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Machine learning is an important task for learning artificial neural networks, and we find in the learning one of the common problems of learning the Artificial Neural Network (ANN) is over-fitting and under-fitting to outlier points. In this paper we performed various methods in avoiding over-fitting and under-fitting; that is penalty and early stopping methods. A comparative study has been presented for the aforementioned methods to evaluate their performance within a range of specific parameters such as; speed of training, over-fitting and under-fitting avoidance, difficulty, capacity, time of training, and their accuracy. Besides these parameters we have included comparison between over-fitting and under-fitting. We found the early stopping method as being better as compared to the penalty method, as it can avoid over-fitting and under-fitting with respect to validation time. Besides we find that Under-fitting neural networks perform poorly on both training and test sets, but Over-fitting networks may do very well on training sets though terribly on test sets.
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ABSTRACT: Machine learning is an important task for learning artificial neural networks, and
we find in the learning one of the common problems of learning the Artificial Neural Network
(ANN) is over-fitting and under-fitting to outlier points. In this paper we performed various
methods in avoiding over-fitting and under-fitting; that is penalty and early stopping methods. A
comparative study has been presented for the aforementioned methods to evaluate their
performance within a range of specific parameters such as; speed of training, over-fitting and
under-fitting avoidance, difficulty, capacity, time of training, and their accuracy. Besides these
parameters we have included comparison between over-fitting and under-fitting. We found the
early stopping method as being better as compared to the penalty method, as it can avoid over-
fitting and under-fitting with respect to validation time. Besides we find that Under-fitting neural
networks perform poorly on both training and test sets, but Over-fitting networks may do very well
on training sets though terribly on test sets.
KEYWORDS: Machine learning, over-fitting, under-fitting, early stopping, penalty method
One of the common problems of the use of ANN is the over-fitting to outlier points (Wang, J.H.;
Jiang, J.H.; Yu, R.Q. 1996). Over-fitting is a key problem in the supervised machine learning
tasks. It is the phenomenon detected when a learning algorithm fits the training data set so well
that noise and the peculiarities of the training data are memorized. According to the result of
learning algorithms performance drops when it is tested in an unknown data set. The amount of
data used for learning process is fundamental in this context. Small data sets are more prone to
over-fitting than large data sets, and despite the complexity of some learning problem, large data
sets can even be affected by over-fitting (Santos, E. M., Sabourin, R., & Maupin, P. 2009). Over-
fitting of the training data leads to deterioration of generalization properties of the model, and
results in its untrustworthy performance when applied to novel measurements. Hence the purpose
of the methods to avoid over-fitting is somehow contradictory to the goal of optimization
algorithms, which aims at finding the best possible solution in parameter space according to pre-
defined objective function and available data. Moreover, different opti mization algorithms may
perform better for simpler or larger ANN architectures. This suggests the importance of proper
coupling of different optimization algorithms; ANN architectures and methods to avoid over-
fitting of real-world data – an issue that is also studied in details in the present paper. We clarified
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Janahanlal Stephen, Harish Rohil and S Vasavi
Copyright © 2015 AET-2014 Organisers.
ISBN: 978-981-09-5247-1
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machine learning in section II, and we described the over-fitting and under-fitting as well as
comparison between them in section III. At section IV and V we described the methods for
avoiding over-fitting and under-fitting.
The ANN learning terminates when error increases for validation data, although it often continues
to decrease for training data set. When error calculated for validation data increases as that for
training data decreases, it is considered as fitting to the noise present in the data instead of signal,
which in other words is considered over–fitting (Sarle, W.S. 1995); (Panchal G, Ganatra A, Shah
P, Panchal D. 2011). Machine-learning research has been making great progress in many
directions, but this paper focuses on four (4) directions. The first direction is the improvement of
classification accuracy by learning ensembles of classifiers, the second refers to the methods for
scaling up supervised learning algorithms, and the third is about reinforcement learning. The
fourth direction talks about the learning of complex stochastic models (Dietterich, T. G. 1997).
This paper further discusses some current open problems.
The over-fitting
Is the one of biggest problem in training neural networks is the over-fitting of training data. That
means that the neural network at the certain time during the training period does not improve its
ability to solve problem anymore. But just starts to learn some random regularity contained in the
set of training patterns. This is equivalent to the empirical observation that error on the test set has
a minimum where the generalization ability of the network is the network is the best before this
error starts to increase again (Wang, J.H.; Jiang, J.H.; Yu, R.Q. 1996); (Gaurang P, Amit G, Parth
S, Devyani P. 2011); (Tom Dietterich. 1995); (Sarle, W.S. 1995); (Lawrence, S., Giles, C.L., Tsoi,
A.-C. 1997). Over-fitting occurs when astatically model describes random error or nose instead of
the underlying relationship (Piotrowski, A.P., Napiorkowski, J.J., 2013); (Chan, K.Y., Kwong,
C.K., Dillon, T.S., Tsim, Y.C. 2011); (Panchal G, Ganatra A, Shah P, Panchal D., 2011);
(Domingos, P. 2000). See Figure 1.
When the model under-fits, the bias is generally high and the variance is low. Over-fitting is
typically characterized by high variance, low bias estimators. In many cases, small increases in
Figure 1.
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bias result in large decreases in variance (Tetko, I. V., Livingstone, D. J., and Luik, A. I. 1995);
(Schaffer, C. 1993); (Loughrey, J., & Cunningham, P. 2005). See figure 2 and Figure 3.
Is the opposite of Over-fitting. This occurs when the model is incapable of capturing the variability
of the data. For example suppose one is training a linear (y= ax+b not polynomial a and b are
constant) classifier on a data set that is a parabola Tom Dietterich. (1995), (van der Aalst, W. M.,
Rubin, V., Verbeek, H. M. W., van Dongen, B. F., Kindler, E., & Günther, C. W. 2010). The
resultant classifier will have no predicative power nor will it able to properly map the training data
(Lawrence, S., Giles, C.L., Tsoi, A.-C. 1997). This is the result of understanding or attempting to
use a model which is too simple to describe a given set of data see figure 4.
Some methods to avoid the problem of over-fitting and under-fitting in supervised machine
There are many methods (Kazushi. M. 2005); (Prechelt L. 1999); (Schittenkopf C, Deco G, Brauer
W. 1997). :
A.Penalty methods:
Map provides a penalty based on P(H).
Minimum description length (MDL) principle.
Structural risk minimization.
Generalization cross-validation.
Hold and cross-validation.
B.Early stopping for training
2 Zero Variance, High Bias
3 High Variance, Low Bias
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Penalty method is the one of important methods to avoid the over-fitting in supervised training
data, the concept of penalty it can be understand as fallow (Ruppert, D. and Carroll, R. J. 2000);
(Babuška, I. 1973). :
Let Etra in be our training set error and Ete st be our test error.
Our real goal is to find the h that minimizes Etest. The problem is that we can’t directly evaluate
Etes t, we can measure Etrain but it is optimistic. Penalty method to find some penalty such that:
Etes t = Etra in + Penalty (1)
Where we directly penalize model complexity. We can also represent the error function as (Wen,
U. P., Lan, K. M., & Shih, H. S. 2009):
Etes t = Etra in + λ (model complexity) (2)
λ is crucial and controls the Bias-Variance Trade-off.
Theoretically the ANN as a highly non-linear model can be used for fitting of any non-linear
function ( 2005). Methods using a penalty term are characterized by
adding a cost term Cλ (W) to the error function where h denotes a learning rate and w the vector
of all weights wi. Training the network by back propagation changes the weights according to the
negative gradient of the error function, and therefore the derivatives are the point of interest.
Usually these derivatives drive’s the weights of the network to zero weight decay terms) and
thereby reduce the number of free parameters they propose the cost function (Haibin, Li., Duan, Z.
2009). The advantage of penalty-term methods is that training and pruning are done in parallel.
Choosing the learning rate A, however, may be tricky. We can try to avoid over-fitting by
maximizing the log likelihood plus a penalty function. For an MLP with one hidden layer, a
suitable penalty to add to minus the log Likelihood might be
Figure 4
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We need to select two constants controlling the penalty, λ1 and λ2. Setting λ1 = λ2 isn’t always
reasonable, since a suitable value for λ1 depends on the measurement units used for the inputs,
whereas a suitable value for λ2 depends on the measurement units for the response. We might try
S-fold cross-validation, but it may not work well, if each training run goes to a dierent local
maximum. So we might use a single split into estimation and validation sets, with no re-training on
the whole training set.
It is one method used for avoid over-fitting and under-fitting and to use early stopping approach,
apart from the training data set, the testing set and the validation set that is required to define
stopping criteria of the method (Prechelt L. 1999). So in this way the data divided to three parts:
1.Part for training
2.Part for validation
3.Part for testing
See figure 5.
It is difficult to decide when it is best to stop training by just looking at the learning curve for
training by itself. It is possible to over-fitting the training data if the training session is not
stopped at the right point (Schittenkopf C, Deco G, Brauer W. 1997). The onset of over-fitting
can be detected through cross validation in which the available data are divided into training,
validation, and testing subsets. The training subset is used for computing the gradient and
updating the network weights. The error on the validation set is monitored during the training
session (Caruana, R., Lawrence, S., & Giles, L. (2001). The validation error will normally
decrease during the initial phase of training (see Fig. 5), as does the error on the training set.
However, when the network begins to over-fitting the data, the error on the validation set will
typically begin to rise. When the validation error increases for a specified number of iterations,
the training is stopped, and the weights at the minimum of the validation error are returned
(GENCAY, R. and QI, M. 2001); (Loughrey, J., & Cunningham, P. 2005). If we have lots of
data and a big model, it’s very expensive to keep retraining it with different amounts of weight
decay. It is much cheaper to start with very small weights and let them grow until the
Time of training
Figure. 5 The time for early stopping to avoid the over-fitting training
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Figure 6 a numerical simulation
performance on the validation set starts getting worse (but don’t get fooled by noise!) (Prechelt
L. 1999). (Schittenkopf C, Deco G, Brauer W. (1997). The capacity of the model is limited
because the weights have not had time to grow big. So this wills guide us to ask the question:
Why early st opping works?
A.When the weights are very small, every hidden unit is in its linear range.
So a net with a large layer of hidden units is linear.
It has no more capacity than a linear net in which the inputs are directly
connected to the outputs.
B.As the weights grow, the hidden units start using their non-linear ranges so the capacity
It is natural to consider early stopping when dealing with big data-sets, since in this context
algorithms should from optimum have computational requirements tailored to the generalization
properties allowed by the data. This is exactly the main property of early stopping regularization.
Figure 6 gives an illustration of this fact in a numerical simulation. Here, we added an explicit
regularization parameter λ as in (Schittenkopf C, Deco G, Brauer W. 1997).
We can tell when to stop gradient descent optimization using a set of validation cases that’s
separate from the cases used to compute the gradient (Prechelt, L. 1998).
Here are the steps:
Randomly divide the training set into an estimation set and a validation set –eg, 80% of
cases in the estimation set and 20% n the validation set.
Randomly initialize the parameters to values near zero.
Repeatedly do the following:
i.Compute the gradient of the log likelihood using the estimation set.
ii.Update the parameters by adding η times the gradient.
iii.Compute the log probability of y given x for the validation cases.
Stop when the average log probability for validation cases is substantially less than
the maximum of the values found previously (definitely getting worse).
Make predictions for test cases using the parameter values from the loop above that gave
the highest average log probability to the validation cases.
Early stopping using a validation set has some advantages over a penalty method, in which you set
λ1 and λ2 using S-fold cross validation, then train again on all the data with the λ1 and λ2 you
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With early stopping:
You only have to train the network once, not once for each setting of λ1 and λ2 you
consider, and then again on all the training data to get the final parameters.
The performance measure from cross-validation applies directly to the actual set of
network parameters you will use, not to values of λ1 and λ2 that may not do the same
thing for a dierent local maximum.
Early stopping also has some problems:
It’s very ad hoc.
It depends on details of the optimization method –eg, results could be dierent with a
dierent η.
In particular, we might want to use a dierent η for w(1)s than for w(2) –sort of like using
dierent λs for a penalty method.
Some of the training data is used only to decide when to stop this seems wasteful.
The avoidance of over-fitting and under-fitting help to improve the performance of machine
learning and for avoiding those problems. We present two of comparatives the first depend on
various parameters and the second Comparative study between Over-fitting VS Under-fitting
(Tetko, I. V., Livingstone, D. J., and Luik, A. I. 1995); (Prechelt L. 1999); (Tom Dietterich. 1995);
(Schittenkopf C, Deco G, Brauer W. 1997); (Schittenkopf C, Deco G, Brauer W. 1997), (Caruana,
R., Lawrence, S., & Giles, L. 2001); (GENCAY, R. and QI, M. 2001); (Ruppert, D. and Carroll, R.
J. 2000); (Babuška, I. 1973); (Loughrey, J., & Cunningham, P. 2005); (Prechelt, L. 1998).
Comparative study between Penalty methods and early stopping
We present in this comparison between early stopping method and penalty methods as follow in
table (1). A comparison based on various parameters derived from some theoretical studies.
Table 1. Comparative by methods
Methods Penalty methods Early stopping
Cost Bad Good
Speed of training Good Bad
Accuracy in avoidance Good Bad
Difficulty Good Bad
Capacity Bad Good
Time of training Good Bad
Influenced by variance Bad Good
Influenced by bias Bad Good
Accuracy in working with another method Bad Good
Small data Bad Good
Big data Good Bad
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Comparative study between Over-fitting VS Under-fitting
When discussing the quality of a model in regards to a set of data two commonly used terms are
over-fitting and under-fitting. A model which is over-fitted is a model which has an excess of
parameters. The added complexity may and often does help the model perform well on a set of
training data but it inhibits prediction of future points. Figure 2 on over-fitting illustrates this.
While it is clear from the picture that we are looking at data generated by a linear function plus
noise (Is the human brain not a powerful machine that it can deter- mine that on the y) the over-
fitted example gains improved accuracy on the training set (the points which it learns on or the
points which are drawn on the graph, while missing the overall message or pattern in the data).
Noise, hidden factors, and difficult high level relations are primary causes of variability in data
that cannot or is difficult to capture with statistical models. Under-fitting is the opposite of over-
fitting. This occurs when the model is incapable of capturing the variability of the data. For
example suppose one is training a LINEAR (y=ax+b not polynomial a and b are constants)
classifier on a data set that is a parabola. The resultant classifier will have no predictive power nor
will it be able to properly map the training data. This is the result of under-fitting, or attempting to
use a model which is too simple to describe a given set of data. Understanding these two
phenomena allows one to thread the needle and go into the space between the two extremes. It is in
this gap where the model has predictive power in the validation set lies.
In this paper we performed a comparative study on various methods of machine learning to avoid
over-fitting and under-fitting and we have found the early stopping method is the best as compared
with penalty method that can avoid over-fitting and under-fitting with taking care to the validation
time. As well as the penalty method it is sensitive to variance and bias while early stopping
method it is less sensitive to variance and bias. Besides we find that under-fitting neural networks
perform poorly on both training and test sets while over-fitting networks may do very well on
training sets but terribly on test sets.
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Haider Khalaf Jabbar Allamy: Haider allamy, is a research scholar in the
Department of Computer Science; Aligarh Muslim University, Aligarh,
India. He joined to his Ph.D course in 28-09-2012. His research interest
includes Artificial Intelligence and Expert system. He did B.Sc Degree in
computer science from University of Basrah, College of Science, Iraq,
M.Sc in computer science from Jamia Hamdard University, Delhi, India.
He is envoy by the University of Misan/Iraq for complete his Ph.D course
in Aligarh Muslim University, Aligarh, India.
Dr. Rafiqul Zaman Khan: Dr. Rafiqul Zaman Khan, is presently working as
a Associate Professor in the Department of Computer Science at Aligarh
Muslim University, Aligarh, India. He received his B.Sc Degree from M.J.P
Rohilkhand University, Bareilly, M.Sc and M.C.A from A.M.U. and Ph.D
(Computer Science) from Jamia Hamdard University. He has 20 years of
Teaching Experience of various reputed International and National
Universities viz King Fahad University of Petroleum & Minerals
(KFUPM), K.S.A, Ittihad University, U.A.E, Pune University, Jamia
Hamdard University and AMU, Aligarh. He worked as a Head of the
Department of Computer Science at Poona College, University of Pune. He
also worked as a Chairman of the Department of Computer Science, AMU, Aligarh. His Research
Interest includes Parallel & Distributed Computing, Gesture Recognition, Expert Systems and
Artificial Intelligence. Presently 04 students are doing PhD under his supervision. He has
published about 50 research papers in International Journals/Conferences. Names of some Journals
of repute in which recently his articles have been published are International Journal of Computer
Applications (ISSN: 0975-8887), U.S.A, Journal of Computer and Information Science (ISSN:
1913-8989), Canada, International Journal of Human Computer Interaction (ISSN: 2180-1347),
Malaysia, and Malaysian Journal of Computer Science(ISSN: 0127-9084), Malaysia. He is the
Member of Advisory Board of International Journal of Emerging Technology and Advanced
Engineering (IJETAE), Editorial Board of International Journal of Advances in Engineering &
Technology (IJAET), International Journal of Computer Science Engineering and Technology
(IJCSET), International Journal in Foundations of Computer Science & technology (IJFCST) and
Journal of Information Technology, and Organizations (JITO).
... Overfitting is one of the biggest problems in training a neural network model [11]. Overfitting means that the neural network model at a certain time during the training period does not improve its ability to solve problem anymore. ...
... Moreover, there are many techniques used to overcome overfitting. For example, dropouts, data augmentation, L1 regularization, L2 regularization, reduce neural network capacity, early stopping [11]. Dropouts have been introduced in section 2.1.3, ...
... Underfitting is the opposite of overfitting [11]. Underfitting occurs when neural network model is not capable to capture the underlying features of the data. ...
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... Since it does not learn well from the training data, it can perform poorly on all data presented. In this case, a graph in which the training loss exceeds the validation loss can be used to diagnose the problem [39], [40]. There are two main techniques for avoiding overfitting, which we will discuss below. ...
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... Overfitting occurs when there is low bias and high variance in the model [16] and arises when the model fits the training data well but not the test data. Figure 1 and Supplementary Fig. 2 depict that the models are trying extremely hard to fit every single data point. ...
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The objective was to develop diabetic retinopathy (DR) risk scores and compute prevalence and incidence probabilities of DR in Indian type 2 diabetes mellitus patients. A double sample of size 388 was collected from the R.G. Centre for Diabetes and Endocrinology, J.N.M.C., A.M.U., Aligarh, India, randomly distributed among training and test sets. DR risk scores of Iran and China were administered on Indian training set. Since prevalence probabilities of DR calculated by Logit model were unacceptable, thus actual data of Iranian and Chinese studies were simulated from their variable characteristics. Ridge regression was selected as optimal by regularization and cross-validation techniques. The yearly incidences of DR from ridge probabilities were determined using absorbing Markov chain. Receiver operating characteristic (ROC) curve and Hosmer Lemeshow test were exerted for model discrimination and calibration. Furthermore, these outcomes were implemented on the test sample. Out of 284 training sample patients, 23 had DR currently. Iranian score with an area of 0.815 (95% CI 0.765–0.859) was the better fit. Ridge coefficients acquired from Chinese simulated data contented the Indian data, providing accurate probabilities and an area of 0.784 (95% CI 0.731–0.830). Validating on test data, ROC curves for current, 1 year and 2 years prediction resulted in areas of 0.819, 0.811 and 0.686. Iranian score and simulated Chinese ridge coefficients for prevalence of DR were the best fit on Indian type 2 diabetes patients. Markov two-state model can be applied to forecast yearly incidence of DR.
... Discriminant functions with enough degree of dimension must warranty non linearity to recognise any data class [7]. Since a major common tasks is fitting a model for data class discrimination, the concepts of over-fitting and under-fitting are relevant [1,8,9]. A too complex model that has been over-fitted (usually non bijective mapping) has too many parameters w.r.t. the number of observations. ...
Conference Paper
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This manuscript presents an analysis of numerical fitting methods used for solving classification problems as discriminant functions in machine learning. Non linear polynomial, exponential, and trigonometric models are mathematically deduced and discussed. Analysis about their pros and cons, and their mathematical modelling are made on what method to chose for what type of highly non linear multi-dimension problems are more suitable to be solved. In this study only deterministic models with analytic solutions are involved, or parameters calculation by numeric methods, which the complete model can subsequently be treated as a theoretical model. Models deduction are summarised and presented as a survey.
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A drawback of the error-back propagation algorithm for a multilayer feed forward neural network is over learning or over fitting. We have discussed this problem, and obtained necessary and sufficient Experiment and conditions for over-learning problem to arise. Using those conditions and the concept of a reproducing, this paper proposes methods for choosing training set which is used to prevent over-learning. For a classifier, besides classification capability, its size is another fundamental aspect. In pursuit of high performance, many classifiers do not take into consideration their sizes and contain numerous both essential and insignificant rules. This, however, may bring adverse situation to classifier, for its efficiency will been put down greatly by redundant rules. Hence, it is necessary to eliminate those unwanted rules. We have discussed various experiments with and without over learning or over fitting problem.
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It is acknowledged that overfitting can oc-cur in feature selection using the wrapper method when there is a limited amount of training data available. It has also been shown that the severity of overfitting is re-lated to the intensity of the search algo-rithm used during this process. In this pa-per we show that two stochastic search tech-niques (Simulated Annealing and Genetic Al-gorithms) that can be used for wrapper-based feature selection are susceptible to overfit-ting in this way. However, because of their stochastic nature, these algorithms can be stopped early to prevent overfitting. We present a framework that implements early-stopping for both of these stochastic search techniques and we show that this is success-ful in reducing the effects of overfitting and in increasing generalisation accuracy in most cases.
Optimization calculation is one of the important application fields in Neural Network. This paper proposes a Neural Network computational method of nonlinear programming problems based on precise F-function for the problems of not catering to network size, computational efficiency and accuracy when solving the constraint nonlinear programming problems in current Neural Network. A low order precise F-function of constraint nonlinear programming problems is served as energy function of Neural Network. The dynamics equation of Neural Network is constructed and the clarification of its stability is given by the most rapid decreasing principle of the energy function. Theoretical analysis and computational example emulation show that the proposed Neural Network dynamics equation globally and precisely constringes a local optimization solution of the original programming problems. In particular, the neural network dynamics equation is easy to be mapped as dynamic circuit, which is a Real-time calculation method for engineering optimization problems.
The paper studies spline fitting with a roughness penalty that adapts to spatial heterogeneity in the regression function. The estimates are pth degree piecewise polynomials with p - 1 continuous derivatives. A large and fixed number of knots is used and smoothing is achieved by putting a quadratic penalty on the jumps of the pth derivative at the knots. To be spatially adaptive, the logarithm of the penalty is itself a linear spline but with relatively few knots and with values at the knots chosen to minimize the generalized cross validation (GCV) criterion. This locally-adaptive spline estimator is compared with other spline estimators in the literature such as cubic smoothing splines and knot-selection techniques for least squares regression. Our estimator can be interpreted as an empirical Bayes estimate for a prior allowing spatial heterogeneity. In cases of spatially heterogeneous regression functions, empirical Bayes confidence intervals using this prior achieve better pointwise coverage probabilities than confidence intervals based on a global-penalty parameter. The method is developed first for univariate models and then extended to additive models. © Australian Statistical Publishing Association Inc. 2000. Published by Blackwell Publishers Ltd.
Artificial neural networks (ANNs) becomes very popular tool in hydrology, especially in rainfall–runoff modelling. However, a number of issues should be addressed to apply this technique to a particular problem in an efficient way, including selection of network type, its architecture, proper optimization algorithm and a method to deal with overfitting of the data. The present paper addresses the last, rarely considered issue, namely comparison of methods to prevent multi-layer perceptron neural networks from overfitting of the training data in the case of daily catchment runoff modelling. Among a number of methods to avoid overfitting the early stopping, the noise injection and the weight decay have been known for about two decades, however only the first one is frequently applied in practice. Recently a new methodology called optimized approximation algorithm has been proposed in the literature.
Validation can be used to detect when overfitting starts during supervised training of a neural network; training is then stopped before convergence to avoid the overfitting ("early stopping"). The exact criterion used for validation-based early stopping, however, is usually chosen in an ad-hoc fashion or training is stopped interactively. This trick describes how to select a stopping criterion in a systematic fashion; it is a trick for either speeding learning procedures or improving generalization, whichever is more important in the particular situation. An empirical investigation on multi-layer perceptrons shows that there exists a tradeoff between training time and generalization: From the given mix of 1296 training runs using different 12 problems and 24 different network architectures I conclude slower stopping criteria allow for small improvements in generalization there: about 4% on average), but cost much more training time there: about factor 4 longer on average).
The ordinary back propagation algorithm for the artificial network is non-robust. It has been shown in a straightforward way that the BP algorithm can be robustified by introducing some kind of transform on the residual term rp. Using the proposed robust BP algorithm, the problem of overfitting to outlier points can effectively be circumvented. The feasibility of the proposed method has been testified by treating simulated examples and a real chemical data.
The strategy of early stopping is a regularization technique based on choosing a stopping time for an iterative algorithm. Focusing on non-parametric regression in a reproducing kernel Hilbert space, we analyze the early stopping strategy for a form of gradient-descent applied to the least-squares loss function. We propose a data-dependent stopping rule that does not involve hold-out or cross-validation data, and we prove upper bounds on the squared error of the resulting function estimate, measured in either the $L^2(P)$ and $L^2(P_n)$ norm. These upper bounds lead to minimax-optimal rates for various kernel classes, including Sobolev smoothness classes and other forms of reproducing kernel Hilbert spaces. We show through simulation that our stopping rule compares favorably to two other stopping rules, one based on hold-out data and the other based on Stein's unbiased risk estimate. We also establish a tight connection between our early stopping strategy and the solution path of a kernel ridge regression estimator.