Implement AUC-ROC with correct tie handling, a numerically stable softmax, and logistic regression trained with cross-entropy and L2 regularization.

AUC is the chance a random positive outscores a random negative, so compute it from average ranks, which counts ties as half.

def auc(y, s):
    order = np.argsort(s, kind="mergesort")
    ss = s[order]
    ranks = np.empty(len(s))
    i = 0
    while i < len(s):          # average rank per tie group
        j = i
        while j + 1 < len(s) and ss[j + 1] == ss[i]:
            j += 1
        ranks[order[i:j + 1]] = (i + j) / 2 + 1
        i = j + 1
    pos = y == 1
    n_pos, n_neg = pos.sum(), (~pos).sum()
    u = ranks[pos].sum() - n_pos * (n_pos + 1) / 2
    return u / (n_pos * n_neg)

def softmax(z):
    z = z - z.max(axis=-1, keepdims=True)
    e = np.exp(z)
    return e / e.sum(axis=-1, keepdims=True)

def fit_logreg(X, y, lam=0.1, lr=0.1, steps=2000):
    w, b = np.zeros(X.shape[1]), 0.0
    for _ in range(steps):
        z = X @ w + b
        p = 0.5 * (1 + np.tanh(z / 2))   # stable sigmoid
        g = p - y
        w -= lr * (X.T @ g / len(y) + lam * w)  # no L2 on b
        b -= lr * g.mean()
    return w, b

Interviewers check ties, overflow on large logits, which subtracting the max prevents, and that the bias isn't regularised.