多元线性回归成本太高

Question

我正在使用 this link、imports-85.data 中提供的数据集进行价格预测。

使用 horsepower、curb-weight、engine-size 和 highway-mpg，我尝试归一化（由于成本高）和运行梯度下降通过实现以下算法：

初始化

data = df[attrs]
m = len(data) # m-training examples
f = len(attrs) # n-features
X = np.hstack((np.ones(shape=(m,1)),np.array(data)))
T = np.zeros(f + 1) # Coefficients of x(0),x(1),...x(n)
norm_price = df.price / 1000
Y = np.array(norm_price)

# Normalization
data['curb-weight'] = (data['curb-weight'] * 0.453592) / 1000    # To kg (e-1000)
data['highway-mpg'] = data['highway-mpg'] * 0.425144    # To km per litre (kml)
data['engine-size'] = data['engine-size'] / 100     # To e-100
data['horsepower'] = data['horsepower'] / 100   # To e-100

col_rename = {
    'curb-weight':'curb-weight-kg(e-1000)',
    'highway-mpg':'highway-kml',
    'engine-size':'engine-size(e-100)',
    'horsepower':'horsepower(e-100)'
}
data.rename(columns=col_rename,inplace=True)

成本计算

def calculateCost():
    global m,T,X
    hypot = (X.dot(T) - Y).transpose().dot(X.dot(T) - Y)
    return hypot / (2 * m)

梯度下降

def gradDescent(threshold,iter = 10000,alpha = 3e-8):
    global T,X,Y,m
    i = 0
    cost = calculateCost()
    cost_hist = [cost]
    while i < iter:
        T = T - (alpha / m) * X.transpose().dot(X.dot(T) - Y)
        cost = calculateCost()
        cost_hist.append(cost)
        i += 1
        if cost <= threshold:
            return cost_hist

我运行使用此实现的梯度下降： Batch Gradient Descent

如果没有规范化，成本将为 118634960.460199。通过规范化，成本将为 118.634960460199

因此，我有几个问题：

我的标准化技术正确吗？
标准化后，成本会有所不同。归一化后的成本阈值如何设置？

Answer 1

我认为您在机器学习的上下文中可能存在误解 'normalization'。根据我对您代码的解释，您的 'normalization' 部分正在进行单位转换。在梯度下降之前，通常应用最大最小缩放或标准缩放，请参阅 scikit learn user guide. These techniques create features with a consistent scale range, so that changes in a single feature do not completely dominate the loss function. This question and this blog post 有更长的讨论。

多元线性回归成本太高

Multivariate Linear Regression Cost Too High

python

machine-learning

linear-regression

gradient-descent

non-linear-regression