This is my learning experience when I read the paper Multi-class AdaBoost. To learn this new algorithm, first I reviewed AdaBoost and its relationship with Forward Stagewise Additive Modeling. Then I realized that multi-AdaBoost is natural extension of AdaBoost.
AdaBoost
Let T(x) denote a weak multi-class classifier that assigns a class label to x, then the AdaBoost algorithm proceeds as follows:
AdaBoost
Pay attention to the weak classifier weight, α, to let α >0, we need to ensure the err<0.5. This can be easily done when it's binary classification problem, because error rate of random guessing is 0.5. But it is difficult to achieve in multi-class case. So it may easily fail in multi-class case.
Forward Stagewise Additive Modeling
First let's take a look at Additive Model:
Additive Model
Typically, {βm, γm} are estimated by minimizing some loss function L, which measures the prediction errors over training data.
Directly optimizing such loss function is often difficult. However, consider it's an additive model, a simple greedy method can be used. We can sequentially optimize the following loss function, then add new base functions to the expansion function f(x) without changing the parameters that have been added.
loss function for each step
Thus the Forward Stagewise Additive Modeling algorithm (denote FSAM) is:
- Initialize f_0(x)=0
- For m = 1 to M :
a) Compute
 = \mathop{\arg\max}{\beta, \gamma} \sum{i=1}^{N} L[y_i,f_{m-1}(x_i) + \beta b(x_i;\gamma)] (2.1) )
b) Set
 = f_{m-1}(x) + \beta_m b(x;\gamma_m) --(2.2))
AdaBoost and Forward Stagewise Additive Modeling
In fact AdaBoost is equivalent to Forward Stagewise Additive Modeling using the exponential loss funtion
] = exp[-yf(x)]. )
Again let T(x) denote a classifier that assigns a class label to x, in this case, T(x)=1 or -1.
Then (2.1) is:
 = \mathop{\arg\max}{\beta, T} \sum{i=1}^{N} \ exp [-y_i[f_{m-1}(x_i) + \beta T(x_i)]])
To simplify the expression, let
}i = exp[-y_i f{m-1}(x_i)] )
Notice that w depends neither on β nor T(x).
Then the expression is
 = \mathop{\arg\max}{\beta, T} \sum{i=1}^{N} w^{(m)}_i exp [-\beta T(x_i)])
It can be rewritten as
 = \mathop{\arg\max}{\beta, T} \ e^{-\beta}\sum{y_i = T(x_i)} w^{(m)}i + e^{\beta}\sum{y_i \neq T(x_i)} w^{(m)}i \ = \mathop{\arg\max}{\beta, T} \ (e^\beta - e^{-\beta}) \sum_{i=1}^N w_i^{(m)} I[y_i \neq T(x_i)] + e{-\beta}\sum_{i=1}N w_i^{(m)} )
Now we can get that
} (1- I[y_i \neq T_m(x_i)]}{\sum_{i=1}^N I[y_i \neq T_m(x_i)]} )
denote err_m as
} I[y_i \neq T_m(x_i) ]} {\sum_{i=1}^N w_i^{(m)}} )
Then
Look familiar, right? It is actually the AdaBoost's weak classifier weight, α_m, except the 1/2. I take the 1/2 as learning rate.
Also recall w_m,
}i = exp[-y_i f{m-1}(x_i)] )
Then
}_i = w_i^{(m)} exp[-\beta_m y_i T_m(x_i)] \= w_i^{(m)} \cdot exp[2\beta_m I[y_i \neq T_m(x_i)] \cdot exp(-\beta_m))
using the fact that
=2 \cdot I[ y_i \neq T_m(x_i)]-1)
compare to the weight in AdaBoost,
weight in AdaBoost
We can know that they are equivalent.
As I mentioned before, AdaBoost can easily fail in multi-class case due to the error rate. To avoid this, a natural way is to do something about α. That's what SAMME does. SAMME is short for Stagewise Additive Modeling using a Multi-class Exponential loss function.
SAMME
Note that SAMME is very similar to AdaBoost, except the term log(K-1). Obviously, when K=2, SAMME is equivalent to AdaBoost. In SAMME, in order for α to be positive, we only need err<(K-1)/K. It's just a little better than random guessing.
SAMME and Forward Stagewise Additive Modeling
We can mimic the steps in AdaBoost and Forward Stagewise Additive Modeling to prove SAMME is equivalent to Forward Stagewise Additive Modeling using a Multi-class Exponential loss function.
In the multi-class classification setting, we can recode the output c with a K-dimensional vector y, as:
y_k = 1 if c == k else -1 / (K - 1), like
y encoding
Given the training data, we wish to find f(x) = (f1(x),...,fK(x))T such that
}\sum{i=1}^nL[y_i,f(x_i)]\subject\ to \quad f_1(x)+f_2(x)+...+f_K(X)=0 )
We consider f(x) has following form:
 = \sum_{m=1}^M\beta_{m}g_{m}(x) )
where β is weight and g(x) is base classifier.
Let's start from the loss function. The multi-class exponential loss function:
=exp[-1/K(y_1f_1+...+y_Kf_K)]\=exp(-\frac{1}{K} y^Tf) )
Then (2.1) in multi-class case is:
 = \mathop{\arg\max}{\beta, g} \sum{i=1}^{N} \ exp [-\frac{1}{K} y_i^T(f_{m-1}(x_i) + \beta g(x_i)] \ =\mathop{\arg\max}{\beta, g} \sum{i=1}^{N}w_i\ exp[-\frac{1}{K}\beta y_i^T g(x_i)] \qquad\qquad(3.1))
where
] )
Notice that every g(x) has a one-to-one correspondence with a multi-class classifier T (x) in the following way:
 =k,\qquad if \quad g_K(x)=1 )
Hence, solving for g_m(x) is equivalent to finding the multi-class classifier T_m(x) that can generate g_m(x).
Recall that in AdaBoost and Forward Stagewise Additive Modeling, we rewrite the expression as
 = \mathop{\arg\max}{\beta, T} \ e^{-\beta}\sum{y_i = T(x_i)} w^{(m)}i + e^{\beta}\sum{y_i \neq T(x_i)} w^{(m)}i \ = \mathop{\arg\max}{\beta, T} \ (e^\beta - e^{-\beta}) \sum_{i=1}^N w_i^{(m)} I[y_i \neq T(x_i)] + e{-\beta}\sum_{i=1}N w_i^{(m)} )
Similarly, we rewrite (3.1) as
 = \mathop{\arg\max}{\beta, T} \ e^{-\frac{\beta}{K-1}}\sum{c_i = T(x_i)} w^{(m)}i + e{\frac{\beta}{(K-1)2}}\sum{c_i \neq T(x_i)} w^{(m)}i \ = \mathop{\arg\max}{\beta, T} \ [e{\frac{\beta}{(K-1)2}} - e^{-\frac{\beta}{K-1}} ]\sum_{i=1}^N w_i^{(m)} I[c_i \neq T(x_i)]+ e{-\frac{\beta}{K-1}}\sum_{i=1}N w_i^{(m)} )
So the solution is
^2}{K} [log \frac{1-err_m}{err_m} +log(K-1) ] )
where error
} I[c_i \neq T_m(x_i) ]} {\sum_{i=1}^N w_i^{(m)}} ).
Look at the β_m, we can ignore (K-1)^2/K, the rest is just the difference between AdaBoost and SAMME. It proves that the extra term log(K − 1) in SAMME is not artificial.
SAMME.R
The SAMME algorithm expects the weak learner to deliver a classifier T (x) ∈ {1, . . . , K}. Analternative is to use real-valued confidence-rated predictions such as weighted probability estimates,to update the additive model, rather than the classifications themselves.
In my opinion, the SAMME.R uses more information than SAMME, and has more robustness.
Python3 Implementation of Multi-class AdaBoost using SAMME and SAMME.R
__author__ = 'Xin'
import numpy as np
from numpy.core.umath_tests import inner1d
from copy import deepcopy
class AdaBoostClassifier(object):
'''
Parameters
-----------
base_estimator: object
The base model from which the boosted ensemble is built.
n_estimators: integer, optional(default=50)
The maximum number of estimators
learning_rate: float, optional(default=1)
algorithm: {'SAMME','SAMME.R'}, optional(default='SAMME.R')
SAMME.R uses predicted probabilities to update wights, while SAMME uses class error rate
random_state: int or None, optional(default=None)
Attributes
-------------
estimators_: list of base estimators
estimator_weights_: array of floats
Weights for each base_estimator
estimator_errors_: array of floats
Classification error for each estimator in the boosted ensemble.
Reference:
1. [multi-adaboost](https://web.stanford.edu/~hastie/Papers/samme.pdf)
2. [scikit-learn:weight_boosting](https://github.com/scikit-learn/
scikit-learn/blob/51a765a/sklearn/ensemble/weight_boosting.py#L289)
'''
def __init__(self, *args, **kwargs):
if kwargs and args:
raise ValueError(
'''AdaBoostClassifier can only be called with keyword
arguments for the following keywords: base_estimator ,n_estimators,
learning_rate,algorithm,random_state''')
allowed_keys = ['base_estimator', 'n_estimators', 'learning_rate', 'algorithm', 'random_state']
keywords_used = kwargs.keys()
for keyword in keywords_used:
if keyword not in allowed_keys:
raise ValueError(keyword + ": Wrong keyword used --- check spelling")
n_estimators = 50
learning_rate = 1
algorithm = 'SAMME.R'
random_state = None
if kwargs and not args:
if 'base_estimator' in kwargs:
base_estimator = kwargs.pop('base_estimator')
else:
raise ValueError('''base_estimator can not be None''')
if 'n_estimators' in kwargs: n_estimators = kwargs.pop('n_estimators')
if 'learning_rate' in kwargs: learning_rate = kwargs.pop('learning_rate')
if 'algorithm' in kwargs: algorithm = kwargs.pop('algorithm')
if 'random_state' in kwargs: random_state = kwargs.pop('random_state')
self.base_estimator_ = base_estimator
self.n_estimators_ = n_estimators
self.learning_rate_ = learning_rate
self.algorithm_ = algorithm
self.random_state_ = random_state
self.estimators_ = list()
self.estimator_weights_ = np.zeros(self.n_estimators_)
self.estimator_errors_ = np.ones(self.n_estimators_)
def _samme_proba(self, estimator, n_classes, X):
"""Calculate algorithm 4, step 2, equation c) of Zhu et al [1].
References
----------
.. [1] J. Zhu, H. Zou, S. Rosset, T. Hastie, "Multi-class AdaBoost", 2009.
"""
proba = estimator.predict_proba(X)
# Displace zero probabilities so the log is defined.
# Also fix negative elements which may occur with
# negative sample weights.
proba[proba < np.finfo(proba.dtype).eps] = np.finfo(proba.dtype).eps
log_proba = np.log(proba)
return (n_classes - 1) * (log_proba - (1. / n_classes)
* log_proba.sum(axis=1)[:, np.newaxis])
def fit(self, X, y):
self.n_samples = X.shape[0]
# There is hidden trouble for classes, here the classes will be sorted.
# So in boost we have to ensure that the predict results have the same classes sort
self.classes_ = np.array(sorted(list(set(y))))
self.n_classes_ = len(self.classes_)
for iboost in range(self.n_estimators_):
if iboost == 0:
sample_weight = np.ones(self.n_samples) / self.n_samples
sample_weight, estimator_weight, estimator_error = self.boost(X, y, sample_weight)
# early stop
if estimator_error == None:
break
# append error and weight
self.estimator_errors_[iboost] = estimator_error
self.estimator_weights_[iboost] = estimator_weight
if estimator_error <= 0:
break
return self
def boost(self, X, y, sample_weight):
if self.algorithm_ == 'SAMME':
return self.discrete_boost(X, y, sample_weight)
elif self.algorithm_ == 'SAMME.R':
return self.real_boost(X, y, sample_weight)
def real_boost(self, X, y, sample_weight):
estimator = deepcopy(self.base_estimator_)
if self.random_state_:
estimator.set_params(random_state=1)
estimator.fit(X, y, sample_weight=sample_weight)
y_pred = estimator.predict(X)
incorrect = y_pred != y
estimator_error = np.dot(incorrect, sample_weight) / np.sum(sample_weight, axis=0)
# if worse than random guess, stop boosting
if estimator_error >= 1.0 - 1 / self.n_classes_:
return None, None, None
y_predict_proba = estimator.predict_proba(X)
# repalce zero
y_predict_proba[y_predict_proba < np.finfo(y_predict_proba.dtype).eps] = np.finfo(y_predict_proba.dtype).eps
y_codes = np.array([-1. / (self.n_classes_ - 1), 1.])
y_coding = y_codes.take(self.classes_ == y[:, np.newaxis])
# for sample weight update
intermediate_variable = (-1. * self.learning_rate_ * (((self.n_classes_ - 1) / self.n_classes_) *
inner1d(y_coding, np.log(
y_predict_proba)))) #dot iterate for each row
# update sample weight
sample_weight *= np.exp(intermediate_variable)
sample_weight_sum = np.sum(sample_weight, axis=0)
if sample_weight_sum <= 0:
return None, None, None
# normalize sample weight
sample_weight /= sample_weight_sum
# append the estimator
self.estimators_.append(estimator)
return sample_weight, 1, estimator_error
def discrete_boost(self, X, y, sample_weight):
estimator = deepcopy(self.base_estimator_)
if self.random_state_:
estimator.set_params(random_state=1)
estimator.fit(X, y, sample_weight=sample_weight)
y_pred = estimator.predict(X)
incorrect = y_pred != y
estimator_error = np.dot(incorrect, sample_weight) / np.sum(sample_weight, axis=0)
# if worse than random guess, stop boosting
if estimator_error >= 1 - 1 / self.n_classes_:
return None, None, None
# update estimator_weight
estimator_weight = self.learning_rate_ * np.log((1 - estimator_error) / estimator_error) + np.log(
self.n_classes_ - 1)
if estimator_weight <= 0:
return None, None, None
# update sample weight
sample_weight *= np.exp(estimator_weight * incorrect)
sample_weight_sum = np.sum(sample_weight, axis=0)
if sample_weight_sum <= 0:
return None, None, None
# normalize sample weight
sample_weight /= sample_weight_sum
# append the estimator
self.estimators_.append(estimator)
return sample_weight, estimator_weight, estimator_error
def predict(self, X):
n_classes = self.n_classes_
classes = self.classes_[:, np.newaxis]
pred = None
if self.algorithm_ == 'SAMME.R':
# The weights are all 1. for SAMME.R
pred = sum(self._samme_proba(estimator, n_classes, X) for estimator in self.estimators_)
else: # self.algorithm == "SAMME"
pred = sum((estimator.predict(X) == classes).T * w
for estimator, w in zip(self.estimators_,
self.estimator_weights_))
pred /= self.estimator_weights_.sum()
if n_classes == 2:
pred[:, 0] *= -1
pred = pred.sum(axis=1)
return self.classes_.take(pred > 0, axis=0)
return self.classes_.take(np.argmax(pred, axis=1), axis=0)
def predict_proba(self, X):
if self.algorithm_ == 'SAMME.R':
# The weights are all 1. for SAMME.R
proba = sum(self._samme_proba(estimator, self.n_classes_, X)
for estimator in self.estimators_)
else: # self.algorithm == "SAMME"
proba = sum(estimator.predict_proba(X) * w
for estimator, w in zip(self.estimators_,
self.estimator_weights_))
proba /= self.estimator_weights_.sum()
proba = np.exp((1. / (n_classes - 1)) * proba)
normalizer = proba.sum(axis=1)[:, np.newaxis]
normalizer[normalizer == 0.0] = 1.0
proba /= normalizer
return proba
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