Post on 11-Feb-2016
description
Distributions of Randomized Backtrack Search
• Key Properties:
• I Erratic behavior of mean
• II Distributions have “heavy tails”.
Median = 1!
samplemean
3500!
Erratic Behavior of Search CostQuasigroup Completion Problem
500
2000
number of runs
1
75%<=30
Number backtracks Number backtracks
Prop
ortio
n of
cas
es S
olve
d
5%>100000
Heavy-Tailed Distributions
• … … infinite variance … infinite meaninfinite variance … infinite mean
• Introduced by Pareto in the 1920’s• --- “probabilistic curiosity.”
• Mandelbrot established the use of heavy-tailed distributions to model real-world fractal phenomena.
• Examples: stock-market, earth-quakes, weather,...
Decay of Distributions
• Standard --- Exponential Decay• e.g. Normal:•
• Heavy-Tailed --- Power Law Decay• e.g. Pareto-Levy:••
Pr[ ] , ,X x Ce x for some C x 2 0 1
Pr[ ] ,X x Cx x 0
Standard Distribution(finite mean & variance)
Power Law Decay
Exponential Decay
Normal, Cauchy, and Levy
Normal - Exponential Decay
Cauchy -Power law DecayLevy -Power law Decay
Tail Probabilities (Standard Normal, Cauchy, Levy)
• c Normal Cauchy Levy0 0.5 0.5 11 0.1587 0.25 0.68272 0.0228 0.1476 0.52053 0.001347 0.1024 0.43634 0.00003167 0.078 0.3829
Example of Heavy Tailed Model(Random Walk)
•Random Walk:•Start at position 0•Toss a fair coin:
–with each head take a step up (+1)–with each tail take a step down (-1)
X --- number of steps the random walk takes to return to position 0.
The record of 10,000 tosses of an ideal coin(Feller)
Zero crossing Long periods without zero crossing
Random Walk
Heavy-tails vs. Non-Heavy-Tails
Normal(2,1000000)
Normal(2,1)
O,1%>200000
50%
2
Median=2
1-F(
x)U
nsol
ved
frac
tion
X - number of steps the walk takes to return to zero (log scale)
How to Check for “Heavy Tails”?
• Log-Log plot of tail of distribution• should be approximately linear.
• Slope gives value of • • infinite mean and infinite varianceinfinite mean and infinite variance
• infinite varianceinfinite variance
1
1 2
466.0
319.0153.0
Number backtracks (log)
(1-F
(x))
(log)
Uns
olve
d fr
actio
n
1 => Infinite mean
Heavy-Tailed Behavior in QCP Domain
18% unsolved
0.002% unsolved
Formal Models of Heavy-Tailed Behavior in
Combinatorial Search
Chen, Gomes, Selman 2001
MotivationMotivation• Research on heavy-tails has been largely based on empirical studies of run time distribution.
• Goal: to provide a formal characterization of tree search models and show under what conditions heavy-tailed distributions can arise.
• Intuition: Heavy-tailed behavior arises:
• from the fact that wrong branching decisions may lead the procedure to explore an exponentially large subtree of the search space that contains no solutions;
• the procedure is characterized by a large variability in the time to find a solution on different runs, which leads to highly different trees from run to run;
••
Balanced vs. ImbalancedBalanced vs. Imbalanced Tree Model Tree Model
• Balanced Tree Model:
• chronological backtrack search model;• fixed variable ordering;• random child selection with no propagation
mechanisms;
(show demo)
221)]([
nnTE
12
122)]([n
nTV
The run time distribution of chronological backtrack search ona complete balanced tree is uniform (therefore not heavy-tailed).Both the expected run time and variance scale exponentially
Balanced Tree ModelBalanced Tree Model
– The expected run time and variance scale exponentially, in the height of the search tree (number of variables);
– The run time distribution is Uniform, (not heavy tailed ).
– Backtrack search on balanced tree model has no restart strategy with exponential polynomial time.
221)]([
nnTE
12
122)]([n
nTV
Chen, Gomes & Selman 01
• How can we improve on the balanced serach tree model?
• Very clever search heuristic that leads quickly to the solution node - but that is hard in general;
• Combination of pruning, propagation, dynamic variable ordering that prune subtrees that do not contain the solution, allowing for runs that are short.
• ---> resulting trees may vary dramatically from run to run.
• T - the number of leaf nodes visited up to and including the successful node; b - branching factor
0)1(][ iippibTP
Formal Model Yielding Heavy-Tailed Behavior
b = 2
(show demo)
• Expected Run Time• (infinite expected time)
• Variance•• (infinite variance)
• Tail
• (heavy-tailed)
•
][1 TEbp
][21 TVb
p
2log2][2
1 LCpbLpLTP
bp
Bounded Heavy-Tailed Behavior
(show demo)
No Heavy-tailed behavior for Proving Optimality
Proving Optimality
Small-World Vs. Heavy-Tailed Behavior
• Does a Small-World topology (Watts & Strogatz) induce heavy-tail behavior?
The constraint graph of a quasigroup exhibits a small-world topology(Walsh 99)
Exploiting Heavy-Tailed Behavior
• Heavy Tailed behavior has been observed in several domains: QCP, Graph Coloring, Planning, Scheduling, Circuit synthesis, Decoding, etc.
• Consequence for algorithm design: • Use restarts or parallel / interleaved
runs to exploit the extreme variance performance.Restarts provably eliminate
heavy-tailed behavior.(Gomes et al. 97, Hoos 99, Horvitz 99, Huberman, Lukose and Hogg 97, Karp et al 96, Luby et al. 93, Rish et al. 97, Wlash 99)
X XX XX
solved10 101010 10
Sequential: 50 +1 = 51 secondsParallel: 10 machines --- 1 second 51 x speedup
Super-linear Speedups
Interleaved (1 machine): 10 x 1 = 10 seconds 5 x speedup
Restarts70%
unsolved
1-F
(x)
Uns
olve
d fr
actio
n
Number backtracks (log)
no restarts
restart every 4 backtracks
250 (62 restarts)
0.001%unsolved
Example of Rapid Restart Speedup(planning)
1000
10000
100000
1000000
1 10 100 1000 10000 100000 1000000
log( cutoff )
log
( bac
ktra
cks
)
20
2000 ~100 restarts
Cutoff (log)
Num
ber
back
trac
ks (l
og)
~10 restarts100000
Sketch of proof of elimination of heavy tails
• Let’s truncate the search procedure• after m backtracks.
• Probability of solving problem with truncated version:
• Run the truncated procedure and restart it repeatedly.
pm X m Pr[ ]
X numberof backtracks to solve the problem
Y total number backtracks with restarts
F Y y pmY m c e c y
Pr[ ] ( ) /1 1
2
Number of starts Y m Geometric pmRe / ~ ( )
Y - does not have Heavy Tails