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### Hardness-Aware Restart Policies

Yongshao Ruan, Eric Horvitz, & Henry Kautz

IJCAI 2003 Workshop on Stochastic Search

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?

No Knowledge of RTD

- Can we do better?
- Information about progress of the current run (looking good?)
- Partial knowledge of RTD

Short

Long

Median run time

Answers- (UAI 2001) – Can predict a particular run’s time to solution (very roughly) based on features of the solver’s 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.

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.

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 RTD’s. 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?

Partitioning ensemble RTD by instance median run-times

Instance median > ensemble median

Ensemble RTD

Instance median < ensemble median

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 }

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’)

Simple Example

- Suppose RTD of each instance is a scaled Pareto controlled by a parameter b Uniform[11, 200]
- Median run time = 2b, so medians are uniformly distributedin [22, 200]
- Cluster into two sub-distributions
- Median 110
- Median > 110
- Dynamic programming solution:

201 ,222 ,234 ,239 ,242 ,244 …

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!

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