Mixing Time CS 70, Summer 2019 Bonus Lecture, 8/9/19 1 / 17 - - PowerPoint PPT Presentation
Mixing Time CS 70, Summer 2019 Bonus Lecture, 8/9/19 1 / 17 Disclaimer: Much handwaving today!! Goal is to get a high level intuition / picture for the concept of mixing time and applications Emphasis is on heuristics rather than rigor 2 /
CS 70, Summer 2019 Bonus Lecture, 8/9/19
1 / 17
Much handwaving today!! Goal is to get a high level intuition / picture for the concept of mixing time and applications Emphasis is on heuristics rather than rigor
2 / 17
Every irreducible, aperiodic Markov chain has a unique stationary distribution.
m πm(1) πm(2) πm(3) πm = π0P m = π0 0.8 0.2 0.3 0.7 0.6 0.4
m
.
Q: How long does it take to get close to the stationary distribution?
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' n'
=
IT
Just one of many ways to measure how close two distributions are. Let P1, P2 be two PMFs. Their TV distance is:
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→
States
{ 1,2 ,
. . . ,n }
/ Phil
thing lighted
I have an irreducible, aperiodic Markov chain. Notation: µ(n) is the distribution at time n, and ⇡ is its (unique) stationary distribution. I want to keep running my chain until: The mixing time tmix(") is the first time this happens. (Omitted fact: The TV distance between µ(n) and ⇡ decreases as n increases.)
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Tv
( UM, Tl )
E
E
←
small
positive
9
number
dist
at
time in stationary
↳
worst
case
starting
di St
.U
l l
with
high prob , at
least
Mixing time analysis sometimes direct! Take a random walk on a complete graph with
What does the transition matrix look like? Mixing Time? Dependence on "?
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a
①
a kn
:
F-
symbtanwtnfn
vertices
.,
.int
= IT1
.for
all
E
. S: All orderings of n cards in a deck. Transitions: Choose card randomly in the deck. Move it to the top. Different strategy called coupling:
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In ! con figs
.Tl
here
is
uniform
all States
.←
assume arena;Hed
.A
T
chosen
start
at
arbitrary
State
.I #
we keep
applying p,
dist
.Stays
at
it
. At each time, each card is labeled coupled or not. Initially, all cards start uncoupled. Pick a random card C. In both decks, take card C and move it to the top. If C isn’t already coupled, mark it as coupled. What happens when we look at each deck individually?
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number number
LOOKS
like
random
move
!
④
a
#
IS
labeled
" coupled ;
it
has
the
same
position
in
both
decks !
Time until all cards get coupled = time until Deck 1 is fully random. For all ✏: tmix(") =
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me
1st 2nd 3rd couple
T2 couple
couple
.'t
Geomfnnt )
Geomftn) "
coupon
collector
., -1T
, -1
. ."
"
Take an n-dimensional hypercube. Stationary distribution of hypercube walk: Try coupling again:
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vertices
n
bit strings F- Cfn
all vertices
same
deg
.:
Every
vertex
self loop Wp 's
transitions
to Nbr
up In
→
truly
random
:
randomize
all
bits
Each coordinate is labeled coupled or not. Initially all coordinates are uncoupled. Pick a random coordinate i. Flip a coin to set the i-th coordinate to 0 or 1. If i-th coordinate is uncoupled, set it to coupled.
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all
word
. randomized ]=
N log N
Use Coupon Collector again! For all ", tmix(") =
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same
as
before
. Complete graph Kn? Path on n vertices? Dumbbell?
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=L
Start
in
middle
.time
NZ
mixing
time
nz
mgaehffengnere
f-
chance getting hmltgefdte
from
pink dot
. We use a measurement called the conductance to quantify the notion of a bottleneck. The conductance of a set A ⊆ S is: Φ(A) = The conductance of the chain M is: Φ(M) =
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A
E
Milk ;
←
VOILA)
min Ast
VOKAKE
Measuring conductance = looking at all subsets
How many subsets, potentially? Alternative: Get lower bound on Φ(M) using second largest eigenvalue of transition matrix. Φ(M) ≥ Eigenvalues are much faster to compute!
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±
2
Monte Carlo: randomized algorithm where the
Use cases:
I sampling from complicated distributions I counting combinatorial objects I Bayesian inference I statistical physics I volume estimation, integration
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Key idea: Design a Markov chain so that its stationary follows the distribution that you want to sample from. Run the chain, wait for it to mix. Runtime depends on...
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+
mix
CE )
!