Automatic Algorithm Configuration based on Local Search EARG presentation December 13, 2006 Frank Hutter

2 Motivation Want to design “best” algorithm A to solve a problem  Many design choices need to be made  Some choices deferred to later: free parameters of algorithm  Set parameters to maximise empirical performance Finding best parameter configuration is non-trivial  Many parameter configurations  Many test instances  Many runs to get realistic estimates for randomised algorithms  Tuning still often done manually, up to 50% of development time Let’s automate tuning!

3 Parameters in different research areas NP-hard problems: tree search  Variable/value heuristics, learning, restarts, … NP-hard problems: local search  Percentage of random steps, tabu length, strength of escape moves, … Nonlinear optimisation: interior point methods  Slack, barrier init, barrier decrease rate, bound multiplier init, … Computer vision: object detection  Locality, smoothing, slack, … Supervised machine learning  NOT model parameters  L1/L2 loss, penalizer, kernel, preprocessing, num. optimizer, … Compiler optimisation, robotics, …

4 Related work Best fixed parameter setting  Search approaches [Minton ’93, ‘96], [Hutter ’04], [Cavazos & O’Boyle ’05], [Adenso-Diaz & Laguna ’06], [Audet & Orban ’06]  Racing algorithms/Bandit solvers [Birattari et al. ’02], [Smith et al. ’04 – ’ 06]  Stochastic Optimisation [Kiefer & Wolfowitz ’52], [Geman & Geman ’84], [Spall ’87] Per instance  Algorithm selection [Knuth ’75], [Rice 1976], [Lobjois and Lemaître ’98], [Leyton-Brown et al. ’02], [Gebruers et al. ’05]  Instance-specific parameter setting [Patterson & Kautz ’02] During algorithm run  Portfolios [Kautz et al. ’02], [Carchrae & Beck ’05], [Gagliolo & Schmidhuber ’05, ’06]  Reactive search [Lagoudakis & Littman ’01, ’02], [Battiti et al ’05], [Hoos ’02]

5 Static Algorithm Configuration (SAC) SAC problem instance: 3-tuple ( D, A,  ), where  D is a distribution of problem instances,  A is a parameterised algorithm, and   is the parameter configuration space of A. Candidate solution: configuration  2 , expected cost C(  ) = E I » D [Cost( A, , I)]  Stochastic Optimisation Problem

6 Static Algorithm Configuration (SAC) SAC problem instance: 3-tuple ( D, A,  ), where  D is a distribution of problem instances,  A is a parameterised algorithm, and   is the parameter configuration space of A. Candidate solution: configuration  2 , expected cost C(  ) = statistic[ CD ( A, , I) ] CD( A, , D ): cost distribution of algorithm A with parameter configuration  across instances from D. Variation due to randomisation of A and variation in instances.  Stochastic Optimisation Problem

7 Parameter tuning in practice Manual approaches are often fairly ad hoc  Full factorial design - Expensive (exponential in number of parameters)  Tweak one parameter at a time - Only optimal if parameters independent  Tweak one parameter at a time until no more improvement possible - Local search ! local minimum Manual approaches are suboptimal  Only find poor parameter configurations  Very long tuning time  Want to automate

8 Simple Local Search in Configuration Space

9 Iterated Local Search (ILS)

10 ILS in Configuration Space

11 What is the objective function? User-defined objective function  E.g. expected runtime across a number of instances  Or expected speedup over a competitor  Or average approximation error  Or anything else BUT: must be able to approximate objective based on a finite (small) number of samples  Statistic is expectation ! sample mean (weak law of large numbers)  Statistic is median ! sample median (converges ??)  Statistic is 90% quantile ! sample 90% quantile (underestimated with small samples !)  Statistic is maximum (supremum) ! cannot generally approximate based on finite sample?

12 Parameter tuning as “pure” optimisation problem Approximate objective function (statistic of a distribution) based on fixed number of N instances  Beam search [Minton ’93, ‘96]  Genetic algorithms [Cavazos & O’Boyle ’05]  Exp. design & local search [Adenso-Diaz & Laguna ‘06]  Mesh adaptive direct search [Audet & Orban ‘06] But how large should N be ?  Too large: take too long to evaluate a configuration  Too small: very noisy approximation ! over-tuning

13 Minimum is a biased estimator Let x 1, …, x n be realisations of rv’s X 1, …, X n Each x i is a priori an unbiased estimator of E[ X i ]  Let x j = min(x 1,…,x n ) This is an unbiased estimator of E[ min(X 1,…,X n ) ] but NOT of E[ X j ] (because we’re conditioning on it being the minimum!) Example  Let X i » Exp( ), i.e. F(x| ) = 1 - exp(- x)  E[ min(X 1,…,X n ) ] = 1/n E[ min(X i ) ]  I.e. if we just take the minimum and report its runtime, we underestimate cost by a factor of n (over-confidence) Similar issues for cross-validation etc

14 Primer: over-confidence for N=1 sample y-axis: runlength of best  found so far Training: approximation based on N=1 sample (1 run, 1 instance) Test: 100 independent runs (on the same instance) for each  Median & quantiles over 100 repetitions of the procedure

15 Over-tuning More training ! worse performance  Training cost monotonically decreases  Test cost can increase Big error in cost approximation leads to over- tuning (on expectation):  Let  * 2  be the optimal parameter configuration  Let  ’ 2  be a suboptimal parameter configuration with better training performance than  *  If the search finds  * before  ’ (and it is the best one so far), and the search then finds  ’, training cost decreases but test cost increases

16 1, 10, 100 training runs (qwh)

17 Over-tuning on uf400 instance

18 Other approaches without over-tuning Another approach  Small N for poor configurations , large N for good ones - Racing algorithms [Birattari et al. ’02, ‘05] - Bandit solvers [Smith et al., ’04 –’06]  But treat all parameter configurations as independent - May work well for small configuration spaces - E.g. SAT4J has over a million possible configurations ! Even a single run each is infeasible My work: combination of approaches  Local search in parameter configuration space  Start with N=1 for each , increase it whenever  re-visited  Does not have to visit all configurations, good ones visited often  Does not suffer from over-tuning

19 Unbiased ILS in parameter space Over-confidence for each  vanishes as N !1 Increase N for good   Start with N=0 for all   Increment N for  whenever  is visited Can proof convergence  Simple property, even applies for round-robin Experiments: SAPS on qwh ParamsILS with N=1 Focused ParamsILS

20 No over-tuning for my approach (qwh)

21 Runlength on simple instance (qwh)

22 SAC problem instances SAPS [Hutter, Tompkins, Hoos ’02]  4 continuous parameters h , ,wp, P smooth i  Each discretised to 7 values ! 7 4 =2,401 configurations Iterated Local Search for MPE [Hutter ’04]  4 discrete + 4 continuous (2 of them conditional)  In total 2,560 configurations Instance distributions  Two single instances, one easy (qwh), one harder (uf)  Heterogeneous distribution with 98 instances from satlib for which SAPS median is 50K-1M steps  Heterogeneous distribution with 50 MPE instances: mixed Cost: average runlength, q90 runtime, approximation error

23 Comparison against CALIBRA

24 Comparison against CALIBRA

25 Comparison against CALIBRA

26 Local Search for SAC: Summary Direct approach for SAC Positive  Incremental homing-in to good configurations  But using the structure of parameter space (unlike bandits)  No distributional assumptions  Natural treatment of conditional parameters Limitations  Does not learn - Which parameters are important? - Unnecessary binary parameter: doubles search space  Requires discretization - Could be relaxed (hybrid scheme etc)