Note that the optional watermark extension is a small IPython notebook plugin that I developed to make the code reproducible. You can just skip the following line(s).
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%load_ext watermark
%watermark -a 'Sebastian Raschka' -u -d -v -p matplotlib,numpy,scipy
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# to install watermark just uncomment the following line:
#%install_ext https://raw.githubusercontent.com/rasbt/watermark/master/watermark.py
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%matplotlib inline
Softmax Regression (synonyms: Multinomial Logistic, Maximum Entropy Classifier, or just Multi-class Logistic Regression) is a generalization of logistic regression that we can use for multi-class classification (under the assumption that the classes are mutually exclusive). In contrast, we use the (standard) Logistic Regression model in binary classification tasks.
Below is a schematic of a Logistic Regression model that we discussed in Chapter 3.
In Softmax Regression (SMR), we replace the sigmoid logistic function by the so-called softmax function $\phi_{softmax}(\cdot)$.
$$P(y=j \mid z^{(i)}) = \phi_{softmax}(z^{(i)}) = \frac{e^{z^{(i)}}}{\sum_{j=0}^{k} e^{z_{k}^{(i)}}},$$where we define the net input z as
$$z = w_1x_1 + ... + w_mx_m + b= \sum_{l=0}^{m} w_l x_l + b= \mathbf{w}^T\mathbf{x} + b.$$(w is the weight vector, $\mathbf{x}$ is the feature vector of 1 training sample, and $b$ is the bias unit.)
Now, this softmax function computes the probability that this training sample $\mathbf{x}^{(i)}$ belongs to class $j$ given the weight and net input $z^{(i)}$. So, we compute the probability $p(y = j \mid \mathbf{x^{(i)}; w}_j)$ for each class label in $j = 1, \ldots, k.$. Note the normalization term in the denominator which causes these class probabilities to sum up to one.
To illustrate the concept of softmax, let us walk through a concrete example. Let's assume we have a training set consisting of 4 samples from 3 different classes (0, 1, and 2)
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import numpy as np
y = np.array([0, 1, 2, 2])
First, we want to encode the class labels into a format that we can more easily work with; we apply one-hot encoding:
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y_enc = (np.arange(np.max(y) + 1) == y[:, None]).astype(float)
print('one-hot encoding:\n', y_enc)
A sample that belongs to class 0 (the first row) has a 1 in the first cell, a sample that belongs to class 2 has a 1 in the second cell of its row, and so forth.
Next, let us define the feature matrix of our 4 training samples. Here, we assume that our dataset consists of 2 features; thus, we create a 4x2 dimensional matrix of our samples and features. Similarly, we create a 2x3 dimensional weight matrix (one row per feature and one column for each class).
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X = np.array([[0.1, 0.5],
[1.1, 2.3],
[-1.1, -2.3],
[-1.5, -2.5]])
W = np.array([[0.1, 0.2, 0.3],
[0.1, 0.2, 0.3]])
bias = np.array([0.01, 0.1, 0.1])
print('Inputs X:\n', X)
print('\nWeights W:\n', W)
print('\nbias:\n', bias)
To compute the net input, we multiply the 4x2 matrix feature matrix X with the 2x3 (n_features x n_classes) weight matrix W, which yields a 4x3 output matrix (n_samples x n_classes) to which we then add the bias unit:
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X = np.array([[0.1, 0.5],
[1.1, 2.3],
[-1.1, -2.3],
[-1.5, -2.5]])
W = np.array([[0.1, 0.2, 0.3],
[0.1, 0.2, 0.3]])
bias = np.array([0.01, 0.1, 0.1])
print('Inputs X:\n', X)
print('\nWeights W:\n', W)
print('\nbias:\n', bias)
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def net_input(X, W, b):
return (X.dot(W) + b)
net_in = net_input(X, W, bias)
print('net input:\n', net_in)
Now, it's time to compute the softmax activation that we discussed earlier:
$$P(y=j \mid z^{(i)}) = \phi_{softmax}(z^{(i)}) = \frac{e^{z^{(i)}}}{\sum_{j=0}^{k} e^{z_{k}^{(i)}}}.$$
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def softmax(z):
return (np.exp(z.T) / np.sum(np.exp(z), axis=1)).T
smax = softmax(net_in)
print('softmax:\n', smax)
As we can see, the values for each sample (row) nicely sum up to 1 now. E.g., we can say that the first sample
[ 0.29450637 0.34216758 0.36332605] has a 29.45% probability to belong to class 0.
Now, in order to turn these probabilities back into class labels, we could simply take the argmax-index position of each row:
[[ 0.29450637 0.34216758 0.36332605] -> 2
[ 0.21290077 0.32728332 0.45981591] -> 2
[ 0.42860913 0.33380113 0.23758974] -> 0
[ 0.44941979 0.32962558 0.22095463]] -> 0
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def to_classlabel(z):
return z.argmax(axis=1)
print('predicted class labels: ', to_classlabel(smax))
As we can see, our predictions are terribly wrong, since the correct class labels are [0, 1, 2, 2]. Now, in order to train our logistic model (e.g., via an optimization algorithm such as gradient descent), we need to define a cost function $J(\cdot)$ that we want to minimize:
which is the average of all cross-entropies over our $n$ training samples. The cross-entropy function is defined as
$$H(T_i, O_i) = -\sum_m T_i \cdot log(O_i).$$Here the $T$ stands for "target" (i.e., the true class labels) and the $O$ stands for output -- the computed probability via softmax; not the predicted class label.
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def cross_entropy(output, y_target):
return - np.sum(np.log(output) * (y_target), axis=1)
xent = cross_entropy(smax, y_enc)
print('Cross Entropy:', xent)
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def cost(output, y_target):
return np.mean(cross_entropy(output, y_target))
J_cost = cost(smax, y_enc)
print('Cost: ', J_cost)
In order to learn our softmax model -- determining the weight coefficients -- via gradient descent, we then need to compute the derivative
$$\nabla \mathbf{w}_j \, J(\mathbf{W}; \mathbf{b}).$$I don't want to walk through the tedious details here, but this cost derivative turns out to be simply:
$$\nabla \mathbf{w}_j \, J(\mathbf{W}; \mathbf{b}) = \frac{1}{n} \sum^{n}_{i=0} \big[\mathbf{x}^{(i)}\ \big(O_i - T_i \big) \big]$$We can then use the cost derivate to update the weights in opposite direction of the cost gradient with learning rate $\eta$:
$$\mathbf{w}_j := \mathbf{w}_j - \eta \nabla \mathbf{w}_j \, J(\mathbf{W}; \mathbf{b})$$for each class $$j \in \{0, 1, ..., k\}$$
(note that $\mathbf{w}_j$ is the weight vector for the class $y=j$), and we update the bias units
$$\mathbf{b}_j := \mathbf{b}_j - \eta \bigg[ \frac{1}{n} \sum^{n}_{i=0} \big(O_i - T_i \big) \bigg].$$As a penalty against complexity, an approach to reduce the variance of our model and decrease the degree of overfitting by adding additional bias, we can further add a regularization term such as the L2 term with the regularization parameter $\lambda$:
L2: $\frac{\lambda}{2} ||\mathbf{w}||_{2}^{2}$,
where
$$||\mathbf{w}||_{2}^{2} = \sum^{m}_{l=0} \sum^{k}_{j=0} w_{i, j}$$so that our cost function becomes
$$J(\mathbf{W}; \mathbf{b}) = \frac{1}{n} \sum_{i=1}^{n} H(T_i, O_i) + \frac{\lambda}{2} ||\mathbf{w}||_{2}^{2}$$and we define the "regularized" weight update as
$$\mathbf{w}_j := \mathbf{w}_j - \eta \big[\nabla \mathbf{w}_j \, J(\mathbf{W}) + \lambda \mathbf{w}_j \big].$$(Please note that we don't regularize the bias term.)
Bringing the concepts together, we could come up with an implementation as follows:
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# Sebastian Raschka 2016
# Implementation of the mulitnomial logistic regression algorithm for
# classification.
# Author: Sebastian Raschka <sebastianraschka.com>
#
# License: BSD 3 clause
import numpy as np
class SoftmaxRegression(object):
"""Softmax regression classifier.
Parameters
------------
eta : float (default: 0.01)
Learning rate (between 0.0 and 1.0)
epochs : int (default: 50)
Passes over the training dataset.
l2_lambda : float
Regularization parameter for L2 regularization.
No regularization if l2_lambda=0.0.
minibatches : int (default: 1)
Divide the training data into *k* minibatches
for accelerated stochastic gradient descent learning.
Gradient Descent Learning if `minibatches` = 1
Stochastic Gradient Descent learning if `minibatches` = len(y)
Minibatch learning if `minibatches` > 1
random_seed : int (default: None)
Set random state for shuffling and initializing the weights.
zero_init_weight : bool (default: False)
If True, weights are initialized to zero instead of small random
numbers following a standard normal distribution with mean=0 and
stddev=1.
Attributes
-----------
w_ : 2d-array, shape=[n_features, n_classes]
Weights after fitting.
cost_ : list
List of floats, the average cross_entropy for each epoch.
"""
def __init__(self, eta=0.01, epochs=50,
l2_lambda=0.0, minibatches=1,
random_seed=None,
zero_init_weight=False,
print_progress=0):
self.random_seed = random_seed
self.eta = eta
self.epochs = epochs
self.l2_lambda = l2_lambda
self.minibatches = minibatches
self.zero_init_weight = zero_init_weight
def _one_hot(self, y, n_labels):
mat = np.zeros((len(y), n_labels))
for i, val in enumerate(y):
mat[i, val] = 1
return mat.astype(float)
def _net_input(self, X, W, b):
return (X.dot(W) + b)
def _softmax(self, z):
return (np.exp(z.T) / np.sum(np.exp(z), axis=1)).T
def _cross_entropy(self, output, y_target):
return - np.sum(np.log(output) * (y_target), axis=1)
def _cost(self, cross_entropy):
return np.mean(cross_entropy)
def _to_classlabels(self, z):
return z.argmax(axis=1)
def fit(self, X, y, init_weights=True, n_classes=None):
"""Learn weight coefficients from training data.
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
Training vectors, where n_samples is the number of samples and
n_features is the number of features.
y : array-like, shape = [n_samples]
Target values.
init_weights : bool (default: True)
(Re)initializes weights to small random floats if True.
n_classes : int (default: None)
A positive integer to declare the number of class labels
if not all class labels are present in a partial training set.
Gets the number of class labels automatically if None.
Ignored if init_weights=False.
Returns
-------
self : object
"""
if init_weights:
if n_classes:
self._n_classes = n_classes
else:
self._n_classes = np.max(y) + 1
self._n_features = X.shape[1]
self.w_ = self._init_weights(
shape=(self._n_features, self._n_classes),
zero_init_weight=self.zero_init_weight,
seed=self.random_seed)
self.b_ = self._init_weights(
shape=self._n_classes,
zero_init_weight=self.zero_init_weight,
seed=self.random_seed)
self.cost_ = []
n_idx = list(range(y.shape[0]))
y_enc = self._one_hot(y, self._n_classes)
# random seed for shuffling
if self.random_seed:
np.random.seed(self.random_seed)
for i in range(self.epochs):
if self.minibatches > 1:
n_idx = np.random.permutation(n_idx)
minis = np.array_split(n_idx, self.minibatches)
for idx in minis:
# givens:
# w_ -> n_feat x n_classes
# b_ -> n_classes
# net_input, softmax and diff -> n_samples x n_classes:
net = self._net_input(X[idx], self.w_, self.b_)
softm = self._softmax(net)
diff = softm - y_enc[idx]
# gradient -> n_features x n_classes
grad = np.dot(X[idx].T, diff)
# update in opp. direction of the cost gradient
self.w_ -= (self.eta * grad +
self.eta * self.l2_lambda * self.w_)
self.b_ -= np.mean(diff, axis=0)
# compute cost of the whole epoch
net = self._net_input(X, self.w_, self.b_)
softm = self._softmax(net)
cross_ent = self._cross_entropy(output=softm, y_target=y_enc)
cost = self._cost(cross_ent)
self.cost_.append(cost)
return self
def predict_proba(self, X):
"""Predict class probabilities of X from the net input.
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
Training vectors, where n_samples is the number of samples and
n_features is the number of features.
Returns
----------
Class probabilties : array-like, shape= [n_samples, n_classes]
"""
net = self._net_input(X, self.w_, self.b_)
softm = self._softmax(net)
return softm
def predict(self, X):
"""Predict class labels of X.
Parameters
----------
X : {array-like, sparse matrix}, shape = [n_samples, n_features]
Training vectors, where n_samples is the number of samples and
n_features is the number of features.
Returns
----------
class_labels : array-like, shape = [n_samples]
Predicted class labels.
"""
probas = self.predict_proba(X)
return self._to_classlabels(probas)
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from mlxtend.data import iris_data
from mlxtend.evaluate import plot_decision_regions
from mlxtend.classifier import SoftmaxRegression
import matplotlib.pyplot as plt
# Loading Data
X, y = iris_data()
X = X[:, [0, 3]] # sepal length and petal width
# standardize
X[:,0] = (X[:,0] - X[:,0].mean()) / X[:,0].std()
X[:,1] = (X[:,1] - X[:,1].mean()) / X[:,1].std()
lr = SoftmaxRegression(eta=0.005, epochs=200, minibatches=1, random_seed=1)
lr.fit(X, y)
plot_decision_regions(X, y, clf=lr)
plt.title('Softmax Regression - Stochastic Gradient Descent')
plt.show()
plt.plot(range(len(lr.cost_)), lr.cost_)
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.show()
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y_pred = lr.predict(X)
print('Last 3 Class Labels: %s' % y_pred[-3:])
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y_pred = lr.predict_proba(X)
print('Last 3 Class Labels:\n %s' % y_pred[-3:])
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from mlxtend.data import iris_data
from mlxtend.evaluate import plot_decision_regions
from mlxtend.classifier import SoftmaxRegression
import matplotlib.pyplot as plt
# Loading Data
X, y = iris_data()
X = X[:, [0, 3]] # sepal length and petal width
# standardize
X[:,0] = (X[:,0] - X[:,0].mean()) / X[:,0].std()
X[:,1] = (X[:,1] - X[:,1].mean()) / X[:,1].std()
lr = SoftmaxRegression(eta=0.005, epochs=200, minibatches=len(y), random_seed=1)
lr.fit(X, y)
plot_decision_regions(X, y, clf=lr)
plt.title('Softmax Regression - Stochastic Gradient Descent')
plt.show()
plt.plot(range(len(lr.cost_)), lr.cost_)
plt.xlabel('Iterations')
plt.ylabel('Cost')
plt.show()