Mixing times via super-fast coupling
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1 s via super-fast coupling Yevgeniy Kovchegov Department of Mathematics Oregon State University
2 Outline 1 About 2 Definition Coupling Result.
3 About Outline 1 About 2 Definition Coupling Result.
4 About Coauthor Paper. This presentation is based on joint work with R.Burton.
5 About History Diaconis and Shahshahani (early 80 s) The mixing time for shuffling a deck of n cards by random transpositions is of order O(n log(n)) with cut-off asymptotics at 1 2 n log(n). Method used: relatively rarified mathematical residential district of representation theory.
6 About The Problem Open problem (Y.Peres) Provide a coupling proof of O(n log(n)) mixing rate. Continuous time: < a, b > has rate 2 n 2, i.e. < a, b > and < b, a > happen with rate 1 n 2 each. Transposition < a, a > has rate 1 n 2.
7 Outline 1 About 2 Definition Coupling Result.
8 Definition Definition of Mixing Time Total variation distance: µ ν TV := 1 µ(x) ν(x) 2 : process {X t } with stationary distribution π. If X t ν t = ν 0 P t, { t mix := inf t : ν t π TV 1 } 4, all ν 0 x
9 Coupling via coupling X t ν 0 P t and Y t µ 0 P t. If T coupling is the coupling time for (X t, Y t ), then ν 0 P t µ 0 P t TV P[T coupling > t] Goal: bound P[T coupling > t] for all µ 0. Here max x ν 0 P t xp t TV ν 0 P t π TV as π(x) ν 0 P t xp t TV ν 0 P t π TV x by convexity of f (x) = ν 0 P t xp t TV.
10 Coupling Notations Notations <, > - transpositions < ( a, ) > - transposition initiated by card a At - the coupled process B t ( ) At, - transpositions in B t
11 Coupling Coupling #1. (Aldous and Fill) a, i : moves card a to location i in both processes, A t and B t. Mixing: order O(n 2 ) instead of O(n log n); E[T coupling ] n d=2 n 2 ( ) π 2 d n 2 Problem: slows down significantly when the number of discrepancies is small enough, Coupling #2. (equivalent to prev.): applying transposition a, i if a is not coupled, applying transposition a, b if a is coupled Coupling #2 can be improved to match O(n log n) Transpositions a, b called label-to-label.
12 Coupling Group invariance X m is a Markov Chain on a discrete group G. The chain is group invariant if for all γ, α S n. In other words, dist(γx m+1 X m = α) = dist(x m+1 X m = γ 1 α) label-to-label a, b can be ignored. Reason: don t change cycle structure. The situation is invariant under label-to-label transpositions. If label-to-label a, b is applied before label-to-location b, i. We do not relabel, and do not reselect i.
13 Vocabulary. association map - hidden association between positions/locations in the top process and positions/locations in the bottom process that will be used to establish the rates for the coupled process
14 Two discrepancies (d = 2) at d 1 and d 2 : A t : b 9 a 8 a B t : a 9 b 8 a d 2 d 1 i 1 Label-to-location coupling: E[T coupling ] = n2 4 too large.
15 Jump a, i 1 of a to random location i 1 at exponential time t 1 : From to A t : b 9 a 8 a B t : a 9 b 8 a d 2 d 1 i 1 A t : b 9 a1 8 a 2... B t : a1 9 b 8 a 2... d 2 d 1 i 1
16 Different way of saying the same: Start with A t : b 9 a1 8 a 2... B t : a1 9 b 8 a 2... d 2 /i 1 i 1 /d 1 d 1 /d 2 where at time t 1 the locations relabel according to d 1 /d 2 i 1 i 1 /d 1 d 1. d 2 /i 1 d 2
17 Different way of saying the same: Jump a, i 1 at time t 1 exponential ( 1 n). From to A t : b 9 a1 8 a 2... B t : a1 9 b 8 a 2... d 2 /i 1 i 1 /d 1 d 1 /d 2 A t : b 9 a1 8 a 2... B t : a1 9 b 8 a 2... d 2 d 1 i 1
18 The Association Map. The following association map will determine jumps of a1. A t : b 9 a1 8 a 2... B t : a1 9 b 8 a 2... Card a1 will jump to position i 2 on the assoc. map at time t 2, even if t 2 < t 1.
19 A t : b 9 a1 8 a 2... B t : a1 9 b 8 a 2... d2 d1 i1 Now i 2 i 1 and ( t 2 exponential (1 1/n) 1 ) n a1, i 1 = a1, a is label-to-label, we can skip. If i 1 = d 1 or d 2, discrepancies cancel at t 1 ; if i2 = d 1 or d 2, discrepancies cancel on the assoc. map at t 2. If t 1 < t 2, assoc. map real picture at t 1, we create one more assoc. map.
20 Case t 2 < t 1, and i2 = d 2. On association map: Start with A t : b 9 a1 8 a 2... B t : a1 9 b 8 a 2... d 2 /i 1 i 1 /d 1 d 1 /d 2 At time t 2 : A t : a1 9 b 8 a 2... B t : a1 9 b 8 a 2... d 2 /i 1 i 1 /d 1 d 1 /d 2
21 Case t 2 < t 1, and i2 = d 2. On association map: At time t 2 : A t : a1 9 b 8 a 2... B t : a1 9 b 8 a 2... d 2 /i 1 i 1 /d 1 d 1 /d 2 At time t 1 : A t : a1 9 b 8 a 2... B t : a1 9 b 8 a 2... d 2 d 1 i 1
22 Case t 2 < t 1, and i2 = d 2. Same evolution, original association: Start with A t : b 9 a 8 a B t : a 9 b 8 a d 2 d 1 i 1 At time t 2 : A t : a1 9 a 8 b 2... B t : a 9 b 8 a d 2 d 1 i 1
23 Case t 2 < t 1, and i2 = d 2. Same evolution, original association: At time t 2 : A t : a1 9 a 8 b 2... B t : a 9 b 8 a d 2 d 1 i 1 At time t 1 : A t : a1 9 b 8 a 2... B t : a1 9 b 8 a 2... d 2 d 1 i 1
24 Faster coupling. The new coupling time for two discrepancies: E[T coupling ] n2 8
25 Chain of association maps. a1, i 2 occurs at t 2 A t :... a2 6 b 9 a1 8 a 2... B t :... a2 6 a1 9 b 8 a 2... i 2 d2 d1 i1 d 1 is i 1/d 1 before t 1, and d 1 after t 1 ; d 2 is d 2/i 1 before t 1, and d 2 after t 1 ; i 1 is d 1/d 2 before t 1, and i 1 after t 1.
26 Chain of association maps. New association map: A t :... a1 6 b 9 a2 8 a 2... B t :... a1 6 a2 9 b 8 a 2... d1 /d 2 d2 /i 2 i 2 /d1 i1 where at t 2, d 1 /d 2 i 2 i 2 /d 1 d 1 d 2 /i 2 d 2.
27 Chain of association maps. a2 will do label-to-location jump w.r.t. the following assoc. map d d i A t :... a1 6 b 9 a2 8 a 2... B t :... a1 6 a2 9 b 8 a 2... i2 d2 d1 i1 1 is i 2/d1 before t 2, and d1 after t 2; 2 is d 2 /i 2 before t 2, and d2 after t 2; 2 is d 1 /d 2 before t 2, and i 2 after t 2. a2, i 3 occurs at t 3 exponential ( (1 2/n) 1 n )
28 Chain of association maps. And so on, creating a chain of k = εn association maps. In case of d = 2 discrepancies, the average time of discrepancy cancelation on one of association maps is E[T 2 ] = n 2 4(k + 1) n 4ε.
29 Result. Result. General d: E[T d ] = n 2 2(k + 1)d n 2εd. Coupling time (all discrepancies): [ 1 E[T coupling ] 2ε + κ (1 κ)(κ ε) for any 0 < ε < κ < 1. ] n log n
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