input: Write a function to implement a logistic regression classifier in Python to classify handwritten digits from the MNIST dataset. The function should take the MNIST dataset as input and return the accuracy of the classifier on the training, validation, and test sets. The MNIST dataset should be preprocessed by appending a column of ones to the feature matrices for intercept terms.
  
  assert LogisticRegression(train_data, valid_data, test_data, 1000, 0.001) == (0.91, 0.90, 0.92)
output: def LogisticRegression(train_data, valid_data, test_data, num_iterations, learning_rate):
      import numpy as np
      from sklearn.datasets import fetch_openml
      from sklearn.preprocessing import OneHotEncoder
  
      def sigmoid(z):
          return 1 / (1 + np.exp(-z))
  
      def compute_cost(X, y, theta):
          m = len(y)
          h = sigmoid(X @ theta)
          cost = (-1/m) * np.sum(y * np.log(h) + (1 - y) * np.log(1 - h))
          return cost
  
      def gradient_descent(X, y, theta, learning_rate, num_iterations):
          m = len(y)
          cost_history = []
          for _ in range(num_iterations):
              h = sigmoid(X @ theta)
              gradient = (1/m) * X.T @ (h - y)
              theta -= learning_rate * gradient
              cost_history.append(compute_cost(X, y, theta))
          return theta, cost_history
  
      def predict(X, theta):
          return np.round(sigmoid(X @ theta))
  
      def accuracy(X, y, theta):
          predictions = predict(X, theta)
          return np.mean(predictions == y)
  
      # Preprocess data
      X_train, y_train = train_data
      X_valid, y_valid = valid_data
      X_test, y_test = test_data
  
      X_train = np.hstack((np.ones((X_train.shape[0], 1)), X_train))
      X_valid = np.hstack((np.ones((X_valid.shape[0], 1)), X_valid))
      X_test = np.hstack((np.ones((X_test.shape[0], 1)), X_test))
  
      # One-hot encode labels
      encoder = OneHotEncoder(sparse=False)
      y_train_encoded = encoder.fit_transform(y_train.reshape(-1, 1))
      y_valid_encoded = encoder.transform(y_valid.reshape(-1, 1))
      y_test_encoded = encoder.transform(y_test.reshape(-1, 1))
  
      # Initialize theta
      theta = np.zeros((X_train.shape[1], y_train_encoded.shape[1]))
  
      # Train model
      theta, _ = gradient_descent(X_train, y_train_encoded, theta, learning_rate, num_iterations)
  
      # Calculate accuracy
      train_accuracy = accuracy(X_train, y_train_encoded, theta)
      valid_accuracy = accuracy(X_valid, y_valid_encoded, theta)
      test_accuracy = accuracy(X_test, y_test_encoded, theta)
  
      return train_accuracy, valid_accuracy, test_accuracy