Recent developments in tree induction for KDD « Towards soft tree induction »

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Recent developments in tree induction for KDD

« Towards soft tree induction »

Louis WEHENKELUniversity of Liège – Belgium

Department of Electrical and Computer Engineering

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A. Supervised learning (notation)

x = (x1,…,xm) vector of input variables (numerical and/or symbolic)

y single output variable Symbolic : classification problem Numeric : regression problem

LS = ((x1,y1),…,(xN,yN)), sample of I/O pairs Learning (or modeling) algorithm

Mapping from sample sp. to hypothesis sp. H Say : y = f(x) + e , where ‘e’ = modeling error « Guess » ‘fLS’ in so as to minimize ‘e’

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Statistical viewpoint x and y are random variables distributed

according to p(x,y) LS is distributed according to pN(x,y) fLS is a random function (selected in H)

e(x) = y – fLS(x) is also a random variable

Given a ‘metric’ to measure the error we can define the best possible model (Bayes model) Regression : fB(x) = E(y|x) Classification : fB(x) = argmaxy P(y|x)

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B. Crisp decision trees (what is it ?)

is big

is small is very big

Yes No

Yes No

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B. Crisp decision trees (what is it ?)

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Tree induction (Overview)

Growing the tree (uses GS, a part of LS) Top down (until all nodes are closed) At each step

Select open node to split (best first, greedy approach) Find best input variable and best question If node can be purified split, otherwise close the node

Pruning the tree (uses PS, rest of LS) Bottom up (until all nodes are contracted) At each step

Select test node to contract (worst first, greedy…) Contract and evaluate

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Tree GrowingDemo : Titanic database Comments

Tree growing is a local process Very efficient Can select relevant input variables Cannot determine appropriate tree shape (Just like real trees…)

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Tree PruningStrategy

To determine appropriate tree shape let tree grow too big (allong all branches), and then reshape it by pruning away irrelevant parts

Tree pruning uses global criterion to determine appropriate shape

Tree pruning is even faster than growing Tree pruning avoids overfitting the data

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Growing – Pruning (graphically)

Tree complexity

Error (GS / PS)

Growing

OverfittingUnderfitting

Final tree

Pruning

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C. Soft trees (what is it ?)

Generalization of crisp trees using continuous splits and aggregation of terminal node predictions

)(x

1y 2y

)(1ˆ)(ˆ)(ˆ ,2,1 xyxyxy

)(, x

x

0

1

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Soft trees (discussion)

Each split is defined by two parameters Position , and width of transition region

Generalize decision/regression trees into a continuous and differentiable model w.r.t. the model parameters Test nodes : jj Terminal nodes : i

Other names (of similar models) Fuzzy trees, continuous trees Tree structured (neural, bayesian) networks Hierarchical models

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Soft trees (Motivations)

Improve performance (w.r.t. crisp trees) Use of a larger hypothesis space Reduced variance and bias Improved optimization (à la backprop)

Improve interpretability More « honest » model Reduced parameter variance Reduced complexity

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D. Plan of the presentation

Bias/Variance tradeoff (in tree induction)

Main techniques to reduce variance

Why soft trees have lower variance

Techniques for learning soft trees

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Concept of variance Learning sample is random Learned model is function of the sample

Model is also random variance Model predictions have variance Model structure / parameters have variance

Variance reduces accuracy and interpretability

Variance can be reduced by various ‘averaging or smoothing’ techniques

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Theoretical explanation Bias, variance and residual error

Residual error Difference between output variable and the best possible model

(i.e. error of the Bayes model) Bias

Difference between the best possible model and the average model produced by algorithm

Variance Average variability of model around average model

Expected error2 : res2+bias2+var

NB: these notions depend on the ‘metric’ used for measuring error

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Regression (locally, at point x)

Find y’=f(x) such that Ey|x{err(y,y’)} is minimum, where err is an error measure.

Usually, err = squared error = (y- y’)2

f(x)=Ey|x{y} minimizes the error at every point x Bayes model is the conditional expectation

y

p(y|x)

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Learning algorithm (1)

Usually, p(y|x) is unknown Use LS = ((x1,y1),…,(xN,yN)), and a learning

algorithm to choose hypothesis in ŷLS(x)=f(LS,x)

At each input point x, the prediction ŷLS(x) is a random variable

Distribution of ŷLS(x) depends on sample size N and on the learning algorithm used

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Learning algorithm (2)

Since LS is randomly drawn, estimation ŷ(x) is a random variable

ŷ

pLS (ŷ(x))

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Good learning algorithm

A good learning algorithm should minimize the average (generalization) error over all learning sets

In regression, the usual error is the mean squared error. So we want to minimize (at each point x)

Err(x)=ELS{Ey|x{(y-ŷLS(x))2}}

There exists a useful additive decomposition of this error into three (positive) terms

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Bias/variance decomposition (1)

Err(x)= Ey|x{(y- Ey|x{y})2} + …

Ey|x{y} = arg miny’ Ey|x{(y- y’)2}} =Bayes model

vary|x{y} = residual error = minimal error

y

vary|x{y}

Ey|x{y}

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Err(x) = vary|x{y} + (Ey|x{y}-ELS{ŷ(x)})2 + …

ELS{ŷ(x)} = average model (w.r.t. LS)

bias2(x) = error between Bayes and average model

ŷEy|x{y} ELS{ŷ(x)}

bias2(x)

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Err(x)= vary|x{y} + bias2(x) + ELS{(ŷ(x)-ELS{ŷ(x)})2}

varLS{ŷ(x)} = variance

ŷ

varLS{ŷ}

ELS{ŷ}

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Local error decomposition Err(x) = vary|x{y} + bias2(x) + varLS{ŷ(x)}

Global error decomposition (take average w.r.t. p(x))EX{Err(x)} = EX{vary|x{y}} + EX{bias2(x)} + EX{varLS{ŷ(x)}}

ŷEy|x{y} ELS{ŷ(x)}

bias2(x)

vary|x{y} varLS{ŷ(x)}

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Illustration (1)

Problem definition: One input x, uniform random variable in [0,1] y=h(x)+ε where εN(0,1)

h(x)=Ey|x{y}

x

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Illustration (2)

Small variance, high bias method

ELS{ŷ(x)}

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Illustration (3)

Small bias, high variance method

ELS{ŷ(x)}

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Illustration (Methods comparison)

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

Experimentation: ELS average over 50 learning sets of size 500 Ex,y average over 2000 cases Estimate variance and bias (+ residual error)

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Illustration (Linear regression)

Very few parameters : small variance Goal function is not linear : high bias

Method Err2 Bias2+Noise Variance

Linear regr. 7.0 6.8 0.2k-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

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Illustration (k-Nearest Neighbors)

Small k : high variance and moderate bias High k : smaller variance but higher bias


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 1.3MLP (10) 2.0 1.2 0.8

MLP (10 – 10) 4.6 1.4 3.2

Regr. Tree 10.2 3.5 6.7

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Illustration (Multilayer Perceptrons)

Small bias Variance increases with the model complexity



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 3.2Regr. Tree 10.2 3.5 6.7

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Illustration (Regression trees)

Small bias, a (complex enough) tree can approximate any non linear function

High variance (see later)



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

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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.

Wrapper techniques: Change the bias/variance trade-off. Universal but destroys some features of the initial

method. Example: bagging.

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Variance reduction: 1 model (1)

General idea: reduce the ability of the learning algorithm to over-fit the LS Pruning

reduces the model complexity explicitly Early stopping

reduces the amount of search Regularization

reduce the size of hypothesis space

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Bias2 error on the learning set, E error on an independent test set

Selection of the optimal level of tuning a priori (not optimal) by cross-validation (less efficient)

E=bias2+var

bias2

var

Fitting

Optimal fitting

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As expected, reduces variance and increases bias

Examples: Post-pruning of regression trees Early stopping of MLP by cross-validation

Method E Bias Variance

Full regr. Tree (488) 10.2 3.5 6.7

Pr. regr. Tree (93) 9.1 4.3 4.8

Full learned MLP 4.6 1.4 3.2

Early stopped MLP 3.8 1.5 2.3

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Variance reduction: bagging (1)

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)

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LS

LS1 LS2 LSk

ŷ1(x) ŷ2(x) ŷk(x)

ŷ(x) = 1/k.(ŷ1(x)+ŷ2(x)+…+ŷk(x))

x

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Application to regression trees

Strong variance reduction without increasing bias (although the model is much more complex than a single tree)


3 Test regr. Tree 14.8 11.1 3.7

Bagged 11.7 10.7 1.0

Full regr. Tree 10.2 3.5 6.7

Bagged 5.3 3.8 1.5

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Dual bagging (1)

Instead of perturbing learning sets to obtain several predictions, directly perturb the test case at the prediction stage

Given a model ŷ(.) and a test case x: Form k attribute vectors by adding Gaussian noise

to x: {x+ε1, x+ε2, …, x+εk}. Average the predictions of the model at these

points to get the prediction at point x:

1/k.(ŷ(x+ε1)+ŷ(x+ε2)+…+ŷ(x+εk)

Noise level λ (variance of Gaussian noise) selected by cross-validation

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Dual bagging (2)

With regression trees:

Smooth the function ŷ(.). Too much noise increases bias

there is a (new) trade-off between bias and variance

Noise level E Bias Variance

0.0 10.2 3.5 6.7

0.2 6.3 3.5 2.8

0.5 5.3 4.4 0.9

2.0 13.3 13.1 0.2

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Dual bagging (classification trees)

λ = 0 error =3.7 %

λ = 1.5 error =4.6 %

λ = 0.3 error =1.4 %

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Variance in tree induction Tree induction is among the ML methods of highest variance

(together with 1-NN) Main reason

Generalization is local Depends on small parts of the learning set

Sources of variance: Discretization of numerical attributes (60 %)

The selected thresholds have a high variance Structure choice (10 %)

Sometimes, attribute scores are very close Estimation at leaf nodes (30 %)

Because of the recursive partitioning, prediction at leaf nodes is based on very small samples of objects

Consequences: Questionable interpretability and higher error rates

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Threshold variance (1)

Test on numerical attributes : [a(o)<ath]

Discretization: find ath which minimizes score Classification: maximize information Regression: minimize residual variance

Score

ath

a(o)

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E m p iric a l o p tim a l th re sh o ld d is trib u tion

N =10 0

S co re c u rv es

800. 900. 1000. 1100. 1200.-- - Thres hold - --

0.0

0.1

0.2

0.3

0.4

Sc ore/N b o f c as es

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E m p irica l o p tim a l th re sh o ld d is trib u tion

N =1 00 0

S co re cu rve s

800. 900. 1000. 1100. 1200.- - - Thres hold -- -

0.0

5.e-2

0.1

0.15

0.2

0.25

S c ore/N b of c as es

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Tree variance DT/RT are among the machine learning

methods which present the highest variance


RT, no test 25.5 25.4 0.1

RT, 1 test 19.0 17.7 1.3

RT, 3 tests 14.8 11.1 3.7

RT, full (250 tests) 10.2 3.5 6.7

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DT variance reduction Pruning:

Necessary to select the right complexity Decreases variance but increases bias : small effect on accuracy

Threshold stabilization: Smoothing of score curves, bootstrap sampling… Reduces parameter variance but has only a slight effect on

accuracy and prediction variance Bagging:

Very efficient at reducing variance But jeopardizes interpretability of trees and computational efficiency

Dual bagging: In terms of variance reduction, similar to bagging Much faster and can be simulated by soft trees

Fuzzy tree induction Build soft trees in a full fledged approach

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Dual tree bagging = Soft treesReformulation of dual bagging as an

explicit soft tree propagation algorithmAlgorithms

Forward-backward propagation in soft trees

Softening of thresholds during learning stage

Some results

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Dual bagging = soft thresholds x+ε<xth sometimes left, sometimes right Multiple ‘crisp’ propagations can be ‘replaced’

by one ‘soft’ propagation E.g. if ε has uniform pdf in [ath- /2,ath+ /2]

then probability of right propagation is as follows

ath

TSleft TSright

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Learning of values

Use of an independent ‘validation’ set and bisection search

One single value can be learned very efficiently (amounts to 10 full tests of a DT/RT on the validation set)

Combination of several values can also be learned with the risk of overfitting (see fuzzy tree induction, in what follows)

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Some results with dual bagging

0

5

10

15

20

25Pe % DT

DT + Dual BaggBaggingBagging + Dual

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Fuzzy tree inductionGeneral ideasLearning Algorithm

Growing Refitting Pruning Backfitting

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General IdeasObviously, soft trees have much lower

variance than crisp trees In the « Dual Bagging » approach,

attribute selection is carried out in a cloassical way, then tests are softened in a post-processing stage

Might be more effective to combine the two methods

Fuzzy tree induction

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Soft trees Samples are handled as fuzzy subsets

Each observation belongs to such a FS with a certain membership degree

SCORE measure is modified Objects are weighted by their membership degree

Output y Denotes the membership degree to a class

Goal of Fuzzy tree induction Provide a smooth model of ‘y’ as a function of the

input variables

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Fuzzy discretization Same as fuzzification Carried out locally, at the tree growing stage

At each test node On the basis of local fuzzy sub-training set

Select attribute, together with discriminator so as to maximize local SCORE

Split in soft way and proceed recursively

Criteria for SCORE Minimal residual variance Maximal (fuzzy) information quantity Etc

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Attaching labels to leaves Basically, for each terminal node, we need to

determine a local estimate ŷi of y During intermediate steps

Use average of ‘y’ in local sub-learning set Direct computation

Refitting of the labels Once the full tree has been grown and at each step of

pruning Determine all values simultaneously To minimize Square Error Amounts to a linear least squares problem Direct solution

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Refitting (Explanation)

A leaf corresponds to a basis function i(x) Product of discriminators encountered on the path

from root Tree prediction is equivalent to a weighted

average of these basis functions y(x) = ŷ1 * 1(x) + ŷ2 * 2(x) + … + ŷk * k(x) the weights ŷi are the labels attached to the

terminal nodes

Refitting amounts to tune the ŷi parameters to minimize square error on training set

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Tree growing and pruningGrow treeRefit leaf labelsPrune tree, while refitting at each stage

leaf labelsTest sequence of pruned trees on

validation setSelect best pruning level

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Backfitting (1)

After growing and pruning, the fuzzy tree structure has been determined

Leaf labels are globally optimal, but not the parameters of the discriminators (tuned locally)

Resulting model has 2 parameters per test node, and 1 parameter per terminal node

The output (and hence Mean square error) of the fuzzy tree is a smooth function of these parameters

The parameters can be optimized, by using a standard LSE technique, e.g. Levenberg-Marquardt

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Backfitting (2)

How to compute the derivatives needed by nonlinear optimization technique Use a modified version of backpropagation to compute

derivates with respect to parameters Yields an efficient algorithm (linear in the size of the tree)

Backfitting starts from tree produced after growing and pruning Already a good approximation of a local optimum Only a small number of iterations are necessary to backfit

Backfitting may also lead to overfitting…

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Summary and conclusions Variance is the problem number one in

decision/regression tree induction It is possible to reduce variance significantly

Bagging and/or tree softening Soft trees have the advantage of preserving interpretability

and computational efficiency Two approaches have been presented to get soft trees

Dual bagging Generic approach Fast and simple Best approach for very large databases

Fuzzy tree induction Similar to ANN type of model, but (more) interpretable Best approach for small learning sets (probably)

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Some references for further reading Variance evaluation/reduction, bagging

Contact : Pierre GEURTS (PhD student) [email protected]

Papers Discretization of continuous attributes for supervised learning -

Variance evaluation and variance reduction. (Invited) L. Wehenkel. Proc. of IFSA'97, International Fuzzy Systems Association World Congress, Prague, June 1997, pp. 381--388.

Investigation and Reduction of Discretization Variance in Decision Tree Induction. Pierre GEURTS and Louis WEHENKEL, Proc. of ECML’2000

Some Enhancements of Decision Tree Bagging. Pierre GEURTS, Proc. of PKDD’2000

Dual Perturb and Combine Algorithm. Pierre GEURTS, Proc. of AI and Statistics 2001.

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See also www.montefiore.ulg.ac.be/services/stochastic/

Fuzzy/soft tree induction Contact : Cristina OLARU (PhD student)

[email protected] Papers

Automatic induction of fuzzy decision trees and its application to power system security assessment. X. Boyen, L. Wehenkel, Int. Journal on Fuzzy Sets and Systems, Vol. 102, No 1, pp. 3-19, 1999.

On neurofuzzy and fuzzy decision trees approaches. C. Olaru, L. Wehenkel. (Invited) Proc. of IPMU'98, 7th Int. Congr. on Information Processing and Management of Uncertainty in Knowledge based Systems, 1998.

Recent developments in tree induction for KDD « Towards soft tree induction »

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