Finite Horizon Dynamic Programming: Getting Value from Spending Symmetry J. Michael Steele University of Pennsylvania The Wharton School Department of Statistics Stochastic Processes and Applications, Buenos Aires, August 8, 2014 J. M. Steele (UPenn, Wharton) On-line Selection August 2015 1
Introduction Some History and Motivation Famous combinatorial problems with long mathematical history on sequences of n real numbers, or permutations of the integers 1 , . . . , n ◮ Erd˝ os and Szekeres (1935): monotone subsequences ◮ Fan Chung (1980): unimodal subsequences ◮ Euler (cf. Stanley, 2010): alternating permutations J. M. Steele (UPenn, Wharton) On-line Selection August 2015 2
Introduction Some History and Motivation 1 Famous combinatorial problems with long mathematical 0.9 history on sequences of n real numbers, or permutations 0.8 0.7 of the integers 1 , . . . , n 0.6 0.5 ◮ Erd˝ os and Szekeres (1935): monotone subsequences 0.4 0.3 ◮ Fan Chung (1980): unimodal subsequences 0.2 ◮ Euler (cf. Stanley, 2010): alternating permutations 0.1 0 0 10 20 30 40 50 60 70 80 90 100 1 Probabilistic version (full-information) 0.9 0.8 ◮ Longest monotone subsequences: Hammersley (1972), 0.7 Kingman (1973), Logan and Shepp (1977), Verˇ sik and 0.6 0.5 Kerov (1977), . . . 0.4 ◮ Longest Unimodal subsequences: Steele (1981) 0.3 0.2 ◮ Longest Alternating subsequences: Widom (2006), 0.1 0 Pemantle (cf. Stanley, 2007), Stanley (2008), Houdr´ e 0 10 20 30 40 50 60 70 80 90 100 and Restrepo (2010) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 J. M. Steele (UPenn, Wharton) On-line Selection August 2015 2
Introduction Some History and Motivation 1 Famous combinatorial problems with long mathematical 0.9 history on sequences of n real numbers, or permutations 0.8 0.7 of the integers 1 , . . . , n 0.6 0.5 ◮ Erd˝ os and Szekeres (1935): monotone subsequences 0.4 0.3 ◮ Fan Chung (1980): unimodal subsequences 0.2 ◮ Euler (cf. Stanley, 2010): alternating permutations 0.1 0 0 10 20 30 40 50 60 70 80 90 100 1 Probabilistic version (full-information) 0.9 0.8 ◮ Longest monotone subsequences: Hammersley (1972), 0.7 Kingman (1973), Logan and Shepp (1977), Verˇ sik and 0.6 0.5 Kerov (1977), . . . 0.4 ◮ Longest Unimodal subsequences: Steele (1981) 0.3 0.2 ◮ Longest Alternating subsequences: Widom (2006), 0.1 0 Pemantle (cf. Stanley, 2007), Stanley (2008), Houdr´ e 0 10 20 30 40 50 60 70 80 90 100 and Restrepo (2010) 1 0.9 0.8 Now ... Study the sequential (on-line) version of these 0.7 problems 0.6 0.5 ◮ Objective: maximize the expected length (number of 0.4 selections) of monotone, unimodal and alternating 0.3 0.2 subsequences 0.1 0 0 10 20 30 40 50 60 70 80 90 100 J. M. Steele (UPenn, Wharton) On-line Selection August 2015 2
Introduction Full-information vs. on-line — Increasing n = 100 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 J. M. Steele (UPenn, Wharton) On-line Selection August 2015 3
Introduction Full-information vs. on-line — Increasing n = 100 I n = 15 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 J. M. Steele (UPenn, Wharton) On-line Selection August 2015 3
Introduction Full-information vs. on-line — Increasing I o n ( π ∗ n = 100 I n = 15 n ) = 14 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 J. M. Steele (UPenn, Wharton) On-line Selection August 2015 3
Introduction Full-information vs. on-line — Unimodal n = 100 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 J. M. Steele (UPenn, Wharton) On-line Selection August 2015 3
Introduction Full-information vs. on-line — Unimodal n = 100 U n = 22 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 J. M. Steele (UPenn, Wharton) On-line Selection August 2015 3
Introduction Full-information vs. on-line — Unimodal U o n ( π ∗ n = 100 U n = 22 n ) = 21 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 90 100 J. M. Steele (UPenn, Wharton) On-line Selection August 2015 3
Introduction Summary View of Means in Some On-Line Selection Problems How Much Better Does a “Prophet” Do Asymptotically? Full Information Real Time Info. Only Realized Bonus Increasing Unimodal Alternating J. M. Steele (UPenn, Wharton) On-line Selection August 2015 4
Introduction Summary View of Means in Some On-Line Selection Problems How Much Better Does a “Prophet” Do Asymptotically? Full Information Real Time Info. Only Realized Bonus √ 2 √ n Increasing 2 n 29% Unimodal Alternating J. M. Steele (UPenn, Wharton) On-line Selection August 2015 4
Introduction Summary View of Means in Some On-Line Selection Problems How Much Better Does a “Prophet” Do Asymptotically? Full Information Real Time Info. Only Realized Bonus √ 2 √ n Increasing 2 n 29% √ 2 √ n Unimodal 2 2 n 29% Alternating J. M. Steele (UPenn, Wharton) On-line Selection August 2015 4
Introduction Summary View of Means in Some On-Line Selection Problems How Much Better Does a “Prophet” Do Asymptotically? Full Information Real Time Info. Only Realized Bonus √ 2 √ n Increasing 2 n 29% √ 2 √ n Unimodal 2 2 n 29% √ Alternating 2 n / 3 (2 − 2) n 12% J. M. Steele (UPenn, Wharton) On-line Selection August 2015 4
Introduction Summary View of Means in Some On-Line Selection Problems How Much Better Does a “Prophet” Do Asymptotically? Full Information Real Time Info. Only Realized Bonus √ 2 √ n Increasing 2 n 29% √ 2 √ n Unimodal 2 2 n 29% √ Alternating 2 n / 3 (2 − 2) n 12% Question: Can one get more detailed information? J. M. Steele (UPenn, Wharton) On-line Selection August 2015 4
Introduction Summary View of Means in Some On-Line Selection Problems How Much Better Does a “Prophet” Do Asymptotically? Full Information Real Time Info. Only Realized Bonus √ 2 √ n Increasing 2 n 29% √ 2 √ n Unimodal 2 2 n 29% √ Alternating 2 n / 3 (2 − 2) n 12% Question: Can one get more detailed information? More precise asymptotics of the means? J. M. Steele (UPenn, Wharton) On-line Selection August 2015 4
Introduction Summary View of Means in Some On-Line Selection Problems How Much Better Does a “Prophet” Do Asymptotically? Full Information Real Time Info. Only Realized Bonus √ 2 √ n Increasing 2 n 29% √ 2 √ n Unimodal 2 2 n 29% √ Alternating 2 n / 3 (2 − 2) n 12% Question: Can one get more detailed information? More precise asymptotics of the means? Any second-order information, i.e. what about the variances? J. M. Steele (UPenn, Wharton) On-line Selection August 2015 4
Introduction Summary View of Means in Some On-Line Selection Problems How Much Better Does a “Prophet” Do Asymptotically? Full Information Real Time Info. Only Realized Bonus √ 2 √ n Increasing 2 n 29% √ 2 √ n Unimodal 2 2 n 29% √ Alternating 2 n / 3 (2 − 2) n 12% Question: Can one get more detailed information? More precise asymptotics of the means? Any second-order information, i.e. what about the variances? Is there hope for a CLT or other distributional result? J. M. Steele (UPenn, Wharton) On-line Selection August 2015 4
Introduction Summary View of Means in Some On-Line Selection Problems How Much Better Does a “Prophet” Do Asymptotically? Full Information Real Time Info. Only Realized Bonus √ 2 √ n Increasing 2 n 29% √ 2 √ n Unimodal 2 2 n 29% √ Alternating 2 n / 3 (2 − 2) n 12% Question: Can one get more detailed information? More precise asymptotics of the means? Any second-order information, i.e. what about the variances? Is there hope for a CLT or other distributional result? There is a CLT for the On-Line Alternating Subsequence Problem (briefly noted in next frame) There has much further work on the On-Line Selection of a Monotone Increasing Subsequence , the original motivating problem. This will get most of our attention. J. M. Steele (UPenn, Wharton) On-line Selection August 2015 4
Introduction CLT for Alternating Sequentially Selected Alternating Series — A CLT Theorem (Arlotto & Steele, AAP 2014) There is a constant σ > 0 such that √ A o n ( π ∗ n ) − n (2 − 2) ⇒ N (0 , 1) . n σ J. M. Steele (UPenn, Wharton) On-line Selection August 2015 5
Introduction CLT for Alternating Sequentially Selected Alternating Series — A CLT Theorem (Arlotto & Steele, AAP 2014) There is a constant σ > 0 such that √ A o n ( π ∗ n ) − n (2 − 2) ⇒ N (0 , 1) . n σ The Mysterious σ ? Its existence is proved but the value is not yet known. J. M. Steele (UPenn, Wharton) On-line Selection August 2015 5
Introduction CLT for Alternating Sequentially Selected Alternating Series — A CLT Theorem (Arlotto & Steele, AAP 2014) There is a constant σ > 0 such that √ A o n ( π ∗ n ) − n (2 − 2) ⇒ N (0 , 1) . n σ The Mysterious σ ? Its existence is proved but the value is not yet known. A Candidate σ ? Yes, but not yet in the bag. J. M. Steele (UPenn, Wharton) On-line Selection August 2015 5
Recommend
More recommend
