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HardnessAware Restart Policies
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(UAI 2001) Can predict a particular run's time to solution (very roughly) based ... Each run is from a different randomly-selected problem. ... – PowerPoint PPT presentation
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Title: HardnessAware Restart Policies
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Hardness-Aware Restart Policies
- Yongshao Ruan, Eric Horvitz, Henry Kautz
IJCAI 2003 Workshop on Stochastic Search
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Randomized Restart Strategies for Backtrack Search
- Simple idea randomize branching heuristic,
restart with new seed if solution is not found in
a reasonable amount of time - Effective on a wide range of structured problems
(Luby 1993, Gomes et al 1997) - Issue when to restart?
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Complete Knowledge of RTD
P(t)
D
t
T
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No Knowledge of RTD
- Can we do better?
- Information about progress of the current run
(looking good?) - Partial knowledge of RTD
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Answers
- (UAI 2001) Can predict a particular runs time
to solution (very roughly) based on features of
the solvers trace during an initial window - (AAAI 2002) Can improve time to solution by
immediately pruning runs that are predicted to be
long - Scenario You know RTD of a problem ensemble.
Each run is from a different randomly-selected
problem. Goal is solve some problem as soon as
possible (i.e., you can skip ones that look
hard). - In general optimal policy is to set cutoff
conditionally on value of observed features.
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Answers (continued)
- (CP 2002) Given partial knowledge about an
ensemble RTD, the optimal strategy uses the
information gained from each failed run to update
its beliefs about the shape of the RTD. - Scenario There is a set of k different problem
ensembles, and you know the ensemble RTD of each.
Nature chooses one of the ensembles at random,
but does not tell you which one. Each run is
from a different randomly-chosen problem from
that ensemble. Your goal is to solve some
problem as soon as possible. - In general cutoffs change for each run.
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Answers (final!)
- (IJCAI 2003 Workshop) The unknown RTD of a
particular problem instance can be approximated
by the RTD of a sub-ensemble - Scenario You are allowed to sample a problem
distribution and consider various ways of
clustering instances that have similar instance
RTDs. Then you are given a new random instance
and must solve it as quickly as possible (i.e.,
you cannot skip over problems!) - Most realistic?
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Partitioning ensemble RTD by instance median
run-times
Instance median gt ensemble median
Ensemble RTD
Instance median lt ensemble median
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MSE versus number of clusters
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Computing the restart strategy
- Assume that the (unknown) RTD of the given
instance is well-approximated by the RTD of one
of the clusters - Strategy depends upon your state of belief about
which cluster that is - Formalize as an POMDP
- State state of belief
- Actions use a particular cutoff K
- Effect solved, not solved
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Solving
- Bellman equation
- Solve by dynamic programming (ouch!)
Optimal expected time to solution from belief
state s
Probability that running with cutoff t in state s
fails (resulting in state s)
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Simple Example
- Suppose RTD of each instance is a scaled Pareto
controlled by a parameter b ? Uniform11, 200 - Median run time 2b, so medians are uniformly
distributed in 22, 200 - Cluster into two sub-distributions
- Median ? 110
- Median gt 110
- Dynamic programming solution
- 201 ,222 ,234 ,239 ,242 ,244
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Empirical Results
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Summary
- Last piece in basic mathematics for optimal
restarts with partial information - See paper for details of incorporating
observations - RTD alone gives log speedup over Luby universal
(still can be significant) - Unlimited potential for speedup with more
accurate run-time predictors!
