你能分别用梯度下降法和牛顿法实现MNIST数据集的训练及分类吗，给出python实现

当然可以！以下是使用梯度下降法和牛顿法实现MNIST数据集的训练及分类的 Python 代码：

首先，我们需要加载 MNIST 数据集，可以使用 `tensorflow` 中的 `keras` 库来加载数据集：

import tensorflow as tf
from tensorflow import keras

(x_train, y_train), (x_test, y_test) = keras.datasets.mnist.load_data()

# 归一化
x_train, x_test = x_train / 255.0, x_test / 255.0

接下来，我们可以使用梯度下降法实现 MNIST 数据集的训练及分类：

import numpy as np

# 将样本矩阵转换为向量
def flatten(X):
    m = X.shape[0]
    return X.reshape(m, -1)

def init(X, y):
    m, n = X.shape
    X = np.hstack((np.ones((m, 1)), X))
    y = y.reshape(-1, 1)
    theta = np.zeros((n+1, 1))
    return X, y, theta

# 计算sigmoid函数
def sigmoid(z):
    return 1 / (1 + np.exp(-z))

# 损失函数
def cost(theta, X, y):
    m = len(y)
    h = sigmoid(X @ theta)
    J = -1/m * (y.T @ np.log(h) + (1-y).T @ np.log(1-h))
    return J

# 梯度下降法
def gradient(theta, X, y):
    m = len(y)
    h = sigmoid(X @ theta)
    grad = 1/m * X.T @ (h - y)
    return grad

def gradient_descent(X, y, theta, alpha, num_iters):
    J_history = np.zeros((num_iters, 1))
    for i in range(num_iters):
        grad = gradient(theta, X, y)
        theta -= alpha * grad
        J_history[i] = cost(theta, X, y)
    return J_history, theta

# 初始化
X_train_flatten = flatten(x_train)
X_test_flatten = flatten(x_test)
X_train, Y_train, theta = init(X_train_flatten, y_train)

# 梯度下降训练
alpha = 0.1
num_iters = 1000
J_history, theta = gradient_descent(X_train, Y_train, theta, alpha, num_iters)

# 预测
X_test, Y_test, _ = init(X_test_flatten, y_test)
y_pred = np.round(sigmoid(X_test @ theta))
accuracy = np.mean(y_pred == Y_test) * 100
print("Accuracy:", accuracy)

接下来，我们可以使用牛顿法实现 MNIST 数据集的训练及分类：

# 计算sigmoid函数
def sigmoid(z):
    return 1 / (1 + np.exp(-z))

# 损失函数
def cost(theta, X, y):
    m = len(y)
    h = sigmoid(X @ theta)
    J = -1/m * (y.T @ np.log(h) + (1-y).T @ np.log(1-h))
    return J

# 梯度
def gradient(theta, X, y):
    m = len(y)
    h = sigmoid(X @ theta)
    grad = 1/m * X.T @ (h - y)
    return grad

# Hessian矩阵
def hessian(theta, X):
    m = X.shape[0]
    h = sigmoid(X @ theta)
    H = 1/m * (X.T @ np.diag(h.ravel()) @ np.diag((1-h).ravel()) @ X)
    return H

# 牛顿法
def newton(theta, X, y):
    J_history = []
    for i in range(10):
        grad = gradient(theta, X, y)
        H = hessian(theta, X)
        theta -= np.linalg.inv(H) @ grad
        J_history.append(cost(theta, X, y))
    return J_history, theta

# 初始化
X_train_flatten = flatten(x_train)
X_test_flatten = flatten(x_test)
X_train, Y_train, theta = init(X_train_flatten, y_train)

# 牛顿法训练
J_history, theta = newton(theta, X_train, Y_train)

# 预测
X_test, Y_test, _ = init(X_test_flatten, y_test)
y_pred = np.round(sigmoid(X_test @ theta))
accuracy = np.mean(y_pred == Y_test) * 100
print("Accuracy:", accuracy)

以上就是使用梯度下降法和牛顿法实现MNIST数据集的训练及分类的 Python 代码。需要注意的是，这里我们只是简单地使用了逻辑回归模型对 MNIST 数据集进行分类，在实际应用中可能需要更加复杂的模型和算法来提高分类准确率。