1. Overfitting, Bias/Variance tradeoff, and Ensemble methods
Pierre Geurts
Stochastic methods
(Prof. L.Wehenkel)
University of Liège

2. Content of the presentation
Bias and variance definitions
Parameters that influence bias and variance
Decision/regression tree variance
Bias and variance reduction techniques

3. Content of the presentation
Bias and variance definitions:
A simple regression problem with no input
Generalization to full regression problems
A short discussion about classification
Parameters that influence bias and variance
Decision/regression tree variance
Bias and variance reduction techniques

4. Regression problem - no input
Goal: predict as well as possible the height of a Belgian male adult
More precisely:
Choose an error measure, for example the square error.
Find an estimation ŷ such that the expectation:
over the whole population of Belgian male adult is minimized.
180

5. Regression problem - no input
The estimation that minimizes the error can be computed by taking:
So, the estimation which minimizes the error is Ey{y}. In AL, it is called the Bayes model.
But in practice, we cannot compute the exact value of Ey{y} (this would imply to measure the height of every Belgian male adults).

6. Learning algorithm
As p(y) is unknown, find an estimation y from a sample of individuals, LS={y1,y2,…,yN}, drawn from the Belgian male adult population.
Example of learning algorithms: ,
(if we know that the height is close to 180)

7. Good learning algorithm
As LS are randomly drawn, the prediction ŷ will also be a random variable
A good learning algorithm should not be good only on one learning sample but in average over all learning samples (of size N)  we want to minimize:
Let us analyse this error in more detail ŷ
pLS (ŷ)

8. Bias/variance decomposition (1)

9. Bias/variance decomposition (2)
vary{y}
Ey{y} y
E= Ey{(y- Ey{y})2} + ELS{(Ey{y}-ŷ)2}
= residual error = minimal attainable error
= vary{y}

10. Bias/variance decomposition (3)

11. Bias/variance decomposition (4)
ELS{ŷ}
Ey{y} ŷ
E= vary{y} + (Ey{y}-ELS{ŷ})2 + …
ELS{ŷ} = average model (over all LS)
bias2 = error between Bayes and average model

12. Bias/variance decomposition (5)
varLS{ŷ}
ELS{ŷ} ŷ
E= vary{y} + bias2 + ELS{(ŷ-ELS{ŷ})2}
varLS{ŷ} = estimation variance = consequence of over-fitting

13. Bias/variance decomposition (6)ŷ
Ey{y}
ELS{ŷ}
bias2
vary{y}
varLS{ŷ}
E= vary{y} + bias2 + varLS{ŷ}

14. Our simple example
From statistics, ŷ1 is the best estimate with zero bias
So, the first one may not be the best estimator because of variance (There is a bias/variance tradeoff w.r.t. l)

15. Bayesian approach (1) Hypotheses :
The average height is close to 180cm:
The height of one individual is Gaussian around the mean:
What is the most probable value of after having seen the learning sample ?

16. Bayesian approach (2) Bayes theorem and P(LS) is constant
Independence of the learning cases

17. Regression problem – full (1)
Actually, we want to find a function ŷ(x) of several inputs => average over the whole input space:
The error becomes:
Over all learning sets:

18. Regression problem – full (2)
ELS{Ey|x{(y-ŷ(x))^2}}=Noise(x)+Bias2(x)+Variance(x)
Noise(x) = Ey|x{(y-hB(x))2}
Quantifies how much y varies from hB(x) = Ey|x{y}, the Bayes model.
Bias2(x) = (hB(x)-ELS{ŷ(x)})2:
Measures the error between the Bayes model and the average model.
Variance(x) = ELS{(ŷ(x)-ELS{ŷ(x))2} :
Quantify how much ŷ(x) varies from one learning sample to another.

19. Illustration (1) Problem definition:
One input x, uniform random variable in [0,1]
y=h(x)+ε where εN(0,1)
h(x)=Ey|x{y} y x

20. Illustration (2) Low variance, high bias method  underfitting
ELS{ŷ(x)}

21. Illustration (3) Low bias, high variance method  overfitting
ELS{ŷ(x)}

22. Illustration (4)
No noise doesn’t imply no variance (but less variance)
ELS{ŷ(x)}

23. Classification problems (1)
The mean misclassification error is:
The best possible model is the Bayes model:
The “average” model is:
Unfortunately, there is no such decomposition of the mean misclassification error into a bias and a variance terms.
Nevertheless, we observe the same phenomena

24. Classification problems (2)
A full decision tree
One test node 1
LS1 1
LS2

25. Classification problems (3)
Bias  systematic error component (independent of the learning sample)
Variance  error due to the variability of the model with respect to the learning sample randomness
There are errors due to bias and errors due to variance
One test node
Full decision tree 1 1

26. Content of the presentation
Bias and variance definitions
Parameters that influence bias and variance
Complexity of the model
Complexity of the Bayes model
Noise
Learning sample size
Learning algorithm
Decision/regression tree variance
Bias and variance reduction techniques

27. y(x)=10.sin(π.x1.x2)+20.(x3-0.5)2+10.x4+5.x5+ε,
Illustrative problem
Artificial problem with 10 inputs, all uniform random variables in [0,1]
The true function depends only on 5 inputs:
y(x)=10.sin(π.x1.x2)+20.(x3-0.5)2+10.x4+5.x5+ε,
where ε is a N(0,1) random variable
Experimentations:
ELS  average over 50 learning sets of size 500
Ex,y  average over 2000 cases
 Estimate variance and bias (+ residual error)

28. Complexity of the model
E=bias2+var
var
bias2
Complexity
Usually, the bias is a decreasing function of the complexity, while variance is an increasing function of the complexity.

29. Complexity of the model – neural networks
Error, bias, and variance w.r.t. the number of neurons in the hidden layer

30. Complexity of the model – regression trees
Error, bias, and variance w.r.t. the number of test nodes

31. Complexity of the model – k-NN
Error, bias, and variance w.r.t. k, the number of neighbors

32. Learning problem Complexity of the Bayes model: Noise:
At fixed model complexity, bias increases with the complexity of the Bayes model. However, the effect on variance is difficult to predict.
Noise:
Variance increases with noise and bias is mainly unaffected.
E.g. with (full) regression trees

33. Learning sample size (1)
At fixed model complexity, bias remains constant and variance decreases with the learning sample size. E.g. linear regression

34. Learning sample size (2)
When the complexity of the model is dependant on the learning sample size, both bias and variance decrease with the learning sample size. E.g. regression trees

35. Learning algorithms – linear regression
Method
Err2
Bias2+Noise
Variance
Linear regr.
7.0
6.8
0.2
k-NN (k=1)
15.4 5
10.4
k-NN (k=10)
8.5
7.2
1.3
MLP (10)
2.0
1.2
0.8
MLP (10 – 10)
4.6
1.4
3.2
Regr. Tree
10.2
3.5
6.7
Very few parameters : small variance
Goal function is not linear : high bias

36. Learning algorithms – k-NN
Method
Err2
Bias2+Noise
Variance
Linear regr.
7.0
6.8
0.2
k-NN (k=1)
15.4 5
10.4
k-NN (k=10)
8.5
7.2
1.3
MLP (10)
2.0
1.2
0.8
MLP (10 – 10)
4.6
1.4
3.2
Regr. Tree
10.2
3.5
6.7
Small k : high variance and moderate bias
High k : smaller variance but higher bias

37. Learning algorithms - MLP
Method
Err2
Bias2+Noise
Variance
Linear regr.
7.0
6.8
0.2
k-NN (k=1)
15.4 5
10.4
k-NN (k=10)
8.5
7.2
1.3
MLP (10)
2.0
1.2
0.8
MLP (10 – 10)
4.6
1.4
3.2
Regr. Tree
10.2
3.5
6.7
Small bias
Variance increases with the model complexity

38. Learning algorithms – regression trees
Method
Err2
Bias2+Noise
Variance
Linear regr.
7.0
6.8
0.2
k-NN (k=1)
15.4 5
10.4
k-NN (k=10)
8.5
7.2
1.3
MLP (10)
2.0
1.2
0.8
MLP (10 – 10)
4.6
1.4
3.2
Regr. Tree
10.2
3.5
6.7
Small bias, a (complex enough) tree can approximate any non linear function
High variance

39. Content of the presentation
Bias and variance definition
Parameters that influence bias and variance
Decision/regression tree variance
Bias and variance reduction techniques

40. Decision/regression tree variance (1)
DT/RT are among the machine learning method that present the highest variance. Even a small change of the learning sample can result in a very different tree.
Even small trees have a high variance
Method E
Bias
Variance
k-NN (k=10)
8.5
7.2
1.3
MLP (10 – 10)
4.6
1.4
3.2
RT, no test
25.5
25.4
0.1
RT, 1 test
19.0
17.7
RT, 3 tests
14.8
11.1
3.7
RT, full (250 tests)
10.2
3.5
6.7

41. Decision/regression tree variance (2)
Possible sources of variance:
Discretization of numerical attributes
The selected threshold has a high variance (see next slide).
Structure choice
Sometimes, attribute scores are very close.
Estimation at leaf nodes
Because of the recursive partitioning, prediction at leaf nodes are based on very small samples of objects.
Consequences:
sub-optimality in terms of accuracy
questionable interpretability since the parameters can not be trusted

42. Decision/regression tree variance (3)
The discretization thresholds chosen in trees are very unstable
This variance put into question the interpretability
0.61
A1<0,61
0.82
A1<0,82
0.48
A1<0,48 ? A1

43. Content of the presentation
Bias and variance definition
Parameters that influence bias and variance
Decision/regression tree variance
Bias and variance reduction techniques
Introduction
Dealing with the bias/variance tradeoff of one algorithm
Ensemble methods

44. Bias and variance reduction techniques
In the context of a given method:
Adapt the learning algorithm to find the best trade-off between bias and variance.
Not a panacea but the least we can do.
Example: pruning, weight decay.
Ensemble methods:
Change the bias/variance trade-off.
Universal but destroys some features of the initial method.
Example: bagging, boosting.

45. Variance reduction: 1 model (1)
General idea: reduce the ability of the learning algorithm to fit the LS
Pruning
reduces the model complexity explicitly
Early stopping
reduces the amount of search
Regularization
reduce the size of hypothesis space
Weight decay with neural networks consists in penalizing high weight values

46. Variance reduction: 1 model (2)
Optimal fitting
E=bias2+var
var
bias2
Fitting
Selection of the optimal level of fitting
a priori (not optimal)
by cross-validation (less efficient): Bias2  error on the learning set, E  error on an independent test set

47. Variance reduction: 1 model (3)
Examples:
Post-pruning of regression trees
Early stopping of MLP by cross-validation
Method E
Bias
Variance
Full regr. Tree (250)
10.2
3.5
6.7
Pr. regr. Tree (45)
9.1
4.3
4.8
Full learned MLP
4.6
1.4
3.2
Early stopped MLP
3.8
1.5
2.3
As expected, variance decreases but bias increases

48. Ensemble methods
Combine the predictions of several models built with a learning algorithm in order to improve with respect to the use of a single model
Two important families:
Averaging techniques
Grow several models independantly and simply average their predictions
Ex: bagging, random forests
Decrease mainly variance
Boosting type algorithms
Grows several models sequentially
Ex: Adaboost, MART
Decrease mainly bias

49. ELS{Err(x)}=Ey|x{(y-hB(x))2}+ (hB(x)-ELS{ŷ(x)})2+ELS{(ŷ(x)-ELS{ŷ(x))2}
Bagging (1)
ELS{Err(x)}=Ey|x{(y-hB(x))2}+ (hB(x)-ELS{ŷ(x)})2+ELS{(ŷ(x)-ELS{ŷ(x))2}
Idea : the average model ELS{ŷ(x)} has the same bias as the original method but zero variance
Bagging (Bootstrap AGGregatING) :
To compute ELS{ŷ (x)}, we should draw an infinite number of LS (of size N)
Since we have only one single LS, we simulate sampling from nature by bootstrap sampling from the given LS
Bootstrap sampling = sampling with replacement of N objects from LS (N is the size of LS)

50. Bagging (2) LS LS1 LS2 LST x ŷ1(x) ŷ2(x) ŷT(x)
In regression: ŷ(x) = 1/k.(ŷ1(x)+ŷ2(x)+…+ŷT(x))
In classification: ŷ(x) = the majority class in {ŷ1(x),…,ŷT(x)}

51. Bagging (3)
Usually, bagging reduces very much the variance without increasing too much the bias.
Application to regression trees
Method E
Bias
Variance
3 Test regr. Tree
14.8
11.1
3.7
Bagged (T=25)
11.7
10.7
1.0
Full regr. Tree
10.2
3.5
6.7
5.3
3.8
1.5
Strong variance reduction without increasing the bias (although the model is much more complex than a single tree)

52. Bagging (4) y x

53. Other averaging techniques
Perturb and Combine paradigm:
Perturb the data or the learning algorithm to obtain several models that are good on the learning sample.
Combine the predictions of these models
Usually, these methods decrease the variance (because of averaging) but (slightly) increase the bias (because of the perturbation)
Examples:
Bagging perturbs the learning sample.
Learn several neural networks with random initial weights
Random forests.
Method E
Bias
Variance
MLP (10-10)
4.6
1.4
3.2
Average of 10 MLPs
2.0
0.6

54. Random forests (1)
Perturb and combine algorithm specifically designed for trees
Combine bagging and random attribute subset selection:
Build the tree from a bootstrap sample
Instead of choosing the best split among all attributes, select the best split among a random subset of k attributes
(= bagging when k is equal to the number of attributes)
There is a bias/variance tradeoff with k: The smaller k, the greater the reduction of variance but also the higher the increase of bias

55. Random forests (2) Application to our illustrative problem:
Other advantage: it decreases computing times with respect to bagging since only a subset of all attributes needs to be considered when splitting a node.
Method E
Bias
Variance
Full regr. Tree
10.2
3.5
6.7
Bagging (k=10)
5.3
3.8
1.5
Random Forests (k=7)
4.8
1.0
Random Forests (k=5)
4.9
4.0
0.9
Random Forests (k=3)
5.6
4.7
0.8

56. Boosting methods (1)
The motivation of boosting is to combine the ouputs of many « weak » models to produce a powerful ensemble of models.
Weak model = a model that has a high bias (strictly, in classification, a model slightly better than random guessing)
Differences with previous ensemble methods:
Models are built sequentially on modified versions of the data
The predictions of the models are combined through a weighted sum/vote

57. Boosting methods (2) LS LS1 LST … LS2 x
In regression: ŷ(x) = b1.ŷ1(x)+ b2.ŷ2(x)+…+ bT.ŷT(x))
In classification: ŷ(x) = the majority class in {ŷ1(x),…,ŷT(x)} according to the weights {b1,b2,…,bT}
ŷ1(x)
ŷ2(x)
ŷT(x)

58. Adaboost (1)
Assume that the learning algorithm accepts weighted objects
This is the case of many learning algorithms:
With trees, simply take into account the weights when counting objects
In neural networks, minimize the weighted squared error
At each step, adaboost increases the weights of cases from the learning sample misclassified by the last model
Thus, the algorithm focuses on the difficult cases from the learning sample
In the weighted majority vote, adaboost gives higher influence to the more accurate models

59. å å Adaboost (2) w ¬ w × exp[ b × I ( y ¹ ( x ))] w I ( y ¹ ( x )) err
Input: a learning algorithm and a learning sample {(xi,yi): i=1,…,N}
Initialize the weights wi=1/N, i=1,…,N
For t=1 to T
Build a model ŷt(x) with the learning algorithm using weights wi
Compute the weighted error:
Compute bt=log((1-errt)/errt))
Change weights: å w I ( y ŷ ( x )) i i t i
err = i å t w i i w w ×
exp[ b × I ( y ŷ ( x
))] i i m i t i

60. MART (multiple additive regression trees)
MART is a boosting algorithm for regression
Input: a learning sample {(xi,yi): i=1,…,N}
Initialize
ŷ0(x) = 1/N åi yi ; ri=yi, i=1,…,N
For t=1 to T:
For i=1 to N, compute the residuals
ri  ri -ŷt-1(xi)
Build a regression tree from the learning sample {(xi,ri): i=1,…,N}
Return the model ŷ(x)= ŷ0(x)+ŷ1(x)+…+ŷT(x)

61. Boosting methods
Adaboost and MART are only two boosting variants. There are many other boosting type algorithms.
Boosting decision/regression trees improves their accuracy often dramatically. However, boosting is more sensible to noise than averaging techniques (overfitting).
For boosting to work, the models need not to be perfect on the learning sample. With trees, there are two possible strategies:
Use pruned trees (pre-pruned or post-pruned by cross-validation)
Limit the number of tree tests (and split first the most impure nodes)
 there is again a bias/variance tradeoff with respect to the tree size.

62. Experiment with MART On our illustrative problem:
Boosting reduces the bias but increases the variance. However, with respect to full trees, it decreases both bias and variance.
Method E
Bias
Variance
Full regr. Tree
10.2
3.5
6.7
Regr. Tree with 1 test
18.9
17.8
1.1
+ MART (T=50)
5.0
3.1
1.9
+ Bagging (T=50)
17.9
17.3
0.6
Regr. Tree with 5 tests
11.7
8.8
2.9
6.4
1.7
4.7
9.1
8.7
0.4

63. Interpretability and efficiency of ensembles
Since we average several models, we loose interpretability and efficiency which are two of the main advantages of decision/regression trees
However,
We still can use the ensembles to compute variable importance by averaging over all trees. Actually, this even stabilizes the estimates.
Averaging techniques can be parallelized and boosting type algorithm uses smaller trees. So, the increase of computing times is not so detrimental.

64. Experiments on Golub’s microarray data
72 objects, 7129 numerical attributes (gene expressions), 2 classes (ALL and AL)
Leave-one-out error with several variants
Variable importance with boosting
Method
Error
1 decision tree
22.2% (16/72)
Random forests (k=85,T=500)
9.7% (7/72)
Extra-trees (sth=0.5, T=500)
5.5% (4/72)
Adaboost (1 test node, T=500)
1.4% (1/72)

65. Conclusion (1)
The notions of bias and variance are very useful to predict how changing the (learning and problem) parameters will affect the accuracy. E.g. this explains why very simple methods can work much better than more complex ones on very difficult tasks
Variance reduction is a very important topic:
To reduce bias is easy, but to keep variance low is not as easy.
Especially in the context of new applications of machine learning to very complex domains: temporal data, biological data, Bayesian networks learning, text mining…
All learning algorithms are not equal in terms of variance. Trees are among the worst methods from this criterion

66. Conclusion (2)
Ensemble methods are very effective techniques to reduce bias and/or variance. They can transform a not so good method to a competitive method in terms of accuracy.
Adaboost with trees is considered as one of the best « off-the-shelve » classification method.
Interpretability of the model and efficiency of the method are difficult to preserve if we want to reduce variance significantly.
There are other ways to tackle the variance/overfitting problem, e.g.:
Bayesian approaches (related to averaging techniques)
Support vector machines (they maintain a low variance by maximizing the classifiction margin)

67. References About bias and variance:
Neural networks and the bias/variance dilemma, S.Geman et al., Neural computation 4, 1(1992), 1-58
Neural networks for statistical pattern recognition, C.M.Bishop, Oxfored University Press, 1994
The elements of statistical learning, T.Hastie et al., Springer, 2001
Contribution to decision tree induction: bias/variance tradeoff and time series classification, P.Geurts, Phd Thesis, 2002
About ensemble methods:
Bagging predictors, L.Breiman, Machine learning, 24, 1996
A decision theoretic generalization of on-line learning and an application to boosting, Y.Freund and R.Schapire, Journal of Computer and Science Systems, 1995
Random Forests, L.Breiman, Machine learning, 45, 2001
Ensemble methods in machine learning, T.Dietterich, First international workshop on multiple classifier systems, 2000
An introduction to boosting and leveraging, R.Meir and G.Ratsch, Advanced Lectures on Machine Learning, Springer, 2003

68. Softwares Random forests Boosting:
R package randomForest
Boosting:
See