用下降搜索法GOLDSTEIN和牛顿方向确定多元向量关于级数1到n的Xi(yi-e^bx/z(1+e^bx))极大似然代码

这是一个比较复杂的问题，需要先给出具体的数学模型和目标函数。以下是一个可能的实现：

假设我们有 $n$ 组数据 $(x_1, y_1), (x_2, y_2), \cdots, (x_n, y_n)$，其中 $x_i$ 和 $y_i$ 都是实数。我们希望找到一个向量 $\mathbf{w}=(w_1, w_2, \cdots, w_n)^T$，使得它能够最大化下面的似然函数：

$$L(\mathbf{w})=\prod_{i=1}^n p(y_i|x_i, \mathbf{w})=\prod_{i=1}^n \frac{1}{z}\frac{e^{w_ix_i}}{1+e^{w_ix_i}}$$

其中 $z=\prod_{i=1}^n (1+e^{w_ix_i})$ 是归一化因子。我们可以将似然函数取对数并加上负号，得到下面的损失函数：

$$J(\mathbf{w})=-\sum_{i=1}^n \log\left(\frac{1}{z}\frac{e^{w_ix_i}}{1+e^{w_ix_i}}\right)=\sum_{i=1}^n \log(1+e^{-w_ix_i})-\log z$$

我们的目标是求出使得 $J(\mathbf{w})$ 最小化的向量 $\mathbf{w}$。为了实现这一点，我们可以使用梯度下降算法或者牛顿法。

首先考虑梯度下降法。我们需要求出损失函数的梯度向量：

$$\nabla J(\mathbf{w})=\left(\frac{\partial J(\mathbf{w})}{\partial w_1}, \frac{\partial J(\mathbf{w})}{\partial w_2}, \cdots, \frac{\partial J(\mathbf{w})}{\partial w_n}\right)^T$$

我们可以使用链式法则求出每个分量的导数：

$$\frac{\partial J(\mathbf{w})}{\partial w_i}=\frac{\partial}{\partial w_i} \log(1+e^{-w_ix_i})-\frac{\partial}{\partial w_i} \log z$$

$$=\frac{-x_ie^{-w_ix_i}}{1+e^{-w_ix_i}}-\frac{1}{z}\frac{\partial z}{\partial w_i}$$

其中：

$$\frac{\partial z}{\partial w_i}=\prod_{j=1}^n (1+e^{w_jx_j})\frac{\partial}{\partial w_i}(1+e^{w_ix_i})$$

$$=\prod_{j=1}^n (1+e^{w_jx_j})\cdot e^{w_ix_i}\cdot \frac{x_i}{1+e^{w_ix_i}}$$

将上式代入前面的式子，得到：

$$\frac{\partial J(\mathbf{w})}{\partial w_i}=-x_i\frac{1}{1+e^{w_ix_i}}+\frac{x_ie^{w_ix_i}}{z}=\frac{x_i}{z}(e^{w_ix_i}-y_i)$$

因此，梯度向量可以写成：

$$\nabla J(\mathbf{w})=\left(\frac{x_1}{z}(e^{w_1x_1}-y_1), \frac{x_2}{z}(e^{w_2x_2}-y_2), \cdots, \frac{x_n}{z}(e^{w_nx_n}-y_n)\right)^T$$

然后我们可以使用标准的梯度下降算法来更新参数：

$$\mathbf{w} \leftarrow \mathbf{w} - \eta \nabla J(\mathbf{w})$$

其中 $\eta$ 是学习率。具体的实现细节可以参考下面的代码：

import numpy as np

# define the objective function
def J(w, x, y):
    z = np.prod(1 + np.exp(w * x))
    return np.sum(np.log(1 + np.exp(-w * x))) - np.log(z)

# define the gradient of the objective function
def grad_J(w, x, y):
    z = np.prod(1 + np.exp(w * x))
    return np.array([x[i] / z * (np.exp(w[i] * x[i]) - y[i]) for i in range(len(x))])

# define the gradient descent algorithm
def gradient_descent(x, y, w0, eta, epsilon, max_iter):
    w = w0
    for i in range(max_iter):
        grad = grad_J(w, x, y)
        w_new = w - eta * grad
        if np.linalg.norm(w_new - w) < epsilon:
            break
        w = w_new
    return w

# generate some data for testing
n = 100
x = np.random.randn(n)
w_true = np.random.randn(n)
y = 1 / (1 + np.exp(-w_true * x)) + 0.01 * np.random.randn(n)

# run the gradient descent algorithm
w0 = np.zeros(n)
eta = 0.1
epsilon = 1e-6
max_iter = 1000
w_est = gradient_descent(x, y, w0, eta, epsilon, max_iter)

# print the results
print("True parameters:", w_true)
print("Estimated parameters:", w_est)

接下来考虑牛顿法。我们需要求出损失函数的一阶导数和二阶导数：

$$\nabla J(\mathbf{w})=\left(\frac{x_1}{z}(e^{w_1x_1}-y_1), \frac{x_2}{z}(e^{w_2x_2}-y_2), \cdots, \frac{x_n}{z}(e^{w_nx_n}-y_n)\right)^T$$

$$\nabla^2 J(\mathbf{w})=\begin{pmatrix} \frac{\partial^2 J(\mathbf{w})}{\partial w_1^2} & \frac{\partial^2 J(\mathbf{w})}{\partial w_1 \partial w_2} & \cdots & \frac{\partial^2 J(\mathbf{w})}{\partial w_1 \partial w_n} \\ \frac{\partial^2 J(\mathbf{w})}{\partial w_2 \partial w_1} & \frac{\partial^2 J(\mathbf{w})}{\partial w_2^2} & \cdots & \frac{\partial^2 J(\mathbf{w})}{\partial w_2 \partial w_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial^2 J(\mathbf{w})}{\partial w_n \partial w_1} & \frac{\partial^2 J(\mathbf{w})}{\partial w_n \partial w_2} & \cdots & \frac{\partial^2 J(\mathbf{w})}{\partial w_n^2} \end{pmatrix}$$

其中：

$$\frac{\partial^2 J(\mathbf{w})}{\partial w_i^2}=\frac{x_i^2 e^{w_ix_i}}{z(1+e^{w_ix_i})^2}$$

$$\frac{\partial^2 J(\mathbf{w})}{\partial w_i \partial w_j}=-\frac{x_ix_je^{w_ix_i}e^{w_jx_j}}{z(1+e^{w_ix_i})(1+e^{w_jx_j})}$$

牛顿法的迭代公式为：

$$\mathbf{w} \leftarrow \mathbf{w} - (\nabla^2 J(\mathbf{w}))^{-1} \nabla J(\mathbf{w})$$

具体的实现细节可以参考下面的代码：

import numpy as np

# define the objective function
def J(w, x, y):
    z = np.prod(1 + np.exp(w * x))
    return np.sum(np.log(1 + np.exp(-w * x))) - np.log(z)

# define the gradient of the objective function
def grad_J(w, x, y):
    z = np.prod(1 + np.exp(w * x))
    return np.array([x[i] / z * (np.exp(w[i] * x[i]) - y[i]) for i in range(len(x))])

# define the Hessian matrix of the objective function
def hess_J(w, x, y):
    z = np.prod(1 + np.exp(w * x))
    hess = np.zeros((len(w), len(w)))
    for i in range(len(w)):
        for j in range(len(w)):
            if i == j:
                hess[i, j] = np.sum(x[i]**2 * np.exp(w[i] * x[i]) / (z * (1 + np.exp(w[i] * x[i]))**2))
            else:
                hess[i, j] = -np.sum(x[i] * x[j] * np.exp(w[i] * x[i]) * np.exp(w[j] * x[j]) / (z * (1 + np.exp(w[i] * x[i])) * (1 + np.exp(w[j] * x[j]))))
    return hess

# define the Newton's method algorithm
def newton_method(x, y, w0, epsilon, max_iter):
    w = w0
    for i in range(max_iter):
        grad = grad_J(w, x, y)
        hess_inv = np.linalg.inv(hess_J(w, x, y))
        w_new = w - np.dot(hess_inv, grad)
        if np.linalg.norm(w_new - w) < epsilon:
            break
        w = w_new
    return w

# generate some data for testing
n = 100
x = np.random.randn(n)
w_true = np.random.randn(n)
y = 1 / (1 + np.exp(-w_true * x)) + 0.01 * np.random.randn(n)

# run the Newton's method algorithm
w0 = np.zeros(n)
epsilon = 1e-6
max_iter = 100
w_est = newton_method(x, y, w0, epsilon, max_iter)

# print the results
print("True parameters:", w_true)
print("Estimated parameters:", w_est)

需要注意的是，牛顿法的收敛速度通常比梯度下降法快，但是每次迭代的计算量更大。因此，在实际应用中需要根据具体情况选择不同的优化算法和参数。