CHAPTER 12 STATISTICAL METHODS FOR OPTIMIZATION IN DISCRETE PROBLEMS Slides for Introduction to Stochastic Search and Optimization (ISSO) by J. C. Spall CHAPTER 12 STATISTICAL METHODS FOR OPTIMIZATION IN DISCRETE PROBLEMS Organization of chapter in ISSO Basic problem in multiple comparisons Finite number of elements in search domain  Tukey-Kramer test “Many-to-one” tests for sharper analysis Measurement noise variance known Measurement noise variance unknown (estimated) Ranking and selection methods

Background Statistical methods used here to solve optimization problem Not just for evaluation purposes Extending standard pairwise t-test to multiple comparisons Let     {1, 2, …, K} be finite search space (K possible options) Optimization problem is to find the j such that  = j Only have noisy measurements of L(i)

Applications with Monte Carlo Simulations Suppose wish to evaluate K possible options in a real system Too difficult to use real system to evaluate options Suppose run Monte Carlo simulation(s) for each of the K options Compare options based on a performance measure (or loss function) L() representing average (mean) performance  represents options that can be varied Monte Carlo simulations produce noisy measurement of loss function L at each option

Statistical Hypothesis Testing Null hypothesis: All options in   {1, 2, …, K} are effectively the same in the sense that L(1) = L(2) = … = L(K) Challenge in multiple comparisons: alternative hypothesis is not unique Contrasts with standard pairwise t-test Analogous to standard t-test, hypothesis testing based on collecting sample values of L(1), L(2), and L(K), forming sample means

Tukey–Kramer Test Tukey (1953) and Kramer (1956) independently developed popular multiple comparisons analogue to standard t-test Recall null hypothesis that all options in   {1, 2, …, K} are effectively the same in the sense that L(1) = L(2) = … = L(K) Tukey–Kramer test forms multiple acceptance intervals for K(K–1)/2 differences ij  Intervals require sample variance calculation based on samples at all K options Null hypothesis is accepted if evidence suggests all differences ij lie in their respective intervals Null hypothesis is rejected if evidence suggests at least one ij lies outside its respective interval

Example: Widths of 95% Acceptance Intervals Increasing with K in Tukey–Kramer Test (n1=n2=…=nK=10)

Example of Tukey-Kramer Test (Example 12.2 in ISSO) Goal: With K = 4, test null hypothesis L(1) = L(2) = L(3) = L(4) based on 10 measurements at each i All (six) differences ij  must lie in acceptance intervals [–1.23, 1.23] Find that 34 = 1.72 Have 34  [–1.23, 1.23] Since at least one ij is not in acceptance interval, reject null hypothesis Conclude at least one i likely better than others Further analysis required to find i that is better

Multiple Comparisons Against One Candidate Assume prior information suggests one of K points is optimal, say m Reduces number of comparisons from K(K–1)/2 differences ij = to only K–1 differences mj Under null hypothesis, L(m)  L(j) for all j Aim to reject null hypothesis Implies that L(m) < L(j) for at least some j Tests based on critical values < 0 for observed differences mj To show that L(m) < L(j) for all j requires additional analysis

Example of Many-to-One Test with Known Variances (Example 12 Example of Many-to-One Test with Known Variances (Example 12.3 in ISSO) Suppose K = 4, m = 2  Need to compute 3 critical values , , and for acceptance regions Valid to take Under Bonferroni/Chebyshev: Under Bonferroni/normal noise: Under Slepian/normal noise: Note tighter (smaller) acceptance regions when assuming normal noise

Widths of 95% Acceptance Intervals (< 0) for Tukey-Kramer and Many-to-One Tests (n1=n2=…=nK=10)

Ranking and Selection: Indifference Zone Methods Consider usual problem of determining best of K possible options, represented 1 , 2 ,…, K Have noisy loss measurements yk(i ) Suppose analyst is willing to accept any i such that L(i) is in indifference zone [L(), L() + ) Analyst can specify  such that P(correct selection of  = )  1  whenever L(i) L()   for all i   Can use independent sampling or common random numbers (see Section 14.5 of ISSO)