Let f be a differentiable and μ-strongly-convex function whose minimum is achieved at x*. Let us assume that the variance on the gradients is controlled: There exists σ > 0 and L≥ 0 such that E; [||Vf; (x)||² | xk] ≤0² + L ||xk − x* ||². Prove the following statements: 1. If σ > 0 and L = 0, SGD with step size ŋk satisfies | E || xo - Ef (zk) - f*] ≤ =0 (1) 2 Στο where Σj=0jxj Zk= (2) ΣΕ In particular, E[f (zk) - f*] converges to 0 if and only if Σ, n; = ∞ and 2. If σ > 0 and L > 0, SGD with a constant step size n satisfies = 0. E||xk+1 - x* ||² ≤ (1 - 2nμ+n²L)*E ||xo-x* ||² + (1 − 2nµ + n²L); - ησε 2μ-nL (3) What is the restriction on the stepsize? 3. Let us observe by definition, SGD with step size n satisfies: ||K+1 – x = ||xk - xu t ng Vi(x)|| − 20k (k – x*, Vf(x)). (4) Derive the optimal step size and comment on it.