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!