-- Modeling complex functions with artificial neural networks
Single-layer neural network recap
Introducing the multi-layer neural network architecture
Activating a neural network via forward propagation
-- Classifying handwritten digits
Obtaining the MNIST dataset
gzip *ubyte.gz -d
import os
import struct
import numpy as np
def load_mnist(path, kind='train'):
"""Load MNIST data from `path`"""
labels_path = os.path.join(path,
'%s-labels-idx1-ubyte'
% kind)
images_path = os.path.join(path,
'%s-images-idx3-ubyte'
% kind)
with open(labels_path, 'rb') as lbpath:
magic, n = struct.unpack('>II',
lbpath.read(8))
labels = np.fromfile(lbpath,
dtype=np.uint8)
with open(images_path, 'rb') as imgpath:
magic, num, rows, cols = struct.unpack(">IIII",
imgpath.read(16))
images = np.fromfile(imgpath,
dtype=np.uint8).reshape(len(labels), 784)
return images, labels
magic, n = struct.unpack('>II', lbpath.read(8))
labels = np.fromfile(lbpath, dtype=np.int8)
X_train, y_train = load_mnist('mnist', kind='train')
print('Rows: %d, columns: %d'
% (X_train.shape[0], X_train.shape[1]))
X_test, y_test = load_mnist('mnist', kind='t10k')
print('Rows: %d, columns: %d'
% (X_test.shape[0], X_test.shape[1]))
import matplotlib.pyplot as plt
fig, ax = plt.subplots(nrows=2, ncols=5, sharex=True, sharey=True,)
ax = ax.flatten()
for i in range(10):
img = X_train[y_train == i][0].reshape(28, 28)
ax[i].imshow(img, cmap='Greys', interpolation='nearest')
ax[0].set_xticks([])
ax[0].set_yticks([])
plt.tight_layout()
plt.show()
fig, ax = plt.subplots(nrows=5,
ncols=5,
sharex=True,
sharey=True,)
ax = ax.flatten()
for i in range(25):
img = X_train[y_train == 7][i].reshape(28, 28)
ax[i].imshow(img, cmap='Greys', interpolation='nearest')
ax[0].set_xticks([])
ax[0].set_yticks([])
plt.tight_layout()
plt.show()
np.savetxt('train_img.csv', X_train,
fmt='%i', delimiter=',')
np.savetxt('train_labels.csv', y_train,
fmt='%i', delimiter=',')
np.savetxt('test_img.csv', X_test,
fmt='%i', delimiter=',')
np.savetxt('test_labels.csv', y_test,
fmt='%i', delimiter=',')
Implementing a multi-layer perceptron
from neuralnet import NeuralNetMLP
import numpy as np
from scipy.special import expit
import sys
class NeuralNetMLP(object):
def __init__(self, n_output, n_features, n_hidden=30,
l1=0.0, l2=0.0, epochs=500, eta=0.001,
alpha=0.0, decrease_const=0.0, shuffle=True,
minibatches=1, random_state=None):
np.random.seed(random_state)
self.n_output = n_output
self.n_features = n_features
self.n_hidden = n_hidden
self.w1, self.w2 = self._initialize_weights()
self.l1 = l1
self.l2 = l2
self.epochs = epochs
self.eta = eta
self.alpha = alpha
self.decrease_const = decrease_const
self.shuffle = shuffle
self.minibatches = minibatches
def _encode_labels(self, y, k):
onehot = np.zeros((k, y.shape[0]))
for idx, val in enumerate(y):
onehot[val, idx] = 1.0
return onehot
def _initialize_weights(self):
w1 = np.random.uniform(-1.0, 1.0,
size=self.n_hidden*(self.n_features + 1))
w1 = w1.reshape(self.n_hidden, self.n_features + 1)
w2 = np.random.uniform(-1.0, 1.0,
size=self.n_output*(self.n_hidden + 1))
w2 = w2.reshape(self.n_output, self.n_hidden + 1)
return w1, w2
def _sigmoid(self, z):
# expit is equivalent to 1.0/(1.0 + np.exp(-z))
return expit(z)
def _sigmoid_gradient(self, z):
sg = self._sigmoid(z)
return sg * (1 - sg)
def _add_bias_unit(self, X, how='column'):
if how == 'column':
X_new = np.ones((X.shape[0], X.shape[1]+1))
X_new[:, 1:] = X
elif how == 'row':
X_new = np.ones((X.shape[0]+1, X.shape[1]))
X_new[1:, :] = X
else:
raise AttributeError('`how` must be `column` or `row`')
return X_new
def _feedforward(self, X, w1, w2):
a1 = self._add_bias_unit(X, how='column')
z2 = w1.dot(a1.T)
a2 = self._sigmoid(z2)
a2 = self._add_bias_unit(a2, how='row')
z3 = w2.dot(a2)
a3 = self._sigmoid(z3)
return a1, z2, a2, z3, a3
def _L2_reg(self, lambda_, w1, w2):
return (lambda_/2.0) * (np.sum(w1[:, 1:] ** 2)\
+ np.sum(w2[:, 1:] ** 2))
def _L1_reg(self, lambda_, w1, w2):
return (lambda_/2.0) * (np.abs(w1[:, 1:]).sum()\
+ np.abs(w2[:, 1:]).sum())
def _get_cost(self, y_enc, output, w1, w2):
term1 = -y_enc * (np.log(output))
term2 = (1 - y_enc) * np.log(1 - output)
cost = np.sum(term1 - term2)
L1_term = self._L1_reg(self.l1, w1, w2)
L2_term = self._L2_reg(self.l2, w1, w2)
cost = cost + L1_term + L2_term
return cost
def _get_gradient(self, a1, a2, a3, z2, y_enc, w1, w2):
# backpropagation
sigma3 = a3 - y_enc
z2 = self._add_bias_unit(z2, how='row')
sigma2 = w2.T.dot(sigma3) * self._sigmoid_gradient(z2)
sigma2 = sigma2[1:, :]
grad1 = sigma2.dot(a1)
grad2 = sigma3.dot(a2.T)
# regularize
grad1[:, 1:] += (w1[:, 1:] * (self.l1 + self.l2))
grad2[:, 1:] += (w2[:, 1:] * (self.l1 + self.l2))
return grad1, grad2
def predict(self, X):
a1, z2, a2, z3, a3 = self._feedforward(X, self.w1, self.w2)
y_pred = np.argmax(z3, axis=0)
return y_pred
def fit(self, X, y, print_progress=False):
self.cost_ = []
X_data, y_data = X.copy(), y.copy()
y_enc = self._encode_labels(y, self.n_output)
delta_w1_prev = np.zeros(self.w1.shape)
delta_w2_prev = np.zeros(self.w2.shape)
for i in range(self.epochs):
# adaptive learning rate
self.eta /= (1 + self.decrease_const*i)
if print_progress:
sys.stderr.write(
'\rEpoch: %d/%d' % (i+1, self.epochs))
sys.stderr.flush()
if self.shuffle:
idx = np.random.permutation(y_data.shape[0])
X_data, y_data = X_data[idx], y_data[idx]
mini = np.array_split(range(
y_data.shape[0]), self.minibatches)
for idx in mini:
# feedforward
a1, z2, a2, z3, a3 = self._feedforward(
X[idx], self.w1, self.w2)
cost = self._get_cost(y_enc=y_enc[:, idx],
output=a3,
w1=self.w1,
w2=self.w2)
self.cost_.append(cost)
# compute gradient via backpropagation
grad1, grad2 = self._get_gradient(a1=a1, a2=a2,
a3=a3, z2=z2,
y_enc=y_enc[:, idx],
w1=self.w1,
w2=self.w2)
# update weights
delta_w1, delta_w2 = self.eta * grad1,\
self.eta * grad2
self.w1 -= (delta_w1 + (self.alpha * delta_w1_prev))
self.w2 -= (delta_w2 + (self.alpha * delta_w2_prev))
delta_w1_prev, delta_w2_prev = delta_w1, delta_w2
return self
nn = NeuralNetMLP(n_output=10,
n_features=X_train.shape[1],
n_hidden=50,
l2=0.1,
l1=0.0,
epochs=1000,
eta=0.001,
alpha=0.001,
decrease_const=0.00001,
shuffle=True,
minibatches=50,
random_state=1)
nn.fit(X_train, y_train, print_progress=True)
plt.plot(range(len(nn.cost_)), nn.cost_)
plt.ylim([0, 2000])
plt.ylabel('Cost')
plt.xlabel('Epochs * 50')
plt.tight_layout()
plt.show()
batches = np.array_split(range(len(nn.cost_)), 1000)
cost_ary = np.array(nn.cost_)
cost_avgs = [np.mean(cost_ary[i]) for i in batches]
plt.plot(range(len(cost_avgs)),
cost_avgs,
color='red')
plt.ylim([0, 2000])
plt.ylabel('Cost')
plt.xlabel('Epochs')
plt.tight_layout()
plt.show()
y_train_pred = nn.predict(X_train)
acc = np.sum(y_train == y_train_pred, axis=0) / X_train.shape[0]
print('Training accuracy: %.2f%%' % (acc * 100))
y_test_pred = nn.predict(X_test)
acc = np.sum(y_test == y_test_pred, axis=0) / X_test.shape[0]
print('Training accuracy: %.2f%%' % (acc * 100))
miscl_img = X_test[y_test != y_test_pred][:25]
correct_lab = y_test[y_test != y_test_pred][:25]
miscl_lab= y_test_pred[y_test != y_test_pred][:25]
fig, ax = plt.subplots(nrows=5,
ncols=5,
sharex=True,
sharey=True,)
ax = ax.flatten()
for i in range(25):
img = miscl_img[i].reshape(28, 28)
ax[i].imshow(img,
cmap='Greys',
interpolation='nearest')
ax[i].set_title('%d) t: %d p: %d'
% (i+1, correct_lab[i], miscl_lab[i]))
ax[0].set_xticks([])
ax[0].set_yticks([])
plt.tight_layout()
plt.show()
-- Training an artificial neural network
Computing the logistic cost function
Training neural networks via backpropagation
-- Developing your intuition for backpropagation
-- Debugging neural networks with gradient checking
def _gradient_checking(self, X, y_enc, w1,
w2, epsilon, grad1, grad2):
""" Apply gradient checking (for debugging only)
Returns
---------
relative_error : float
Relative error between the numerically
approximated gradients and the backpropagated gradients.
"""
num_grad1 = np.zeros(np.shape(w1))
epsilon_ary1 = np.zeros(np.shape(w1))
for i in range(w1.shape[0]):
for j in range(w1.shape[1]):
epsilon_ary1[i, j] = epsilon
a1, z2, a2, z3, a3 = self._feedforward(
X,
w1 - epsilon_ary1,
w2)
cost1 = self._get_cost(y_enc,
a3,
w1-epsilon_ary1,
w2)
a1, z2, a2, z3, a3 = self._feedforward(
X,
w1 + epsilon_ary1,
w2)
cost2 = self._get_cost(y_enc,
a3,
w1 + epsilon_ary1,
w2)
num_grad1[i, j] = (cost2 - cost1) / (2 * epsilon)
epsilon_ary1[i, j] = 0
num_grad2 = np.zeros(np.shape(w2))
epsilon_ary2 = np.zeros(np.shape(w2))
for i in range(w2.shape[0]):
for j in range(w2.shape[1]):
epsilon_ary2[i, j] = epsilon
a1, z2, a2, z3, a3 = self._feedforward(
X,
w1,
w2 - epsilon_ary2)
cost1 = self._get_cost(y_enc,
a3,
w1,
w2 - epsilon_ary2)
a1, z2, a2, z3, a3 = self._feedforward(
X,
w1,
w2 + epsilon_ary2)
cost2 = self._get_cost(y_enc,
a3,
w1,
w2 + epsilon_ary2)
num_grad2[i, j] = (cost2 - cost1) / (2 * epsilon)
epsilon_ary2[i, j] = 0
num_grad = np.hstack((num_grad1.flatten(),
num_grad2.flatten()))
grad = np.hstack((grad1.flatten(), grad2.flatten()))
norm1 = np.linalg.norm(num_grad - grad)
norm2 = np.linalg.norm(num_grad)
norm3 = np.linalg.norm(grad)
relative_error = norm1 / (norm2 + norm3)
return relative_error
class MLPGradientCheck(object):
[...]
def fit(self, X, y, print_progress=False):
[...]
# compute gradient via backpropagation
grad1, grad2 = self._get_gradient(
a1=a1,
a2=a2,
a3=a3,
z2=z2,
y_enc=y_enc[:, idx],
w1=self.w1,
w2=self.w2)
## start gradient checking
grad_diff = self._gradient_checking(
X=X[idx],
y_enc=y_enc[:, idx],
w1=self.w1,
w2=self.w2,
epsilon=1e-5,
grad1=grad1,
grad2=grad2)
if grad_diff <= 1e-7:
print('Ok: %s' % grad_diff)
elif grad_diff <= 1e-4:
print('Warning: %s' % grad_diff)
else:
print('PROBLEM: %s' % grad_diff)
## end gradient checking
# update weights; [alpha * delta_w_prev]
# for momentum learning
delta_w1 = self.eta * grad1
delta_w2 = self.eta * grad2
self.w1 -= (delta_w1 +\
(self.alpha * delta_w1_prev))
self.w2 -= (delta_w2 +\
(self.alpha * delta_w2_prev))
delta_w1_prev = delta_w1
delta_w2_prev = delta_w2
return self
nn_check = MLPGradientCheck(n_output=10,
n_features=X_train.shape[1],
n_hidden=10,
l2=0.0,
l1=0.0,
epochs=10,
eta=0.001,
alpha=0.0,
decrease_const=0.0,
minibatches=1,
random_state=1)
nn_check.fit(X_train[:5], y_train[:5], print_progress=False)
-- Convergence in neural networks
-- Other neural network architectures
Convolutional Neural Networks
Recurrent Neural Networks
A few last words about neural network implementation
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