Problem

At Stripe, you need a tiny training routine for a linear pricing model. Implement batch gradient descent for linear regression with an intercept (bias).

Specification

Write train_linear_regression(X, y, lr, epochs) where:

• X is a 2D list of floats with shape (n_samples, n_features)
• y is a list of floats with length n_samples
• lr is a positive float learning rate
• epochs is a positive integer

Return a dictionary:

• {"weights": [bias, w1, ..., wd], "preds": [p0, ..., p(n-1)]} where preds[i] is the prediction for X[i] using the final weights.

Raise ValueError if:

1. X or y is empty, or len(X) != len(y)
2. X is ragged (rows have different lengths) or has zero features

Examples

Example 1 Input: X = [[1.0],[2.0],[3.0]], y = [2.0,4.0,6.0], lr = 0.1, epochs = 2000 Output: weights ≈ [0.0, 2.0], preds ≈ [2.0, 4.0, 6.0] Explanation: Perfect line y = 2x.

Example 2 Input: X = [[0.0],[1.0]], y = [1.0,3.0], lr = 0.1, epochs = 2000 Output: weights ≈ [1.0, 2.0], preds ≈ [1.0, 3.0] Explanation: Line y = 1 + 2x.

Constraints

• 1 <= n_samples <= 2 * 10^4
• 1 <= n_features <= 50
• 1 <= epochs <= 10^4
• 0 < lr <= 1
• All values in X and y are finite floats in [-1e6, 1e6]