Optimal Sequential Selection Monotone, Alternating, and Unimodal Subsequences: With Additional Reflections on Shepp at 75

Transcription

1 Optimal Sequential Selection Monotone, Alternating, and Unimodal Subsequences: With Additional Reflections on Shepp at 75 J. Michael Steele Columbia University, June, 2011 J.M. Steele () Subsequence Selection June, / 15

2 Two Strange Reflections First in Decimal Points Integers... made binary... then reflected in the decimal point : {1, 2, 3, 4, 5, 6,...} {1, 10, 11, 100, 101, 110,...} {1, 10, 11, 100, 101, 110,...} {0.1, 0.01, 0.11, 0.001, 0.101, 0.011,...} {0.1, 0.01, 0.11, 0.001, 0.101, 0.011,...} {1/2, 1/4, 3/4, 1/8, 5/8, 3/8,...} Called the Van der Corput Sequence: Very Small Discrepancy: O(log n /n) compared to O(1/ n) for i.i.d. Uniform Samples Equidistributed so it has some obvious baked-in pseudo-random properties. More amusingly, it has some less obvious pseudo-random properties to be mentioned later. Much loved by me: An Example of an Interesting Object J.M. Steele () Subsequence Selection June, / 15

3 The Second Reflection Reflection in One s Birthday Larry Shepp was born September 9, less 1936 gives us 75 Now Reflect: That takes us back 75 years from 1936 That lands us in was no more (or less) remote from Shepp s birth as Shepp s birth is remote from the present The US Civil war began in April No one had any idea what it would become. Riemann was 35 and would live five more years. Wilbur was born five years later and Orville five years after that. Their first flight would be 42 years in the future. Queen Victoria ruled England and India and would continue to do so for 36 more years. Christmas trees were almost unheard of but would become customary ten years later. Germany as a state would come into existence in ten years later. One had to wait 42 years until the Dow Jones Industrial Average would be in print. J.M. Steele () Subsequence Selection June, / 15

4 Acknowledgements Back to the Present: Up-Front Acknowledgements Alessandro Arlotto (Univ. Pennsylvania, Wharton) Lawrence A. Shepp (Univ. Pennsylvania, Wharton) Robert Chen (Univ. Miami) J.M. Steele () Subsequence Selection June, / 15

5 Introduction to the Main Story Optimal Sequential Selection of Subsequences Increasing Unimodal Alternating J.M. Steele () Subsequence Selection June, / 15

6 Introduction to the Main Story Increasing, Unimodal and Alternating plots () 1 / 15 J.M. Steele () Subsequence Selection June, / 15

7 Introduction to the Main Story On-line vs. full-information 1 n = 100, J.M. Steele () Subsequence Selection June, / 15

8 Introduction to the Main Story On-line vs. full-information 1 n = 100, U o n (π n ) = 21, J.M. Steele () Subsequence Selection June, / 15

9 Introduction to the Main Story On-line vs. full-information 1 n = 100, U o n (π n ) = 21, U n = J.M. Steele () Subsequence Selection June, / 15

10 Main Results Increasing Subsequences: Beginning with the Classics Theorem There is a policy π Π(n) such that E[I o n (π )] = sup π Π(n) E[I o n (π)], and for such an optimal policy one has so, in particular, one has (2n) 1/2 (8n) 1/4 2 < E[I o n (π )] < (2n) 1/2 for all n 1. E[I o n (π )] (2n) 1/2 as n. Asymptotic behavior: Samuels and Steele (1981) Upper bound: Bruss and Robertson (1991), Gnedin (1999) Lower bound: Rhee and Talagrand (1991) Well Trod Ground but with Something New: Different proof for the upper-bound; Variance bounds J.M. Steele () Subsequence Selection June, / 15

11 Main Results Unimodal Subsequences: More Complex but Still Analogous Theorem There is a policy π Π(n) such that E[U o n (π )] = sup E[Un o (π)], π Π(n) and for such an optimal policy there is a constant C such that So, in particular, one has 2n 1/2 Cn 1/4 < E[U o n (π )] < 2n 1/2 for all n 1. E[U o n (π )] 2n 1/2 as n. J.M. Steele () Subsequence Selection June, / 15

12 Main Results Alternating Subsequences: Something Quite Different Theorem (Asymptotic Selection Rate for Large Samples) For each n = 1, 2,..., there is a policy π n Π such that E[A o n(π n )] = sup π Π E[A o n(π)], and for such an optimal policy one has for all n 1 that (2 2)n E[A o n(π n )] (2 2)n + C, where C is a constant with C < In particular, one has E[A o n(π n )] (2 2)n as n. Theorem (Expected Selection Size in Geometric Samples) For each 0 < ρ < 1, there is a π Π, such that E[A o N(π )] = sup π Π E[A o N(π)], and for such an optimal policy one has E[A o N(π )] = ρ + ρ 2 ρ(1 ρ) (2 2)(1 ρ) 1 (2 2)EN as ρ 1. J.M. Steele () Subsequence Selection June, / 15

13 Proof of the Expected Length of Alternating Subsequences Proof of the Expected Length of Alternating Subsequences (sketch) Finite-horizon Bellman equation: { svi+1,n (s, 0) + 1 v i,n (s, r) = max {v s i+1,n(s, 0), 1 + v i+1,n (x, 1)} dx if r = 0 (1 s)v i+1,n (s, 1) + s max {v 0 i+1,n(s, 1), 1 + v i+1,n (x, 0)} dx if r = 1 Reflection identity: v i,n (s, 0) = v i,n (1 s, 1) for all 1 i n and all s [0, 1]. Flipped finite-horizon Bellman equation: v i,n (y) = yv i+1,n (y) + 1 Flipped infinite-horizon Bellman equation: v(y) = ρyv(y) + y 1 y max {v i+1,n (y), 1 + v i+1,n (1 x)} dx. max {ρv(y), 1 + ρv(1 x)} dx. Threshold-policy for infinite-horizon: f (y) = max{ξ 0, y}, ξ 0 [0, 1/2) Solve for v( ) and obtain v(0) = v(ξ 0) = ρ + ρ 2. ρ(1 ρ) J.M. Steele () Subsequence Selection June, / 15

14 Proof of the Expected Length of Alternating Subsequences Proof of the Expected Length of Alternating Subsequences (sketch) Finite-horizon lower bound: use the infinite-horizon threshold policy. Finite-horizon upper bound: use the finite-horizon optimal threshold functions {f1,n,..., fn 2,n} and regenerate this selection process over an infinite horizon. The value of E[A o N(π )] then gives the desired upper bound. J.M. Steele () Subsequence Selection June, / 15

15 Big Picture The News You Can Use First Soften The Ground: Study the more symmetrical infinite horizon (or other smoothed ) problem variations Second Have the Courage (and Techniques) to Return to Finite n: Typically, it is the finite n problem that interests us most. We can try to return to finite n by exploiting suboptimality. This works well enough for means but more refined information (such as variance) requires much more work. Open Problems: CLT for Monotone and Finite n? CLT for Alternating (even good variance asymptotics) Richer Understanding of Martingale connections with the Bellman Equation Richer Understanding of Bellman Equation asymptotics (and max-type integral equations) J.M. Steele () Subsequence Selection June, / 15

16 Thank you! J.M. Steele () Subsequence Selection June, / 15

17 References References F. Thomas Bruss and James B. Robertson. Wald s lemma for sums of order statistics of i.i.d. random variables. Adv. in Appl. Probab., 23(3): , ISSN doi: / Alexander V. Gnedin. Sequential selection of an increasing subsequence from a sample of random size. J. Appl. Probab., 36(4): , ISSN WanSoo Rhee and Michel Talagrand. A note on the selection of random variables under a sum constraint. J. Appl. Probab., 28(4): , ISSN Stephen M. Samuels and J. Michael Steele. Optimal sequential selection of a monotone sequence from a random sample. Ann. Probab., 9(6): , ISSN J.M. Steele () Subsequence Selection June, / 15