Microsoft Cognitive Toolkit 简明教程
CNTK - Logistic Regression Model
本章讨论如何在CNTK中构建逻辑回归模型。
Basics of Logistic Regression model
逻辑回归是最简单的机器学习技术之一，特别用于二分类技术。换而言之，在变量值可以是两个分类值之一的情况下创建预测模型。逻辑回归最简单的例子之一是根据一个人的年龄、声音、头发等预测一个人是男性还是女性。
Example
让我们借助另一个示例从数学角度了解逻辑回归的概念-
假设我们想根据申请人 debt , income 和 credit rating 预测贷款申请的信用价值；0 表示拒绝，1 表示批准。我们使用 X1 表示债务，使用 X2 表示收入，使用 X3 表示信用评级。
在逻辑回归中，我们为每个特征确定一个重量值（由 w 表示），并为每个特征确定一个单个偏差值（由 b 表示）。
现在假设
X1 = 3.0
X2 = -2.0
X3 = 1.0现在假设我们如下确定重量和偏差（bias）
W1 = 0.65, W2 = 1.75, W3 = 2.05 and b = 0.33现在，对于预测类别，我们需要应用以下公式
Z = (X1*W1)+(X2*W2)+(X3+W3)+b
i.e. Z = (3.0)*(0.65) + (-2.0)*(1.75) + (1.0)*(2.05) + 0.33
= 0.83接下来，我们需要计算 P = 1.0/(1.0 + exp(-Z)) 。这里，exp() 函数是欧拉数。
P = 1.0/(1.0 + exp(-0.83)
= 0.6963P 值可以解释为类别为 1 的概率。如果 P < 0.5，则预测为类别 = 0，否则预测（P >= 0.5）为类别 = 1。
要确定重量和偏差的值，我们必须获取一组训练数据，其中包含已知的输入预测变量值和已知的正确类别标签值。之后，我们可以使用一个算法（通常为梯度下降）来找到重量和偏差的值。
LR model implementation example
对于此 LR 模型，我们将使用以下数据集
1.0, 2.0, 0
3.0, 4.0, 0
5.0, 2.0, 0
6.0, 3.0, 0
8.0, 1.0, 0
9.0, 2.0, 0
1.0, 4.0, 1
2.0, 5.0, 1
4.0, 6.0, 1
6.0, 5.0, 1
7.0, 3.0, 1
8.0, 5.0, 1要在 CNTK 中启动该 LR 模型实现，我们需要首先导入以下包
import numpy as np
import cntk as C程序的结构采用 main() 函数，如下所示
def main():
print("Using CNTK version = " + str(C.__version__) + "\n")现在，我们需要按照如下方式将训练数据加载到内存中
data_file = ".\\dataLRmodel.txt"
print("Loading data from " + data_file + "\n")
features_mat = np.loadtxt(data_file, dtype=np.float32, delimiter=",", skiprows=0, usecols=[0,1])
labels_mat = np.loadtxt(data_file, dtype=np.float32, delimiter=",", skiprows=0, usecols=[2], ndmin=2)现在，我们将创建一个训练程序，该程序将创建一个适合训练数据的逻辑回归模型
features_dim = 2
labels_dim = 1
X = C.ops.input_variable(features_dim, np.float32)
y = C.input_variable(labels_dim, np.float32)
W = C.parameter(shape=(features_dim, 1)) # trainable cntk.Parameter
b = C.parameter(shape=(labels_dim))
z = C.times(X, W) + b
p = 1.0 / (1.0 + C.exp(-z))
model = p现在，我们需要按照如下方式创建 Lerner 和培训人员
ce_error = C.binary_cross_entropy(model, y) # CE a bit more principled for LR
fixed_lr = 0.010
learner = C.sgd(model.parameters, fixed_lr)
trainer = C.Trainer(model, (ce_error), [learner])
max_iterations = 4000LR Model training
一旦我们创建了 LR 模型，接下来，就该开始训练过程了
np.random.seed(4)
N = len(features_mat)
for i in range(0, max_iterations):
row = np.random.choice(N,1) # pick a random row from training items
trainer.train_minibatch({ X: features_mat[row], y: labels_mat[row] })
if i % 1000 == 0 and i > 0:
mcee = trainer.previous_minibatch_loss_average
print(str(i) + " Cross-entropy error on curr item = %0.4f " % mcee)现在，借助以下代码，我们可以打印模型重量和偏差
np.set_printoptions(precision=4, suppress=True)
print("Model weights: ")
print(W.value)
print("Model bias:")
print(b.value)
print("")
if __name__ == "__main__":
main()Training a Logistic Regression model - Complete example
import numpy as np
import cntk as C
def main():
print("Using CNTK version = " + str(C.__version__) + "\n")
data_file = ".\\dataLRmodel.txt" # provide the name and the location of data file
print("Loading data from " + data_file + "\n")
features_mat = np.loadtxt(data_file, dtype=np.float32, delimiter=",", skiprows=0, usecols=[0,1])
labels_mat = np.loadtxt(data_file, dtype=np.float32, delimiter=",", skiprows=0, usecols=[2], ndmin=2)
features_dim = 2
labels_dim = 1
X = C.ops.input_variable(features_dim, np.float32)
y = C.input_variable(labels_dim, np.float32)
W = C.parameter(shape=(features_dim, 1)) # trainable cntk.Parameter
b = C.parameter(shape=(labels_dim))
z = C.times(X, W) + b
p = 1.0 / (1.0 + C.exp(-z))
model = p
ce_error = C.binary_cross_entropy(model, y) # CE a bit more principled for LR
fixed_lr = 0.010
learner = C.sgd(model.parameters, fixed_lr)
trainer = C.Trainer(model, (ce_error), [learner])
max_iterations = 4000
np.random.seed(4)
N = len(features_mat)
for i in range(0, max_iterations):
row = np.random.choice(N,1) # pick a random row from training items
trainer.train_minibatch({ X: features_mat[row], y: labels_mat[row] })
if i % 1000 == 0 and i > 0:
mcee = trainer.previous_minibatch_loss_average
print(str(i) + " Cross-entropy error on curr item = %0.4f " % mcee)
np.set_printoptions(precision=4, suppress=True)
print("Model weights: ")
print(W.value)
print("Model bias:")
print(b.value)
if __name__ == "__main__":
main()Prediction using trained LR Model
一旦训练了 LR 模型，我们就可以按照如下方式使用它进行预测
首先，我们的评估程序导入 numpy 包，并将训练数据加载到特征矩阵和类别标签矩阵中，方式与我们上面实现的训练程序相同
import numpy as np
def main():
data_file = ".\\dataLRmodel.txt" # provide the name and the location of data file
features_mat = np.loadtxt(data_file, dtype=np.float32, delimiter=",",
skiprows=0, usecols=(0,1))
labels_mat = np.loadtxt(data_file, dtype=np.float32, delimiter=",",
skiprows=0, usecols=[2], ndmin=2)接下来，是时候设置由我们的训练程序确定的权重和偏差的值了
print("Setting weights and bias values \n")
weights = np.array([0.0925, 1.1722], dtype=np.float32)
bias = np.array([-4.5400], dtype=np.float32)
N = len(features_mat)
features_dim = 2接下来，我们的评估程序将通过如下遍历每个训练项来计算逻辑回归概率 -
print("item pred_prob pred_label act_label result")
for i in range(0, N): # each item
x = features_mat[i]
z = 0.0
for j in range(0, features_dim):
z += x[j] * weights[j]
z += bias[0]
pred_prob = 1.0 / (1.0 + np.exp(-z))
pred_label = 0 if pred_prob < 0.5 else 1
act_label = labels_mat[i]
pred_str = ‘correct’ if np.absolute(pred_label - act_label) < 1.0e-5 \
else ‘WRONG’
print("%2d %0.4f %0.0f %0.0f %s" % \ (i, pred_prob, pred_label, act_label, pred_str))现在让我们演示如何进行预测 -
x = np.array([9.5, 4.5], dtype=np.float32)
print("\nPredicting class for age, education = ")
print(x)
z = 0.0
for j in range(0, features_dim):
z += x[j] * weights[j]
z += bias[0]
p = 1.0 / (1.0 + np.exp(-z))
print("Predicted p = " + str(p))
if p < 0.5: print("Predicted class = 0")
else: print("Predicted class = 1")Complete prediction evaluation program
import numpy as np
def main():
data_file = ".\\dataLRmodel.txt" # provide the name and the location of data file
features_mat = np.loadtxt(data_file, dtype=np.float32, delimiter=",",
skiprows=0, usecols=(0,1))
labels_mat = np.loadtxt(data_file, dtype=np.float32, delimiter=",",
skiprows=0, usecols=[2], ndmin=2)
print("Setting weights and bias values \n")
weights = np.array([0.0925, 1.1722], dtype=np.float32)
bias = np.array([-4.5400], dtype=np.float32)
N = len(features_mat)
features_dim = 2
print("item pred_prob pred_label act_label result")
for i in range(0, N): # each item
x = features_mat[i]
z = 0.0
for j in range(0, features_dim):
z += x[j] * weights[j]
z += bias[0]
pred_prob = 1.0 / (1.0 + np.exp(-z))
pred_label = 0 if pred_prob < 0.5 else 1
act_label = labels_mat[i]
pred_str = ‘correct’ if np.absolute(pred_label - act_label) < 1.0e-5 \
else ‘WRONG’
print("%2d %0.4f %0.0f %0.0f %s" % \ (i, pred_prob, pred_label, act_label, pred_str))
x = np.array([9.5, 4.5], dtype=np.float32)
print("\nPredicting class for age, education = ")
print(x)
z = 0.0
for j in range(0, features_dim):
z += x[j] * weights[j]
z += bias[0]
p = 1.0 / (1.0 + np.exp(-z))
print("Predicted p = " + str(p))
if p < 0.5: print("Predicted class = 0")
else: print("Predicted class = 1")
if __name__ == "__main__":
main()Output
设置权重和偏差值。
Item pred_prob pred_label act_label result
0 0.3640 0 0 correct
1 0.7254 1 0 WRONG
2 0.2019 0 0 correct
3 0.3562 0 0 correct
4 0.0493 0 0 correct
5 0.1005 0 0 correct
6 0.7892 1 1 correct
7 0.8564 1 1 correct
8 0.9654 1 1 correct
9 0.7587 1 1 correct
10 0.3040 0 1 WRONG
11 0.7129 1 1 correct
Predicting class for age, education =
[9.5 4.5]
Predicting p = 0.526487952
Predicting class = 1