Train a binary classifier that is linear regression with one sigmoid bolted on, without sklearn. The skeleton matches [[algo/mle-interview/linear-regression-scratch|linear regression]] — only the prediction function and the gradient constant differ.

Input

X  # (n, d) real-valued features. A list of lists or a numpy array
y  # (n,)  0 or 1

If X arrives one-dimensional, treat it as (n, 1).

Model

Prepend a column of ones to the design matrix — w[0] is the intercept and the weight vector has length d + 1.

$\sigma(z) = \frac{1}{1 + e^{-z}}, \qquad \hat{p} = \sigma(Aw)$

Training rule

Exactly two things differ from linear regression. Miss either one and every number changes.

• The prediction is sigmoid(Aw), not Aw.

• The log-loss gradient has no constant 2:

  $\nabla_w L = \frac{1}{n}A^\top(\sigma(Aw) - y)$

Everything else is the same — start from the zero vector and apply w = w - lr * grad exactly epochs times.

Return

• fit_logistic_regression(X, y, lr, epochs) → a list of length d+1
• Round every float to 6 decimal places.

There is no randomness — the same input always produces the same numbers.

Implement fit_logistic_regression(X, y, lr, epochs).

• Return a list of length d+1, w[0] is the intercept, start from the zero vector.
• The prediction is sigmoid(Aw) and the gradient is A.T @ (sigmoid(Aw) - y) / n — no constant 2. Copying it straight from linear regression is exactly where this goes wrong.
• Round each element with round(v, 6).