CNTK - 邏輯迴歸模型

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本章介紹如何在 CNTK 中構建邏輯迴歸模型。

邏輯迴歸模型基礎

邏輯迴歸是最簡單的機器學習技術之一，尤其適用於二元分類。換句話說，在預測變數的值只能是兩個分類值之一的情況下建立預測模型。邏輯迴歸最簡單的例子之一是根據一個人的年齡、聲音、頭髮等預測該人是男性還是女性。

示例

讓我們藉助另一個示例從數學上理解邏輯迴歸的概念：

假設我們想根據申請人的**負債、收入**和**信用評級**來預測貸款申請的信用價值；0 表示拒絕，1 表示批准。我們用 X1 表示負債，用 X2 表示收入，用 X3 表示信用評級。

在邏輯迴歸中，我們為每個特徵確定一個權重值，用**w**表示，以及一個偏差值，用**b**表示。

現在假設：

X1 = 3.0
X2 = -2.0
X3 = 1.0

並且假設我們確定權重和偏差如下：

W1 = 0.65, W2 = 1.75, W3 = 2.05 and b = 0.33

現在，為了預測類別，我們需要應用以下公式：

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

接下來，我們需要計算**P = 1.0/(1.0 + exp(-Z))**。這裡，exp() 函式是尤拉數。

P = 1.0/(1.0 + exp(-0.83)
= 0.6963

P 值可以解釋為類別為 1 的機率。如果 P < 0.5，則預測類別 = 0，否則預測 (P >= 0.5) 類別 = 1。

為了確定權重和偏差的值，我們必須獲得一組訓練資料，其中包含已知的輸入預測變數值和已知的正確類別標籤值。之後，我們可以使用一種演算法（通常是梯度下降）來查詢權重和偏差的值。

LR 模型實現示例

對於此 LR 模型，我們將使用以下資料集：

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 中開始此 LR 模型的實現，我們需要首先匯入以下包：

import numpy as np
import cntk as C

程式的結構如下，使用 main() 函式：

def main():
print("Using CNTK version = " + str(C.__version__) + "\n")

現在，我們需要將訓練資料載入到記憶體中，如下所示：

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)

現在，我們將建立一個訓練程式，該程式建立一個與訓練資料相容的邏輯迴歸模型：

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

現在，我們需要建立學習器和訓練器，如下所示：

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

LR 模型訓練

一旦我們建立了 LR 模型，接下來就是開始訓練過程：

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)

現在，藉助以下程式碼，我們可以列印模型權重和偏差：

np.set_printoptions(precision=4, suppress=True)
print("Model weights: ")
print(W.value)
print("Model bias:")
print(b.value)
print("")
if __name__ == "__main__":
main()

訓練邏輯迴歸模型 -完整示例

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()

輸出

Using CNTK version = 2.7
1000 cross entropy error on curr item = 0.1941
2000 cross entropy error on curr item = 0.1746
3000 cross entropy error on curr item = 0.0563
Model weights:
[-0.2049]
   [0.9666]]
Model bias:
[-2.2846]

使用訓練好的 LR 模型進行預測

一旦 LR 模型經過訓練，我們就可以像下面這樣使用它進行預測：

首先，我們的評估程式匯入 numpy 包並將訓練資料載入到特徵矩陣和類別標籤矩陣中，與我們上面實現的訓練程式相同：

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)

接下來，是時候設定訓練程式確定的權重和偏差的值了：

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

接下來，我們的評估程式將透過遍歷每個訓練項來計算邏輯迴歸機率，如下所示：

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))

現在讓我們演示如何進行預測：

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")

完整的預測評估程式

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()

輸出

設定權重和偏差值。

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
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