[Machine learning][Part4] 多维矩阵下的梯度下降线性预测模型的实现
目录
模型初始化信息：
模型实现：
多变量损失函数：
多变量梯度下降实现：
多变量梯度实现：
多变量梯度下降实现：
之前部分实现的梯度下降线性预测模型中的training example只有一个特征属性：房屋面积，这显然是不符合实际情况的，这里增加特征属性的数量再实现一次梯度下降线性预测模型。
这里回顾一下梯度下降线性模型的实现方法：
- 实现线性模型：f = w*x + b，模型参数w,b待定
- 寻找最优的w,b组合：
（1）引入衡量模型优劣的cost function：J(w,b) ——损失函数或者代价函数
（2）损失函数值最小的时候，模型最接近实际情况：通过梯度下降法来寻找最优w,b组合
模型初始化信息：
- 新的房子的特征有：房子面积、卧室数、楼层数、房龄共4个特征属性。
| Size (sqft) | Number of Bedrooms | Number of floors | Age of Home | Price (1000s dollars) |
|---|---|---|---|---|
| 2104 | 5 | 1 | 45 | 460 |
| 1416 | 3 | 2 | 40 | 232 |
| 852 | 2 | 1 | 35 | 17 |
上面表中的训练样本有3个，输入特征矩阵模型为：
具体代码实现为，X_train是输入矩阵，y_train是输出矩阵
X_train = np.array([[2104, 5, 1, 45], [1416, 3, 2, 40],[852, 2, 1, 35]])
y_train = np.array([460, 232, 178])模型参数w,b矩阵：
代码实现：w中的每一个元素对应房屋的一个特征属性
b_init = 785.1811367994083
w_init = np.array([ 0.39133535, 18.75376741, -53.36032453, -26.42131618])模型实现：
def predict(x, w, b): """single predict using linear regressionArgs:x (ndarray): Shape (n,) example with multiple featuresw (ndarray): Shape (n,) model parameters b (scalar): model parameter Returns:p (scalar): prediction"""p = np.dot(x, w) + b return p 多变量损失函数：
J(w,b)为：
代码实现为:
def compute_cost(X, y, w, b): """compute costArgs:X (ndarray (m,n)): Data, m examples with n featuresy (ndarray (m,)) : target valuesw (ndarray (n,)) : model parameters b (scalar) : model parameterReturns:cost (scalar): cost"""m = X.shape[0]cost = 0.0for i in range(m): f_wb_i = np.dot(X[i], w) + b #(n,)(n,) = scalar (see np.dot)cost = cost + (f_wb_i - y[i])**2 #scalarcost = cost / (2 * m) #scalar return cost多变量梯度下降实现：
多变量梯度实现：
def compute_gradient(X, y, w, b): """Computes the gradient for linear regression Args:X (ndarray (m,n)): Data, m examples with n featuresy (ndarray (m,)) : target valuesw (ndarray (n,)) : model parameters b (scalar) : model parameterReturns:dj_dw (ndarray (n,)): The gradient of the cost w.r.t. the parameters w. dj_db (scalar): The gradient of the cost w.r.t. the parameter b. """m,n = X.shape #(number of examples, number of features)dj_dw = np.zeros((n,))dj_db = 0.for i in range(m): err = (np.dot(X[i], w) + b) - y[i] for j in range(n): dj_dw[j] = dj_dw[j] + err * X[i, j] dj_db = dj_db + err dj_dw = dj_dw / m dj_db = dj_db / m return dj_db, dj_dw多变量梯度下降实现：
def gradient_descent(X, y, w_in, b_in, cost_function, gradient_function, alpha, num_iters): """Performs batch gradient descent to learn theta. Updates theta by taking num_iters gradient steps with learning rate alphaArgs:X (ndarray (m,n)) : Data, m examples with n featuresy (ndarray (m,)) : target valuesw_in (ndarray (n,)) : initial model parameters b_in (scalar) : initial model parametercost_function : function to compute costgradient_function : function to compute the gradientalpha (float) : Learning ratenum_iters (int) : number of iterations to run gradient descentReturns:w (ndarray (n,)) : Updated values of parameters b (scalar) : Updated value of parameter """# An array to store cost J and w's at each iteration primarily for graphing laterJ_history = []w = copy.deepcopy(w_in) #avoid modifying global w within functionb = b_infor i in range(num_iters):# Calculate the gradient and update the parametersdj_db,dj_dw = gradient_function(X, y, w, b) ##None# Update Parameters using w, b, alpha and gradientw = w - alpha * dj_dw ##Noneb = b - alpha * dj_db ##None# Save cost J at each iterationif i<100000: # prevent resource exhaustion J_history.append( cost_function(X, y, w, b))# Print cost every at intervals 10 times or as many iterations if < 10if i% math.ceil(num_iters / 10) == 0:print(f"Iteration {i:4d}: Cost {J_history[-1]:8.2f} ")return w, b, J_history #return final w,b and J history for graphing梯度下降算法测试：
# initialize parameters
initial_w = np.zeros_like(w_init)
initial_b = 0.
# some gradient descent settings
iterations = 1000
alpha = 5.0e-7
# run gradient descent
w_final, b_final, J_hist = gradient_descent(X_train, y_train, initial_w, initial_b,compute_cost, compute_gradient, alpha, iterations)
print(f"b,w found by gradient descent: {b_final:0.2f},{w_final} ")
m,_ = X_train.shape
for i in range(m):print(f"prediction: {np.dot(X_train[i], w_final) + b_final:0.2f}, target value: {y_train[i]}")# plot cost versus iteration
fig, (ax1, ax2) = plt.subplots(1, 2, constrained_layout=True, figsize=(12, 4))
ax1.plot(J_hist)
ax2.plot(100 + np.arange(len(J_hist[100:])), J_hist[100:])
ax1.set_title("Cost vs. iteration"); ax2.set_title("Cost vs. iteration (tail)")
ax1.set_ylabel('Cost') ; ax2.set_ylabel('Cost')
ax1.set_xlabel('iteration step') ; ax2.set_xlabel('iteration step')
plt.show()结果为：
可以看到，右图中损失函数在traning次数结束之后还一直在下降，没有找到最佳的w,b组合。具体解决方法，后面会有更新。
完整的代码为：
import copy, math
import numpy as np
import matplotlib.pyplot as pltnp.set_printoptions(precision=2) # reduced display precision on numpy arraysX_train = np.array([[2104, 5, 1, 45], [1416, 3, 2, 40], [852, 2, 1, 35]])
y_train = np.array([460, 232, 178])b_init = 785.1811367994083
w_init = np.array([ 0.39133535, 18.75376741, -53.36032453, -26.42131618])def predict(x, w, b):"""single predict using linear regressionArgs:x (ndarray): Shape (n,) example with multiple featuresw (ndarray): Shape (n,) model parametersb (scalar): model parameterReturns:p (scalar): prediction"""p = np.dot(x, w) + breturn pdef compute_cost(X, y, w, b):"""compute costArgs:X (ndarray (m,n)): Data, m examples with n featuresy (ndarray (m,)) : target valuesw (ndarray (n,)) : model parametersb (scalar) : model parameterReturns:cost (scalar): cost"""m = X.shape[0]cost = 0.0for i in range(m):f_wb_i = np.dot(X[i], w) + b # (n,)(n,) = scalar (see np.dot)cost = cost + (f_wb_i - y[i]) ** 2 # scalarcost = cost / (2 * m) # scalarreturn costdef compute_gradient(X, y, w, b):"""Computes the gradient for linear regressionArgs:X (ndarray (m,n)): Data, m examples with n featuresy (ndarray (m,)) : target valuesw (ndarray (n,)) : model parametersb (scalar) : model parameterReturns:dj_dw (ndarray (n,)): The gradient of the cost w.r.t. the parameters w.dj_db (scalar): The gradient of the cost w.r.t. the parameter b."""m, n = X.shape # (number of examples, number of features)dj_dw = np.zeros((n,))dj_db = 0.for i in range(m):err = (np.dot(X[i], w) + b) - y[i]for j in range(n):dj_dw[j] = dj_dw[j] + err * X[i, j]dj_db = dj_db + errdj_dw = dj_dw / mdj_db = dj_db / mreturn dj_db, dj_dwdef gradient_descent(X, y, w_in, b_in, cost_function, gradient_function, alpha, num_iters):"""Performs batch gradient descent to learn theta. Updates theta by takingnum_iters gradient steps with learning rate alphaArgs:X (ndarray (m,n)) : Data, m examples with n featuresy (ndarray (m,)) : target valuesw_in (ndarray (n,)) : initial model parametersb_in (scalar) : initial model parametercost_function : function to compute costgradient_function : function to compute the gradientalpha (float) : Learning ratenum_iters (int) : number of iterations to run gradient descentReturns:w (ndarray (n,)) : Updated values of parametersb (scalar) : Updated value of parameter"""# An array to store cost J and w's at each iteration primarily for graphing laterJ_history = []w = copy.deepcopy(w_in) # avoid modifying global w within functionb = b_infor i in range(num_iters):# Calculate the gradient and update the parametersdj_db, dj_dw = gradient_function(X, y, w, b) ##None# Update Parameters using w, b, alpha and gradientw = w - alpha * dj_dw ##Noneb = b - alpha * dj_db ##None# Save cost J at each iterationif i < 100000: # prevent resource exhaustionJ_history.append(cost_function(X, y, w, b))# Print cost every at intervals 10 times or as many iterations if < 10if i % math.ceil(num_iters / 10) == 0:print(f"Iteration {i:4d}: Cost {J_history[-1]:8.2f} ")return w, b, J_history # return final w,b and J history for graphing# initialize parameters
initial_w = np.zeros_like(w_init)
initial_b = 0.
# some gradient descent settings
iterations = 1000
alpha = 5.0e-7
# run gradient descent
w_final, b_final, J_hist = gradient_descent(X_train, y_train, initial_w, initial_b,compute_cost, compute_gradient,alpha, iterations)
print(f"b,w found by gradient descent: {b_final:0.2f},{w_final} ")
m,_ = X_train.shape
for i in range(m):print(f"prediction: {np.dot(X_train[i], w_final) + b_final:0.2f}, target value: {y_train[i]}")# plot cost versus iteration
fig, (ax1, ax2) = plt.subplots(1, 2, constrained_layout=True, figsize=(12, 4))
ax1.plot(J_hist)
ax2.plot(100 + np.arange(len(J_hist[100:])), J_hist[100:])
ax1.set_title("Cost vs. iteration"); ax2.set_title("Cost vs. iteration (tail)")
ax1.set_ylabel('Cost') ; ax2.set_ylabel('Cost')
ax1.set_xlabel('iteration step') ; ax2.set_xlabel('iteration step')
plt.show()