深度学习Note.5(机器学习2）

多项式回归

1.与线性回归联系：

与线性回归大体相似，代码部分复用性高，不同点：公式中为x的次方，可能要规范化处理。

2.paddle的API

paddle.sin(x, name=None)

功能：计算输入的正弦值。

输入：输入Tensor

输出：Tensor,输入的sin值
paddle.ones(shape, dtype=None)

功能：创建建指定形状且值全为1的Tensor

输入：输出结果形状

输出：值全为1的Tensor
paddle.multiply(x, y, name=None)

功能：逐元素相乘

输入：2个Tensor

输出:Tensor,相乘结果
paddle.concat(x, axis=0, name=None)

功能：对输入沿axis轴进行联结

输入：待联结的Tensorlist域者Tensortuple和运算轴

输出：联结后的Tensor

3.模型--sin(2*pi*x）

3. 1数据集构建

import math

# sin函数: sin(2 * pi * x)
def sin(x):
    y = paddle.sin(2 * math.pi * x)
    return y

用前面定义的create_toy_data函数来构建训练和测试数据\

import math
import paddle
import numpy as np
from matplotlib import pyplot as plt

def sin(x):
    y = paddle.sin(2 * math.pi * x)
    return y

def create_toy_data(func, interval, sample_num, noise=0.0, add_outlier=False, outlier_ratio=0.01):
    X = paddle.rand(shape=[sample_num]) * (interval[1] - interval[0]) + interval[0]
    y = func(X)
    epsilon = paddle.normal(0, noise, shape=[y.shape[0]])
    y += epsilon
    y_np = y.numpy()  # 转换为 NumPy 数组
    
    if add_outlier:
        outlier_num = max(1, int(len(y_np) * outlier_ratio))
        outlier_idx = paddle.randint(len(y_np), shape=[outlier_num]).numpy()
        y_np[outlier_idx] *= 5  # 直接操作 NumPy 数组
    
    return X.numpy(), y_np

# 生成数据
func = sin
interval = (0, 1)
train_num = 15
test_num = 10
noise = 0.5

X_train, y_train = create_toy_data(func=func, interval=interval, sample_num=train_num, noise=noise)
X_test, y_test = create_toy_data(func=func, interval=interval, sample_num=test_num, noise=noise)

# 生成用于绘图的基准数据（转换为 NumPy）
X_underlying = paddle.linspace(interval[0], interval[1], num=100).numpy()
y_underlying = sin(paddle.to_tensor(X_underlying)).numpy()

# 绘制图像
plt.rcParams['figure.figsize'] = (8.0, 6.0)
plt.scatter(X_train, y_train, facecolor="none", edgecolor='#e4007f', s=50, label="train data")
plt.plot(X_underlying, y_underlying, c='#000000', label=r"$\sin(2\pi x)$")
plt.legend(fontsize='x-large')
plt.savefig('ml-vis2.pdf')
plt.show()

结果：

3.2模型构建

def polynomial_basis_function(x, degree = 2):
    """
    输入：
       - x: tensor, 输入的数据，shape=[N,1]
       - degree: int, 多项式的阶数
       example Input: [[2], [3], [4]], degree=2
       example Output: [[2^1, 2^2], [3^1, 3^2], [4^1, 4^2]]
       注意：本案例中,在degree>=1时不生成全为1的一列数据；degree为0时生成形状与输入相同，全1的Tensor
    输出：
       - x_result： tensor
    """
    
    if degree==0:
        return paddle.ones(shape = x.shape,dtype='float32') 

    x_tmp = x
    x_result = x_tmp

    for i in range(2, degree+1):
        x_tmp = paddle.multiply(x_tmp,x) # 逐元素相乘
        x_result = paddle.concat((x_result,x_tmp),axis=-1)

    return x_result

# 简单测试
data = [[2], [3], [4]]
X = paddle.to_tensor(data = data,dtype='float32')
degree = 3
transformed_X = polynomial_basis_function(X,degree=degree)
print("转换前：",X)
print("阶数为",degree,"转换后：",transformed_X)

转换前： Tensor(shape=[3, 1], dtype=float32, place=CPUPlace, stop_gradient=True,
       [[2.],
        [3.],
        [4.]])
阶数为 3 转换后： Tensor(shape=[3, 3], dtype=float32, place=CPUPlace, stop_gradient=True,
       [[2. , 4. , 8. ],
        [3. , 9. , 27.],
        [4. , 16., 64.]])

3.3模型训练

plt.rcParams['figure.figsize'] = (12.0, 8.0)

for i, degree in enumerate([0, 1, 3, 8]): # []中为多项式的阶数
    model = Linear(degree)
    X_train_transformed = polynomial_basis_function(X_train.reshape([-1,1]), degree)
    X_underlying_transformed = polynomial_basis_function(X_underlying.reshape([-1,1]), degree)
    
    model = optimizer_lsm(model,X_train_transformed,y_train.reshape([-1,1])) #拟合得到参数

    y_underlying_pred = model(X_underlying_transformed).squeeze()

    print(model.params)
    
    # 绘制图像
    plt.subplot(2, 2, i + 1)
    plt.scatter(X_train, y_train, facecolor="none", edgecolor='#e4007f', s=50, label="train data")
    plt.plot(X_underlying, y_underlying, c='#000000', label=r"$\sin(2\pi x)$")
    plt.plot(X_underlying, y_underlying_pred, c='#f19ec2', label="predicted function")
    plt.ylim(-2, 1.5)
    plt.annotate("M={}".format(degree), xy=(0.95, -1.4))

#plt.legend(bbox_to_anchor=(1.05, 0.64), loc=2, borderaxespad=0.)
plt.legend(loc='lower left', fontsize='x-large')
plt.savefig('ml-vis3.pdf')
plt.show()

分析：当阶数太小，拟合曲线简单，欠拟合。

当阶数太大，拟合曲线复杂，过拟合。

3.4模型评估

# 训练误差和测试误差
training_errors = []
test_errors = []
distribution_errors = []

# 遍历多项式阶数
for i in range(9):
    model = Linear(i)

    X_train_transformed = polynomial_basis_function(X_train.reshape([-1,1]), i) 
    X_test_transformed = polynomial_basis_function(X_test.reshape([-1,1]), i) 
    X_underlying_transformed = polynomial_basis_function(X_underlying.reshape([-1,1]), i)
    
    optimizer_lsm(model,X_train_transformed,y_train.reshape([-1,1]))
    
    y_train_pred = model(X_train_transformed).squeeze()
    y_test_pred = model(X_test_transformed).squeeze()
    y_underlying_pred = model(X_underlying_transformed).squeeze()

    train_mse = mean_squared_error(y_true=y_train, y_pred=y_train_pred).item()
    training_errors.append(train_mse)

    test_mse = mean_squared_error(y_true=y_test, y_pred=y_test_pred).item()
    test_errors.append(test_mse)

    #distribution_mse = mean_squared_error(y_true=y_underlying, y_pred=y_underlying_pred).item()
    #distribution_errors.append(distribution_mse)

print ("train errors: \n",training_errors)
print ("test errors: \n",test_errors)
#print ("distribution errors: \n", distribution_errors)

# 绘制图片
plt.rcParams['figure.figsize'] = (8.0, 6.0)
plt.plot(training_errors, '-.', mfc="none", mec='#e4007f', ms=10, c='#e4007f', label="Training")
plt.plot(test_errors, '--', mfc="none", mec='#f19ec2', ms=10, c='#f19ec2', label="Test")
#plt.plot(distribution_errors, '-', mfc="none", mec="#3D3D3F", ms=10, c="#3D3D3F", label="Distribution")
plt.legend(fontsize='x-large')
plt.xlabel("degree")
plt.ylabel("MSE")
plt.savefig('ml-mse-error.pdf')
plt.show()

当阶数较低的时候，模型的表示能力有限，训练误差和测试误差都很高，代表模型欠拟合；

当阶数较高的时候，模型表示能力强，但将训练数据中的噪声也作为特征进行学习，一般情况下训练误差继续降低而测试误差显著升高，代表模型过拟合。

如何解决？

引入正则化方法，通过向误差函数中添加一个惩罚项来避免系数倾向于较大的取值

degree = 8 # 多项式阶数
reg_lambda = 0.0001 # 正则化系数

X_train_transformed = polynomial_basis_function(X_train.reshape([-1,1]), degree)
X_test_transformed = polynomial_basis_function(X_test.reshape([-1,1]), degree)
X_underlying_transformed = polynomial_basis_function(X_underlying.reshape([-1,1]), degree)

model = Linear(degree) 

optimizer_lsm(model,X_train_transformed,y_train.reshape([-1,1]))

y_test_pred=model(X_test_transformed).squeeze()
y_underlying_pred=model(X_underlying_transformed).squeeze()

model_reg = Linear(degree) 

optimizer_lsm(model_reg,X_train_transformed,y_train.reshape([-1,1]),reg_lambda=reg_lambda)

y_test_pred_reg=model_reg(X_test_transformed).squeeze()
y_underlying_pred_reg=model_reg(X_underlying_transformed).squeeze()

mse = mean_squared_error(y_true = y_test, y_pred = y_test_pred).item()
print("mse:",mse)
mes_reg = mean_squared_error(y_true = y_test, y_pred = y_test_pred_reg).item()
print("mse_with_l2_reg:",mes_reg)

# 绘制图像
plt.scatter(X_train, y_train, facecolor="none", edgecolor="#e4007f", s=50, label="train data")
plt.plot(X_underlying, y_underlying, c='#000000', label=r"$\sin(2\pi x)$")
plt.plot(X_underlying, y_underlying_pred, c='#e4007f', linestyle="--", label="$deg. = 8$")
plt.plot(X_underlying, y_underlying_pred_reg, c='#f19ec2', linestyle="-.", label="$deg. = 8, \ell_2 reg$")
plt.ylim(-1.5, 1.5)
plt.annotate("lambda={}".format(reg_lambda), xy=(0.82, -1.4))
plt.legend(fontsize='large')
plt.savefig('ml-vis4.pdf')
plt.show()