逻辑回归

逻辑回归将直线弯成S曲线，用概率回答是非问题。

类型: 构建 语言: Python 前置条件: Phase 2 第1-2课（什么是ML、线性回归） 时间: ~90 分钟

学习目标

• 使用sigmoid函数和二元交叉熵损失从零实现逻辑回归
• 计算并解释精确率、召回率、F1分数和混淆矩阵
• 解释为什么MSE不适用于分类，以及为什么二元交叉熵产生凸代价曲面
• 构建softmax回归模型进行多类分类，并评估阈值调优的权衡

问题

你想根据肿瘤大小预测它是恶性还是良性。你尝试线性回归。它输出0.3或1.7或-0.5这样的数字。这些意味着什么？1.7是”非常恶性”吗？-0.5是”非常良性”吗？线性回归输出无界的数字。分类需要0和1之间的有界概率，以及一个明确的决定：是或否。

逻辑回归解决了这个问题。它取相同的线性组合(wx + b)并通过sigmoid函数，将任何数字压缩到(0, 1)范围。输出是一个概率。你设定阈值（通常0.5）并做出决定。

这是实践中使用最广泛的算法之一。尽管名字中有”回归”，逻辑回归是分类算法，不是回归算法。名称来自它使用的逻辑(sigmoid)函数。

概念

为什么线性回归不适用于分类

想象根据学习时间预测通过/未通过(1/0)。线性回归拟合一条线：

hours:  1   2   3   4   5   6   7   8   9   10
actual: 0   0   0   0   1   1   1   1   1   1

线性拟合可能在1小时处产生-0.2的预测，在10小时处产生1.3的预测。这些值不是概率。它们低于0或高于1。更糟的是，一个异常值（学习了50小时的人）会拖动整条线，改变所有人的预测。

分类需要一个函数：

• 输出0到1之间的值（概率）
• 创建急剧过渡（决策边界）
• 不被远离边界的异常值扭曲

Sigmoid函数

Sigmoid函数恰好做到了这一点：

sigmoid(z) = 1 / (1 + e^(-z))

特性：

• 当z很大且为正时，sigmoid(z)接近1
• 当z很大且为负时，sigmoid(z)接近0
• 当z = 0时，sigmoid(z) = 0.5
• 输出始终在0和1之间
• 函数处处平滑可微

导数有方便的形式：sigmoid’(z) = sigmoid(z) * (1 - sigmoid(z))。这使得梯度计算高效。

逻辑回归 = 线性模型 + Sigmoid

模型计算z = wx + b（与线性回归相同），然后应用sigmoid：

flowchart LR
    X[输入特征 x] --> L["线性: z = wx + b"]
    L --> S["Sigmoid: p = 1/(1+e^-z)"]
    S --> D{"p >= 0.5?"}
    D -->|是| P[预测1]
    D -->|否| N[预测0]

输出p被解释为P(y=1 | x)，即输入属于类别1的概率。决策边界是wx + b = 0的地方，此时sigmoid输出恰好0.5。

二元交叉熵损失

你不能用MSE做逻辑回归。MSE加sigmoid会创建一个非凸的代价曲面，有许多局部最小值。改用二元交叉熵（对数损失）：

Loss = -(1/n) * sum(y * log(p) + (1-y) * log(1-p))

为什么这有效：

• 当y=1且p接近1：log(1) = 0，损失接近0（正确，低代价）
• 当y=1且p接近0：log(0)趋近负无穷，损失巨大（错误，高代价）
• 当y=0且p接近0：log(1) = 0，损失接近0（正确，低代价）
• 当y=0且p接近1：log(0)趋近负无穷，损失巨大（错误，高代价）

这个损失函数对逻辑回归是凸的，保证有单一全局最小值。

逻辑回归的梯度下降

二元交叉熵加sigmoid的梯度有简洁的形式：

dL/dw = (1/n) * sum((p - y) * x)
dL/db = (1/n) * sum(p - y)

这些看起来与线性回归梯度完全相同。区别在于p = sigmoid(wx + b)而不是p = wx + b。sigmoid引入了非线性，但梯度更新规则保持不变。

flowchart TD
    A[初始化 w=0, b=0] --> B[前向传播: z = wx+b, p = sigmoid(z)]
    B --> C[计算损失: 二元交叉熵]
    C --> D["计算梯度: dw = (1/n) * sum((p-y)*x)"]
    D --> E[更新: w = w - lr*dw, b = b - lr*db]
    E --> F{收敛了吗？}
    F -->|否| B
    F -->|是| G[模型训练完成]

决策边界

对于2D输入（两个特征），决策边界是线：

w1*x1 + w2*x2 + b = 0

一侧的点被分类为1，另一侧为0。逻辑回归总是产生线性决策边界。如果需要弯曲的边界，你要么添加多项式特征，要么使用非线性模型。

使用Softmax的多类分类

二元逻辑回归处理两个类别。对于k个类别，使用softmax函数：

softmax(z_i) = e^(z_i) / sum(e^(z_j) for all j)

每个类别有自己的权重向量。模型为每个类别计算一个分数z_i，然后softmax将分数转换为总和为1的概率。预测类别是概率最高的那个。

损失函数变为分类交叉熵：

Loss = -(1/n) * sum(sum(y_k * log(p_k)))

其中y_k对真实类别为1，对其他所有类别为0（one-hot编码）。

评估指标

仅靠准确率不够。对于95%负样本和5%正样本的数据集，总是预测负的模型获得95%准确率但毫无用处。

混淆矩阵：

精确率：在所有预测为正的样本中，有多少实际为正？

Precision = TP / (TP + FP)

召回率（灵敏度）：在所有实际为正的样本中，我们捕获了多少？

Recall = TP / (TP + FN)

F1分数：精确率和召回率的调和平均。平衡两个指标。

F1 = 2 * (Precision * Recall) / (Precision + Recall)

何时优先考虑：

• 精确率：当假正例代价高时（垃圾邮件过滤器，你不想阻止合法邮件）
• 召回率：当假负例代价高时（癌症筛查，你不想漏掉肿瘤）
• F1：当你需要一个平衡的单一指标时

动手构建

步骤1：Sigmoid函数和数据生成

import random
import math

def sigmoid(z):
    z = max(-500, min(500, z))
    return 1.0 / (1.0 + math.exp(-z))


random.seed(42)
N = 200
X = []
y = []

for _ in range(N // 2):
    X.append([random.gauss(2, 1), random.gauss(2, 1)])
    y.append(0)

for _ in range(N // 2):
    X.append([random.gauss(5, 1), random.gauss(5, 1)])
    y.append(1)

combined = list(zip(X, y))
random.shuffle(combined)
X, y = zip(*combined)
X = list(X)
y = list(y)

print(f"Generated {N} samples (2 classes, 2 features)")
print(f"Class 0 center: (2, 2), Class 1 center: (5, 5)")
print(f"First 5 samples:")
for i in range(5):
    print(f"  Features: [{X[i][0]:.2f}, {X[i][1]:.2f}], Label: {y[i]}")

步骤2：从零实现逻辑回归

class LogisticRegression:
    def __init__(self, n_features, learning_rate=0.01):
        self.weights = [0.0] * n_features
        self.bias = 0.0
        self.lr = learning_rate
        self.loss_history = []

    def predict_proba(self, x):
        z = sum(w * xi for w, xi in zip(self.weights, x)) + self.bias
        return sigmoid(z)

    def predict(self, x, threshold=0.5):
        return 1 if self.predict_proba(x) >= threshold else 0

    def compute_loss(self, X, y):
        n = len(y)
        total = 0.0
        for i in range(n):
            p = self.predict_proba(X[i])
            p = max(1e-15, min(1 - 1e-15, p))
            total += y[i] * math.log(p) + (1 - y[i]) * math.log(1 - p)
        return -total / n

    def fit(self, X, y, epochs=1000, print_every=200):
        n = len(y)
        n_features = len(X[0])
        for epoch in range(epochs):
            dw = [0.0] * n_features
            db = 0.0
            for i in range(n):
                p = self.predict_proba(X[i])
                error = p - y[i]
                for j in range(n_features):
                    dw[j] += error * X[i][j]
                db += error
            for j in range(n_features):
                self.weights[j] -= self.lr * (dw[j] / n)
            self.bias -= self.lr * (db / n)
            loss = self.compute_loss(X, y)
            self.loss_history.append(loss)
            if epoch % print_every == 0:
                print(f"  Epoch {epoch:4d} | Loss: {loss:.4f} | w: [{self.weights[0]:.3f}, {self.weights[1]:.3f}] | b: {self.bias:.3f}")
        return self

    def accuracy(self, X, y):
        correct = sum(1 for i in range(len(y)) if self.predict(X[i]) == y[i])
        return correct / len(y)


split = int(0.8 * N)
X_train, X_test = X[:split], X[split:]
y_train, y_test = y[:split], y[split:]

print("\n=== Training Logistic Regression ===")
model = LogisticRegression(n_features=2, learning_rate=0.1)
model.fit(X_train, y_train, epochs=1000, print_every=200)

print(f"\nTrain accuracy: {model.accuracy(X_train, y_train):.4f}")
print(f"Test accuracy:  {model.accuracy(X_test, y_test):.4f}")
print(f"Weights: [{model.weights[0]:.4f}, {model.weights[1]:.4f}]")
print(f"Bias: {model.bias:.4f}")

步骤3：从零实现混淆矩阵和指标

class ClassificationMetrics:
    def __init__(self, y_true, y_pred):
        self.tp = sum(1 for t, p in zip(y_true, y_pred) if t == 1 and p == 1)
        self.tn = sum(1 for t, p in zip(y_true, y_pred) if t == 0 and p == 0)
        self.fp = sum(1 for t, p in zip(y_true, y_pred) if t == 0 and p == 1)
        self.fn = sum(1 for t, p in zip(y_true, y_pred) if t == 1 and p == 0)

    def accuracy(self):
        total = self.tp + self.tn + self.fp + self.fn
        return (self.tp + self.tn) / total if total > 0 else 0

    def precision(self):
        denom = self.tp + self.fp
        return self.tp / denom if denom > 0 else 0

    def recall(self):
        denom = self.tp + self.fn
        return self.tp / denom if denom > 0 else 0

    def f1(self):
        p = self.precision()
        r = self.recall()
        return 2 * p * r / (p + r) if (p + r) > 0 else 0

    def print_confusion_matrix(self):
        print(f"\n  Confusion Matrix:")
        print(f"                  Predicted")
        print(f"                  Pos   Neg")
        print(f"  Actual Pos     {self.tp:4d}  {self.fn:4d}")
        print(f"  Actual Neg     {self.fp:4d}  {self.tn:4d}")

    def print_report(self):
        self.print_confusion_matrix()
        print(f"\n  Accuracy:  {self.accuracy():.4f}")
        print(f"  Precision: {self.precision():.4f}")
        print(f"  Recall:    {self.recall():.4f}")
        print(f"  F1 Score:  {self.f1():.4f}")


y_pred_test = [model.predict(x) for x in X_test]
print("\n=== Classification Report (Test Set) ===")
metrics = ClassificationMetrics(y_test, y_pred_test)
metrics.print_report()

步骤4：决策边界分析

print("\n=== Decision Boundary ===")
w1, w2 = model.weights
b = model.bias
print(f"Decision boundary: {w1:.4f}*x1 + {w2:.4f}*x2 + {b:.4f} = 0")
if abs(w2) > 1e-10:
    print(f"Solved for x2:     x2 = {-w1/w2:.4f}*x1 + {-b/w2:.4f}")

print("\nSample predictions near the boundary:")
test_points = [
    [3.0, 3.0],
    [3.5, 3.5],
    [4.0, 4.0],
    [2.5, 2.5],
    [5.0, 5.0],
]
for point in test_points:
    prob = model.predict_proba(point)
    pred = model.predict(point)
    print(f"  [{point[0]}, {point[1]}] -> prob={prob:.4f}, class={pred}")

步骤5：使用Softmax的多类分类

class SoftmaxRegression:
    def __init__(self, n_features, n_classes, learning_rate=0.01):
        self.n_features = n_features
        self.n_classes = n_classes
        self.lr = learning_rate
        self.weights = [[0.0] * n_features for _ in range(n_classes)]
        self.biases = [0.0] * n_classes

    def softmax(self, scores):
        max_score = max(scores)
        exp_scores = [math.exp(s - max_score) for s in scores]
        total = sum(exp_scores)
        return [e / total for e in exp_scores]

    def predict_proba(self, x):
        scores = [
            sum(self.weights[k][j] * x[j] for j in range(self.n_features)) + self.biases[k]
            for k in range(self.n_classes)
        ]
        return self.softmax(scores)

    def predict(self, x):
        probs = self.predict_proba(x)
        return probs.index(max(probs))

    def fit(self, X, y, epochs=1000, print_every=200):
        n = len(y)
        for epoch in range(epochs):
            grad_w = [[0.0] * self.n_features for _ in range(self.n_classes)]
            grad_b = [0.0] * self.n_classes
            total_loss = 0.0
            for i in range(n):
                probs = self.predict_proba(X[i])
                for k in range(self.n_classes):
                    target = 1.0 if y[i] == k else 0.0
                    error = probs[k] - target
                    for j in range(self.n_features):
                        grad_w[k][j] += error * X[i][j]
                    grad_b[k] += error
                true_prob = max(probs[y[i]], 1e-15)
                total_loss -= math.log(true_prob)
            for k in range(self.n_classes):
                for j in range(self.n_features):
                    self.weights[k][j] -= self.lr * (grad_w[k][j] / n)
                self.biases[k] -= self.lr * (grad_b[k] / n)
            if epoch % print_every == 0:
                print(f"  Epoch {epoch:4d} | Loss: {total_loss / n:.4f}")
        return self

    def accuracy(self, X, y):
        correct = sum(1 for i in range(len(y)) if self.predict(X[i]) == y[i])
        return correct / len(y)


random.seed(42)
X_3class = []
y_3class = []

centers = [(1, 1), (5, 1), (3, 5)]
for label, (cx, cy) in enumerate(centers):
    for _ in range(50):
        X_3class.append([random.gauss(cx, 0.8), random.gauss(cy, 0.8)])
        y_3class.append(label)

combined = list(zip(X_3class, y_3class))
random.shuffle(combined)
X_3class, y_3class = zip(*combined)
X_3class = list(X_3class)
y_3class = list(y_3class)

split_3 = int(0.8 * len(X_3class))
X_train_3 = X_3class[:split_3]
y_train_3 = y_3class[:split_3]
X_test_3 = X_3class[split_3:]
y_test_3 = y_3class[split_3:]

print("\n=== Multi-class Softmax Regression (3 classes) ===")
softmax_model = SoftmaxRegression(n_features=2, n_classes=3, learning_rate=0.1)
softmax_model.fit(X_train_3, y_train_3, epochs=1000, print_every=200)
print(f"\nTrain accuracy: {softmax_model.accuracy(X_train_3, y_train_3):.4f}")
print(f"Test accuracy:  {softmax_model.accuracy(X_test_3, y_test_3):.4f}")

print("\nSample predictions:")
for i in range(5):
    probs = softmax_model.predict_proba(X_test_3[i])
    pred = softmax_model.predict(X_test_3[i])
    print(f"  True: {y_test_3[i]}, Predicted: {pred}, Probs: [{', '.join(f'{p:.3f}' for p in probs)}]")

步骤6：阈值调优

print("\n=== Threshold Tuning ===")
print("Default threshold: 0.5. Adjusting the threshold trades precision for recall.\n")

thresholds = [0.3, 0.4, 0.5, 0.6, 0.7]
print(f"{'Threshold':>10} {'Accuracy':>10} {'Precision':>10} {'Recall':>10} {'F1':>10}")
print("-" * 52)

for t in thresholds:
    y_pred_t = [1 if model.predict_proba(x) >= t else 0 for x in X_test]
    m = ClassificationMetrics(y_test, y_pred_t)
    print(f"{t:>10.1f} {m.accuracy():>10.4f} {m.precision():>10.4f} {m.recall():>10.4f} {m.f1():>10.4f}")

实际使用

现在用scikit-learn做同样的事。

from sklearn.linear_model import LogisticRegression as SklearnLR
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
from sklearn.metrics import confusion_matrix, classification_report
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
import numpy as np

np.random.seed(42)
X_0 = np.random.randn(100, 2) + [2, 2]
X_1 = np.random.randn(100, 2) + [5, 5]
X_sk = np.vstack([X_0, X_1])
y_sk = np.array([0] * 100 + [1] * 100)

X_tr, X_te, y_tr, y_te = train_test_split(X_sk, y_sk, test_size=0.2, random_state=42)

scaler = StandardScaler()
X_tr_sc = scaler.fit_transform(X_tr)
X_te_sc = scaler.transform(X_te)

lr = SklearnLR()
lr.fit(X_tr_sc, y_tr)
y_pred = lr.predict(X_te_sc)

print("=== Scikit-learn Logistic Regression ===")
print(f"Accuracy:  {accuracy_score(y_te, y_pred):.4f}")
print(f"Precision: {precision_score(y_te, y_pred):.4f}")
print(f"Recall:    {recall_score(y_te, y_pred):.4f}")
print(f"F1:        {f1_score(y_te, y_pred):.4f}")
print(f"\nConfusion Matrix:\n{confusion_matrix(y_te, y_pred)}")
print(f"\nClassification Report:\n{classification_report(y_te, y_pred)}")

你的从零实现产生相同的决策边界和指标。Scikit-learn添加了求解器选项（liblinear、lbfgs、saga）、自动正则化、多类策略（one-vs-rest、multinomial）和数值稳定性优化。

交付成果

本课程产出：

• code/logistic_regression.py - 从零实现逻辑回归及指标

练习

1. 生成一个非线性可分的数据集（如两个同心圆）。训练逻辑回归并观察其失败。然后添加多项式特征(x1^2, x2^2, x1*x2)再训练。展示准确率提升。
2. 为3类softmax模型实现多类混淆矩阵。计算每个类别的精确率和召回率。哪个类别最难分类？
3. 从零构建ROC曲线。对0到1的100个阈值值，计算真正例率和假正例率。用梯形法则计算AUC（曲线下面积）。

关键术语