## Understanding the role of noise in stochastic local search: Analysis and experiments (2008)

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Venue: | Artificial Intelligence |

Citations: | 8 - 4 self |

### BibTeX

@ARTICLE{Mengshoel08understandingthe,

author = {Ole J. Mengshoel},

title = {Understanding the role of noise in stochastic local search: Analysis and experiments},

journal = {Artificial Intelligence},

year = {2008},

pages = {990}

}

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### Abstract

Stochastic local search (SLS) algorithms have recently been proven to be among the best approaches to solving computationally hard problems. SLS algorithms typically have a number of parameters, optimized empirically, that characterize and determine their performance. In this article, we focus on the noise parameter. The theoretical foundation of SLS, including an understanding of how to the optimal noise varies with problem di ¢ culty, is lagging compared to the strong empirical results obtained using these algorithms. A purely empirical approach to understanding and optimizing SLS noise, as problem instances vary, can be very computationally intensive. To complement existing experimental results, we formulate and analyze several Markov chain models of SLS. In particular, we compute expected hitting times and show that they are rational functions for individual problem instances as well as their mixtures. Expected hitting time curves are analytical counterparts to noise response curves reported in the experimental literature. Hitting time analysis using polynomials and convex functions is also discussed. In addition, we present examples and experimental results illustrating the impact of varying noise probability on SLS run time. In experiments, where most probable explanations in Bayesian networks are computed, we use synthetic problem instances as well as problem instances from applications. We believe that our results provide an improved theoretical understanding of the role of noise in stochastic local search, thereby providing a foundation for further progress in this area. 1

### Citations

7440 |
Probabilistie Reasoning in Intelligent Systems: Networks of Plausible Inference
- Pearl
- 1988
(Show Context)
Citation Context ...sume that the reader is familiar with fundamental de…nitions and results from the areas of graph theory, probability theory, and statistics; and in particular Markov chains [23] and Bayesian networks =-=[39]-=-. Some of the most pertinent concepts are brie‡y reviewed in this section. A direct and natural way to analyze an SLS algorithm’s operation on a problem instance is as a discrete time Markov chain wit... |

1342 |
Local computations with probabilities on graphical structures and their application to expert systems
- Lauritzen, Spiegelhalter
- 1988
(Show Context)
Citation Context ...easured in terms of upper bounds on minimal maximal clique size (or treewidth) [29, 33]. Upper bounds on optimal clique tree size and optimal maximal clique size can be computed using tree clustering =-=[24]-=-. In this article, we use the C=V -ratio directly to characterize the di¢ culty of synthetic BNs for SLS. There is a clear relationship between our Markov chain approach and observed SLS run times. We... |

1280 |
Information Theory, Inference and Learning Algorithms
- MacKay
- 2003
(Show Context)
Citation Context ...], is essentially Gibbs sampling in Bayesian networks. Even though the Gibbs sampler in many respects is general, it is quite di¤erent from most SLS approaches in that it typically operates in cycles =-=[25]-=-. A cyclic Gibbs sampler iterates systematically over all non-evidence nodes in a BN. SLS algorithms, on the other hand, are generally more opportunistic and do not operate on such …xed schedules. Sto... |

1262 |
Error bounds for convolution codes and an asymptotically optimal decoding algorithm
- Viterbi
- 1967
(Show Context)
Citation Context ...iform noise SU (p) and SGS with guided noise SG(p), were used. Further, a BN-speci…c optimal initialization algorithm — either forward simulation [13] or a randomized variant of the Viterbi algorithm =-=[31, 49]-=- — was used in I for each application BN. Each SGS variant — SU (p) or SG(p) — was tested with four di¤erent BNs — Mildew, Munin1, Pir3, and Water — giving a total of eight conditions as shown in Figu... |

700 | A new method for solving hard satisfiability problems - Selman, Levesque, et al. - 1992 |

604 |
The computational complexity of probabilistic inference using Bayesian belief network
- Cooper
- 1990
(Show Context)
Citation Context ...l BN problems of interest are hard. Exact MPE computation is NP-hard [47] and the problem of relative approximation of an MPE also been shown NP-hard [1]. Belief updating is computationally hard also =-=[4, 40]-=-. i=1 i=1 43 Stochastic Local Search We discuss a simpli…ed variant of SLS, SimpleSLS, based on the seminal WalkSAT architecture [16,44,45]. SimpleSLS is not intended to be competitive with state-of-... |

377 | Noise strategies for improving local search
- Selman, Kautz, et al.
- 1994
(Show Context)
Citation Context ...ndom variable; the analysis is then di¤erent and we shall not follow this route in this article. Our approach is closely related to previous stochastic local search (SLS) algorithms for satis…ability =-=[11, 16, 18, 43, 45, 46]-=- and Bayesian network problems [22, 27, 30, 36, 37]. It is somewhat related to previous research on stochastic simulation and guided local search. Stochastic simulation, which can be used to compute M... |

327 | Boosting Combinatorial Search Through Randomization
- Gomes, Selman, et al.
- 1998
(Show Context)
Citation Context ... V, and Theorem 28 can be adapted correspondingly to re‡ect this. Another technique, namely restarts or using MAX-FLIPS < 1, has been shown to be bene…cial in SLS [38] as well as in systematic search =-=[10]-=-. In many cases, restarts play a central role in SLS and using a close to optimal MAX-FLIPS value is essential for strong performance [38,43]; for TMCs this was observed in Section 5.3. In recent rese... |

290 |
and easy distributions of SAT problems
- Hard
- 1992
(Show Context)
Citation Context ...tional logic or the number of non-root nodes in BNs. For V > 0, the ratio C=V is well-de…ned and has turned out to be useful in predicting inference di¢ culty for randomly generated problem instances =-=[33,34]-=-. An easy-hard-easy pattern in inference di¢ culty as a function of the C=V -ratio has been observed for SAT [34]. For BNs, an easy-hard-harder pattern has been established when inference di¢ culty is... |

252 |
Stochastic Local Search: Foundations & Applications
- Hoos, Stutzle
- 2004
(Show Context)
Citation Context ... Introduction The stochastic local search (SLS) approach has proven to be highly competitive for solving a range of hard computational problems including satis…ability of propositional logic formulas =-=[11, 18, 45, 46]-=- as well as computing the most probable explanation [22, 27, 30] and the maximum a posteriori hypothesis [36, 37] in Bayesian networks. While the details of di¤erent SLS algorithms vary [18], by de…ni... |

233 | On the hardness of approximate reasoning
- Roth
- 1996
(Show Context)
Citation Context ...s, language understanding, intelligent data analysis, error correction coding, and biological analysis. Many interesting computational BN problems, including MPE computation, are NPcomplete or harder =-=[37,40,47]-=-, hence heuristic methods such as SLS are of interest. In this work we study the problem of computing the most probable explanation (MPE) in Bayesian networks. We use an SLS algorithm known as stochas... |

224 | Hard and easy distributions for SAT problems
- Mitchell, Selman, et al.
- 1992
(Show Context)
Citation Context ...itional logic or the number of non-root nodes in BNs. For V>0, the ratio C/V is welldefined and has turned out to be useful in predicting inference difficulty for randomly generated problem instances =-=[33,34]-=-. An easy–hard–easy pattern in inference difficulty as a function of the C/V -ratio has been observed for SAT [34]. For BNs, an easy–hard–harder pattern has been established when inference difficulty ... |

223 | Domain-independent extensions to GSAT: solving large structured satisfiability problems
- Selman, Kautz
- 1993
(Show Context)
Citation Context ...ty or noise. In this article we focus on noise during local search rather than, say, noisy initialization. Empirically, it turns out that noise has a dramatic impact on the run time of SLS algorithms =-=[14, 17, 26, 44, 45]-=-. Intuitively, there is a fundamental trade-o¤ between using high and low levels of noise in SLS. Let 0 p 1 represent the probability of taking a noise step. The argument for using low noise p is that... |

218 | The gambler’s ruin problem, genetic algorithms and the sizing of populations
- Harik, Cantu-Paz, et al.
- 1997
(Show Context)
Citation Context ...onsiderations, the trap Markov chain is an analytically derived model (see above as well as earlier work [6]). Markov chains have also been used in the analysis of genetic and evolutionary algorithms =-=[3, 9, 12, 48]-=-. Generally, this analysis emphasizes population-level e¤ects and is quite di¤erent from our analysis in this article. In particular, we do not know of any analysis of genetic or evolutionary algorith... |

195 |
Propagating Uncertainty Bayesian Networks by Probabilistic Logic Sampling
- Henrion
- 1988
(Show Context)
Citation Context ...d were: Initialization portfolio I : Uniform initialization IU versus guided initialization IG. Here, uniform means initialization uniformly at random while guided means the use of forward simulation =-=[13]-=-. Search portfolio S: Uniform noise SU (p) versus guided noise SG(p). The four conditions investigated were: (1) - IU and SU (p); (2) - IG and SU (p); (3) - IU and SG(p); and (4) - IG and SG(p). Figur... |

189 | Evidence for invariance in local search
- McAllester, Selman, et al.
- 1997
(Show Context)
Citation Context ...ty or noise. In this article we focus on noise during local search rather than, say, noisy initialization. Empirically, it turns out that noise has a dramatic impact on the run time of SLS algorithms =-=[14, 17, 26, 44, 45]-=-. Intuitively, there is a fundamental trade-o¤ between using high and low levels of noise in SLS. Let 0 p 1 represent the probability of taking a noise step. The argument for using low noise p is that... |

148 |
Analyzing deception in trap functions
- Deb, Goldberg
- 1993
(Show Context)
Citation Context ...sformed search space in order to obtain the number of correct bits; clearly d = 1:::1. To simplify notation, we often gloss over the transformations, and say that b = 1:::1 without loss of generality =-=[6]-=-. See Figure 7 for concrete examples. Using conditional expectations, one obtains from De…nition 3 the following well-known result. Theorem 5 (Expected hitting time) Let M be a Markov chain with state... |

131 | Algorithms for the Satisfiability (SAT) Problem: A Survey
- Gu, Purdom, et al.
- 1996
(Show Context)
Citation Context ...Introduction The stochastic local search (SLS) approach has proven to be highly competitive for solving a range of hard computational problems including satisfiability of propositional logic formulas =-=[11,18,45,46]-=- as well as computing the most probable explanation [22,27,30] and the maximum a posteriori hypothesis [36,37] in Bayesian networks. While the details of different SLS algorithms vary [18], by definit... |

111 |
Finite Markov chain analysis of genetic algorithms
- Goldberg, Segrest
- 1987
(Show Context)
Citation Context ...onsiderations, the trap Markov chain is an analytically derived model (see above as well as earlier work [6]). Markov chains have also been used in the analysis of genetic and evolutionary algorithms =-=[3, 9, 12, 48]-=-. Generally, this analysis emphasizes population-level e¤ects and is quite di¤erent from our analysis in this article. In particular, we do not know of any analysis of genetic or evolutionary algorith... |

99 |
Finding MAPs for Belief Networks is NP-hard
- Shimony, E
- 1994
(Show Context)
Citation Context ...s, language understanding, intelligent data analysis, error correction coding, and biological analysis. Many interesting computational BN problems, including MPE computation, are NPcomplete or harder =-=[37,40,47]-=-, hence heuristic methods such as SLS are of interest. In this work we study the problem of computing the most probable explanation (MPE) in Bayesian networks. We use an SLS algorithm known as stochas... |

90 |
MUNIN—A causal probabilistic network for interpretation of electromyographic findings
- Andreassen, Woldbye, et al.
- 1987
(Show Context)
Citation Context ... Munin1, Pir3, and Water. The Mildew BN is for determining the amount of fungicides to use to counter-act mildew attacks on wheat. The Munin1 network is a medical BN from the …eld of electromyography =-=[2]-=-. The Pir3 BN is for information …ltering for the purpose of battle…eld situation awareness [21,32]. The Water BN models the biological processes of water puri…cation. The purpose of these experiments... |

89 |
Applications of a general propagation algorithm for probabilistic expert systems
- Dawid
- 1992
(Show Context)
Citation Context ...e generated. The existence of satisfying assignments was checked by processing the BNs, with clamped leaf nodes, using the Hugin tree clustering system, which implements the tree clustering algorithm =-=[5, 24]-=-. The SGS system [27, 30, 31] with no restarts, uniform initialization in I, and search portfolio S(p) := f(p,UN) , (1 p,GM)g was employed, with varying noise probability. Following standard methodolo... |

73 | An adaptive noise mechanism for walksat
- Hoos
- 2002
(Show Context)
Citation Context ...ty or noise. In this article we focus on noise during local search rather than, say, noisy initialization. Empirically, it turns out that noise has a dramatic impact on the run time of SLS algorithms =-=[14, 17, 26, 44, 45]-=-. Intuitively, there is a fundamental trade-o¤ between using high and low levels of noise in SLS. Let 0 p 1 represent the probability of taking a noise step. The argument for using low noise p is that... |

66 | A bayesian approach to tackling hard computational problems - Horvitz, Ruan, et al. - 2001 |

65 | Local search algorithms for sat: An empirical evaluation
- Hoos, Stützle
(Show Context)
Citation Context ...high, to p + p, when operating under conditions of uncertainty about p and the problem instance distribution. Similar recommendations have in fact already been made based on experimental observations =-=[17]-=-. We believe these results shed additional light on the signi…cant impact of varying noise as observed in experiments [17, 26, 45]. When one does not know much about the problem instance distribution ... |

60 | Local search characteristics of incomplete SAT procedures
- Schuurmans, Southey
- 2000
(Show Context)
Citation Context ...ndom variable; the analysis is then di¤erent and we shall not follow this route in this article. Our approach is closely related to previous stochastic local search (SLS) algorithms for satis…ability =-=[11, 16, 18, 43, 45, 46]-=- and Bayesian network problems [22, 27, 30, 36, 37]. It is somewhat related to previous research on stochastic simulation and guided local search. Stochastic simulation, which can be used to compute M... |

47 | Towards a characterisation of the behaviour of stochastic local search algorithms for SAT
- Hoos, Stützle
- 1999
(Show Context)
Citation Context ...ons P (p)=Q(p), where P (p) and Q(p) are polynomials. This explicitly provides a functional 1form corresponding to noise response curves previously only established empirically in the SLS literature =-=[7,8,14,16,26]-=-. We also consider the use of polynomials and convex functions. Convexity is important because local optimality implies global optimality, a dramatic simpli…cation compared to unrestricted optimizatio... |

42 | Tuning local search for satisfiability testing - Parkes, Walser - 1996 |

39 | Using weighted MAX-SAT engines to solve MPE - Park |

34 | Stochastic local search for Bayesian networks
- Kask, Dechter
- 1999
(Show Context)
Citation Context ...to be highly competitive for solving a range of hard computational problems including satis…ability of propositional logic formulas [11, 18, 45, 46] as well as computing the most probable explanation =-=[22, 27, 30]-=- and the maximum a posteriori hypothesis [36, 37] in Bayesian networks. While the details of di¤erent SLS algorithms vary [18], by de…nition they all use stochasticity or noise. In this article we foc... |

34 | Complexity Results and Approximation Strategies for MAP Explanations
- Park, Darwiche
(Show Context)
Citation Context ...computational problems including satis…ability of propositional logic formulas [11, 18, 45, 46] as well as computing the most probable explanation [22, 27, 30] and the maximum a posteriori hypothesis =-=[36, 37]-=- in Bayesian networks. While the details of di¤erent SLS algorithms vary [18], by de…nition they all use stochasticity or noise. In this article we focus on noise during local search rather than, say,... |

33 | Restart policies with dependence among runs: A dynamic programming approach
- Ruan, Horvitz, et al.
(Show Context)
Citation Context ...s essential for strong performance [38,43]; for TMCs this was observed in Section 5.3. In recent research, an approach to dynamically optimizing the SLS restart parameter MAX-FLIPS has been developed =-=[41, 42]-=-, based on learned Bayesian networks that predict inference run times [19]. In general, it is consequently important to distinguish between expected run time and expected hitting time. We now formally... |

28 |
Approximating MAPs for belief networks in NPhard and other theorems. Arti¯cial Intelligence. 102
- Abdelbar, Hedetniemi
- 1998
(Show Context)
Citation Context ...ted belief updating [39]. Many of the computational BN problems of interest are hard. Exact MPE computation is NP-hard [47] and the problem of relative approximation of an MPE also been shown NP-hard =-=[1]-=-. Belief updating is computationally hard also [4, 40]. i=1 i=1 43 Stochastic Local Search We discuss a simpli…ed variant of SLS, SimpleSLS, based on the seminal WalkSAT architecture [16,44,45]. Simp... |

26 |
A Markov chain analysis on a genetic algorithm
- Suzuki
- 1993
(Show Context)
Citation Context ...onsiderations, the trap Markov chain is an analytically derived model (see above as well as earlier work [6]). Markov chains have also been used in the analysis of genetic and evolutionary algorithms =-=[3,9,12,48]-=-. Generally, this analysis emphasizes population-level effects and is quite different from our analysis in this article. In particular, we do not know of any analysis of genetic or evolutionary algori... |

24 | Approximating map using local search
- Park, Darwiche
- 2001
(Show Context)
Citation Context ...computational problems including satis…ability of propositional logic formulas [11, 18, 45, 46] as well as computing the most probable explanation [22, 27, 30] and the maximum a posteriori hypothesis =-=[36, 37]-=- in Bayesian networks. While the details of di¤erent SLS algorithms vary [18], by de…nition they all use stochasticity or noise. In this article we focus on noise during local search rather than, say,... |

17 |
adding more constraints makes a problem easier for hill-climbing algorithms: Analysing landscapes of csp’s
- Why
- 1997
(Show Context)
Citation Context ...p state j c 2 f0; 1g` . Our concept of trap is related to search space traps [11, 15] as well as search space reachability analysis, where states from which a solution is not reachable are determined =-=[51]-=-. It is easy to show that the trap size of a binary trap function is as follows. Lemma 22 Let f(b; `; z) be a binary trap function. The trap size jT j of f is given by jT j = P z 1 i=0 0 if z = o ` i ... |

17 | Efficient stochastic local search for MPE solving
- Hutter, Hoos, et al.
- 2005
(Show Context)
Citation Context ...approach of combining greedy and noisy search [22], and we do not investigate stochastic simulation in this article. There is also another class of local search algorithms, called guided local search =-=[20,35]-=-, which emphasizes changing the cost function rather than employing noise. Guided local search algorithms such as GLS [35] and GLS+ [20] are clearly very interesting, however our focus in this article... |

15 | Markov chain models of parallel genetic algorithms
- Cantú-Paz
- 2000
(Show Context)
Citation Context ...onsiderations, the trap Markov chain is an analytically derived model (see above as well as earlier work [6]). Markov chains have also been used in the analysis of genetic and evolutionary algorithms =-=[3, 9, 12, 48]-=-. Generally, this analysis emphasizes population-level e¤ects and is quite di¤erent from our analysis in this article. In particular, we do not know of any analysis of genetic or evolutionary algorith... |

14 | Modeling, Analysis, Design, and Control of Stochastic Systems - Kulkarni - 1999 |

11 |
Efficient Bayesian Network Inference: Genetic Algorithms, Stochastic Local Search, and Abstraction
- Mengshoel
- 1999
(Show Context)
Citation Context ...hms [17], hence we investigate one SLS system in depth. Stochastic greedy search (SGS), which can simulate SimpleSLS, computes MPEs and supports multiple search operators and initialization operators =-=[27,30]-=-. SGS has these input parameters: - a BN; f - MPE probability Pr (xi ); I - a portfolio of initialization algorithms; S - a portfolio of search algorithms; MAXFLIPS - the number of ‡ips before a resta... |

11 | A new method for solving hard satisability problems - Selman, Levesque, et al. - 1992 |

11 |
Finding the M most probable configurations in arbitrary graphical models
- Yanover, Weiss
- 2004
(Show Context)
Citation Context ...en multiple MPEs exist in a BN. Computation of the M most probable explanations, for M � 1 and where explanations do not have the same probability, is a generalization that has also been investigated =-=[50]-=-. Following Pearl, we sometimes denote computing an MPE as belief revision, while computing the marginal distribution over a BN node is also denoted belief updating [39]. Many of the computational BN ... |

10 |
CoRAVEN: Modelling and design of a multimedia intelligent infrastructure for collaborative intelligence analysis
- Jones, Hayes, et al.
- 1998
(Show Context)
Citation Context ...ounter-act mildew attacks on wheat. The Munin1 network is a medical BN from the …eld of electromyography [2]. The Pir3 BN is for information …ltering for the purpose of battle…eld situation awareness =-=[21,32]-=-. The Water BN models the biological processes of water puri…cation. The purpose of these experiments was to investigate the e¤ect of varying p, and also investigate SLS strategies that go beyond unif... |

9 | Controlled generation of hard and easy Bayesian networks: Impact on maximal clique tree in tree clustering
- Mengshoel, Wilkins, et al.
- 2006
(Show Context)
Citation Context ...tional logic or the number of non-root nodes in BNs. For V > 0, the ratio C=V is well-de…ned and has turned out to be useful in predicting inference di¢ culty for randomly generated problem instances =-=[33,34]-=-. An easy-hard-easy pattern in inference di¢ culty as a function of the C=V -ratio has been observed for SAT [34]. For BNs, an easy-hard-harder pattern has been established when inference di¢ culty is... |

7 | Designing resource-bounded reasoners using Bayesian networks: System health monitoring and diagnosis - Mengshoel - 2007 |

7 |
How adding more constraints makes a problem easier for hill-climbing algorithms: Analyzing landscapes of CSPs
- Yokoo
- 1997
(Show Context)
Citation Context ...ard are shown in Fig. 4. Our concept of trap is related to search space traps [11,15] as well as search space reachability analysis, where states from which a solution is not reachable are determined =-=[51]-=-. It is easy to show that the trap size of a binary trap function is as follows. Lemma 22. Let f(b; ℓ, z) be a binary trap function. The trap size |T | of f is given by { 0 if z = o |T |= ) if z � 1. ... |

6 | A mixture-model for the behaviour of SLS algorithms for SAT, in
- Hoos
- 2002
(Show Context)
Citation Context ...eful application of noise helps in escaping such traps. Trap functions are closely related to search space traps — portions of the search space that are attractive to SLS but do not contain solutions =-=[11, 15]-=-. Our results also include experiments. In this area, we used the stochastic local search algorithm SGS (stochastic greedy search) for computing MPEs in Bayesian networks [27,30]. Using SGS, we experi... |

5 | An empirical analysis of some heuristic features for local search in LPG
- Gerevini, Saetti, et al.
- 2004
(Show Context)
Citation Context ...e Work The use of randomization, in the form of noisy initialization and noisy local search steps, has empirically been shown to have a dramatic and positive impact on the performance of local search =-=[7,8,14,17,18,26,27,30,45, 46]-=-. Consequently, stochastic local search (SLS) algorithms are currently strong performers in several areas of automated reasoning, including the problems of satis…ability (SAT) in propositional logic [... |

5 |
Initialization and restart in stochastic local search: Computing a most probable explanation in Bayesian networks
- Mengshoel, Wilkins, et al.
(Show Context)
Citation Context ...ce of satisfying assignments was checked by processing the BNs, with clamped leaf nodes, using the Hugin tree clustering system, which implements the tree clustering algorithm [5, 24]. The SGS system =-=[27, 30, 31]-=- with no restarts, uniform initialization in I, and search portfolio S(p) := f(p,UN) , (1 p,GM)g was employed, with varying noise probability. Following standard methodology, experiments with SGS were... |

5 | Macroscopic models of clique tree growth for Bayesian networks
- Mengshoel
(Show Context)
Citation Context ... has been observed for SAT [34]. For BNs, an easy–hard–harder pattern has been established when inference difficulty is measured in terms of upper bounds on minimal maximal clique size (or treewidth) =-=[29,33]-=-. Upper bounds on optimal clique tree size and optimal maximal clique size can be computed using tree clustering [24]. In this article, we use the C/V -ratio directly to characterize the difficulty of... |