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Hardness-Aware Restart Policies

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

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Hardness-Aware Restart Policies

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  1. Hardness-Aware Restart Policies Yongshao Ruan, Eric Horvitz, & Henry Kautz IJCAI 2003 Workshop on Stochastic Search

  2. 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?

  3. Complete Knowledge of RTD P(t) D T* t

  4. No Knowledge of RTD • Can we do better? • Information about progress of the current run (looking good?) • Partial knowledge of RTD

  5. Observation horizon 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.

  6. 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.

  7. 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?

  8. Partitioning ensemble RTD by instance median run-times Instance median > ensemble median Ensemble RTD Instance median < ensemble median

  9. MSE versus number of clusters

  10. 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 }

  11. 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’)

  12. 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 …

  13. Empirical Results

  14. 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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