A Gentle Introduction to Gradient Boosting
Cheng Li
chengli@ccs.neu.edu
College of Computer and Information Science
Northeastern University
Gradient Boosting
I
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a powerful machine learning algorithm
it can do
I
I
I
I
regression
classification
ranking
won Track 1 of the Yahoo Learning to Rank Challenge
Our implementation of Gradient Boosting is available at
https://github.com/cheng-li/pyramid
Outline of the Tutorial
1 What is Gradient Boosting
2 A brief history
3 Gradient Boosting for regression
4 Gradient Boosting for classification
5 A demo of Gradient Boosting
6 Relationship between Adaboost and Gradient Boosting
7 Why it works
Note: This tutorial focuses on the intuition. For a formal
treatment, see [Friedman, 2001]
What is Gradient Boosting
Gradient Boosting = Gradient Descent + Boosting
Adaboost
Figure: AdaBoost. Source: Figure 1.1 of [Schapire and Freund, 2012]
What is Gradient Boosting
Gradient Boosting = Gradient Descent + Boosting
Adaboost
Figure: AdaBoost. Source: Figure 1.1 of [Schapire and Freund, 2012]
P
I
Fit an additive model (ensemble)
stage-wise manner.
I
In each stage, introduce a weak learner to compensate the
shortcomings of existing weak learners.
I
In Adaboost,“shortcomings” are identified by high-weight data
points.
t
ρt ht (x) in a forward
What is Gradient Boosting
Gradient Boosting = Gradient Descent + Boosting
Adaboost
H(x) =
X
ρt ht (x)
t
Figure: AdaBoost. Source: Figure 1.2 of [Schapire and Freund, 2012]
What is Gradient Boosting
Gradient Boosting = Gradient Descent + Boosting
Gradient Boosting
P
I
Fit an additive model (ensemble)
stage-wise manner.
I
In each stage, introduce a weak learner to compensate the
shortcomings of existing weak learners.
I
In Gradient Boosting,“shortcomings” are identified by
gradients.
I
Recall that, in Adaboost,“shortcomings” are identified by
high-weight data points.
I
Both high-weight data points and gradients tell us how to
improve our model.
t
ρt ht (x) in a forward
What is Gradient Boosting
Why and how did researchers invent Gradient Boosting?
A Brief History of Gradient Boosting
I
Invent Adaboost, the first successful boosting algorithm
[Freund et al., 1996, Freund and Schapire, 1997]
I
Formulate Adaboost as gradient descent with a special loss
function[Breiman et al., 1998, Breiman, 1999]
I
Generalize Adaboost to Gradient Boosting in order to handle
a variety of loss functions
[Friedman et al., 2000, Friedman, 2001]
Gradient Boosting for Regression
Gradient Boosting for Different Problems
Difficulty:
regression ===> classification ===> ranking
Gradient Boosting for Regression
Let’s play a game...
You are given (x1 , y1 ), (x2 , y2 ), ..., (xn , yn ), and the task is to fit a
model F (x) to minimize square loss.
Suppose your friend wants to help you and gives you a model F .
You check his model and find the model is good but not perfect.
There are some mistakes: F (x1 ) = 0.8, while y1 = 0.9, and
F (x2 ) = 1.4 while y2 = 1.3... How can you improve this model?
Gradient Boosting for Regression
Let’s play a game...
You are given (x1 , y1 ), (x2 , y2 ), ..., (xn , yn ), and the task is to fit a
model F (x) to minimize square loss.
Suppose your friend wants to help you and gives you a model F .
You check his model and find the model is good but not perfect.
There are some mistakes: F (x1 ) = 0.8, while y1 = 0.9, and
F (x2 ) = 1.4 while y2 = 1.3... How can you improve this model?
Rule of the game:
I
You are not allowed to remove anything from F or change any
parameter in F .
Gradient Boosting for Regression
Let’s play a game...
You are given (x1 , y1 ), (x2 , y2 ), ..., (xn , yn ), and the task is to fit a
model F (x) to minimize square loss.
Suppose your friend wants to help you and gives you a model F .
You check his model and find the model is good but not perfect.
There are some mistakes: F (x1 ) = 0.8, while y1 = 0.9, and
F (x2 ) = 1.4 while y2 = 1.3... How can you improve this model?
Rule of the game:
I
You are not allowed to remove anything from F or change any
parameter in F .
I
You can add an additional model (regression tree) h to F , so
the new prediction will be F (x) + h(x).
Gradient Boosting for Regression
Simple solution:
You wish to improve the model such that
F (x1 ) + h(x1 ) = y1
F (x2 ) + h(x2 ) = y2
...
F (xn ) + h(xn ) = yn
Gradient Boosting for Regression
Simple solution:
Or, equivalently, you wish
h(x1 ) = y1 − F (x1 )
h(x2 ) = y2 − F (x2 )
...
h(xn ) = yn − F (xn )
Gradient Boosting for Regression
Simple solution:
Or, equivalently, you wish
h(x1 ) = y1 − F (x1 )
h(x2 ) = y2 − F (x2 )
...
h(xn ) = yn − F (xn )
Can any regression tree h achieve this goal perfectly?
Gradient Boosting for Regression
Simple solution:
Or, equivalently, you wish
h(x1 ) = y1 − F (x1 )
h(x2 ) = y2 − F (x2 )
...
h(xn ) = yn − F (xn )
Can any regression tree h achieve this goal perfectly?
Maybe not....
Gradient Boosting for Regression
Simple solution:
Or, equivalently, you wish
h(x1 ) = y1 − F (x1 )
h(x2 ) = y2 − F (x2 )
...
h(xn ) = yn − F (xn )
Can any regression tree h achieve this goal perfectly?
Maybe not....
But some regression tree might be able to do this approximately.
Gradient Boosting for Regression
Simple solution:
Or, equivalently, you wish
h(x1 ) = y1 − F (x1 )
h(x2 ) = y2 − F (x2 )
...
h(xn ) = yn − F (xn )
Can any regression tree h achieve this goal perfectly?
Maybe not....
But some regression tree might be able to do this approximately.
How?
Gradient Boosting for Regression
Simple solution:
Or, equivalently, you wish
h(x1 ) = y1 − F (x1 )
h(x2 ) = y2 − F (x2 )
...
h(xn ) = yn − F (xn )
Can any regression tree h achieve this goal perfectly?
Maybe not....
But some regression tree might be able to do this approximately.
How?
Just fit a regression tree h to data
(x1 , y1 − F (x1 )), (x2 , y2 − F (x2 )), ..., (xn , yn − F (xn ))
Gradient Boosting for Regression
Simple solution:
Or, equivalently, you wish
h(x1 ) = y1 − F (x1 )
h(x2 ) = y2 − F (x2 )
...
h(xn ) = yn − F (xn )
Can any regression tree h achieve this goal perfectly?
Maybe not....
But some regression tree might be able to do this approximately.
How?
Just fit a regression tree h to data
(x1 , y1 − F (x1 )), (x2 , y2 − F (x2 )), ..., (xn , yn − F (xn ))
Congratulations, you get a better model!
Gradient Boosting for Regression
Simple solution:
yi − F (xi ) are called residuals. These are the parts that existing
model F cannot do well.
The role of h is to compensate the shortcoming of existing model
F.
Gradient Boosting for Regression
Simple solution:
yi − F (xi ) are called residuals. These are the parts that existing
model F cannot do well.
The role of h is to compensate the shortcoming of existing model
F.
If the new model F + h is still not satisfactory,
Gradient Boosting for Regression
Simple solution:
yi − F (xi ) are called residuals. These are the parts that existing
model F cannot do well.
The role of h is to compensate the shortcoming of existing model
F.
If the new model F + h is still not satisfactory, we can add another
regression tree...
Gradient Boosting for Regression
Simple solution:
yi − F (xi ) are called residuals. These are the parts that existing
model F cannot do well.
The role of h is to compensate the shortcoming of existing model
F.
If the new model F + h is still not satisfactory, we can add another
regression tree...
We are improving the predictions of training data, is the procedure
also useful for test data?
Gradient Boosting for Regression
Simple solution:
yi − F (xi ) are called residuals. These are the parts that existing
model F cannot do well.
The role of h is to compensate the shortcoming of existing model
F.
If the new model F + h is still not satisfactory, we can add another
regression tree...
We are improving the predictions of training data, is the procedure
also useful for test data?
Yes! Because we are building a model, and the model can be
applied to test data as well.
Gradient Boosting for Regression
Simple solution:
yi − F (xi ) are called residuals. These are the parts that existing
model F cannot do well.
The role of h is to compensate the shortcoming of existing model
F.
If the new model F + h is still not satisfactory, we can add another
regression tree...
We are improving the predictions of training data, is the procedure
also useful for test data?
Yes! Because we are building a model, and the model can be
applied to test data as well.
How is this related to gradient descent?
Gradient Boosting for Regression
Gradient Descent
Minimize a function by moving in the opposite direction of the
gradient.
∂J
θi := θi − ρ
∂θi
Figure: Gradient Descent. Source:
http://en.wikipedia.org/wiki/Gradient_descent
Gradient Boosting for Regression
How is this related to gradient descent?
Loss function L(y , F (x)) =P
(y − F (x))2 /2
We want to minimize J = i L(yi , F (xi )) by adjusting
F (x1 ), F (x2 ), ..., F (xn ).
Notice that F (x1 ), F (x2 ), ..., F (xn ) are just some numbers. We can
treat F (xi ) as parameters and take derivatives
P
∂ i L(yi , F (xi ))
∂L(yi , F (xi ))
∂J
=
=
= F (xi ) − yi
∂F (xi )
∂F (xi )
∂F (xi )
So we can interpret residuals as negative gradients.
yi − F (xi ) = −
∂J
∂F (xi )
Gradient Boosting for Regression
How is this related to gradient descent?
F (xi ) := F (xi ) + h(xi )
F (xi ) := F (xi ) + yi − F (xi )
∂J
F (xi ) := F (xi ) − 1
∂F (xi )
∂J
θi := θi − ρ
∂θi
Gradient Boosting for Regression
How is this related to gradient descent?
For regression with square loss,
residual ⇔ negative gradient
fit h to residual ⇔ fit h to negative gradient
update F based on residual ⇔ update F based on negative gradient
Gradient Boosting for Regression
How is this related to gradient descent?
For regression with square loss,
residual ⇔ negative gradient
fit h to residual ⇔ fit h to negative gradient
update F based on residual ⇔ update F based on negative gradient
So we are actually updating our model using gradient descent!
Gradient Boosting for Regression
How is this related to gradient descent?
For regression with square loss,
residual ⇔ negative gradient
fit h to residual ⇔ fit h to negative gradient
update F based on residual ⇔ update F based on negative gradient
So we are actually updating our model using gradient descent!
It turns out that the concept of gradients is more general and
useful than the concept of residuals. So from now on, let’s stick
with gradients. The reason will be explained later.
Gradient Boosting for Regression
Regression with square Loss
Let us summarize the algorithm we just derived using the concept
of gradients. Negative gradient:
−g (xi ) = −
∂L(yi , F (xi ))
= yi − F (xi )
∂F (xi )
Pn
y
i
start with an initial model, say, F (x) = i=1
n
iterate until converge:
calculate negative gradients −g (xi )
fit a regression tree h to negative gradients −g (xi )
F := F + ρh, where ρ = 1
Gradient Boosting for Regression
Regression with square Loss
Let us summarize the algorithm we just derived using the concept
of gradients. Negative gradient:
−g (xi ) = −
∂L(yi , F (xi ))
= yi − F (xi )
∂F (xi )
Pn
y
i
start with an initial model, say, F (x) = i=1
n
iterate until converge:
calculate negative gradients −g (xi )
fit a regression tree h to negative gradients −g (xi )
F := F + ρh, where ρ = 1
The benefit of formulating this algorithm using gradients is that it
allows us to consider other loss functions and derive the
corresponding algorithms in the same way.
Gradient Boosting for Regression
Loss Functions for Regression Problem
Why do we need to consider other loss functions? Isn’t square loss
good enough?
Gradient Boosting for Regression
Loss Functions for Regression Problem
Square loss is:
X Easy to deal with mathematically
× Not robust to outliers
Outliers are heavily punished because the error is squared.
Example:
yi
0.5
1.2
2
5*
F (xi )
0.6
1.4
1.5
1.7
L = (y − F )2 /2 0.005 0.02 0.125 5.445
Consequence?
Gradient Boosting for Regression
Loss Functions for Regression Problem
Square loss is:
X Easy to deal with mathematically
× Not robust to outliers
Outliers are heavily punished because the error is squared.
Example:
yi
0.5
1.2
2
5*
F (xi )
0.6
1.4
1.5
1.7
L = (y − F )2 /2 0.005 0.02 0.125 5.445
Consequence?
Pay too much attention to outliers. Try hard to incorporate
outliers into the model. Degrade the overall performance.
Gradient Boosting for Regression
Loss Functions for Regression Problem
I
Absolute loss (more robust to outliers)
L(y , F ) = |y − F |
Gradient Boosting for Regression
Loss Functions for Regression Problem
I
Absolute loss (more robust to outliers)
L(y , F ) = |y − F |
I
Huber loss (more robust to outliers)
(
1
(y − F )2
|y − F | ≤ δ
L(y , F ) = 2
δ(|y − F | − δ/2) |y − F | > δ
Gradient Boosting for Regression
Loss Functions for Regression Problem
I
Absolute loss (more robust to outliers)
L(y , F ) = |y − F |
I
Huber loss (more robust to outliers)
(
1
(y − F )2
|y − F | ≤ δ
L(y , F ) = 2
δ(|y − F | − δ/2) |y − F | > δ
yi
F (xi )
Square loss
Absolute loss
Huber loss(δ = 0.5)
0.5
0.6
0.005
0.1
0.005
1.2
1.4
0.02
0.2
0.02
2
1.5
0.125
0.5
0.125
5*
1.7
5.445
3.3
1.525
Gradient Boosting for Regression
Regression with Absolute Loss
Negative gradient:
−g (xi ) = −
∂L(yi , F (xi ))
= sign(yi − F (xi ))
∂F (xi )
Pn
y
i
start with an initial model, say, F (x) = i=1
n
iterate until converge:
calculate gradients −g (xi )
fit a regression tree h to negative gradients −g (xi )
F := F + ρh
Gradient Boosting for Regression
Regression with Huber Loss
Negative gradient:
∂L(yi , F (xi ))
−g (xi ) = −
∂F (xi )
(
yi − F (xi )
|yi − F (xi )| ≤ δ
=
δsign(yi − F (xi )) |yi − F (xi )| > δ
Pn
y
i
start with an initial model, say, F (x) = i=1
n
iterate until converge:
calculate negative gradients −g (xi )
fit a regression tree h to negative gradients −g (xi )
F := F + ρh
Gradient Boosting for Regression
Regression with loss function L: general procedure
Give any differentiable loss function L
Pn
y
i
start with an initial model, say F (x) = i=1
n
iterate until converge:
i ,F (xi ))
calculate negative gradients −g (xi ) = − ∂L(y
∂F (xi )
fit a regression tree h to negative gradients −g (xi )
F := F + ρh
Gradient Boosting for Regression
Regression with loss function L: general procedure
Give any differentiable loss function L
Pn
y
i
start with an initial model, say F (x) = i=1
n
iterate until converge:
i ,F (xi ))
calculate negative gradients −g (xi ) = − ∂L(y
∂F (xi )
fit a regression tree h to negative gradients −g (xi )
F := F + ρh
In general,
negative gradients 6⇔ residuals
We should follow negative gradients rather than residuals. Why?
Gradient Boosting for Regression
Negative Gradient vs Residual: An Example
Huber loss
(
L(y , F ) =
1
2 (y
− F )2
|y − F | ≤ δ
δ(|y − F | − δ/2) |y − F | > δ
Update by Negative Gradient:
(
yi − F (xi )
|yi − F (xi )| ≤ δ
h(xi ) = −g (xi ) =
δsign(yi − F (xi )) |yi − F (xi )| > δ
Update by Residual:
h(xi ) = yi − F (xi )
Difference: negative gradient pays less attention to outliers.
Gradient Boosting for Regression
Summary of the Section
P
I
Fit an additive model F =
manner.
I
In each stage, introduce a new regression tree h to
compensate the shortcomings of existing model.
I
The“shortcomings” are identified by negative gradients.
I
For any loss function, we can derive a gradient boosting
algorithm.
I
Absolute loss and Huber loss are more robust to outliers than
square loss.
t
ρt ht in a forward stage-wise
Things not covered
How to choose a proper learning rate for each gradient boosting
algorithm. See [Friedman, 2001]
Gradient Boosting for Classification
Problem
Recognize the given hand written capital letter.
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Multi-class classification
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26 classes. A,B,C,...,Z
Data Set
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http://archive.ics.uci.edu/ml/datasets/Letter+
Recognition
I
20000 data points, 16 features
Gradient Boosting for Classification
Feature Extraction
1
2
3
4
5
6
7
8
horizontal position of box
vertical position of box
width of box
height of box
total number on pixels
mean x of on pixels in box
mean y of on pixels in box
mean x variance
9
10
11
12
13
14
15
16
mean y variance
mean x y correlation
mean of x * x * y
mean of x * y * y
mean edge count left to right
correlation of x-ege with y
mean edge count bottom to top
correlation of y-ege with x
Feature Vector= (2, 1, 3, 1, 1, 8, 6, 6, 6, 6, 5, 9, 1, 7, 5, 10)
Label = G
Gradient Boosting for Classification
Model
I
26 score functions (our models): FA , FB , FC , ..., FZ .
I
FA (x) assigns a score for class A
I
scores are used to calculate probabilities
e FA (x)
PA (x) = PZ
Fc (x)
c=A e
e FB (x)
PB (x) = PZ
Fc (x)
c=A e
...
e FZ (x)
PZ (x) = PZ
Fc (x)
c=A e
I
predicted label = class that has the highest probability
Loss Function for each data point
Step 1 turn the label yi into a (true) probability distribution Yc (xi )
For example: y5 =G,
YA (x5 ) = 0, YB (x5 ) = 0, ..., YG (x5 ) = 1, ..., YZ (x5 ) = 0
Gradient Boosting for Classification
Figure: true probability distribution
Loss Function for each data point
Step 1 turn the label yi into a (true) probability distribution Yc (xi )
For example: y5 =G,
YA (x5 ) = 0, YB (x5 ) = 0, ..., YG (x5 ) = 1, ..., YZ (x5 ) = 0
Step 2 calculate the predicted probability distribution Pc (xi ) based on
the current model FA , FB , ..., FZ .
PA (x5 ) = 0.03, PB (x5 ) = 0.05, ..., PG (x5 ) = 0.3, ..., PZ (x5 ) =
0.05
Gradient Boosting for Classification
Figure: predicted probability distribution based on current model
Loss Function for each data point
Step 1 turn the label yi into a (true) probability distribution Yc (xi )
For example: y5 =G,
YA (x5 ) = 0, YB (x5 ) = 0, ..., YG (x5 ) = 1, ..., YZ (x5 ) = 0
Step 2 calculate the predicted probability distribution Pc (xi ) based on
the current model FA , FB , ..., FZ .
PA (x5 ) = 0.03, PB (x5 ) = 0.05, ..., PG (x5 ) = 0.3, ..., PZ (x5 ) =
0.05
Step 3 calculate the difference between the true probability
distribution and the predicted probability distribution.
Here we use KL-divergence
Gradient Boosting for Classification
Goal
I
minimize the total loss (KL-divergence)
I
for each data point, we wish the predicted probability
distribution to match the true probability distribution as
closely as possible
Gradient Boosting for Classification
Figure: true probability distribution
Gradient Boosting for Classification
Figure: predicted probability distribution at round 0
Gradient Boosting for Classification
Figure: predicted probability distribution at round 1
Gradient Boosting for Classification
Figure: predicted probability distribution at round 2
Gradient Boosting for Classification
Figure: predicted probability distribution at round 10
Gradient Boosting for Classification
Figure: predicted probability distribution at round 20
Gradient Boosting for Classification
Figure: predicted probability distribution at round 30
Gradient Boosting for Classification
Figure: predicted probability distribution at round 40
Gradient Boosting for Classification
Figure: predicted probability distribution at round 50
Gradient Boosting for Classification
Figure: predicted probability distribution at round 100
Gradient Boosting for Classification
Goal
I
minimize the total loss (KL-divergence)
I
for each data point, we wish the predicted probability
distribution to match the true probability distribution as
closely as possible
I
we achieve this goal by adjusting our models FA , FB , ..., FZ .
Gradient Boosting for Regression: Review
Regression with loss function L: general procedure
Give any differentiable loss function L
start with an initial model F
iterate until converge:
i ,F (xi ))
calculate negative gradients −g (xi ) = − ∂L(y
∂F (xi )
fit a regression tree h to negative gradients −g (xi )
F := F + ρh
Gradient Boosting for Classification
Differences
I
FA , FB , ..., FZ vs F
I
a matrix of parameters to optimize vs a column of parameters
to optimize
FA (x1 ) FB (x1 ) ... FZ (x1 )
FA (x2 ) FB (x2 ) ... FZ (x2 )
...
...
...
...
FA (xn ) FB (xn ) ... FZ (xn )
a matrix of gradients vs a column of gradients
∂L
∂L
... FZ∂L
FA (x1 )
FB (x1 )
(x1 )
∂L
∂L
∂L
...
FA (x2 )
FB (x2 )
FZ (x2 )
...
...
...
...
∂L
∂L
∂L
...
FA (xn )
FB (xn )
FZ (xn )
I
Gradient Boosting for Classification
start with initial models FA , FB , FC , ..., FZ
iterate until converge:
calculate negative gradients for class A: −gA (xi ) = − ∂F∂L
A (xi )
calculate negative gradients for class B: −gB (xi ) = − ∂F∂L
B (xi )
...
calculate negative gradients for class Z:−gZ (xi ) = − ∂F∂L
Z (xi )
fit a regression tree hA to negative gradients −gA (xi )
fit a regression tree hB to negative gradients −gB (xi )
...
fit a regression tree hZ to negative gradients −gZ (xi )
FA := FA + ρA hA
FB := FB + ρB hB
...
FZ := FZ + ρZ hZ
Gradient Boosting for Classification
start with initial models FA , FB , FC , ..., FZ
iterate until converge:
calculate negative gradients for class A: −gA (xi ) = YA (xi ) − PA (xi )
calculate negative gradients for class B: −gB (xi ) = YB (xi ) − PB (xi )
...
calculate negative gradients for class Z:−gZ (xi ) = YZ (xi ) − PZ (xi )
fit a regression tree hA to negative gradients −gA (xi )
fit a regression tree hB to negative gradients −gB (xi )
...
fit a regression tree hZ to negative gradients −gZ (xi )
FA := FA + ρA hA
FB := FB + ρB hB
...
FZ := FZ + ρZ hZ
Gradient Boosting for Classification
round 0
i
1
2
3
4
5
...
y
T
I
D
N
G
...
i
1
2
3
4
5
...
i
1
2
3
4
5
...
i
y
1
2
3
4
5
...
T
I
D
N
G
...
y
T
I
D
N
G
...
YA −
PA
-0.04
-0.04
-0.04
-0.04
-0.04
...
PA
0.04
0.04
0.04
0.04
0.04
...
PB
0.04
0.04
0.04
0.04
0.04
...
YB −
PB
-0.04
-0.04
-0.04
-0.04
-0.04
...
YA
0
0
0
0
0
...
y
T
I
D
N
G
...
FA
0
0
0
0
0
...
PC
0.04
0.04
0.04
0.04
0.04
...
YC −
PC
-0.04
-0.04
-0.04
-0.04
-0.04
...
YB
0
0
0
0
0
...
FB
0
0
0
0
0
...
PD
0.04
0.04
0.04
0.04
0.04
...
YD −
PD
-0.04
-0.04
0.96
-0.04
-0.04
...
YC
0
0
0
0
0
...
FC
0
0
0
0
0
...
YD
0
0
1
0
0
...
YE
0
0
0
0
0
...
FD
0
0
0
0
0
...
YF
0
0
0
0
0
...
FE
0
0
0
0
0
...
FF
0
0
0
0
0
...
PE
0.04
0.04
0.04
0.04
0.04
...
PF
0.04
0.04
0.04
0.04
0.04
...
PG
0.04
0.04
0.04
0.04
0.04
...
YE −
PE
-0.04
-0.04
-0.04
-0.04
-0.04
...
YF −
PF
-0.04
-0.04
-0.04
-0.04
-0.04
...
YG −
PG
-0.04
-0.04
-0.04
-0.04
0.96
...
YG
0
0
0
0
1
...
FG
0
0
0
0
0
...
PH
0.04
0.04
0.04
0.04
0.04
...
YH −
PH
-0.04
-0.04
-0.04
-0.04
-0.04
...
YH
0
0
0
0
0
...
FH
0
0
0
0
0
...
PI
0.04
0.04
0.04
0.04
0.04
...
YI
0
1
0
0
0
...
YJ
0
0
0
0
0
...
YK
0
0
0
0
0
...
YL
0
0
0
0
0
...
YM
0
0
0
0
0
...
YN
0
0
0
1
0
...
YO
0
0
0
0
0
...
YP
0
0
0
0
0
...
YQ
0
0
0
0
0
...
FJ
0
0
0
0
0
...
FK
0
0
0
0
0
...
FL
0
0
0
0
0
...
FM
0
0
0
0
0
...
FN
0
0
0
0
0
...
FO
0
0
0
0
0
...
FP
0
0
0
0
0
...
FQ
0
0
0
0
0
...
FI
0
0
0
0
0
...
PJ
0.04
0.04
0.04
0.04
0.04
...
YI −
PI
-0.04
0.96
-0.04
-0.04
-0.04
...
PK
0.04
0.04
0.04
0.04
0.04
...
YJ −
PJ
-0.04
-0.04
-0.04
-0.04
-0.04
...
PL
0.04
0.04
0.04
0.04
0.04
...
YK −
PK
-0.04
-0.04
-0.04
-0.04
-0.04
...
PM
0.04
0.04
0.04
0.04
0.04
...
YL −
PL
-0.04
-0.04
-0.04
-0.04
-0.04
...
PN
0.04
0.04
0.04
0.04
0.04
...
PO
0.04
0.04
0.04
0.04
0.04
...
YM −
PM
-0.04
-0.04
-0.04
-0.04
-0.04
...
YN −
PN
-0.04
-0.04
-0.04
0.96
-0.04
...
PP
0.04
0.04
0.04
0.04
0.04
...
YR
0
0
0
0
0
...
FR
0
0
0
0
0
...
PQ
0.04
0.04
0.04
0.04
0.04
...
YO −
PO
-0.04
-0.04
-0.04
-0.04
-0.04
...
YS
0
0
0
0
0
...
YT
1
0
0
0
0
...
FS
0
0
0
0
0
...
FT
0
0
0
0
0
...
PR
0.04
0.04
0.04
0.04
0.04
...
YP −
PP
-0.04
-0.04
-0.04
-0.04
-0.04
...
YU
0
0
0
0
0
...
FU
0
0
0
0
0
...
PS
0.04
0.04
0.04
0.04
0.04
...
YQ −
PQ
-0.04
-0.04
-0.04
-0.04
-0.04
...
YV
0
0
0
0
0
...
FV
0
0
0
0
0
...
PT
0.04
0.04
0.04
0.04
0.04
...
YR −
PR
-0.04
-0.04
-0.04
-0.04
-0.04
...
YW
0
0
0
0
0
...
YX
0
0
0
0
0
...
YY
0
0
0
0
0
...
FW
0
0
0
0
0
...
FX
0
0
0
0
0
...
FY
0
0
0
0
0
...
PU
0.04
0.04
0.04
0.04
0.04
...
PV
0.04
0.04
0.04
0.04
0.04
...
YS −
PS
-0.04
-0.04
-0.04
-0.04
-0.04
...
YT −
PT
0.96
-0.04
-0.04
-0.04
-0.04
...
YZ
0
0
0
0
0
...
FZ
0
0
0
0
0
...
PW
0.04
0.04
0.04
0.04
0.04
...
YU −
PU
-0.04
-0.04
-0.04
-0.04
-0.04
...
PX
0.04
0.04
0.04
0.04
0.04
...
PY
0.04
0.04
0.04
0.04
0.04
...
YV −
PV
-0.04
-0.04
-0.04
-0.04
-0.04
...
PZ
0.04
0.04
0.04
0.04
0.04
...
YW −
PW
-0.04
-0.04
-0.04
-0.04
-0.04
...
YX −
PX
-0.04
-0.04
-0.04
-0.04
-0.04
...
YY −
PY
-0.04
-0.04
-0.04
-0.04
-0.04
...
YZ −
PZ
-0.04
-0.04
-0.04
-0.04
-0.04
...
Gradient Boosting for Classification
(
0.98
hA (x) =
−0.07
(
−0.07
hB (x) =
0.22
feature 10 of x ≤ 2.0
feature 10 of x > 2.0
feature 15 of x ≤ 8.0
feature 15 of x > 8.0
...
(
−0.07 feature 8 of x ≤ 8.0
hZ (x) =
0.82
feature 8 of x > 8.0
FA := FA + ρA hA
FB := FB + ρB hB
...
FZ := FZ + ρZ hZ
Gradient Boosting for Classification
round 1
i
1
2
3
4
5
...
i
1
2
3
4
5
...
y
T
I
D
N
G
...
FA
-0.08
-0.08
-0.08
-0.08
-0.08
...
i
1
2
3
4
5
...
y
T
I
D
N
G
...
i
y
1
2
3
4
5
...
T
I
D
N
G
...
YA −
PA
-0.04
-0.04
-0.04
-0.04
-0.04
...
FB
-0.07
0.23
0.23
-0.07
0.23
...
PA
0.04
0.04
0.04
0.04
0.04
...
y
T
I
D
N
G
...
FC
-0.06
-0.06
-0.06
-0.06
-0.06
...
PB
0.04
0.05
0.05
0.04
0.05
...
YB −
PB
-0.04
-0.05
-0.05
-0.04
-0.05
...
YA
0
0
0
0
0
...
FD
-0.07
-0.07
-0.07
-0.07
-0.07
...
PC
0.04
0.04
0.04
0.04
0.04
...
YC −
PC
-0.04
-0.04
-0.04
-0.04
-0.04
...
YB
0
0
0
0
0
...
PD
0.04
0.04
0.04
0.04
0.04
...
YD −
PD
-0.04
-0.04
0.96
-0.04
-0.04
...
YC
0
0
0
0
0
...
YD
0
0
1
0
0
...
YE
0
0
0
0
0
...
YF
0
0
0
0
0
...
FE
-0.02
-0.02
-0.02
-0.02
-0.02
...
FF
-0.02
-0.02
-0.02
-0.02
-0.02
...
FG
-0.08
0.16
-0.08
0.16
0.16
...
PE
0.04
0.04
0.04
0.04
0.04
...
PF
0.04
0.04
0.04
0.04
0.04
...
PG
0.04
0.05
0.04
0.05
0.05
...
YE −
PE
-0.04
-0.04
-0.04
-0.04
-0.04
...
YF −
PF
-0.04
-0.04
-0.04
-0.04
-0.04
...
YG −
PG
-0.04
-0.05
-0.04
-0.05
0.95
...
YG
0
0
0
0
1
...
FH
-0.02
-0.02
-0.02
-0.02
-0.02
...
PH
0.04
0.04
0.04
0.04
0.04
...
YH −
PH
-0.04
-0.04
-0.04
-0.04
-0.04
...
YH
0
0
0
0
0
...
YI
0
1
0
0
0
...
FI
-0.03
-0.03
-0.03
-0.03
-0.03
...
PI
0.04
0.04
0.04
0.04
0.04
...
YJ
0
0
0
0
0
...
FJ
-0.03
-0.03
-0.03
-0.03
-0.03
...
PJ
0.04
0.04
0.04
0.04
0.04
...
YI −
PI
-0.04
0.96
-0.04
-0.04
-0.04
...
YK
0
0
0
0
0
...
FK
-0.06
-0.06
-0.06
0.26
-0.06
...
PK
0.04
0.04
0.04
0.05
0.04
...
YJ −
PJ
-0.04
-0.04
-0.04
-0.04
-0.04
...
YL
0
0
0
0
0
...
PL
0.04
0.04
0.04
0.04
0.04
...
YK −
PK
-0.04
-0.04
-0.04
-0.05
-0.04
...
YM
0
0
0
0
0
...
FL
-0.04
-0.04
-0.04
-0.04
-0.04
...
PM
0.04
0.04
0.04
0.04
0.04
...
YL −
PL
-0.04
-0.04
-0.04
-0.04
-0.04
...
YN
0
0
0
1
0
...
YO
0
0
0
0
0
...
FM
-0.08
-0.08
-0.08
-0.08
-0.08
...
YP
0
0
0
0
0
...
YQ
0
0
0
0
0
...
FN
-0.08
-0.08
-0.08
0.3
-0.08
...
FO
-0.07
-0.07
-0.07
-0.07
-0.07
...
PN
0.04
0.04
0.04
0.05
0.04
...
PO
0.04
0.04
0.04
0.04
0.04
...
PP
0.04
0.04
0.04
0.04
0.04
...
YM −
PM
-0.04
-0.04
-0.04
-0.04
-0.04
...
YN −
PN
-0.04
-0.04
-0.04
0.95
-0.04
...
YR
0
0
0
0
0
...
YS
0
0
0
0
0
...
FP
-0.07
-0.07
-0.07
-0.07
-0.07
...
PQ
0.04
0.04
0.04
0.04
0.04
...
YO −
PO
-0.04
-0.04
-0.04
-0.04
-0.04
...
YT
1
0
0
0
0
...
FQ
-0.02
-0.02
-0.02
-0.02
-0.02
...
PR
0.04
0.04
0.04
0.04
0.04
...
YP −
PP
-0.04
-0.04
-0.04
-0.04
-0.04
...
YU
0
0
0
0
0
...
FR
-0.04
-0.04
-0.04
-0.04
-0.04
...
PS
0.04
0.04
0.04
0.04
0.04
...
YQ −
PQ
-0.04
-0.04
-0.04
-0.04
-0.04
...
YV
0
0
0
0
0
...
YW
0
0
0
0
0
...
FS
-0.04
-0.04
-0.04
-0.04
-0.04
...
PT
0.07
0.04
0.04
0.04
0.04
...
YR −
PR
-0.04
-0.04
-0.04
-0.04
-0.04
...
YX
0
0
0
0
0
...
FT
0.59
-0.07
-0.07
-0.07
-0.07
...
YY
0
0
0
0
0
...
FU
-0.01
-0.01
-0.01
-0.01
-0.01
...
PU
0.04
0.04
0.04
0.04
0.04
...
PV
0.04
0.04
0.04
0.04
0.04
...
YS −
PS
-0.04
-0.04
-0.04
-0.04
-0.04
...
YT −
PT
0.93
-0.04
-0.04
-0.04
-0.04
...
YZ
0
0
0
0
0
...
FV
-0.07
-0.07
-0.07
-0.07
-0.07
...
FW
-0.07
-0.07
-0.07
-0.07
-0.07
...
FX
-0.05
-0.05
-0.05
-0.05
-0.05
...
PX
0.04
0.04
0.04
0.04
0.04
...
PY
0.04
0.04
0.04
0.04
0.04
...
PZ
0.04
0.04
0.04
0.04
0.04
...
PW
0.04
0.04
0.04
0.04
0.04
...
YU −
PU
-0.04
-0.04
-0.04
-0.04
-0.04
...
YV −
PV
-0.04
-0.04
-0.04
-0.04
-0.04
...
YW −
PW
-0.04
-0.04
-0.04
-0.04
-0.04
...
FY
-0.06
-0.06
-0.06
-0.06
-0.06
...
YX −
PX
-0.04
-0.04
-0.04
-0.04
-0.04
...
FZ
-0.07
-0.07
-0.07
-0.07
-0.07
...
YY −
PY
-0.04
-0.04
-0.04
-0.04
-0.04
...
YZ −
PZ
-0.04
-0.04
-0.04
-0.04
-0.04
...
Gradient Boosting for Classification
(
0.37
hA (x) =
−0.07
(
−0.07
hB (x) =
0.22
feature 10 of x ≤ 2.0
feature 10 of x > 2.0
feature 14 of x ≤ 5.0
feature 14 of x > 5.0
...
(
−0.07 feature 8 of x ≤ 8.0
hZ (x) =
0.35
feature 8 of x > 8.0
FA := FA + ρA hA
FB := FB + ρB hB
...
FZ := FZ + ρZ hZ
Gradient Boosting for Classification
round 2
i
1
2
3
4
5
...
i
1
2
3
4
5
...
y
T
I
D
N
G
...
FA
-0.15
-0.15
-0.15
-0.15
-0.15
...
i
1
2
3
4
5
...
y
T
I
D
N
G
...
i
y
1
2
3
4
5
...
T
I
D
N
G
...
YA −
PA
-0.03
-0.04
-0.04
-0.03
-0.03
...
FB
-0.14
0.16
0.16
-0.14
0.16
...
PA
0.03
0.04
0.04
0.03
0.03
...
y
T
I
D
N
G
...
FC
-0.12
-0.12
-0.12
-0.12
-0.12
...
PB
0.03
0.05
0.05
0.03
0.05
...
YB −
PB
-0.03
-0.05
-0.05
-0.03
-0.05
...
YA
0
0
0
0
0
...
FD
-0.14
-0.14
-0.14
-0.14
-0.14
...
PC
0.03
0.04
0.04
0.03
0.04
...
YC −
PC
-0.03
-0.04
-0.04
-0.03
-0.04
...
YB
0
0
0
0
0
...
PD
0.03
0.04
0.04
0.03
0.04
...
YD −
PD
-0.03
-0.04
0.96
-0.03
-0.04
...
YC
0
0
0
0
0
...
YD
0
0
1
0
0
...
YE
0
0
0
0
0
...
YF
0
0
0
0
0
...
FE
-0.03
-0.03
-0.03
-0.03
-0.03
...
FF
0.28
-0.08
-0.08
-0.08
-0.08
...
FG
-0.14
0.33
0.1
0.1
0.33
...
PE
0.03
0.04
0.04
0.04
0.04
...
PF
0.04
0.04
0.04
0.04
0.04
...
PG
0.03
0.06
0.05
0.04
0.06
...
YE −
PE
-0.03
-0.04
-0.04
-0.04
-0.04
...
YF −
PF
-0.04
-0.04
-0.04
-0.04
-0.04
...
YG −
PG
-0.03
-0.06
-0.05
-0.04
0.94
...
YG
0
0
0
0
1
...
FH
-0.04
-0.04
-0.04
-0.04
-0.04
...
PH
0.03
0.04
0.04
0.04
0.04
...
YH −
PH
-0.03
-0.04
-0.04
-0.04
-0.04
...
YH
0
0
0
0
0
...
YI
0
1
0
0
0
...
FI
1.49
-0.07
-0.07
-0.07
-0.07
...
PI
0.15
0.04
0.04
0.04
0.04
...
YJ
0
0
0
0
0
...
FJ
-0.07
-0.07
-0.07
-0.07
-0.07
...
PJ
0.03
0.04
0.04
0.04
0.04
...
YI −
PI
-0.15
0.96
-0.04
-0.04
-0.04
...
YK
0
0
0
0
0
...
FK
-0.11
-0.11
-0.11
0.46
-0.11
...
PK
0.03
0.04
0.04
0.06
0.04
...
YJ −
PJ
-0.03
-0.04
-0.04
-0.04
-0.04
...
YL
0
0
0
0
0
...
PL
0.03
0.04
0.04
0.04
0.04
...
YK −
PK
-0.03
-0.04
-0.04
-0.06
-0.04
...
YM
0
0
0
0
0
...
FL
-0.08
-0.08
-0.08
-0.08
-0.08
...
PM
0.03
0.04
0.04
0.03
0.04
...
YL −
PL
-0.03
-0.04
-0.04
-0.04
-0.04
...
YN
0
0
0
1
0
...
YO
0
0
0
0
0
...
FM
-0.14
-0.14
-0.14
-0.14
-0.14
...
YP
0
0
0
0
0
...
YQ
0
0
0
0
0
...
FN
-0.17
-0.17
-0.17
0.5
-0.17
...
FO
-0.13
-0.13
-0.13
-0.13
-0.13
...
PN
0.03
0.04
0.04
0.06
0.03
...
PO
0.03
0.04
0.04
0.03
0.04
...
PP
0.03
0.04
0.04
0.03
0.04
...
YM −
PM
-0.03
-0.04
-0.04
-0.03
-0.04
...
YN −
PN
-0.03
-0.04
-0.04
0.94
-0.03
...
YR
0
0
0
0
0
...
YS
0
0
0
0
0
...
FP
-0.13
-0.13
-0.13
-0.13
-0.13
...
PQ
0.03
0.04
0.04
0.04
0.04
...
YO −
PO
-0.03
-0.04
-0.04
-0.03
-0.04
...
YT
1
0
0
0
0
...
FQ
-0.04
-0.04
-0.04
-0.04
-0.04
...
PR
0.03
0.04
0.05
0.04
0.05
...
YP −
PP
-0.03
-0.04
-0.04
-0.03
-0.04
...
YU
0
0
0
0
0
...
FR
-0.11
-0.11
0.19
-0.11
0.19
...
PS
0.03
0.04
0.04
0.04
0.04
...
YQ −
PQ
-0.03
-0.04
-0.04
-0.04
-0.04
...
YV
0
0
0
0
0
...
YW
0
0
0
0
0
...
FS
-0.07
-0.07
-0.07
-0.07
-0.07
...
PT
0.09
0.04
0.04
0.04
0.04
...
YR −
PR
-0.03
-0.04
-0.05
-0.04
-0.05
...
YX
0
0
0
0
0
...
FT
1.05
-0.11
-0.11
-0.11
-0.11
...
YY
0
0
0
0
0
...
FU
0.19
-0.07
-0.07
-0.07
-0.07
...
PU
0.04
0.04
0.04
0.04
0.04
...
PV
0.04
0.04
0.04
0.03
0.04
...
YS −
PS
-0.03
-0.04
-0.04
-0.04
-0.04
...
YT −
PT
0.91
-0.04
-0.04
-0.04
-0.04
...
YZ
0
0
0
0
0
...
FV
0.25
-0.15
-0.15
-0.15
-0.15
...
FW
-0.16
-0.16
-0.16
0.25
-0.16
...
FX
-0.09
-0.09
-0.09
-0.09
-0.09
...
PX
0.03
0.04
0.04
0.04
0.04
...
PY
0.05
0.04
0.04
0.03
0.04
...
PZ
0.03
0.04
0.04
0.03
0.04
...
PW
0.03
0.04
0.04
0.05
0.03
...
YU −
PU
-0.04
-0.04
-0.04
-0.04
-0.04
...
YV −
PV
-0.04
-0.04
-0.04
-0.03
-0.04
...
YW −
PW
-0.03
-0.04
-0.04
-0.05
-0.03
...
FY
0.33
-0.13
-0.13
-0.13
-0.13
...
YX −
PX
-0.03
-0.04
-0.04
-0.04
-0.04
...
FZ
-0.14
-0.14
-0.14
-0.14
-0.14
...
YY −
PY
-0.05
-0.04
-0.04
-0.03
-0.04
...
YZ −
PZ
-0.03
-0.04
-0.04
-0.03
-0.04
...
Gradient Boosting for Classification
round 100
i
1
2
3
4
5
...
i
1
2
3
4
5
...
y
T
I
D
N
G
...
FA
-3.26
-1.64
-2.45
-3.95
-3.14
...
i
1
2
3
4
5
...
y
T
I
D
N
G
...
i
y
1
2
3
4
5
...
T
I
D
N
G
...
YA −
PA
-0
-0.01
-0.01
-0
-0
...
FB
-2.7
-1.09
0.18
-3.38
-0.04
...
PA
0
0.01
0.01
0
0
...
YB −
PB
-0
-0.01
-0.11
-0
-0.03
...
y
T
I
D
N
G
...
YA
0
0
0
0
0
...
FC
-2.2
-2.29
-3.01
-0.22
-2.37
...
PB
0
0.01
0.11
0
0.03
...
FD
-2.22
-1.8
0.18
-0.94
-0.78
...
PC
0
0
0
0.02
0
...
YC −
PC
-0
-0
-0
-0.02
-0
...
YB
0
0
0
0
0
...
PD
0
0.01
0.11
0.01
0.01
...
YD −
PD
-0
-0.01
0.89
-0.01
-0.01
...
YC
0
0
0
0
0
...
YD
0
0
1
0
0
...
FE
-2.48
0.45
-2.79
-1.33
0.02
...
PE
0
0.06
0.01
0.01
0.03
...
YE −
PE
-0
-0.06
-0.01
-0.01
-0.03
...
YE
0
0
0
0
0
...
YF
0
0
0
0
0
...
YG
0
0
0
0
1
...
FF
-0.31
-0.43
-1.7
-1.38
-2.68
...
FG
-2.77
2.14
-2.21
-1.22
2.6
...
FH
-1.19
-1.56
0.43
-0.12
-1.48
...
PF
0.01
0.02
0.02
0.01
0
...
PG
0
0.32
0.01
0.01
0.42
...
PH
0
0.01
0.14
0.03
0.01
...
YF −
PF
-0.01
-0.02
-0.02
-0.01
-0
...
YG −
PG
-0
-0.32
-0.01
-0.01
0.58
...
YH −
PH
-0
-0.01
-0.14
-0.03
-0.01
...
YH
0
0
0
0
0
...
YI
0
1
0
0
0
...
FI
2.77
1.19
-1.12
-2.33
-1.93
...
PI
0.13
0.12
0.03
0
0
...
YJ
0
0
0
0
0
...
YK
0
0
0
0
0
...
FJ
0.1
1.09
0.32
-3.13
0.42
...
PJ
0.01
0.11
0.12
0
0.05
...
YI −
PI
-0.13
0.88
-0.03
-0
-0
...
FK
-1.49
-1.5
0.67
0.58
-1.44
...
PK
0
0.01
0.17
0.05
0.01
...
YJ −
PJ
-0.01
-0.11
-0.12
-0
-0.05
...
YL
0
0
0
0
0
...
YM
0
0
0
0
0
...
FL
-1.02
-0.5
-2.16
-0.65
-1.45
...
PL
0
0.02
0.01
0.02
0.01
...
YK −
PK
-0
-0.01
-0.17
-0.05
-0.01
...
PM
0
0
0
0.02
0
...
YL −
PL
-0
-0.02
-0.01
-0.02
-0.01
...
YN
0
0
0
1
0
...
YO
0
0
0
0
0
...
FM
-1.64
-3.64
-2.91
-0.25
-3.36
...
YP
0
0
0
0
0
...
FN
-0.8
-3.98
-2.76
2.96
-3.98
...
PN
0
0
0.01
0.59
0
...
PO
0
0.03
0.01
0
0.01
...
YM −
PM
-0
-0
-0
-0.02
-0
...
YN −
PN
-0
-0
-0.01
0.41
-0
...
YQ
0
0
0
0
0
...
YR
0
0
0
0
0
...
FO
-2.4
-0.39
-1.92
-2.84
-0.94
...
PP
0
0
0
0
0
...
YS
0
0
0
0
0
...
FP
-3.57
-2.3
-3.04
-1.82
-3.42
...
PQ
0
0.16
0.02
0.04
0.19
...
YO −
PO
-0
-0.03
-0.01
-0
-0.01
...
YT
1
0
0
0
0
...
FQ
-0.9
1.42
-1.47
0.19
1.84
...
PR
0
0.02
0.05
0.05
0.13
...
YP −
PP
-0
-0
-0
-0
-0
...
YU
0
0
0
0
0
...
FR
-2.45
-0.59
-0.48
0.55
1.44
...
PS
0.01
0.05
0.02
0.01
0.06
...
YQ −
PQ
-0
-0.16
-0.02
-0.04
-0.19
...
YV
0
0
0
0
0
...
YW
0
0
0
0
0
...
FS
-0.2
0.27
-1.48
-1.22
0.62
...
YX
0
0
0
0
0
...
FT
4.61
-2.88
-1.25
-1.25
-1.25
...
YY
0
0
0
0
0
...
YZ
0
0
0
0
0
...
FU
0.5
-1.96
-2.25
0.45
-1.33
...
FV
-0.71
-1.67
-3.23
-1.8
-4.41
...
FW
-1.21
-4.38
-4.38
0.11
-4.71
...
PW
0
0
0
0.03
0
...
PX
0.01
0
0.11
0.02
0
...
PY
0.01
0
0
0.01
0
...
PT
0.79
0
0.03
0.01
0.01
...
PU
0.01
0.01
0.01
0.05
0.01
...
PV
0
0.01
0
0.01
0
...
YR −
PR
-0
-0.02
-0.05
-0.05
-0.13
...
YS −
PS
-0.01
-0.05
-0.02
-0.01
-0.06
...
YT −
PT
0.21
-0
-0.03
-0.01
-0.01
...
YU −
PU
-0.01
-0.01
-0.01
-0.05
-0.01
...
YV −
PV
-0
-0.01
-0
-0.01
-0
...
FX
-0.24
-2.06
0.17
-0.69
-2.62
...
FY
0.49
-2.95
-2.95
-1.6
-2.15
...
FZ
-1.66
-1.76
-2.65
-3.78
-1.09
...
PZ
0
0.01
0.01
0
0.01
...
YW −
PW
-0
-0
-0
-0.03
-0
...
YX −
PX
-0.01
-0
-0.11
-0.02
-0
...
YY −
PY
-0.01
-0
-0
-0.01
-0
...
YZ −
PZ
-0
-0.01
-0.01
-0
-0.01
...
Things not covered
I
How to choose proper learning rates. See [Friedman, 2001]
I
Other possible loss functions. See [Friedman, 2001]
References I
Breiman, L. (1999).
Prediction games and arcing algorithms.
Neural computation, 11(7):1493–1517.
Breiman, L. et al. (1998).
Arcing classifier (with discussion and a rejoinder by the
author).
The annals of statistics, 26(3):801–849.
Freund, Y. and Schapire, R. E. (1997).
A decision-theoretic generalization of on-line learning and an
application to boosting.
Journal of computer and system sciences, 55(1):119–139.
Freund, Y., Schapire, R. E., et al. (1996).
Experiments with a new boosting algorithm.
In ICML, volume 96, pages 148–156.
References II
Friedman, J., Hastie, T., and Tibshirani, R. (2000).
Special invited paper. additive logistic regression: A statistical
view of boosting.
Annals of statistics, pages 337–374.
Friedman, J. H. (2001).
Greedy function approximation: a gradient boosting machine.
Annals of Statistics, pages 1189–1232.
Schapire, R. E. and Freund, Y. (2012).
Boosting: Foundations and Algorithms.
MIT Press.