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.

Suppose that consumer has the following utility function: U(X,Y) = X2 + Y2. Which of the following is correct? Preferences are convex and indifference curves are bowed outward from the origin since Law of Diminishing Marginal Utility fails to hold. Preferences are non-convex and indifference curves are bowed outward from the origin since Law of Diminishing Marginal Rate of Substitution fails to hold. Preferences are convex and indifference curves are bowed inward towards the origin since Law of Diminishing Marginal Utility holds. Preferences are convex and indifference curves are bowed inward towards the origin since Law of Diminishing Marginal Rate of Substitution holds.

4. Let X - N(u, o²). The characteristic function (ch.f.) of X is given by $(w) = ejHw-o?w?/2, W E R, where j = v-1 is the imaginary unit. Use the ch.f. to find the mean and variance of X.