51)
7 0.086( 29) 0.077( 31)
8 0.079( 33) 0.062( 35)
9 0.072( 37) 0.055( 39)
10 0.059( 41) 0.047( 43)
12 0.047( 49) 0.042( 51)
15 0.039( 61) 0.032( 63)
20 0.025( 81) 0.018( 83)
25 0.021(101) 0.016(103)
30 0.018(121) 0.015(123)
40 0.012(161) 0.015(163)
50 0.008(201) 0.014(203)
For an MLP with only one hidden layer (column 0), Gaussian hills and valleys
require a large number of hidden units to approximate well. When there is
one unit in the second hidden layer, the network can represent one hill or
valley easily, which is what happens with three to six units in the first
hidden layer. But having only one unit in the second hidden layer is of
little benefit for learning two hills or valleys. Using two units in the
second hidden layer enables the network to approximate two hills or valleys
easily; in this example, only four units are required in the first hidden
layer to get an excellent fit. Each additional unit in the second hidden
layer enables the network to learn another hill or valley with a relatively
small number of units in the first hidden layer, as explained by Chester
(1990). In this example, having three or four units in the second hidden
layer helps little, and actually produces a worse approximation when there
are four units in the first hidden layer due to problems with local minima.
Unfortunately, using two hidden layers exacerbates the problem of local
minima, and it is important to use lots of random initializations or other
methods for global optimization. Local minima with two hidden layers can
have extreme spikes or blades even when the number of weights is much
smaller than the number of training cases. One of the few advantages of
standard backprop is that it is so slow that spikes and blades will not
become very sharp for practical training times.
More than two hidden layers can be useful in certain architectures such as
cascade correlation (Fahlman and Lebiere 1990) and in special applications,
such as the two-spirals problem (Lang and Witbrock 1988) and ZIP code
recognition (Le Cun et al. 1989).
RBF networks are most often used with a single hidden layer. But an extra,
linear hidden layer before the radial hidden layer enables the network to
ignore irrelevant inputs (see How do MLPs compare with RBFs?") The linear
hidden layer allows the RBFs to take elliptical, rather than radial
(circular), shapes in the space of the inputs. Hence the linear layer gives
you an elliptical basis function (EBF) network. In the hill and valley
example, an ORBFUN network requires nine hidden units (37 weights) to get
the test RMSE below .01, but by adding a linear hidden layer, you can get an
essentially perfect fit with three linear units followed by two radial units
(20 weights).
References:
Bishop, C.M. (1995), Neural Networks for Pattern Recognition, Oxford:
Oxford University Press.
Chester, D.L. (1990), "Why Two Hidden Layers are Better than One,"
IJCNN-90-WASH-DC, Lawrence Erlbaum, 1990, volume 1, 265-268.
Fahlman, S.E. and Lebiere, C. (1990), "The Cascade Correlation Learning
Architecture," NIPS2, 524-532,
ftp://archive.cis.ohio-state.edu/pub/neuroprose/fahlman.cascor-tr.ps.Z.
Hornik, K., Stinchcombe, M. and White, H. (1989), "Multilayer feedforward
networks are universal approximators," Neural Networks, 2, 359-366.
Hornik, K. (1993), "Some new results on neural network approximation,"
Neural Networks, 6, 1069-1072.
Lang, K.J. and Witbrock, M.J. (1988), "Learning to tell two spirals
apart," in Touretzky, D., Hinton, G., and Sejnowski, T., eds.,
Procedings of the 1988 Connectionist Models Summer School, San Mateo,
CA: Morgan Kaufmann.
Le Cun, Y., Boser, B., Denker, J.s., Henderson, D., Howard, R.E.,
Hubbard, W., and Jackel, L.D. (1989), "Backpropagation applied to
handwritten ZIP code recognition", Neural Computation, 1, 541-551.
McCullagh, P. and Nelder, J.A. (1989) Generalized Linear Models, 2nd
ed., London: Chapman & Hall.
Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge:
Cambridge University Press.
Sontag, E.D. (1992), "Feedback stabilization using two-hidden-layer
nets", IEEE Transactions on Neural Networks, 3, 981-990.
------------------------------------------------------------------------
Subject: How many hidden units should I use?
=============================================
The best number of hidden units depends in a complex way on:
o the numbers of input and output units
o the number of training cases
o the amount of noise in the targets
o the complexity of the function or classification to be learned
o the architecture
o the type of hidden unit activation function
o the training algorithm
o regularization
In most situations, there is no way to determine the best number of hidden
units without training several networks and estimating the generalization
error of each. If you have too few hidden units, you will get high training
error and high generalization error due to underfitting and high statistical
bias. If you have too many hidden units, you may get low training error but
still have high generalization error due to overfitting and high variance.
Geman, Bienenstock, and Doursat (1992) discuss how the number of hidden
units affects the bias/variance trade-off.
Some books and articles offer "rules of thumb" for choosing an architecture;
for example:
o "A rule of thumb is for the size of this [hidden] layer to be somewhere
between the input layer size ... and the output layer size ..." (Blum,
1992, p. 60).
o "To calculate the number of hidden nodes we use a general rule of:
(Number of inputs + outputs) * (2/3)" (from the FAQ for a commercial
neural network software company).
o "you will never require more than twice the number of hidden units as you
have inputs" in an MLP with one hidden layer (Swingler, 1996, p. 53). See
the section in Part 4 of the FAQ on The Worst books for the source of
this myth.)
o "How large should the hidden layer be? One rule of thumb is that it
should never be more than twice as large as the input layer." (Berry and
Linoff, 1997, p. 323).
o "Typically, we specify as many hidden nodes as dimensions [principal
components] needed to capture 70-90% of the variance of the input data
set." (Boger and Guterman, 1997)
These rules of thumb are nonsense because they ignore the number of training
cases, the amount of noise in the targets, and the complexity of the
function. Even if you restrict consideration to minimizing training error on
data with lots of training cases and no noise, it is easy to construct
counterexamples that disprove these rules of thumb. For example:
o There are 100 Boolean inputs and 100 Boolean targets. Each target is a
conjunction of some subset of inputs. No hidden units are needed.
o There are two continuous inputs X and Y which take values uniformly
distributed on a square [0,8] by [0,8]. Think of the input space as a
chessboard, and number the squares 1 to 64. The categorical target
variable C is the square number, so there are 64 output units. For
example, you could generate the data as follows (this is the SAS
programming language, but it should be easy to translate into any other
language):
data chess;
step = 1/4;
do x = step/2 to 8-step/2 by step;
do y = step/2 to 8-step/2 by step;
c = 8*floor(x) + floor(y) + 1;
output;
end;
end;
run;
No hidden units are needed.
o The classic two-spirals problem has two continuous inputs and a Boolean
classification target. The data can be generated as follows:
data spirals;
pi = arcos(-1);
do i = 0 to 96;
angle = i*pi/16.0;
radius = 6.5*(104-i)/104;
x = radius*cos(angle);
y = radius*sin(angle);
c = 1;
output;
x = -x;
y = -y;
c = 0;
output;
end;
run;
With one hidden layer, about 50 tanh hidden units are needed. Many NN
programs may actually need closer to 100 hidden units to get zero
training error.
o There is one continuous input X that takes values on [0,100]. There is
one continuous target Y = sin(X). Getting a good approximation to Y
requires about 20 to 25 tanh hidden units. Of course, 1 sine hidden unit
would do the job.
Some rules of thumb relate the total number of trainable weights in the
network to the number of training cases. A typical recommendation is that
the number of weights should be no more than 1/30 of the number of training
cases. Such rules are only concerned with overfitting and are at best crude
approximations. Also, these rules do not apply when regularization is used.
It is true that without regularization, if the number of training cases is
much larger (but no one knows exactly how much larger) than the number of
weights, you are unlikely to get overfitting, but you may suffer from
underfitting. For a noise-free quantitative target variable, twice as many
training cases as weights may be more than enough to avoid overfitting. For
a very noisy categorical target variable, 30 times as many training cases as
weights may not be enough to avoid overfitting.
An intelligent choice of the number of hidden units depends on whether you
are using early stopping or some other form of regularization. If not, you
must simply try many networks with different numbers of hidden units,
estimate the generalization error for each one, and choose the network with
the minimum estimated generalization error. For examples using statistical
criteria to choose the number of hidden units, see
ftp://ftp.sas.com/pub/neural/dojo/dojo.html.
Using conventional optimization algorithms (see "What are conjugate
gradients, Levenberg-Marquardt, etc.?"), there is little point in trying a
network with more weights than training cases, since such a large network is
likely to overfit.
Using standard online backprop, however, Lawrence, Giles, and Tsoi (1996,
1997) have shown that it can be difficult to reduce training error to a
level near the globally optimal value, even when using more weights than
training cases. But increasing the number of weights makes it easier for
standard backprop to find a good local optimum, so using "oversize" networks
can reduce both training error and generalization error.
If you are using early stopping, it is essential to use lots of hidden units
to avoid bad local optima (Sarle 1995). There seems to be no upper limit on
the number of hidden units, other than that imposed by computer time and
memory requirements. Weigend (1994) makes this assertion, but provides only
one example as evidence. Tetko, Livingstone, and Luik (1995) provide
simulation studies that are more convincing. Similar results were obtained
in conjunction with the simulations in Sarle (1995), but those results are
not reported in the paper for lack of space. On the other hand, there seems
to be no advantage to using more hidden units than you have training cases,
since bad local minima do not occur with so many hidden units.
If you are using weight decay or Bayesian estimation, you can also use lots
of hidden units (Neal 1996). However, it is not strictly necessary to do so,
because other methods are available to avoid local minima, such as multiple
random starts and simulated annealing (such methods are not safe to use with
early stopping). You can use one network with lots of hidden units, or you
can try different networks with different numbers of hidden units, and
choose on the basis of estimated generalization error. With weight decay or
MAP Bayesian estimation, it is prudent to keep the number of weights less
than half the number of training cases.
Bear in mind that with two or more inputs, an MLP with one hidden layer
containing only a few units can fit only a limited variety of target
functions. Even simple, smooth surfaces such as a Gaussian bump in two
dimensions may require 20 to 50 hidden units for a close approximation.
Networks with a smaller number of hidden units often produce spurious ridges
and valleys in the output surface (see Chester 1990 and "How do MLPs compare
with RBFs?") Training a network with 20 hidden units will typically require
anywhere from 150 to 2500 training cases if you do not use early stopping or
regularization. Hence, if you have a smaller training set than that, it is
usually advisable to use early stopping or regularization rather than to
restrict the net to a small number of hidden units.
Ordinary RBF networks containing only a few hidden units also produce
peculiar, bumpy output functions. Normalized RBF networks are better at
approximating simple smooth surfaces with a small number of hidden units
(see How do MLPs compare with RBFs?).
There are various theoretical results on how fast approximation error
decreases as the number of hidden units increases, but the conclusions are
quite sensitive to the assumptions regarding the function you are trying to
approximate. See p. 178 in Ripley (1996) for a summary. According to a
well-known result by Barron (1993), in a network with I inputs and H units
in a single hidden layer, the root integrated squared error (RISE) will
decrease at least as fast as H^(-1/2) under some quite complicated
smoothness assumptions. Ripley cites another paper by DeVore et al. (1989)
that says if the function has only R bounded derivatives, RISE may decrease
as slowly as H^(-R/I). For some examples with from 1 to 4 hidden layers
see How many hidden layers should I use?" and "How do MLPs compare with
RBFs?"
For learning a finite training set exactly, bounds for the number of hidden
units required are provided by Elisseeff and Paugam-Moisy (1997).
References:
Barron, A.R. (1993), "Universal approximation bounds for superpositions
of a sigmoid function," IEEE Transactions on Information Theory, 39,
930-945.
Berry, M.J.A., and Linoff, G. (1997), Data Mining Techniques, NY: John
Wiley & Sons.
Blum, A. (1992), Neural Networks in C++, NY: Wiley.
Boger, Z., and Guterman, H. (1997), "Knowledge extraction from artificial
neural network models," IEEE Systems, Man, and Cybernetics Conference,
Orlando, FL.
Chester, D.L. (1990), "Why Two Hidden Layers are Better than One,"
IJCNN-90-WASH-DC, Lawrence Erlbaum, 1990, volume 1, 265-268.
DeVore, R.A., Howard, R., and Micchelli, C.A. (1989), "Optimal nonlinear
approximation," Manuscripta Mathematica, 63, 469-478.
Elisseeff, A., and Paugam-Moisy, H. (1997), "Size of multilayer networks
for exact learning: analytic approach," in Mozer, M.C., Jordan, M.I., and
Petsche, T., (eds.) Advances in Neural Information Processing Systems 9,
Cambrideg, MA: The MIT Press, pp.162-168.
Geman, S., Bienenstock, E. and Doursat, R. (1992), "Neural Networks and
the Bias/Variance Dilemma", Neural Computation, 4, 1-58.
Lawrence, S., Giles, C.L., and Tsoi, A.C. (1996), "What size neural
network gives optimal generalization? Convergence properties of
backpropagation," Technical Report UMIACS-TR-96-22 and CS-TR-3617,
Institute for Advanced Computer Studies, University of Maryland, College
Park, MD 20742,
http://www.neci.nj.nec.com/homepages/lawrence/papers/minima-tr96/minima-tr96.html
Lawrence, S., Giles, C.L., and Tsoi, A.C. (1997), "Lessons in Neural
Network Training: Overfitting May be Harder than Expected," Proceedings
of the Fourteenth National Conference on Artificial Intelligence,
AAAI-97, AAAI Press, Menlo Park, California, pp. 540-545,
http://www.neci.nj.nec.com/homepages/lawrence/papers/overfitting-aaai97/overfitting-aaai97-bib.html
Neal, R. M. (1996) Bayesian Learning for Neural Networks, New York:
Springer-Verlag, ISBN 0-387-94724-8.
Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge:
Cambridge University Press,
Sarle, W.S. (1995), "Stopped Training and Other Remedies for
Overfitting," Proceedings of the 27th Symposium on the Interface of
Computing Science and Statistics, 352-360,
ftp://ftp.sas.com/pub/neural/inter95.ps.Z (this is a very large
compressed postscript file, 747K, 10 pages)
Swingler, K. (1996), Applying Neural Networks: A Practical Guide,
London: Academic Press.
Tetko, I.V., Livingstone, D.J., and Luik, A.I. (1995), "Neural Network
Studies. 1. Comparison of Overfitting and Overtraining," J. Chem. Info.
Comp. Sci., 35, 826-833.
Weigend, A. (1994), "On overfitting and the effective number of hidden
units," Proceedings of the 1993 Connectionist Models Summer School,
335-342.
------------------------------------------------------------------------
Subject: How can generalization error be estimated?
====================================================
There are many methods for estimating generalization error.
Single-sample statistics: AIC, SBC, MDL, FPE, Mallows' C_p, etc.
In linear models, statistical theory provides several simple estimators
of the generalization error under various sampling assumptions
(Darlington 1968; Efron and Tibshirani 1993; Miller 1990). These
estimators adjust the training error for the number of weights being
estimated, and in some cases for the noise variance if that is known.
These statistics can also be used as crude estimates of the
generalization error in nonlinear models when you have a "large" training
set. Correcting these statistics for nonlinearity requires substantially
more computation (Moody 1992), and the theory does not always hold for
neural networks due to violations of the regularity conditions.
Among the simple generalization estimators that do not require the noise
variance to be known, Schwarz's Bayesian Criterion (known as both SBC and
BIC; Schwarz 1978; Judge et al. 1980; Raftery 1995) often works well for
NNs (Sarle 1995, 1999). AIC and FPE tend to overfit with NNs. Rissanen's
Minimum Description Length principle (MDL; Rissanen 1978, 1987, 1999) is
closely related to SBC. A special issue of Computer Journal contains
several articles on MDL, which can be found online at
http://www3.oup.co.uk/computer_journal/hdb/Volume_42/Issue_04/
Several articles on SBC/BIC are available at the University of
Washigton's web site at http://www.stat.washington.edu/tech.reports
For classification problems, the formulas are not as simple as for
regression with normal noise. See Efron (1986) regarding logistic
regression.
Split-sample or hold-out validation.
The most commonly used method for estimating generalization error in
neural networks is to reserve part of the data as a "test" set, which
must not be used in any way during training. The test set must be a
representative sample of the cases that you want to generalize to. After
training, run the network on the test set, and the error on the test set
provides an unbiased estimate of the generalization error, provided that
the test set was chosen randomly. The disadvantage of split-sample
validation is that it reduces the amount of data available for both
training and validation. See Weiss and Kulikowski (1991).
Cross-validation (e.g., leave one out).
Cross-validation is an improvement on split-sample validation that allows
you to use all of the data for training. The disadvantage of
cross-validation is that you have to retrain the net many times. See
"What are cross-validation and bootstrapping?".
Bootstrapping.
Bootstrapping is an improvement on cross-validation that often provides
better estimates of generalization error at the cost of even more
computing time. See "What are cross-validation and bootstrapping?".
If you use any of the above methods to choose which of several different
networks to use for prediction purposes, the estimate of the generalization
error of the best network will be optimistic. For example, if you train
several networks using one data set, and use a second (validation set) data
set to decide which network is best, you must use a third (test set) data
set to obtain an unbiased estimate of the generalization error of the chosen
network. Hjorth (1994) explains how this principle extends to
cross-validation and bootstrapping.
References:
Darlington, R.B. (1968), "Multiple Regression in Psychological Research
and Practice," Psychological Bulletin, 69, 161-182.
Efron, B. (1986), "How biased is the apparent error rate of a prediction
rule?" J. of the American Statistical Association, 81, 461-470.
Efron, B. and Tibshirani, R.J. (1993), An Introduction to the Bootstrap,
London: Chapman & Hall.
Hjorth, J.S.U. (1994), Computer Intensive Statistical Methods:
Validation, Model Selection, and Bootstrap, London: Chapman & Hall.
Miller, A.J. (1990), Subset Selection in Regression, London: Chapman &
Hall.
Moody, J.E. (1992), "The Effective Number of Parameters: An Analysis of
Generalization and Regularization in Nonlinear Learning Systems", in
Moody, J.E., Hanson, S.J., and Lippmann, R.P., Advances in Neural
Information Processing Systems 4, 847-854.
Raftery, A.E. (1995), "Bayesian Model Selection in Social Research," in
Marsden, P.V. (ed.), Sociological Methodology 1995, Cambridge, MA:
Blackwell, ftp://ftp.stat.washington.edu/pub/tech.reports/bic.ps.z or
http://www.stat.washington.edu/tech.reports/bic.ps
Rissanen, J. (1978), "Modelling by shortest data description,"
Automatica, 14, 465-471.
Rissanen, J. (1987), "Stochastic complexity" (with discussion), J. of the
Royal Statistical Society, Series B, 49, 223-239.
Rissanen, J. (1999), "Hypothesis Selection and Testing by the MDL
Principle," Computer Journal, 42, 260-269,
http://www3.oup.co.uk/computer_journal/hdb/Volume_42/Issue_04/
Sarle, W.S. (1995), "Stopped Training and Other Remedies for
Overfitting," Proceedings of the 27th Symposium on the Interface of
Computing Science and Statistics, 352-360,
ftp://ftp.sas.com/pub/neural/inter95.ps.Z (this is a very large
compressed postscript file, 747K, 10 pages)
Sarle, W.S. (1999), "Donoho-Johnstone Benchmarks: Neural Net Results,"
ftp://ftp.sas.com/pub/neural/dojo/dojo.html
Weiss, S.M. & Kulikowski, C.A. (1991), Computer Systems That Learn,
Morgan Kaufmann.
------------------------------------------------------------------------
Subject: What are cross-validation and bootstrapping?
======================================================
Cross-validation and bootstrapping are both methods for estimating
generalization error based on "resampling" (Weiss and Kulikowski 1991; Efron
and Tibshirani 1993; Hjorth 1994; Plutowski, Sakata, and White 1994; Shao
and Tu 1995). The resulting estimates of generalization error are often used
for choosing among various models, such as different network architectures.
Cross-validation
++++++++++++++++
In k-fold cross-validation, you divide the data into k subsets of
(approximately) equal size. You train the net k times, each time leaving
out one of the subsets from training, but using only the omitted subset to
compute whatever error criterion interests you. If k equals the sample
size, this is called "leave-one-out" cross-validation. "Leave-v-out" is a
more elaborate and expensive version of cross-validation that involves
leaving out all possible subsets of v cases.
Note that cross-validation is quite different from the "split-sample" or
"hold-out" method that is commonly used for early stopping in NNs. In the
split-sample method, only a single subset (the validation set) is used to
estimate the generalization error, instead of k different subsets; i.e.,
there is no "crossing". While various people have suggested that
cross-validation be applied to early stopping, the proper way of doing so is
not obvious.
The distinction between cross-validation and split-sample validation is
extremely important because cross-validation is markedly superior for small
data sets; this fact is demonstrated dramatically by Goutte (1997) in a
reply to Zhu and Rohwer (1996). For an insightful discussion of the
limitations of cross-validatory choice among several learning methods, see
Stone (1977).
Jackknifing
+++++++++++
Leave-one-out cross-validation is also easily confused with jackknifing.
Both involve omitting each training case in turn and retraining the network
on the remaining subset. But cross-validation is used to estimate
generalization error, while the jackknife is used to estimate the bias of a
statistic. In the jackknife, you compute some statistic of interest in each
subset of the data. The average of these subset statistics is compared with
the corresponding statistic computed from the entire sample in order to
estimate the bias of the latter. You can also get a jackknife estimate of
the standard error of a statistic. Jackknifing can be used to estimate the
bias of the training error and hence to estimate the generalization error,
but this process is more complicated than leave-one-out cross-validation
(Efron, 1982; Ripley, 1996, p. 73).
Choice of cross-validation method
+++++++++++++++++++++++++++++++++
Cross-validation can be used simply to estimate the generalization error of
a given model, or it can be used for model selection by choosing one of
several models that has the smallest estimated generalization error. For
example, you might use cross-validation to choose the number of hidden
units, or you could use cross-validation to choose a subset of the inputs
(subset selection). A subset that contains all relevant inputs will be
called a "good" subsets, while the subset that contains all relevant inputs
but no others will be called the "best" subset. Note that subsets are "good"
and "best" in an asymptotic sense (as the number of training cases goes to
infinity). With a small training set, it is possible that a subset that is
smaller than the "best" subset may provide better generalization error.
Leave-one-out cross-validation often works well for estimating
generalization error for continuous error functions such as the mean squared
error, but it may perform poorly for discontinuous error functions such as
the number of misclassified cases. In the latter case, k-fold
cross-validation is preferred. But if k gets too small, the error estimate
is pessimistically biased because of the difference in training-set size
between the full-sample analysis and the cross-validation analyses. (For
model-selection purposes, this bias can actually help; see the discussion
below of Shao, 1993.) A value of 10 for k is popular for estimating
generalization error.
Leave-one-out cross-validation can also run into trouble with various
model-selection methods. Again, one problem is lack of continuity--a small
change in the data can cause a large change in the model selected (Breiman,
1996). For choosing subsets of inputs in linear regression, Breiman and
Spector (1992) found 10-fold and 5-fold cross-validation to work better than
leave-one-out. Kohavi (1995) also obtained good results for 10-fold
cross-validation with empirical decision trees (C4.5). Values of k as small
as 5 or even 2 may work even better if you analyze several different random
k-way splits of the data to reduce the variability of the cross-validation
estimate.
Leave-one-out cross-validation also has more subtle deficiencies for model
selection. Shao (1995) showed that in linear models, leave-one-out
cross-validation is asymptotically equivalent to AIC (and Mallows' C_p), but
leave-v-out cross-validation is asymptotically equivalent to Schwarz's
Bayesian criterion (called SBC or BIC) when v =
n[1-1/(log(n)-1)], where n is the number of training cases. SBC
provides consistent subset-selection, while AIC does not. That is, SBC will
choose the "best" subset with probability approaching one as the size of the
training set goes to infinity. AIC has an asymptotic probability of one of
choosing a "good" subset, but less than one of choosing the "best" subset
(Stone, 1979). Many simulation studies have also found that AIC overfits
badly in small samples, and that SBC works well (e.g., Hurvich and Tsai,
1989; Shao and Tu, 1995). Hence, these results suggest that leave-one-out
cross-validation should overfit in small samples, but leave-v-out
cross-validation with appropriate v should do better. However, when true
models have an infinite number of parameters, SBC is not efficient, and
other criteria that are asymptotically efficient but not consistent for
model selection may produce better generalization (Hurvich and Tsai, 1989).
Shao (1993) obtained the surprising result that for selecting subsets of
inputs in a linear regression, the probability of selecting the "best" does
not converge to 1 (as the sample size n goes to infinity) for leave-v-out
cross-validation unless the proportion v/n approaches 1. At first glance,
Shao's result seems inconsistent with the analysis by Kearns (1997) of
split-sample validation, which shows that the best generalization is
obtained with v/n strictly between 0 and 1, with little sensitivity to the
precise value of v/n for large data sets. But the apparent conflict is due
to the fundamentally different properties of cross-validation and
split-sample validation.
To obtain an intuitive understanding of Shao (1993), let's review some
background material on generalization error. Generalization error can be
broken down into three additive parts, noise variance + estimation variance
+ squared estimation bias. Noise variance is the same for all subsets of
inputs. Bias is nonzero for subsets that are not "good", but it's zero for
all "good" subsets, since we are assuming that the function to be learned is
linear. Hence the generalization error of "good" subsets will differ only in
the estimation variance. The estimation variance is (2p/t)s^2 where p
is the number of inputs in the subset, t is the training set size, and s^2
is the noise variance. The "best" subset is better than other "good" subsets
only because the "best" subset has (by definition) the smallest value of p.
But the t in the denominator means that differences in generalization error
among the "good" subsets will all go to zero as t goes to infinity.
Therefore it is difficult to guess which subset is "best" based on the
generalization error even when t is very large. It is well known that
unbiased estimates of the generalization error, such as those based on AIC,
FPE, and C_p, do not produce consistent estimates of the "best" subset
(e.g., see Stone, 1979).
In leave-v-out cross-validation, t=n-v. The differences of the
cross-validation estimates of generalization error among the "good" subsets
contain a factor 1/t, not 1/n. Therefore by making t small enough (and
thereby making each regression based on t cases bad enough), we can make
the differences of the cross-validation estimates large enough to detect. It
turns out that to make t small enough to guess the "best" subset
consistently, we have to have t/n go to 0 as n goes to infinity.
The crucial distinction between cross-validation and split-sample validation
is that with cross-validation, after guessing the "best" subset, we train
the linear regression model for that subset using all n cases, but with
split-sample validation, only t cases are ever used for training. If our
main purpose were really to choose the "best" subset, I suspect we would
still have to have t/n go to 0 even for split-sample validation. But
choosing the "best" subset is not the same thing as getting the best
generalization. If we are more interested in getting good generalization
than in choosing the "best" subset, we do not want to make our regression
estimate based on only t cases as bad as we do in cross-validation, because
in split-sample validation that bad regression estimate is what we're stuck
with. So there is no conflict between Shao and Kearns, but there is a
conflict between the two goals of choosing the "best" subset and getting the
best generalization in split-sample validation.
Bootstrapping
+++++++++++++
Bootstrapping seems to work better than cross-validation in many cases
(Efron, 1983). In the simplest form of bootstrapping, instead of repeatedly
analyzing subsets of the data, you repeatedly analyze subsamples of the
data. Each subsample is a random sample with replacement from the full
sample. Depending on what you want to do, anywhere from 50 to 2000
subsamples might be used. There are many more sophisticated bootstrap
methods that can be used not only for estimating generalization error but
also for estimating confidence bounds for network outputs (Efron and
Tibshirani 1993). For estimating generalization error in classification
problems, the .632+ bootstrap (an improvement on the popular .632 bootstrap)
is one of the currently favored methods that has the advantage of performing
well even when there is severe overfitting. Use of bootstrapping for NNs is
described in Baxt and White (1995), Tibshirani (1996), and Masters (1995).
However, the results obtained so far are not very thorough, and it is known
that bootstrapping does not work well for some other methodologies such as
empirical decision trees (Breiman, Friedman, Olshen, and Stone, 1984;
Kohavi, 1995), for which it can be excessively optimistic.
For further information
+++++++++++++++++++++++
Cross-validation and bootstrapping become considerably more complicated for
time series data; see Hjorth (1994) and Snijders (1988).
More information on jackknife and bootstrap confidence intervals is
available at ftp://ftp.sas.com/pub/neural/jackboot.sas (this is a plain-text
file).
References:
Baxt, W.G. and White, H. (1995) "Bootstrapping confidence intervals for
clinical input variable effects in a network trained to identify the
presence of acute myocardial infarction", Neural Computation, 7, 624-638.
Breiman, L. (1996), "Heuristics of instability and stabilization in model
selection," Annals of Statistics, 24, 2350-2383.
Breiman, L., Friedman, J.H., Olshen, R.A. and Stone, C.J. (1984),
Classification and Regression Trees, Belmont, CA: Wadsworth.
Breiman, L., and Spector, P. (1992), "Submodel selection and evaluation
in regression: The X-random case," International Statistical Review, 60,
291-319.
Dijkstra, T.K., ed. (1988), On Model Uncertainty and Its Statistical
Implications, Proceedings of a workshop held in Groningen, The
Netherlands, September 25-26, 1986, Berlin: Springer-Verlag.
Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling
Plans, Philadelphia: SIAM.
Efron, B. (1983), "Estimating the error rate of a prediction rule:
Improvement on cross-validation," J. of the American Statistical
Association, 78, 316-331.
Efron, B. and Tibshirani, R.J. (1993), An Introduction to the Bootstrap,
London: Chapman & Hall.
Efron, B. and Tibshirani, R.J. (1997), "Improvements on cross-validation:
The .632+ bootstrap method," J. of the American Statistical Association,
92, 548-560.
Goutte, C. (1997), "Note on free lunches and cross-validation," Neural
Computation, 9, 1211-1215,
ftp://eivind.imm.dtu.dk/dist/1997/goutte.nflcv.ps.gz.
Hjorth, J.S.U. (1994), Computer Intensive Statistical Methods Validation,
Model Selection, and Bootstrap, London: Chapman & Hall.
Hurvich, C.M., and Tsai, C.-L. (1989), "Regression and time series model
selection in small samples," Biometrika, 76, 297-307.
Kearns, M. (1997), "A bound on the error of cross validation using the
approximation and estimation rates, with consequences for the
training-test split," Neural Computation, 9, 1143-1161.
Kohavi, R. (1995), "A study of cross-validation and bootstrap for
accuracy estimation and model selection," International Joint Conference
on Artificial Intelligence (IJCAI), pp. ?,
http://robotics.stanford.edu/users/ronnyk/
Masters, T. (1995) Advanced Algorithms for Neural Networks: A C++
Sourcebook, NY: John Wiley and Sons, ISBN 0-471-10588-0
Plutowski, M., Sakata, S., and White, H. (1994), "Cross-validation
estimates IMSE," in Cowan, J.D., Tesauro, G., and Alspector, J. (eds.)
Advances in Neural Information Processing Systems 6, San Mateo, CA:
Morgan Kaufman, pp. 391-398.
Ripley, B.D. (1996) Pattern Recognition and Neural Networks, Cambridge:
Cambridge University Press.
Shao, J. (1993), "Linear model selection by cross-validation," J. of the
American Statistical Association, 88, 486-494.
Shao, J. (1995), "An asymptotic theory for linear model selection,"
Statistica Sinica ?.
Shao, J. and Tu, D. (1995), The Jackknife and Bootstrap, New York:
Springer-Verlag.
Snijders, T.A.B. (1988), "On cross-validation for predictor evaluation in
time series," in Dijkstra (1988), pp. 56-69.
Stone, M. (1977), "Asymptotics for and against cross-validation,"
Biometrika, 64, 29-35.
Stone, M. (1979), "Comments on model selection criteria of Akaike and
Schwarz," J. of the Royal Statistical Society, Series B, 41, 276-278.
Tibshirani, R. (1996), "A comparison of some error estimates for neural
network models," Neural Computation, 8, 152-163.
Weiss, S.M. and Kulikowski, C.A. (1991), Computer Systems That Learn,
Morgan Kaufmann.
Zhu, H., and Rohwer, R. (1996), "No free lunch for cross-validation,"
Neural Computation, 8, 1421-1426.
------------------------------------------------------------------------
Subject: How to compute prediction and confidence
=================================================
intervals (error bars)?
=======================
(This answer is only about half finished. I will get around to the other
half eventually.)
In addition to estimating over-all generalization error, it is often useful
to be able to estimate the accuracy of the network's predictions for
individual cases.
Let:
Y = the target variable
y_i = the value of Y for the ith case
X = the vector of input variables
x_i = the value of X for the ith case
N = the noise in the target variable
n_i = the value of N for the ith case
m(X) = E(Y|X) = the conditional mean of Y given X
w = a vector of weights for a neural network
w^ = the weight obtained via training the network
p(X,w) = the output of a neural network given input X and weights w
p_i = p(x_i,w)
L = the number of training (learning) cases, (y_i,x_i), i=1, ..., L
Q(w) = the objective function
Assume the data are generated by the model:
Y = m(X) + N
E(N|X) = 0
N and X are independent
The network is trained by attempting to minimize the objective function
Q(w), which, for example, could be the sum of squared errors or the
negative log likelihood based on an assumed family of noise distributions.
Given a test input x_0, a 100c% prediction interval for y_0 is an
interval [LPB_0,UPB_0] such that Pr(LPB_0 <= y_0 <=
UPB_0) = c, where c is typically .95 or .99, and the probability is
computed over repeated random selection of the training set and repeated
observation of Y given the test input x_0. A 100c% confidence interval
for p_0 is an interval [LCB_0,UCB_0] such that Pr(LCB_0 <=
p_0 <= UCB_0) = c, where again the probability is computed over
repeated random selection of the training set. Note that p_0 is a
nonrandom quantity, since x_0 is given. A confidence interval is narrower
than the corresponding prediction interval, since the prediction interval
must include variation due to noise in y_0, while the confidence interval
does not. Both intervals include variation due to sampling of the training
set and possible variation in the training process due, for example, to
random initial weights and local minima of the objective function.
Traditional statistical methods for nonlinear models depend on several
assumptions (Gallant, 1987):
1. The inputs for the training cases are either fixed or obtained by simple
random sampling or some similarly well-behaved process.
2. Q(w) has continuous first and second partial derivatives with respect
to w over some convex, bounded subset S_W of the weight space.
3. Q(w) has a unique global minimum at w^, which is an interior point of
S_W.
4. The model is well-specified, which requires (a) that there exist weights
w$ in the interior of S_W such that m(x) = p(x,w$), and (b)
that the assumptions about the noise distribution are correct. (Sorry
about the w$ notation, but I'm running out of plain text symbols.)
These traditional methods are based on a linear approximation to p(x,w)
in a neighborhood of w$, yielding a quadratic approximation to Q(w).
Hence the Hessian of Q(w) (the square matrix of second-order partial
derivatives with respect to w) frequently appears in these methods.
Assumption (3) is not satisfied for neural nets, because networks with
hidden units always have multiple global minima, and the global minima are
often improper. Hence, confidence intervals for the weights cannot be
obtained using standard Hessian-based methods. However, Hwang and Ding
(1997) have shown that confidence intervals for predicted values can be
obtained because the predicted values are statistically identified even
though the weights are not.
Cardell, Joerding, and Li (1994) describe a more serious violation of
assumption (3), namely that that for some m(x), no finite global minimum
exists. In such situations, it may be possible to use regularization methods
such as weight decay to obtain valid confidence intervals (De Veaux, Schumi,
Schweinsberg, and Ungar, 1998), but more research is required on this
subject, since the derivation in the cited paper assumes a finite global
minimum.
For large samples, the sampling variability in w^ can be approximated in
various ways:
o Fisher's information matrix, which is the expected value of the Hessian
of Q(w) divided by L, can be used when Q(w) is the negative log
likelihood (Spall, 1998).
o The delta method, based on the Hessian of Q(w) or the Gauss-Newton
approximation using the cross-product Jacobian of Q(w), can also be
used when Q(w) is the negative log likelihood (Tibshirani, 1996; Hwang
and Ding, 1997; De Veaux, Schumi, Schweinsberg, and Ungar, 1998).
o The sandwich estimator, a more elaborate Hessian-based method, relaxes
assumption (4) (Gallant, 1987; White, 1989; Tibshirani, 1996).
o Bootstrapping can be used without knowing the form of the noise
distribution and takes into account variability introduced by local
minima in training, but requires training the network many times on
different resamples of the training set (Tibshirani, 1996; Heskes 1997).
References:
Cardell, N.S., Joerding, W., and Li, Y. (1994), "Why some feedforward
networks cannot learn some polynomials," Neural Computation, 6, 761-766.
De Veaux,R.D., Schumi, J., Schweinsberg, J., and Ungar, L.H. (1998),
"Prediction intervals for neural networks via nonlinear regression,"
Technometrics, 40, 273-282.
Gallant, A.R. (1987) Nonlinear Statistical Models, NY: Wiley.
Heskes, T. (1997), "Practical confidence and prediction intervals," in
Mozer, M.C., Jordan, M.I., and Petsche, T., (eds.) Advances in Neural
Information Processing Systems 9, Cambrideg, MA: The MIT Press, pp.
176-182.
Hwang, J.T.G., and Ding, A.A. (1997), "Prediction intervals for
artificial neural networks," J. of the American Statistical Association,
92, 748-757.
Nix, D.A., and Weigend, A.S. (1995), "Learning local error bars for
nonlinear regression," in Tesauro, G., Touretzky, D., and Leen, T.,
(eds.) Advances in Neural Information Processing Systems 7, Cambridge,
MA: The MIT Press, pp. 489-496.
Spall, J.C. (1998), "Resampling-based calculation of the information
matrix in nonlinear statistical models," Proceedings of the 4th Joint
Conference on Information Sciences, October 23-28, Research Triangle
PArk, NC, USA, Vol 4, pp. 35-39.
Tibshirani, R. (1996), "A comparison of some error estimates for neural
network models," Neural Computation, 8, 152-163.
White, H. (1989), "Some Asymptotic Results for Learning in Single Hidden
Layer Feedforward Network Models", J. of the American Statistical Assoc.,
84, 1008-1013.
------------------------------------------------------------------------
Next part is part 4 (of 7). Previous part is part 2.
--
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.