gradient boosted regression trees - uliege · gradient boosted regression trees peter prettenhofer...

Gradient Boosted Regression Trees

scikit

Peter Prettenhofer (@pprett)DataRobot

Gilles Louppe (@glouppe)Universite de Liege, Belgium

https://twitter.com/pprett

https://twitter.com/glouppe

Motivation

Outline

1 Basics

2 Gradient Boosting

3 Gradient Boosting in scikit-learn

4 Case Study: California housing

About us

Peter

• @pprett

• Python & ML ∼ 6 years

• sklearn dev since 2010

Gilles

• @glouppe

• PhD student (Liege,Belgium)

• sklearn dev since 2011Chief tree hugger

https://twitter.com/pprett

https://twitter.com/glouppe

Outline

1 Basics

2 Gradient Boosting



Machine Learning 101

• Data comes as...

• A set of examples {(xi , yi )|0 ≤ i < n samples}, with

• Feature vector x ∈ Rn features, and

• Response y ∈ R (regression) or y ∈ {−1, 1} (classification)

• Goal is to...

• Find a function y = f (x)

• Such that error L(y , y) on new (unseen) x is minimal

Classification and Regression Trees [Breiman et al, 1984]

MedInc <= 5.04

MedInc <= 3.07 MedInc <= 6.82

AveRooms <= 4.31 AveOccup <= 2.37

1.62 1.16 2.79 1.88

AveOccup <= 2.74 MedInc <= 7.82

3.39 2.56 3.73 4.57

sklearn.tree.DecisionTreeClassifier|Regressor

Function approximation with Regression Trees

0 2 4 6 8 10x

8

6

4

2

0

2

4

6

8

10y

ground truthRT max_depth=1RT max_depth=3RT max_depth=20

Deprecated

• Nowadays seldom used alone

• Ensembles: Random Forest, Bagging, or Boosting(see sklearn.ensemble)

Outline

1 Basics

2 Gradient Boosting



Gradient Boosted Regression Trees

Advantages

• Heterogeneous data (features measured on different scale)

• Supports different loss functions (e.g. huber)

• Automatically detects (non-linear) feature interactions

Disadvantages

• Requires careful tuning

• Slow to train (but fast to predict)

• Cannot extrapolate

Boosting

AdaBoost [Y. Freund & R. Schapire, 1995]

• Ensemble: each member is an expert on the errors of itspredecessor

• Iteratively re-weights training examples based on errors

2 1 0 1 2 3x0

2

1

0

1

2

x1

2 1 0 1 2 3x0

2 1 0 1 2 3x0

2 1 0 1 2 3x0

sklearn.ensemble.AdaBoostClassifier|Regressor

Huge success

• Viola-Jones Face Detector (2001)

• Freund & Schapire won the Godel prize 2003

Gradient Boosting [J. Friedman, 1999]

Statistical view on boosting

• ⇒ Generalization of boosting to arbitrary loss functions

Residual fitting

2 6 10x

2.0

1.5

1.0

0.5

0.0

0.5

1.0

1.5

2.0

2.5

y

Ground truth

2 6 10x

∼

tree 1

2 6 10x

+

tree 2

2 6 10x

+

tree 3

sklearn.ensemble.GradientBoostingClassifier|Regressor

Functional Gradient Descent

Least Squares Regression

• Squared loss: L(yi , f (xi )) = (yi − f (xi ))2

• The residual ∼ the (negative) gradient ∂L(yi , f (xi ))∂f (xi )

Steepest Descent

• Regression trees approximate the (negative) gradient

• Each tree is a successive gradient descent step

4 3 2 1 0 1 2 3 4y−f(x)

0

1

2

3

4

5

6

7

8

L(y,f

(x))

Squared errorAbsolute errorHuber error

4 3 2 1 0 1 2 3 4y ·f(x)

0

1

2

3

4

5

6

7

8

L(y,f

(x))

Zero-one lossLog lossExponential loss

Outline

1 Basics

2 Gradient Boosting



GBRT in scikit-learn

How to use it

>>> from sklearn.ensemble import GradientBoostingClassifier

>>> from sklearn.datasets import make_hastie_10_2

>>> X, y = make_hastie_10_2(n_samples=10000)

>>> est = GradientBoostingClassifier(n_estimators=200, max_depth=3)

>>> est.fit(X, y)

...

>>> # get predictions

>>> pred = est.predict(X)

>>> est.predict_proba(X)[0] # class probabilities

array([ 0.67, 0.33])

Implementation

• Written in pure Python/Numpy (easy to extend).

• Builds on top of sklearn.tree.DecisionTreeRegressor (Cython).

• Custom node splitter that uses pre-sorting (better for shallow trees).

Examplefrom sklearn.ensemble import GradientBoostingRegressor

est = GradientBoostingRegressor(n_estimators=2000, max_depth=1).fit(X, y)

for pred in est.staged_predict(X):

plt.plot(X[:, 0], pred, color=’r’, alpha=0.1)

0 2 4 6 8 10x

8

6

4

2

0

2

4

6

8

10

y

High bias - low variance

Low bias - high variance

ground truthRT max_depth=1RT max_depth=3GBRT max_depth=1

Model complexity & Overfittingtest_score = np.empty(len(est.estimators_))

for i, pred in enumerate(est.staged_predict(X_test)):

test_score[i] = est.loss_(y_test, pred)

plt.plot(np.arange(n_estimators) + 1, test_score, label=’Test’)

plt.plot(np.arange(n_estimators) + 1, est.train_score_, label=’Train’)

0 200 400 600 800 1000n_estimators

0.0

0.5

1.0

1.5

2.0

Err

or

Lowest test error

train-test gap

TestTrain

Regularization

GBRT provides a number of knobs to controloverfitting

• Tree structure

• Shrinkage

• Stochastic Gradient Boosting

Regularization: Tree structure

• The max depth of the trees controls the degree of features interactions

• Use min samples leaf to have a sufficient nr. of samples per leaf.

Regularization: Shrinkage

• Slow learning by shrinking tree predictions with 0 < learning rate <= 1

• Lower learning rate requires higher n estimators

0 200 400 600 800 1000n_estimators

0.0

0.5

1.0

1.5

2.0

Err

or

Requires more trees

Lower test error

Test Train Test learning_rate=0.1

Train learning_rate=0.1

Regularization: Stochastic Gradient Boosting• Samples: random subset of the training set (subsample)

• Features: random subset of features (max features)

• Improved accuracy – reduced runtime

0 200 400 600 800 1000n_estimators

0.0

0.5

1.0

1.5

2.0

Err

or

Even lower test error

Subsample alone does poorly

Train Test Train subsample=0.5, learning_rate=0.1

Test subsample=0.5, learning_rate=0.1

Hyperparameter tuning

1. Set n estimators as high as possible (eg. 3000)

2. Tune hyperparameters via grid search.

from sklearn.grid_search import GridSearchCV

param_grid = {’learning_rate’: [0.1, 0.05, 0.02, 0.01],

’max_depth’: [4, 6],

’min_samples_leaf’: [3, 5, 9, 17],

’max_features’: [1.0, 0.3, 0.1]}

est = GradientBoostingRegressor(n_estimators=3000)

gs_cv = GridSearchCV(est, param_grid).fit(X, y)

# best hyperparameter setting

gs_cv.best_params_

3. Finally, set n estimators even higher and tunelearning rate.

Outline

1 Basics

2 Gradient Boosting



Case Study

California Housing dataset

• Predict log(medianHouseValue)

• Block groups in 1990 census

• 20.640 groups with 8 features(median income, median age, lat,lon, ...)

• Evaluation: Mean absolute erroron 80/20 split

Challenges

• Heterogeneous features

• Non-linear interactions

Predictive accuracy & runtime

Train time [s] Test time [ms] MAE

Mean - - 0.4635Ridge 0.006 0.11 0.2756SVR 28.0 2000.00 0.1888RF 26.3 605.00 0.1620GBRT 192.0 439.00 0.1438

0 500 1000 1500 2000 2500 3000n_estimators

0.0

0.1

0.2

0.3

0.4

0.5

err

or

TestTrain

Model interpretation

Which features are important?

>>> est.feature_importances_

array([ 0.01, 0.38, ...])

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18Relative importance

HouseAge

Population

AveBedrms

Latitude

AveOccup

Longitude

AveRooms

MedInc

Model interpretationWhat is the effect of a feature on the response?

from sklearn.ensemble import partial_dependence import as pd

features = [’MedInc’, ’AveOccup’, ’HouseAge’, ’AveRooms’,

(’AveOccup’, ’HouseAge’)]

fig, axs = pd.plot_partial_dependence(est, X_train, features,

feature_names=names)

1.5 3.0 4.5 6.0 7.5MedInc

0.4

0.2

0.0

0.2

0.4

0.6

Part

ial dependence

2.0 2.5 3.0 3.5 4.0 4.5AveOccup

0.4

0.2

0.0

0.2

0.4

0.6

Part

ial dependence

10 20 30 40 50 60HouseAge

0.4

0.2

0.0

0.2

0.4

0.6

Part

ial dependence

4 5 6 7 8AveRooms

0.4

0.2

0.0

0.2

0.4

0.6

Part

ial dependence

2.0 2.5 3.0 3.5 4.0AveOccup

10

20

30

40

50

House

Age

-0.1

2

-0.0

5

0.0

2

0.09

0.1

6

0.23

Partial dependence of house value on nonlocation featuresfor the California housing dataset

Model interpretation

Automatically detects spatial effects

longitude

lati

tude

-1.54

-1.22

-0.91

-0.60

-0.28

0.03

0.34

0.66

0.97

part

ial dep.

on m

edia

n h

ouse

valu

e

longitudela

titu

de

-0.15

-0.07

0.01

0.09

0.17

0.25

0.33

0.41

0.49

0.57

part

ial dep.

on m

edia

n h

ouse

valu

e

Summary

• Flexible non-parametric classification and regression technique

• Applicable to a variety of problems

• Solid, battle-worn implementation in scikit-learn

Thanks! Questions?

Benchmarks

0.00.20.40.60.81.01.2

Err

or

gbmsklearn-0.15

0.00.51.01.52.02.53.0

Tra

in t

ime

Arc

ene

Bost

on

Calif

orn

ia

Covty

pe

Exam

ple

10

.2

Expedia

Madelo

n

Sola

r

Spam

YahooLT

RC

bio

resp

dataset

0.0

0.2

0.4

0.6

0.8

1.0

Test

tim

e

Tipps & Tricks 1

Input layout

Use dtype=np.float32 to avoid memory copies and fortan layout for slight

runtime benefit.

X = np.asfortranarray(X, dtype=np.float32)

Tipps & Tricks 2

Feature interactionsGBRT automatically detects feature interactions but often explicit interactionshelp.

Trees required to approximate X1 − X2: 10 (left), 1000 (right).

x 0.00.2

0.40.6

0.81.0

y

0.00.2

0.40.6

0.81.0

x -

y

0.3

0.2

0.1

0.0

0.1

0.2

0.3

x 0.00.20.40.60.81.0y

0.00.2

0.40.6

0.81.0

x -

y

1.0

0.5

0.0

0.5

1.0

Tipps & Tricks 3

Categorical variables

Sklearn requires that categorical variables are encoded as numerics. Tree-based

methods work well with ordinal encoding:

df = pd.DataFrame(data={’icao’: [’CRJ2’, ’A380’, ’B737’, ’B737’]})

# ordinal encoding

df_enc = pd.DataFrame(data={’icao’: np.unique(df.icao,

return_inverse=True)[1]})

X = np.asfortranarray(df_enc.values, dtype=np.float32)

gradient boosted regression trees - uliege · gradient boosted regression trees peter prettenhofer...

Documents