Standard (Batch)

The problem, from first principles

06-directional-derivatives establishes that the gradient points in the direction of steepest ascent — but it stops there, never actually using that fact to find a minimum. Gradient descent is the direct, obvious consequence: if the gradient points toward the steepest increase, walking a small step in the opposite direction decreases the function, and repeating that over and over eventually reaches a minimum. This question implements the loop itself, not just the formula.

From theory to code

Implement gradient_descent_step(x, gradient_fn, learning_rate), performing a single update, then gradient_descent(x0, gradient_fn, learning_rate, num_steps), running the full loop and returning every intermediate position. The signatures and docstrings are already in the editor.

Constraints

• x may be a scalar float or a NumPy array (the same update rule applies elementwise either way).
• gradient_fn takes x and returns the gradient at that point, same shape as x.
• gradient_descent returns a list of length num_steps + 1: the starting point x0, followed by the position after each of the num_steps updates.

Hints