In the gradient descent algorithm, $>0$ is the learning rate. In practice, we may anneal $,$ meaning that we start from a relatively large $,$ but decrease it gradually.

Show that cannot be decreased too fast. If is decreased too fast, even if it is strictly positive, the gradient descent algorithm may not converge to the optimum of a convex function.

Hint: Show a specific loss and an annealing scheduler such that the gradient descent algorithm fails to converge to the optimum.