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).
%load_ext watermark %watermark -a 'Sebastian Raschka' -u -d -v -p matplotlib,numpy,scipy
Sebastian Raschka last updated: 2016年06月05日 CPython 3.5.1 IPython 4.2.0 matplotlib 1.5.1 numpy 1.11.0 scipy 0.17.0
# to install watermark just uncomment the following line: #%install_ext https://raw.githubusercontent.com/rasbt/watermark/master/watermark.py
%matplotlib inline
Bonus Material - Softmax Regression¶
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)
- $x_0 \rightarrow \text{class }0$
- $x_1 \rightarrow \text{class }1$
- $x_2 \rightarrow \text{class }2$
- $x_3 \rightarrow \text{class }2$
importnumpyasnp 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:
y_enc = (np.arange(np.max(y) + 1) == y[:, None]).astype(float) print('one-hot encoding:\n', y_enc)
one-hot encoding: [[ 1. 0. 0.] [ 0. 1. 0.] [ 0. 0. 1.] [ 0. 0. 1.]]
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).
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)
Inputs X: [[ 0.1 0.5] [ 1.1 2.3] [-1.1 -2.3] [-1.5 -2.5]] Weights W: [[ 0.1 0.2 0.3] [ 0.1 0.2 0.3]] bias: [ 0.01 0.1 0.1 ]
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:
$$\mathbf{Z} = \mathbf{X}\mathbf{W} + \mathbf{b}.$$
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)
Inputs X: [[ 0.1 0.5] [ 1.1 2.3] [-1.1 -2.3] [-1.5 -2.5]] Weights W: [[ 0.1 0.2 0.3] [ 0.1 0.2 0.3]] bias: [ 0.01 0.1 0.1 ]
defnet_input(X, W, b): return (X.dot(W) + b) net_in = net_input(X, W, bias) print('net input:\n', net_in)
net input: [[ 0.07 0.22 0.28] [ 0.35 0.78 1.12] [-0.33 -0.58 -0.92] [-0.39 -0.7 -1.1 ]]
Now, it's time to compute the softmax activation that we discussed earlier:
$$P(y=j \mid z^{(i)}) = \phi_{softmax}(z^{(i)}) = \frac{e^{z^{(i)}}}{\sum_{j=0}^{k} e^{z_{k}^{(i)}}}.$$
defsoftmax(z): return (np.exp(z.T) / np.sum(np.exp(z), axis=1)).T smax = softmax(net_in) print('softmax:\n', smax)
softmax: [[ 0.29450637 0.34216758 0.36332605] [ 0.21290077 0.32728332 0.45981591] [ 0.42860913 0.33380113 0.23758974] [ 0.44941979 0.32962558 0.22095463]]
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
defto_classlabel(z): return z.argmax(axis=1) print('predicted class labels: ', to_classlabel(smax))
predicted class labels: [2 2 0 0]
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:
$$J(\mathbf{W}; \mathbf{b}) = \frac{1}{n} \sum_{i=1}^{n} H(T_i, O_i),$$
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.
defcross_entropy(output, y_target): return - np.sum(np.log(output) * (y_target), axis=1) xent = cross_entropy(smax, y_enc) print('Cross Entropy:', xent)
Cross Entropy: [ 1.22245465 1.11692907 1.43720989 1.50979788]
defcost(output, y_target): return np.mean(cross_entropy(output, y_target)) J_cost = cost(smax, y_enc) print('Cost: ', J_cost)
Cost: 1.32159787159
In order to learn our softmax model -- determining the weight coefficients -- via gradient descent, we then need to compute the derivative
$$\nabla \mathbf{w}_j ,円 J(\mathbf{W}; \mathbf{b}).$$
I don't want to walk through the tedious details here, but this cost derivative turns out to be simply:
$$\nabla \mathbf{w}_j ,円 J(\mathbf{W}; \mathbf{b}) = \frac{1}{n} \sum^{n}_{i=0} \big[\mathbf{x}^{(i)}\ \big(O_i - T_i \big) \big]$$
We can then use the cost derivate to update the weights in opposite direction of the cost gradient with learning rate $\eta$:
$$\mathbf{w}_j := \mathbf{w}_j - \eta \nabla \mathbf{w}_j ,円 J(\mathbf{W}; \mathbf{b})$$
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},ドル
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
$$\mathbf{w}_j := \mathbf{w}_j - \eta \big[\nabla \mathbf{w}_j ,円 J(\mathbf{W}) + \lambda \mathbf{w}_j \big].$$
(Please note that we don't regularize the bias term.)
SoftmaxRegression Code¶
Bringing the concepts together, we could come up with an implementation as follows:
# Sebastian Raschka 2016 # Implementation of the mulitnomial logistic regression algorithm for # classification. # Author: Sebastian Raschka <sebastianraschka.com> # # License: BSD 3 clause importnumpyasnp fromtimeimport time #from .._base import _BaseClassifier #from .._base import _BaseMultiClass classSoftmaxRegression(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. Prior to each epoch, the dataset is shuffled if `minibatches > 1` to prevent cycles in stochastic gradient descent. l2 : float Regularization parameter for L2 regularization. No regularization if l2=0.0. minibatches : int (default: 1) The number of minibatches for gradient-based optimization. If 1: Gradient Descent learning If len(y): Stochastic Gradient Descent (SGD) online learning If 1 < minibatches < len(y): SGD Minibatch learning 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. random_seed : int (default: None) Set random state for shuffling and initializing the weights. Attributes ----------- w_ : 2d-array, shape={n_features, 1} Model weights after fitting. b_ : 1d-array, shape={1,} Bias unit after fitting. cost_ : list List of floats, the average cross_entropy for each epoch. """ def__init__(self, eta=0.01, epochs=50, l2=0.0, minibatches=1, n_classes=None, random_seed=None): self.eta = eta self.epochs = epochs self.l2 = l2 self.minibatches = minibatches self.n_classes = n_classes self.random_seed = random_seed def_fit(self, X, y, init_params=True): if init_params: if self.n_classes is None: self.n_classes = np.max(y) + 1 self._n_features = X.shape[1] self.b_, self.w_ = self._init_params( weights_shape=(self._n_features, self.n_classes), bias_shape=(self.n_classes,), random_seed=self.random_seed) self.cost_ = [] y_enc = self._one_hot(y=y, n_labels=self.n_classes, dtype=np.float) for i in range(self.epochs): for idx in self._yield_minibatches_idx( n_batches=self.minibatches, data_ary=y, shuffle=True): # 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] mse = np.mean(diff, axis=0) # 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 * self.w_) self.b_ -= (self.eta * np.sum(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 deffit(self, X, y, init_params=True): """Learn model 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_params : bool (default: True) Re-initializes model parametersprior to fitting. Set False to continue training with weights from a previous model fitting. Returns ------- self : object """ if self.random_seed is not None: np.random.seed(self.random_seed) self._fit(X=X, y=y, init_params=init_params) self._is_fitted = True return self def_predict(self, X): probas = self.predict_proba(X) return self._to_classlabels(probas) defpredict(self, X): """Predict targets from 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 ---------- target_values : array-like, shape = [n_samples] Predicted target values. """ if not self._is_fitted: raise AttributeError('Model is not fitted, yet.') return self._predict(X) defpredict_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_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): L2_term = self.l2 * np.sum(self.w_ ** 2) cross_entropy = cross_entropy + L2_term return 0.5 * np.mean(cross_entropy) def_to_classlabels(self, z): return z.argmax(axis=1) def_init_params(self, weights_shape, bias_shape=(1,), dtype='float64', scale=0.01, random_seed=None): """Initialize weight coefficients.""" if random_seed: np.random.seed(random_seed) w = np.random.normal(loc=0.0, scale=scale, size=weights_shape) b = np.zeros(shape=bias_shape) return b.astype(dtype), w.astype(dtype) def_one_hot(self, y, n_labels, dtype): """Returns a matrix where each sample in y is represented as a row, and each column represents the class label in the one-hot encoding scheme. Example: y = np.array([0, 1, 2, 3, 4, 2]) mc = _BaseMultiClass() mc._one_hot(y=y, n_labels=5, dtype='float') np.array([[1., 0., 0., 0., 0.], [0., 1., 0., 0., 0.], [0., 0., 1., 0., 0.], [0., 0., 0., 1., 0.], [0., 0., 0., 0., 1.], [0., 0., 1., 0., 0.]]) """ mat = np.zeros((len(y), n_labels)) for i, val in enumerate(y): mat[i, val] = 1 return mat.astype(dtype) def_yield_minibatches_idx(self, n_batches, data_ary, shuffle=True): indices = np.arange(data_ary.shape[0]) if shuffle: indices = np.random.permutation(indices) if n_batches > 1: remainder = data_ary.shape[0] % n_batches if remainder: minis = np.array_split(indices[:-remainder], n_batches) minis[-1] = np.concatenate((minis[-1], indices[-remainder:]), axis=0) else: minis = np.array_split(indices, n_batches) else: minis = (indices,) for idx_batch in minis: yield idx_batch def_shuffle_arrays(self, arrays): """Shuffle arrays in unison.""" r = np.random.permutation(len(arrays[0])) return [ary[r] for ary in arrays]
Example 1 - Gradient Descent¶
frommlxtend.dataimport iris_data frommlxtend.plottingimport plot_decision_regions importmatplotlib.pyplotasplt # 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=0.01, epochs=10, minibatches=1, random_seed=0) lr.fit(X, y) plot_decision_regions(X, y, clf=lr) plt.title('Softmax Regression - Gradient Descent') plt.show() plt.plot(range(len(lr.cost_)), lr.cost_) plt.xlabel('Iterations') plt.ylabel('Cost') plt.show()
Continue training for another 800 epochs by calling the fit method with init_params=False.
lr.epochs = 800 lr.fit(X, y, init_params=False) 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()
Predicting Class Labels¶
y_pred = lr.predict(X) print('Last 3 Class Labels: %s' % y_pred[-3:])
Last 3 Class Labels: [2 2 2]
Predicting Class Probabilities¶
y_pred = lr.predict_proba(X) print('Last 3 Class Labels:\n%s' % y_pred[-3:])
Last 3 Class Labels: [[ 1.22579350e-09 1.22074818e-02 9.87792517e-01] [ 1.84962350e-12 3.99742930e-04 9.99600257e-01] [ 3.10919973e-07 1.37047172e-01 8.62952517e-01]]
Example 2 - Stochastic Gradient Descent¶
frommlxtend.dataimport iris_data frommlxtend.plottingimport plot_decision_regions frommlxtend.classifierimport SoftmaxRegression importmatplotlib.pyplotasplt # 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=0.05, epochs=200, minibatches=len(y), random_seed=0) 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()