Mixing Times of Markov Chains: Self-Organizing Lists & Biased Permutations

1. Mixing Times of Markov Chains for Self-Organizing Lists and Biased Permutations PrateekBhakta, Sarah Miracle, Dana Randall and Amanda Streib

2. Sampling Permutations - pick a pair of adjacent cards uniformly at random - put j ahead of i with probability pj,i= 1- pi,j n-1 5 n 1 7 ... i j ... 2 6 3 This is related to the “Move-Ahead-One Algorithm” for self-organizing lists.

3. Is M always rapidly mixing? M is not always fast . . . If pi,j ≥ ½ ∀ i < j we say the chain is positively biased. • Q: If the {pij} are positively biased, is M always rapidly mixing? • Conjecture[Fill]: If {pij} are positively biased and monotone then M is rapidly mixing.

4. What is already known? • Uniform bias: If pi,j = ½ ∀ i, j then M mixes in θ(n3 log n) time. [Aldous ‘83, Wilson ‘04] • Constant bias:If pi,j = p > ½ ∀ i < j, then M mixes in θ(n2) time. [Benjamini et al. ’04, Greenberg et al. ‘09] • Linear extensions of a partial order: If pi,j = ½ or 1 ∀ i < j, then Mmixes in O(n3 log n) time. [Bubley, Dyer ‘98]

5. Our Results [Bhakta, M., Randall, Streib] • M is fastfor two new classes • “Choose your weapon” • “League hierarchies” • Both classes extend the uniform and constant biascases • M can be slow even when the {pij} are positively biased

6. Talk Outline • Background • New Classes of Bias where Mis fast • Choose your Weapon • League Hierarchies • Mcan be slow even when the {pij} are positively biased

7. Definition:Given e, the mixing timeis t(e)= maxmin {t: Δx(t’) <e, t’ ≥ t}. A Choose Your Weapon Given parameters ½ ≤r1, … ,rn-1≤ 1. Thm 1: Let pi,j = ri ∀i < j. Then Mis rapidly mixing. Definition: The variation distance is Δx(t) = ½ ∑ |Pt(x,y) – π(y)|. yÎΩ x A Markov chain is rapidly mixingif t(e) is poly(n, log(e-1)).

8. Minvcan swap pairs that are not nearest neighbors • Maintains the same stationary distribution • Allowed moves are based on inversion tables Choose Your Weapon Given parameters ½ ≤r1, … ,rn-1≤ 1. Thm 1: Let pi,j = ri ∀i < j. Then Mis rapidly mixing. • Proof sketch: • A. Define auxiliary Markov chain Minv • B. Show Minvis rapidly mixing • C. Compare the mixing times of Mand Minv

9. Inversion Table Iσ: 3 3 2 3 0 0 0 Inversion Tables Permutation σ: 5 6 3 1 2 7 4 Iσ(2) Iσ(1) Iσ(i) = # elements j > i appearing before i in σ The map I is a bijection from Snto T ={(x1,x2,...,xn): 0 ≤ xi ≤ n-i}.

10. 3 3 2 3 0 0 0 Permutation σ: Inversion Table Iσ: 5 6 3 1 2 7 4 Inversion Tables Iσ(i) = # elements j > i appearing before i in σ Minvon Permutations Minv on Inversion Tables - choose a card iuniformly - choose a column i uniformly - w.p. ri: subtract 1 from xi (if xi > 0) - swap element i with the first j>ito the left w.p. ri - swap element i with the first j>i to the right w.p. 1-ri - w.p. 1- ri: add 1 to xi (if xi < n-i)

11. 3 3 2 3 0 0 0 Permutation σ: Inversion Table Iσ: 5 6 3 1 2 7 4 Inversion Tables Minv on Inversion Tables - choose a column i uniformly - w.p. ri: subtract 1 from xi (if xi>0) - w.p. 1- ri: add 1 to xi (if xi<n-i) Minvis just a product of n independent biased random walks Minvis rapidly mixing. ß

12. Talk Outline • Background • New Classes of Bias where Mis fast • Choose your Weapon • League Hierarchies • M can be slow even when the {pij} are positively biased

13. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then M is rapidly mixing.

14. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then M is rapidly mixing. A-League B-League

15. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then M is rapidly mixing. 1st Tier 2ndTier

16. Mtreecan swap pairs that are not nearest neighbors • Maintains the same stationary distribution • Allowed moves are based on the binary tree T League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2:Let pi,j = qi^jfor all i< j. Then M is rapidly mixing*. • Proof sketch: • A. Define auxiliary Markov chain Mtree • B. Show Mtree is rapidly mixing • C. Compare the mixing times of Mand Mtree

17. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then M is rapidly mixing.

18. Theorem 2: Proof sketch Let T be a binary tree with leaves labeled {1,…,n}. Given qv ≥ 1/2 for each internal vertex v. Thm 2: Let pi,j = qi^jfor all i< j. Then M is rapidly mixing.

19. Theorem 2: Proof sketch Markov chain Mtree allows a transposition if it corresponds to an ASEP move on one of the internal vertices. q2^4 1 - q2^4

20. Talk Outline • Background • New Classes of Bias where M is fast • Choose your Weapon • League Hierarchies • M can be slow even when the {pij} are positively biased

21. 1 n n n n n 1 n 1 2 6 0 7 0 1 3 0 8 9 0 4 1 5 1 0 10 1’s and 0’s or < i < j 1 if i < j ≤ +1 2 2 2 2 2 2 But…..M can be slow Thm 3: There are examples of positively biased {pij} for which M is slowly mixing. 1. Reduce to “biased staircase walks” { always in order pij = 1 2 3 ... ... n Permutation σ:

22. n Each choice of pij where i ≤ < j n n 2 2 2 Slow Mixing Example Thm 3: There are examples of positively biased {p} for which M is slowly mixing. 1. Reduce to biased staircase walks 2. Define bias on individual cells (non-uniform growth proc.) { 1 if i < j ≤ or < i < j pij = M= n2/3 1/2+1/n2 if i+(n-j+1) < M 1- δ otherwise 1-δ 1/2 + 1/n2 determines the bias on square (i, n-j+1) (“fluctuating bias”) [Greenberg, Pascoe, Randall]

23. Slow Mixing Example Thm 3: There are examples of positively biased {p} for which M is slowly mixing. 1. Reduce to biased staircase walks 2. Define bias on individual cells 3. Show that there is a “bad cut” in the state space W S S Implies that Mcan take exponential time to reach stationarity. Therefore biased permutations can be slow too!

24. Open Problems Is M always rapidly mixing when {pi,j} are positively biased and satisfy a monotonicity condition? (i.e., pi,j is decreasing in i and j) 2. When does bias speed up or slow down a chain?

25. Thank you!