Finding Ground States of SK Spin Glasses with hboa and GAs

Transcription

1 Finding Ground States of Sherrington-Kirkpatrick Spin Glasses with hboa and GAs Martin Pelikan, Helmut G. Katzgraber, & Sigismund Kobe Missouri Estimation of Distribution Algorithms Laboratory (MEDAL) University of Missouri, St. Louis, MO Theoretische Physik ETH Zürich, Switzerland Institut fr Theoretische Physik Technische Universität Dresden, Germany

2 Background Background Spin glasses are prototypical models for disordered systems. Important topic in theoretical physics for several decades. Popular also as test problem for evolutionary algorithms Can generate many random instances of varying difficulty. Highly multimodal landscape. Strong interactions between variables. Similarities with other difficult NP-complete problems. Usually spins arranged on 2D or 3D lattices, but only few studies for the infinitely dimensional SK spin glass. Yet the infinitely dimensional systems are most difficult and interesting.

3 Purpose Purpose Develop and test a robust approach to reliably solving large instances of SK spin glass and other NP complete problems. Don t compromise problem size or reliability. Two target areas Computational physics. Optimization.

4 Outline 1. Sherrington-Kirkpatrick (SK) spin glass. 2. Branch and bound for SK spin glass. 3. Approaches to reliable solution of large SK instances. 4. Future work. 5. Summary and conclusions.

5 SK Spin Glass SK spin glass (Sherrington & Kirkpatrick, 1978) Contains n spins s 1, s 2,..., s n. Ising spin can be in two states: +1 or 1. All pairs of spins interact. Interaction of spins s i and s j specified by real-valued coupling J i,j. Spin glass instance is defined by set of couplings {J i,j }. Spin configuration is defined by the values of spins {s i }.

6 Ground States of SK Spin Glasses Energy Energy of a spin configuration C is given by H(C) = i<j J i,j s i s j Ground states are spin configurations that minimize energy. Finding ground states of SK instances is NP-complete. Compare with other standard spin glass types 2D: Spin interacts with only 4 neighbors in 2D lattice. 3D: Spin interacts with only 6 neighbors in 3D lattice. SK: Spin interacts with all other spins. 2D is polynomially solvable; 3D and SK are NP-complete.

7 Random Instances of SK Spin Glass Random spin glass instances Spin glass models usually studied over large sets of random instances. Two most common distributions for couplings Gaussian: N(0, 1). ±J: +1 or 1 with equal probability. Sometimes a distance metric is imposed and coupling strength decreases with distance. Instances used in this work We use Gaussian couplings from N(0, 1).

8 Branch and Bound for SK Spin Glass Basic idea Traverse the entire search space (try all spin configurations). Each level decides on one spin (+1 or -1). Each leaf encodes a unique spin configuration. Branches that lead to provably suboptimal solutions are cut. Why? BB is inefficient, but can verify the global optimum.

9 Iterative Branch and Bound Basic idea Hartwig, Daske, and Kobe (1984). Reduce the system to consider only first i spins. Solve for i = 2 to i = n with step 1. Use previous results to provide better bounds. Denote best energy for for first i spins by f i. Lower bound on best energy for first j spins given by j 1 fj fj 1 J i,j. i=1 Effects of iterative approach We must solve n 1 problems instead of 1. But the overall performance much better.

10 Current Situation and Goal Current situation We have BB which is guaranteed solve small instances. We have hboa and other evolutionary algorithms which can solve larger instances but we need to set Population size. Number of generations. Goal Find reliable optima of relatively large instances. Don t stick with small problems because of BB. Don t compromise reliability by guessing EA parameters wildly.

11 Basic Approach Step 1: Branch and bound Generate many instances for small problems solvable with BB. Solve each instance with iterative BB. Step 2: hboa with optimal settings Apply hboa to each new instance. Find accurate statistical model for hboa parameters. Use model to predict sufficient parameters for larger problems. Step 3: Going to larger problems Apply hboa with the conservative settings from step 2 to find reliable global optima of larger instances. Go to step 2 (to get to larger and larger problems).

12 Step 1: Solve Small Problems with BB Prepare instances Generate 10,000 random SK instances for n = 20 to 80. This gives a total of 310,000 unique problem instances. Solve each instance with BB to find global optimum.

13 Step 2: Run hboa and Analyze Parameters Basic setup hboa with default parameters. Only population size and number of generations tuned. Deterministic 1-bit hill climber improves all solutions. Maximum number of generations is set to n. Population size set with bisection for each instance (10 successes in 10 independent runs). Analysis Total of 3,100,000 hboa runs to analyze. Analyze the distribution of the following Population size. Number of generations. Number of evaluations. Number of flips of hill climber.

14 Step 2: Results Population size appears to follow log-normal distribution. Number of generations is very small in all cases n = 20 n = Frequency Frequency Population size Population size

15 Step 2: Results Estimate parameters of pop. size distribution for each n. Derive upper bound from 0.001% tail of the distribution, which sould solve % instances. Find a fit of this upper bound. Predict pop. size for larger problems (up to n = 200). Population size Fit of % percentile Power law fit 95% prediction bounds percentile Problem size Population size Prediction for larger instances Power law fit 95% prediction bounds Problem size

16 Step 3: Find Reliable Optima of Larger Instances Starting point Predicted bound on pop. size to solve % instances. Prepare larger instances Generate 1,000 instances for n = 100 to 200. For each instance Use estimated upper bound of the population size. Use maximum number of generations of n. Make 10 hboa runs on each instance to find global optimum. Record the best solution found. All runs should agree.

17 Step 2 Revisited: Run hboa and Analyze Parameters Run and analyze hboa Run hboa for n = 100 to 200 as for smaller instances. Repeat bisection 10 times for each instance. Analysis Total of 2,100,000 successful hboa runs. Do the analysis as for smaller problems.

18 Step 2 Revisited: Results Estimate parameters of pop. size distribution for each n. Derive upper bound from 0.001% tail of the distribution. Find a fit of this upper bound. Predict pop. size for larger problems (up to n = 300). Population size Fit of % percentile Power law fit 95% prediction bounds percentile Problem size Population size Prediction for larger instances Power law fit 95% prediction bounds Problem size

19 So How Does It Work? How does it work? Incrementally increase problem size. Set parameters using model based on smaller problems. If distributions are easy to model and the growth of different parameters can be fit reliably, this allows us to reliably solve large instances even when no complete algorithm is tractable. Ultimate goal Go to problems with 4,000 spins or so. Important Don t make too big steps to ensure tractability and reliability.

20 hboa Results for n Mean number of evaluations Problem size

21 Other Approaches: Fit Distribution Parameters Basic idea Fit distribution of a quantity (e.g. pop. size). Fit a model to the parameters of the distribution. Estimate parameters for larger problems from the fit. Compute tails from estimated parameters. Population size Log mean Power law fit for mean Log standard deviation Power law fit for std. dev Problem size Number of iterations Log mean Power law fit (mean) Log standard deviation Power law fit (std. dev.) Problem size

22 Other Approaches: Population Doubling Basic idea Related to parameter-less genetic algorithms. Start with a reasonable population size. Make 10 runs (can change). Double the population and repeat. Terminate doubling when All 10 runs result in the same solution. Last couple of rounds resulted in the same solution.

23 Comparison: hboa vs. GA (Uniform Crossover) Number of evaluations Number of flips Num. GA (U) evals. / num. hboa evals Number of spins Num. GA (U) flips / num. hboa flips Number of spins

24 Comparison: Uniform vs. Two-Point Crossover Number of evaluations Number of flips Num. GA (U) evals. / num. GA (2P) evals Number of spins Num. GA (U) flips / num. GA (2P) flips Number of spins

25 Conclusions and Future Work Conclusions The proposed approaches hold big promise for reliable solution of extremely large problems. The proposed approaches can be used with other optimization techniques which require adequate parameter settings. SK spin glass closely related to other difficult problems, such as protein folding. Future work Compare hboa & GA to other techniques Extremal optimization (EO). Hysteretic optimization (HO). Create efficient hybrids of hboa, GA, EO, HO, and BB. Apply other efficiency enhancement techniques. Further increase problem size to 1,000 4,000 and more.

26 Acknowledgments Acknowledgments NSF; NSF CAREER grant ECS U.S. Air Force, AFOSR; FA University of Missouri; High Performance Computing Collaboratory sponsored by Information Technology Services; Research Award; Research Board.