用下降搜索法GOLDSTEIN和牛顿方向确定多元向量关于级数i是1到n的xi[yi- e^bx/(1+e^bx)],其中B是多元向量极大似然代码Python代码

以下是使用下降搜索法GOLDSTEIN和牛顿方向确定多元向量关于级数i是1到n的xi[yi- e^bx/(1+e^bx)]的Python代码：

import numpy as np

def logistic(x, b):
    return 1 / (1 + np.exp(-np.dot(x, b)))

def likelihood_grad(x, y, b):
    p = logistic(x, b)
    return np.dot(x.T, y - p)

def likelihood_hessian(x, b):
    p = logistic(x, b)
    w = p * (1 - p)
    return np.dot(x.T * w, x)

def goldstein_search(x, y, b, grad, direction):
    alpha = 1
    c = 1e-4
    rho = 0.9
    while True:
        new_b = b + alpha * direction
        f_new = np.sum(np.log(1 + np.exp(np.dot(x, new_b)))) - np.dot(y, np.dot(x, new_b))
        f = np.sum(np.log(1 + np.exp(np.dot(x, b)))) - np.dot(y, np.dot(x, b))
        if f_new <= f + c * alpha * np.dot(grad, direction):
            return alpha
        else:
            alpha *= rho

def newton_direction(x, y, b):
    grad = likelihood_grad(x, y, b)
    hessian = likelihood_hessian(x, b)
    return np.linalg.solve(hessian, grad)

def gradient_descent(x, y, b_init, max_iter=1000, tol=1e-6):
    b = b_init
    for i in range(max_iter):
        grad = likelihood_grad(x, y, b)
        if np.linalg.norm(grad) < tol:
            break
        direction = newton_direction(x, y, b)
        alpha = goldstein_search(x, y, b, grad, direction)
        b += alpha * direction
    return b

# Example usage:
x = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
y = np.array([1, 0, 1])
b_init = np.zeros(3)
b = gradient_descent(x, y, b_init)
print(b)

这段代码实现了使用下降搜索法GOLDSTEIN和牛顿方向来确定多元向量关于级数i是1到n的xi[yi- e^bx/(1+e^bx)]的最大似然估计。其中，logistic函数是sigmoid函数，用于计算模型输出的概率；likelihood_grad函数和likelihood_hessian函数分别计算对数似然函数的梯度和海森矩阵；goldstein_search函数使用GOLDSTEIN搜索法来寻找合适的步长；newton_direction函数使用牛顿法来计算搜索方向；gradient_descent函数使用梯度下降法来最小化对数似然函数，返回最大似然估计的系数向量。