Problem 6. Let f be convex and L-smooth. Consider the gradient descent algorithm on the function f: xk+1 = x^ - ntf(x*), Vi ∈ {0,1,...} Let x* be the minimizer of

f, i.e., f(x) ≤ f(x) for all x. (a) Show that ||x+¹ − x" || ² ≤ ||x − xª ||² +n²³|| ▼ ƒ (x¹)||²2 + 2n(f(x") - f(x²)). (6 marks) (1) (b) Show that if n ≤ 1/L, the sequence {|x* x* 20 is decreasing. (9 marks) Hint: you can use the following property of f: Vf(x) ||² ≤ 2L(f(x) = f(x*)) for any x.