基于Flux.jl实现逻辑回归：求解最优参数θ并完成测试数据预测

Alright, let's break down how to tackle this logistic regression task step by step. Since you already have the loss function defined, our core focus will be minimizing it to find the optimal parameter θ, then using that θ to predict admission results for your test data. No fancy nonlinear networks needed here—stick to the linear logistic regression setup.

1. Data Preprocessing First #

Before diving into optimization, we need to get our data ready:

• Add a bias term: Your input features (the two subject scores) should be augmented with a column of 1s. This corresponds to the θ₀ parameter in the logistic regression model, so if your raw feature matrix is X_raw (shape (m, 2) where m is the number of students), your processed X will be (m, 3) with the first column all 1s.
• Align dimensions: Make sure your label vector y (admission results, 0 or 1) has a shape that matches the input—usually (m, 1) or (m,) depending on your implementation.
• Optional (but recommended) feature scaling: Normalize your subject scores (e.g., using Z-score normalization) to help optimization algorithms converge faster, especially if you're using gradient descent.

2. Minimize the Loss Function to Get Optimal θ #

Since your loss function is already defined, you can use one of two common approaches to find the θ that minimizes it:

Option A: Gradient Descent (Manual Implementation) #

If you want to build the update loop yourself:

1. Compute the gradient of the loss function with respect to θ. For standard logistic regression cross-entropy loss, the gradient is (1/m) * X.T @ (sigmoid(X@θ) - y) (where m is the number of samples).
2. Iteratively update θ using the rule: θ = θ - α * gradient, where α is your learning rate.
3. Stop iterating when the loss stops decreasing significantly (or after a fixed number of epochs).

Example code snippet:

import numpy as np

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

# Assume your pre-defined loss function is called compute_loss
def compute_gradient(theta, X, y):
    m = len(y)
    h = sigmoid(X @ theta)
    gradient = (1/m) * X.T @ (h - y)
    return gradient

# Initialize θ to zeros
theta = np.zeros(X.shape[1])
alpha = 0.01  # Adjust this based on convergence behavior
epochs = 10000

for _ in range(epochs):
    gradient = compute_gradient(theta, X, y)
    theta -= alpha * gradient
    # Optional: Print loss every N epochs to monitor progress

Option B: Use a Pre-built Optimizer (Easier & More Efficient) #

For a quicker, more robust solution, use scipy.optimize.minimize—it handles the optimization heavy lifting for you, even if you don't want to compute gradients manually (though you can provide them if you want).

Example code snippet:

import numpy as np
from scipy.optimize import minimize

# Your pre-defined loss function (example shown here if you need reference)
def compute_loss(theta, X, y):
    m = len(y)
    h = sigmoid(X @ theta)
    loss = (-1/m) * np.sum(y * np.log(h) + (1 - y) * np.log(1 - h))
    return loss

# Initialize θ
theta_init = np.zeros(X.shape[1])

# Run optimization
result = minimize(compute_loss, theta_init, args=(X, y), method='BFGS')
optimal_theta = result.x  # This is your best θ

3. Predict Admission Results for Test Data #

Once you have optimal_theta, predicting is straightforward:

1. Preprocess your test data the same way you did the training data—add the bias term column of 1s.
2. Compute the sigmoid output for each test sample: this gives the probability of admission.
3. Apply a threshold (typically 0.5) to convert probabilities to binary labels (1 for admitted, 0 for not admitted).

Example code:

# Preprocess test features
X_test_processed = np.hstack((np.ones((X_test_raw.shape[0], 1)), X_test_raw))

# Calculate admission probabilities
admission_probs = sigmoid(X_test_processed @ optimal_theta)

# Convert to binary predictions
y_pred = np.where(admission_probs >= 0.5, 1, 0)

Quick Notes to Keep in Mind #

• If using gradient descent, tune your learning rate α carefully: too small and convergence is slow; too large and the loss might oscillate or diverge.
• Double-check that your loss function is correctly implemented (it should return a scalar value, and decrease as θ gets better).
• If you have missing values in your dataset, handle them first (e.g., impute with mean/median, or drop samples) before starting optimization.

内容的提问来源于stack exchange，提问作者Akshay Sharma