SlideShare a Scribd company logo

Efficient Hill Climber for Multi-Objective Pseudo-Boolean Optimization

J
jfrchicanog

1) The document proposes an efficient hill climber algorithm for multi-objective pseudo-boolean optimization problems. 2) It computes scores that represent the change in fitness from moving to neighboring solutions, and updates these scores incrementally as the solution moves rather than recomputing from scratch. 3) The scores can be decomposed and updated in constant time by analyzing the variable interaction graph to identify variables that do not interact.

Technology
1 of 30
Download to read offline
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
1 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background Contribution Experiments Conclusions & Future Work Efficient Hill Climber for Multi-Objective Pseudo-Boolean Optimization Francisco Chicano, Darrell Whitley, Renato Tinós
2 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work r = 1 n r = 2 n 2 r = 3 n 3 r n r Ball Pr i=1 n i S1( r = 1 n r = 2 n 2 r = 3 n 3 r n r Ball Pr i=1 n i • Considering binary strings of length n and Hamming distance… Solutions in a ball of radius r r=1 r=2 r=3 Ball of radius r Previous work Research Question How many solutions at Hamming distance r? If r << n : Θ (nr)
3 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • We want to find improving moves in a ball of radius r around solution x • What is the computational cost of this exploration? • By complete enumeration: O (nr) if the fitness evaluation is O(1) • In previous work we proposed a way to find improving moves in ball of radius r in O(1) (constant time independent of n) Improving moves in a ball of radius r r Ball of radius r Previous work Research Question Chicano, Whitley, Sutton, GECCO 2014: 437-444
4 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Research Question Ball of radius r Previous work Research Question
5 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • Definition: • where f(i) only depends on k variables (k-bounded epistasis) • We will also assume that the variables are arguments of at most c subfunctions • Example (m=4, n=4, k=2): • Is this set of functions too small? Is it interesting? • Max-kSAT is a k-bounded pseudo-Boolean optimization problem • NK-landscapes is a (K+1)-bounded pseudo-Boolean optimization problem • Any compressible pseudo-Boolean function can be reduced to a quadratic pseudo-Boolean function (e.g., Rosenberg, 1975) Mk Landscape (Whitley, GECCO2015: 927-934) Pseudo-Boolean functions Scores Update Decomposition The family of k-bounded pseudo-Boolean Optimization problems have also been described as an embedded landscape. An embedded landscape [3] with bounded epistasis k is de- fined as a function f(x) that can be written as the sum of m subfunctions, each one depending at most on k input variables. That is: f(x) = mX i=1 f(i) (x), (1) where the subfunctions f(i) depend only on k components of x. Embedded Landscapes generalize NK-landscapes and the MAX-kSAT problem. We will consider in this paper that the number of subfunctions is linear in n, that is m 2 O(n). For NK-landscapes m = n and is a common assumption in MAX-kSAT that m 2 O(n). 3. SCORES IN THE HAMMING BALL For v, x 2 Bn , and a pseudo-Boolean function f : Bn ! R, we denote the Score of x with respect to move v as Sv(x), defined as follows:1 Sv(x) = f(x v) f(x), (2) 1 We omit the function f in Sv(x) to simplify the notation. S(l) v (x) = Equation (5) cl change in the mov f(l) the Score of th this subfunction w On the other hand we only need to c changed variables t acterized by the m we can write (3) as S 3.1 Scores De The Score values tion than just the c in that ball. Let us balls of radius r = xj are two variabl ments of any subfu f = + + +f(1)(x) f(2)(x) f(3)(x) f(4)(x) x1 x2 x3 x4
6 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • Based on Mk Landscapes Going Multi-Objective: Vector Mk Landscape Pseudo-Boolean functions Scores Update Decomposition x1 x2 x3 x4 x5 f (1) 1 f (2) 1 f (3) 1 f (4) 1 f (5) 1 f (1) 2 f (2) 2 f (3) 2 (a) Vector Mk Landscape x3x4x5 f1 f2
7 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • Let us represent a potential move of the current solution with a binary vector v having 1s in the positions that should be flipped • The score of move v for solution x is the difference in the fitness value of the neighboring and the current solution • Scores are useful to identify improving moves: if Sv(x) > 0, v is an improving move • We keep all the scores in the score vector Scores: definition Current solution, x Neighboring solution, y Move, v 01110101010101001 01111011010101001 00001110000000000 01110101010101001 00110101110101111 01000000100000110 01110101010101001 01000101010101001 00110000000000000 Pseudo-Boolean functions Scores Update Decomposition Sv(x) = f(x v) f(x)
8 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive) Scores update r Selected improving move Update the score vector Pseudo-Boolean functions Scores Update Decomposition
9 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive) • How can we do it less expensive? • We have still O(nr) scores to update! • … thanks to two key facts: • We don’t need all the O(nr) scores to know if there is an improving move • From the ones we need, we only have to update a constant number of them and we can do each update in constant time Key facts for efficient scores update r Pseudo-Boolean functions Scores Update Decomposition
10 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Examples: 1 and 4 f(1) f(2) f(3) f(4) x1 x2 x3 x4 Ball Pr i=1 n i S1(x) = f(x 1) f(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) f(1) f(2) f(3) f(4) x1 x2 x3 x4 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) Pseudo-Boolean functions Scores Update Decomposition
11 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Example: 1,4 f(1) f(2) f(3) f(4) x1 x2 x3 x4 r Ball Pr i=1 n i S1(x) = f(x 1) f(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) n i S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1(x) = f(x 1) f(x) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) Ball Pr i=1 n i S1(x) = Sv(x) = f(x v) f(x) = S4(x) = S1(x) = f(x 1) f(x) x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) We don’t need to store S1,4(x) since can be computed from others If none of 1 and 4 are improving moves, 1,4 will not be an improving move Pseudo-Boolean functions Scores Update Decomposition
12 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Example: 1,2 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) f(1) f(2) f(3) x1 x2 x3 x4 f(1) f(2) f(3) x1 x2 x3 x4 f(1) x1 x2 Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) x1 and x2 “interact” Pseudo-Boolean functions Scores Update Decomposition
13 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Decomposition rule for scores • When can we decompose a score as the sum of lower order scores? • … when the variables in the move can be partitioned in subsets of variables that DON’T interact • Let us define the Co-occurrence Graph (a.k.a. Variable Interaction Graph, VIG) f(1) f(2) f(3) f(4) x1 x2 x3 x4 There is an edge between two variables if there exists a function that depends on both variables (they “interact”) x4 x3 x1 x2 Pseudo-Boolean functions Scores Update Decomposition
14 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work What is an improving move in MO? • An improving move is one that provides a solution dominating the current one • We define strong and weak improving moves. • We are interested in strong improving moves Taking improving moves Algorithm f2 f1 Assuming maximization Can we safely take only strong improving moves?
15 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work We need to take weak improving moves • We could miss some higher order strong improving moves if we don’t take weak improving moves S2 S1 stored stored not stored current solution Sv(x) = f(x v) f(x) Sv1[v2 (x) = Sv1 (x) + Sv2 (x) Taking improving moves Algorithm
16 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Cycling • We can make the hill climber to cycle if we take weak improving moves S2 S1 stored current solution S2 S1 stored current solution Taking improving moves Algorithm
17 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Solution: weights for the scores • Given a weight vector w with wi > 0 S2 S1 w 1st strong improving 2nd w-improving 2nd w-improving A w-improving move is one with w · Sv(x) > 0 Taking improving moves Algorithm
18 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work MNK Landscape Problem Results Source Code Why NKq and not NK? Floating point precision • An MNK Landscape is a multi-objective pseudo-Boolean problem where each objective is an NK Landscape (Aguirre, Tanaka, CEC 2004: 196-203) • An NK-landscape is a pseudo-Boolean optimization problem with objective function: where each subfunction f(l) depends on variable xl and K other variables •The subfunctions are randomly generated and the values are taken in the range [0,1] • In NKq-landscapes the subfunctions take integer values in the range [0,q-1] • We use NKq-landscapes in the experiments f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) f(x) = NX l=1 f(l) (x) 1 • In the adjacent model the variables are consecutive f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4
19 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Runtime 0 100 200 300 400 500 600 700 800 900 1000 1100 10 20 30 40 50 60 70 80 90 100 Averagetimepermove(microseconds) N (number of variables in thousands) r=1, d=2 r=1, d=3 r=2, d=2 r=2, d=3 r=3, d=2 r=3, d=3 Fig. 2. Average time per move in µs for the Multi-Start Hill Climber based on Algo- Neighborhood size: 166 trillion Problem Results Source Code
20 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Quality 600000 610000 620000 630000 640000 650000 660000 670000 610000 630000 650000 f2 f1 r=1 r=2 r=3 • 50% Empirical Attainment Functions (Knowles, ISDA 2005: 552-557) Problem Results Source Code
21 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Source Code https://github.com/jfrchicanog/EfficientHillClimbers Problem Results Source Code
22 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Conclusions and Future Work Conclusions & Future Work • We can efficiently identify improving moves in a ball of radius r around a solution in constant time (independent of n) • The space required to store the information (scores) is linear in the size of the problem n Conclusions • We can also deal with constrained problems (GECCO 2016) • Generalize to other search spaces • Combine with high-level algorithms Future Work
23 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Acknowledgements Efficient Hill Climber for Multi-Objective Pseudo-Boolean Optimization
24 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • Whitley and Chen proposed an O(1) approximated steepest descent for MAX-kSAT and NK-landscapes based on Walsh decomposition • For k-bounded pseudo-Boolean functions its complexity is O(k2 2k) • Chen, Whitley, Hains and Howe reduced the time required to identify improving moves to O(k3) using partial derivatives • Szeider proved that the exploration of a ball of radius r in MAX-kSAT and kSAT can be done in O(n) if each variable appears in a bounded number of clauses Previous work Ball of radius r Improving moves Previous work Research Question D. Whitley and W. Chen. Constant time steepest descent local search with lookahead for NK-landscapes and MAX-kSAT. GECCO 2012: 1357–1364 W. Chen, D. Whitley, D. Hains, and A. Howe. Second order partial derivatives for NK-landscapes. GECCO 2013: 503–510 S. Szeider. The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optimization, 8(1):139–145, 2011
25 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores to store • In terms of the VIG a score can be decomposed if the subgraph containing the variables in the move is NOT connected • The number of these scores (up to radius r) is O((3kc)r n) • Details of the proof in the paper • With a linear amount of information we can explore a ball of radius r containing O(nr) solutions x4 x3 x1 x2 x4 x3 x1 x2 S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f S1,2(x) 6= S1(x) + S2(x) l=1 l=1 S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) We need to store the scores of moves whose variables form a connected subgraph of the VIG Pseudo-Boolean functions Scores Update Decomposition
26 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores to update • Let us assume that x4 is flipped • Which scores do we need to update? • Those that need to evaluate f(3) and f(4) f(1) f(2) f(3) f(4) x1 x2 x3 x4 x4 x3 x1 x2 • The scores of moves containing variables adjacent or equal to x4 in the VIG Main idea Decomposition of scores Constant time update
27 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores to update and time required • The number of neighbors of a variable in the VIG is bounded by c k • The number of stored scores in which a variable appears is the number of spanning trees of size less than or equal to r with the variable at the root • This number is constant • The update of each score implies evaluating a constant number of functions that depend on at most k variables (constant), so it requires constant time x4 x3 x1 x2 f(1) f(2) f(3) f(4) x1 x2 x3 x4 O( b(k) (3kc)r |v| ) b(k) is a bound for the time to evaluate any subfunction Main idea Decomposition of scores Constant time update
28 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Results: checking the time in the random model • Random model: the number of subfunctions in which a variable appears, c, is not bounded by a constant NKq-landscapes • Random model • N=1,000 to 12,000 • K=1 to 4 • q=2K+1 • r=1 to 4 • 30 instances per conf. K=3 r=1 r=2 r=3 0 2000 4000 6000 8000 10000 12000 0 50 100 150 200 n TimeHsL r=1 r=2 r=3 0 2000 4000 6000 8000 10000 12000 0 100000 200000 300000 400000 N Scoresstoredinmemory NKq-landscapes Sanity check Random model Next improvement
29 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores Problem Formulation Landscape Theory Decomposition SAT Transf. Results f(1) f(2) f(3) x1 x2 x3 x4 f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4 f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4
30 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores Problem Formulation Landscape Theory Decomposition SAT Transf. Results f(1) x1 x2 f(1) f(2) f(3) x1 x2 x3 x4 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x)

More Related Content

PDF
Optimal L-shaped matrix reordering, aka graph's core-periphery
Francesco Tudisco
 
PDF
k-MLE: A fast algorithm for learning statistical mixture models
Frank Nielsen
 
PDF
BEADS : filtrage asymétrique de ligne de base (tendance) et débruitage pour d...
Laurent Duval
 
PDF
Core–periphery detection in networks with nonlinear Perron eigenvectors
Francesco Tudisco
 
PDF
Nodal Domain Theorem for the p-Laplacian on Graphs and the Related Multiway C...
Francesco Tudisco
 
PDF
Patch Matching with Polynomial Exponential Families and Projective Divergences
Frank Nielsen
 
PDF
Polyhedral computations in computational algebraic geometry and optimization
Vissarion Fisikopoulos
 
PDF
A nonlinear approximation of the Bayesian Update formula
Alexander Litvinenko
 
Optimal L-shaped matrix reordering, aka graph's core-periphery
Francesco Tudisco
 
k-MLE: A fast algorithm for learning statistical mixture models
Frank Nielsen
 
BEADS : filtrage asymétrique de ligne de base (tendance) et débruitage pour d...
Laurent Duval
 
Core–periphery detection in networks with nonlinear Perron eigenvectors
Francesco Tudisco
 
Nodal Domain Theorem for the p-Laplacian on Graphs and the Related Multiway C...
Francesco Tudisco
 
Patch Matching with Polynomial Exponential Families and Projective Divergences
Frank Nielsen
 
Polyhedral computations in computational algebraic geometry and optimization
Vissarion Fisikopoulos
 
A nonlinear approximation of the Bayesian Update formula
Alexander Litvinenko
 

What's hot (20)

PDF
Divergence clustering
Frank Nielsen
 
PDF
QMC: Operator Splitting Workshop, A New (More Intuitive?) Interpretation of I...
The Statistical and Applied Mathematical Sciences Institute
 
PDF
The dual geometry of Shannon information
Frank Nielsen
 
PDF
Nonconvex Compressed Sensing with the Sum-of-Squares Method
Tasuku Soma
 
PDF
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
The Statistical and Applied Mathematical Sciences Institute
 
PDF
Computational Information Geometry: A quick review (ICMS)
Frank Nielsen
 
PDF
Divergence center-based clustering and their applications
Frank Nielsen
 
PDF
The low-rank basis problem for a matrix subspace
Tasuku Soma
 
PDF
Regret Minimization in Multi-objective Submodular Function Maximization
Tasuku Soma
 
PDF
CLIM Fall 2017 Course: Statistics for Climate Research, Geostats for Large Da...
The Statistical and Applied Mathematical Sciences Institute
 
PDF
Tailored Bregman Ball Trees for Effective Nearest Neighbors
Frank Nielsen
 
PDF
A series of maximum entropy upper bounds of the differential entropy
Frank Nielsen
 
PDF
H2O World - Consensus Optimization and Machine Learning - Stephen Boyd
Sri Ambati
 
PDF
CLIM Fall 2017 Course: Statistics for Climate Research, Guest lecture: Data F...
The Statistical and Applied Mathematical Sciences Institute
 
PDF
On the Jensen-Shannon symmetrization of distances relying on abstract means
Frank Nielsen
 
PDF
CLIM Fall 2017 Course: Statistics for Climate Research, Estimating Curves and...
The Statistical and Applied Mathematical Sciences Institute
 
PDF
Expectation propagation
Dong Guo
 
PDF
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
The Statistical and Applied Mathematical Sciences Institute
 
PDF
Classification with mixtures of curved Mahalanobis metrics
Frank Nielsen
 
PDF
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
The Statistical and Applied Mathematical Sciences Institute
 
Divergence clustering
Frank Nielsen
 
QMC: Operator Splitting Workshop, A New (More Intuitive?) Interpretation of I...
The Statistical and Applied Mathematical Sciences Institute
 
The dual geometry of Shannon information
Frank Nielsen
 
Nonconvex Compressed Sensing with the Sum-of-Squares Method
Tasuku Soma
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
The Statistical and Applied Mathematical Sciences Institute
 
Computational Information Geometry: A quick review (ICMS)
Frank Nielsen
 
Divergence center-based clustering and their applications
Frank Nielsen
 
The low-rank basis problem for a matrix subspace
Tasuku Soma
 
Regret Minimization in Multi-objective Submodular Function Maximization
Tasuku Soma
 
CLIM Fall 2017 Course: Statistics for Climate Research, Geostats for Large Da...
The Statistical and Applied Mathematical Sciences Institute
 
Tailored Bregman Ball Trees for Effective Nearest Neighbors
Frank Nielsen
 
A series of maximum entropy upper bounds of the differential entropy
Frank Nielsen
 
H2O World - Consensus Optimization and Machine Learning - Stephen Boyd
Sri Ambati
 
CLIM Fall 2017 Course: Statistics for Climate Research, Guest lecture: Data F...
The Statistical and Applied Mathematical Sciences Institute
 
On the Jensen-Shannon symmetrization of distances relying on abstract means
Frank Nielsen
 
CLIM Fall 2017 Course: Statistics for Climate Research, Estimating Curves and...
The Statistical and Applied Mathematical Sciences Institute
 
Expectation propagation
Dong Guo
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
The Statistical and Applied Mathematical Sciences Institute
 
Classification with mixtures of curved Mahalanobis metrics
Frank Nielsen
 
Program on Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applie...
The Statistical and Applied Mathematical Sciences Institute
 
Ad

Viewers also liked (16)

PPTX
Números reales
Ana Trujillo
 
PPTX
(youthlab indo) Indonesian animology: Trends on youth anime community
youth laboratory indonesia
 
DOCX
MANAJEMEN DAN PENDOKUMENTASIANASUHAN KEBIDANAN PADA BAYI NY“M” DENGAN ASFIKSI...
Warnet Raha
 
PPT
Hotelería y Turismo
Karla Armijos
 
PPTX
Fútbol Al Máximo
carlosagr96
 
PPTX
Generational intelligence- a new HR challenge
Abdul Razak
 
PPTX
Radio nahndic3a1
Eduardo Rojas
 
PPTX
Presentacion nueva
angelicafranco
 
DOCX
JDLHBedCheckDoc
Sabrina Searles
 
PDF
MANAJEMEN DAN PENDOKUMENTASIAN ASUHAN KEBIDANAN INTRANATAL PADA NY”Y” DENGAN ...
Warnet Raha
 
DOCX
MANAJEMEN DAN PENDOKUMENTASIANASUHAN KEBIDANAN PADA BAYI NY“M” DENGAN ASFIKSI...
Warnet Raha
 
PPTX
Designing peer review for digital media instruction
Cheryl Ball
 
PPTX
HV 233, Los inicios históricos de las Verapaces, Erick Reyes Andrade
123apn1
 
DOCX
Manfaat dan pengaruh gadget
sanrumi221098
 
PPTX
Amistad
Kevin Barrera
 
PPTX
Master armero
Ahatzin
 
Números reales
Ana Trujillo
 
(youthlab indo) Indonesian animology: Trends on youth anime community
youth laboratory indonesia
 
MANAJEMEN DAN PENDOKUMENTASIANASUHAN KEBIDANAN PADA BAYI NY“M” DENGAN ASFIKSI...
Warnet Raha
 
Hotelería y Turismo
Karla Armijos
 
Fútbol Al Máximo
carlosagr96
 
Generational intelligence- a new HR challenge
Abdul Razak
 
Radio nahndic3a1
Eduardo Rojas
 
Presentacion nueva
angelicafranco
 
JDLHBedCheckDoc
Sabrina Searles
 
MANAJEMEN DAN PENDOKUMENTASIAN ASUHAN KEBIDANAN INTRANATAL PADA NY”Y” DENGAN ...
Warnet Raha
 
MANAJEMEN DAN PENDOKUMENTASIANASUHAN KEBIDANAN PADA BAYI NY“M” DENGAN ASFIKSI...
Warnet Raha
 
Designing peer review for digital media instruction
Cheryl Ball
 
HV 233, Los inicios históricos de las Verapaces, Erick Reyes Andrade
123apn1
 
Manfaat dan pengaruh gadget
sanrumi221098
 
Amistad
Kevin Barrera
 
Master armero
Ahatzin
 
Ad

Similar to Efficient Hill Climber for Multi-Objective Pseudo-Boolean Optimization (20)

PDF
Efficient Identification of Improving Moves in a Ball for Pseudo-Boolean Prob...
jfrchicanog
 
PDF
Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems
jfrchicanog
 
PDF
hillclimbing in Artificial intelligence.pdf
ratnababum
 
PPTX
Local Search Algorithms and Optimization problems.pptx
gsuginlal
 
PPTX
cs-171-05-LocalSearch.pptx
VijayKumarLokanadam
 
PDF
04-local-search.pdf
Dr. Naushad Varish
 
PPT
local-search and optimization slides.ppt
anitasaroj4
 
PPT
local-search algorithms in Artificial intelligence .ppt
AnushyaR6
 
PPTX
Heuristic search
NivethaS35
 
PDF
Erwin. e. obermayer k._schulten. k. _1992: self-organising maps_stationary st...
ArchiLab 7
 
PPTX
Informed Search Techniques new kirti L 8.pptx
Kirti Verma
 
PPT
vdocuments.mx_chapter-3-heuristic-search-techniques-56a314b01c908.ppt
dhanushkumarks2021it
 
PPTX
Hill-climbing #2
Mohamed Gad
 
PPT
Chapter 03 - artifi HEURISTIC SEARCH.ppt
jenyisblessed
 
PPT
Heuristic search problem-solving str.ppt
SahilSharma872516
 
PPT
Heuristc Search Techniques
Jismy .K.Jose
 
PPTX
Heuristic search
NivethaS35
 
PPTX
Heuristic or informed search
HamzaJaved64
 
PPTX
CS4700-LS_v6.pptx
jpradha86
 
PPTX
Topology Matters in Communication
cseiitgn
 
Efficient Identification of Improving Moves in a Ball for Pseudo-Boolean Prob...
jfrchicanog
 
Efficient Hill Climber for Constrained Pseudo-Boolean Optimization Problems
jfrchicanog
 
hillclimbing in Artificial intelligence.pdf
ratnababum
 
Local Search Algorithms and Optimization problems.pptx
gsuginlal
 
cs-171-05-LocalSearch.pptx
VijayKumarLokanadam
 
04-local-search.pdf
Dr. Naushad Varish
 
local-search and optimization slides.ppt
anitasaroj4
 
local-search algorithms in Artificial intelligence .ppt
AnushyaR6
 
Heuristic search
NivethaS35
 
Erwin. e. obermayer k._schulten. k. _1992: self-organising maps_stationary st...
ArchiLab 7
 
Informed Search Techniques new kirti L 8.pptx
Kirti Verma
 
vdocuments.mx_chapter-3-heuristic-search-techniques-56a314b01c908.ppt
dhanushkumarks2021it
 
Hill-climbing #2
Mohamed Gad
 
Chapter 03 - artifi HEURISTIC SEARCH.ppt
jenyisblessed
 
Heuristic search problem-solving str.ppt
SahilSharma872516
 
Heuristc Search Techniques
Jismy .K.Jose
 
Heuristic search
NivethaS35
 
Heuristic or informed search
HamzaJaved64
 
CS4700-LS_v6.pptx
jpradha86
 
Topology Matters in Communication
cseiitgn
 

More from jfrchicanog (20)

PDF
Seminario-taller: Introducción a la Ingeniería del Software Guiada or Búsqueda
jfrchicanog
 
PDF
Combinando algoritmos exactos y heurísticos para problemas en ISGB
jfrchicanog
 
PDF
Quasi-Optimal Recombination Operator
jfrchicanog
 
PDF
Uso de CMSA para resolver el problema de selección de requisitos
jfrchicanog
 
PDF
Enhancing Partition Crossover with Articulation Points Analysis
jfrchicanog
 
PDF
Search-Based Software Project Scheduling
jfrchicanog
 
PDF
Dos estrategias de búsqueda anytime basadas en programación lineal entera par...
jfrchicanog
 
PDF
Mixed Integer Linear Programming Formulation for the Taxi Sharing Problem
jfrchicanog
 
PDF
Descomposición en Landscapes Elementales del Problema de Diseño de Redes de R...
jfrchicanog
 
PDF
Optimización Multi-objetivo Basada en Preferencias para la Planificación de P...
jfrchicanog
 
PDF
Resolviendo in problema multi-objetivo de selección de requisitos mediante re...
jfrchicanog
 
PDF
On the application of SAT solvers for Search Based Software Testing
jfrchicanog
 
PDF
Elementary Landscape Decomposition of the Hamiltonian Path Optimization Problem
jfrchicanog
 
PDF
Recent Research in Search Based Software Testing
jfrchicanog
 
PDF
Problem Understanding through Landscape Theory
jfrchicanog
 
PDF
Searching for Liveness Property Violations in Concurrent Systems with ACO
jfrchicanog
 
PDF
Finding Safety Errors with ACO
jfrchicanog
 
PDF
Elementary Landscape Decomposition of Combinatorial Optimization Problems
jfrchicanog
 
PDF
Elementary Landscape Decomposition of Combinatorial Optimization Problems
jfrchicanog
 
PDF
Elementary Landscape Decomposition of the Quadratic Assignment Problem
jfrchicanog
 
Seminario-taller: Introducción a la Ingeniería del Software Guiada or Búsqueda
jfrchicanog
 
Combinando algoritmos exactos y heurísticos para problemas en ISGB
jfrchicanog
 
Quasi-Optimal Recombination Operator
jfrchicanog
 
Uso de CMSA para resolver el problema de selección de requisitos
jfrchicanog
 
Enhancing Partition Crossover with Articulation Points Analysis
jfrchicanog
 
Search-Based Software Project Scheduling
jfrchicanog
 
Dos estrategias de búsqueda anytime basadas en programación lineal entera par...
jfrchicanog
 
Mixed Integer Linear Programming Formulation for the Taxi Sharing Problem
jfrchicanog
 
Descomposición en Landscapes Elementales del Problema de Diseño de Redes de R...
jfrchicanog
 
Optimización Multi-objetivo Basada en Preferencias para la Planificación de P...
jfrchicanog
 
Resolviendo in problema multi-objetivo de selección de requisitos mediante re...
jfrchicanog
 
On the application of SAT solvers for Search Based Software Testing
jfrchicanog
 
Elementary Landscape Decomposition of the Hamiltonian Path Optimization Problem
jfrchicanog
 
Recent Research in Search Based Software Testing
jfrchicanog
 
Problem Understanding through Landscape Theory
jfrchicanog
 
Searching for Liveness Property Violations in Concurrent Systems with ACO
jfrchicanog
 
Finding Safety Errors with ACO
jfrchicanog
 
Elementary Landscape Decomposition of Combinatorial Optimization Problems
jfrchicanog
 
Elementary Landscape Decomposition of Combinatorial Optimization Problems
jfrchicanog
 
Elementary Landscape Decomposition of the Quadratic Assignment Problem
jfrchicanog
 

Recently uploaded (20)

PDF
Network Security Unit 5.pdf for BCA BBA.
Serpent6
 
PDF
Encapsulation_ Review paper, used for researhc scholars
gurumoop
 
PPTX
1. Introduction to Computer Programming.pptx
bonakiduza1024
 
PPTX
A Presentation on Artificial Intelligence
memembruh336
 
PDF
Building Integrated photovoltaic BIPV_UPV.pdf
SandeepSingh286037
 
PPTX
Spectroscopy.pptx food analysis technology
nawahassan45
 
PDF
TokAI - TikTok AI Agent : The First AI Application That Analyzes 10,000+ Vira...
SOFTTECHHUB
 
PDF
Blue Purple Modern Animated Computer Science Presentation.pdf.pdf
Akar16
 
PPTX
MYSQL Presentation for SQL database connectivity
Swati270511
 
PDF
The Rise and Fall of 3GPP – Time for a Sabbatical?
3G4G
 
PDF
Assigned Numbers - 2025 - Bluetooth® Document
Bloombase
 
PDF
Build a system with the filesystem maintained by OSTree @ COSCUP 2025
Jian-Hong Pan
 
PPT
Teaching material agriculture food technology
LiaRayya
 
PPTX
Big Data Technologies - Introduction.pptx
kuthubussaman1
 
PDF
A comparative analysis of optical character recognition models for extracting...
IAESIJAI
 
PDF
Electronic commerce courselecture one. Pdf
OmerMohamed64
 
PDF
Spectral efficient network and resource selection model in 5G networks
IAESIJAI
 
PPTX
Tartificialntelligence_presentation.pptx
ry7r52mzb4
 
PDF
Approach and Philosophy of On baking technology
30anthuong17
 
PDF
Mobile App Security Testing_ A Comprehensive Guide.pdf
flufftailshop
 
Network Security Unit 5.pdf for BCA BBA.
Serpent6
 
Encapsulation_ Review paper, used for researhc scholars
gurumoop
 
1. Introduction to Computer Programming.pptx
bonakiduza1024
 
A Presentation on Artificial Intelligence
memembruh336
 
Building Integrated photovoltaic BIPV_UPV.pdf
SandeepSingh286037
 
Spectroscopy.pptx food analysis technology
nawahassan45
 
TokAI - TikTok AI Agent : The First AI Application That Analyzes 10,000+ Vira...
SOFTTECHHUB
 
Blue Purple Modern Animated Computer Science Presentation.pdf.pdf
Akar16
 
MYSQL Presentation for SQL database connectivity
Swati270511
 
The Rise and Fall of 3GPP – Time for a Sabbatical?
3G4G
 
Assigned Numbers - 2025 - Bluetooth® Document
Bloombase
 
Build a system with the filesystem maintained by OSTree @ COSCUP 2025
Jian-Hong Pan
 
Teaching material agriculture food technology
LiaRayya
 
Big Data Technologies - Introduction.pptx
kuthubussaman1
 
A comparative analysis of optical character recognition models for extracting...
IAESIJAI
 
Electronic commerce courselecture one. Pdf
OmerMohamed64
 
Spectral efficient network and resource selection model in 5G networks
IAESIJAI
 
Tartificialntelligence_presentation.pptx
ry7r52mzb4
 
Approach and Philosophy of On baking technology
30anthuong17
 
Mobile App Security Testing_ A Comprehensive Guide.pdf
flufftailshop
 

Efficient Hill Climber for Multi-Objective Pseudo-Boolean Optimization

  • 1. 1 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background Contribution Experiments Conclusions & Future Work Efficient Hill Climber for Multi-Objective Pseudo-Boolean Optimization Francisco Chicano, Darrell Whitley, Renato Tinós
  • 2. 2 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work r = 1 n r = 2 n 2 r = 3 n 3 r n r Ball Pr i=1 n i S1( r = 1 n r = 2 n 2 r = 3 n 3 r n r Ball Pr i=1 n i • Considering binary strings of length n and Hamming distance… Solutions in a ball of radius r r=1 r=2 r=3 Ball of radius r Previous work Research Question How many solutions at Hamming distance r? If r << n : Θ (nr)
  • 3. 3 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • We want to find improving moves in a ball of radius r around solution x • What is the computational cost of this exploration? • By complete enumeration: O (nr) if the fitness evaluation is O(1) • In previous work we proposed a way to find improving moves in ball of radius r in O(1) (constant time independent of n) Improving moves in a ball of radius r r Ball of radius r Previous work Research Question Chicano, Whitley, Sutton, GECCO 2014: 437-444
  • 4. 4 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Research Question Ball of radius r Previous work Research Question
  • 5. 5 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • Definition: • where f(i) only depends on k variables (k-bounded epistasis) • We will also assume that the variables are arguments of at most c subfunctions • Example (m=4, n=4, k=2): • Is this set of functions too small? Is it interesting? • Max-kSAT is a k-bounded pseudo-Boolean optimization problem • NK-landscapes is a (K+1)-bounded pseudo-Boolean optimization problem • Any compressible pseudo-Boolean function can be reduced to a quadratic pseudo-Boolean function (e.g., Rosenberg, 1975) Mk Landscape (Whitley, GECCO2015: 927-934) Pseudo-Boolean functions Scores Update Decomposition The family of k-bounded pseudo-Boolean Optimization problems have also been described as an embedded landscape. An embedded landscape [3] with bounded epistasis k is de- fined as a function f(x) that can be written as the sum of m subfunctions, each one depending at most on k input variables. That is: f(x) = mX i=1 f(i) (x), (1) where the subfunctions f(i) depend only on k components of x. Embedded Landscapes generalize NK-landscapes and the MAX-kSAT problem. We will consider in this paper that the number of subfunctions is linear in n, that is m 2 O(n). For NK-landscapes m = n and is a common assumption in MAX-kSAT that m 2 O(n). 3. SCORES IN THE HAMMING BALL For v, x 2 Bn , and a pseudo-Boolean function f : Bn ! R, we denote the Score of x with respect to move v as Sv(x), defined as follows:1 Sv(x) = f(x v) f(x), (2) 1 We omit the function f in Sv(x) to simplify the notation. S(l) v (x) = Equation (5) cl change in the mov f(l) the Score of th this subfunction w On the other hand we only need to c changed variables t acterized by the m we can write (3) as S 3.1 Scores De The Score values tion than just the c in that ball. Let us balls of radius r = xj are two variabl ments of any subfu f = + + +f(1)(x) f(2)(x) f(3)(x) f(4)(x) x1 x2 x3 x4
  • 6. 6 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • Based on Mk Landscapes Going Multi-Objective: Vector Mk Landscape Pseudo-Boolean functions Scores Update Decomposition x1 x2 x3 x4 x5 f (1) 1 f (2) 1 f (3) 1 f (4) 1 f (5) 1 f (1) 2 f (2) 2 f (3) 2 (a) Vector Mk Landscape x3x4x5 f1 f2
  • 7. 7 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • Let us represent a potential move of the current solution with a binary vector v having 1s in the positions that should be flipped • The score of move v for solution x is the difference in the fitness value of the neighboring and the current solution • Scores are useful to identify improving moves: if Sv(x) > 0, v is an improving move • We keep all the scores in the score vector Scores: definition Current solution, x Neighboring solution, y Move, v 01110101010101001 01111011010101001 00001110000000000 01110101010101001 00110101110101111 01000000100000110 01110101010101001 01000101010101001 00110000000000000 Pseudo-Boolean functions Scores Update Decomposition Sv(x) = f(x v) f(x)
  • 8. 8 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive) Scores update r Selected improving move Update the score vector Pseudo-Boolean functions Scores Update Decomposition
  • 9. 9 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • The key idea is to compute the scores from scratch once at the beginning and update their values as the solution moves (less expensive) • How can we do it less expensive? • We have still O(nr) scores to update! • … thanks to two key facts: • We don’t need all the O(nr) scores to know if there is an improving move • From the ones we need, we only have to update a constant number of them and we can do each update in constant time Key facts for efficient scores update r Pseudo-Boolean functions Scores Update Decomposition
  • 10. 10 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Examples: 1 and 4 f(1) f(2) f(3) f(4) x1 x2 x3 x4 Ball Pr i=1 n i S1(x) = f(x 1) f(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) f(1) f(2) f(3) f(4) x1 x2 x3 x4 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) Pseudo-Boolean functions Scores Update Decomposition
  • 11. 11 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Example: 1,4 f(1) f(2) f(3) f(4) x1 x2 x3 x4 r Ball Pr i=1 n i S1(x) = f(x 1) f(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) n i S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1(x) = f(x 1) f(x) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) Ball Pr i=1 n i S1(x) = Sv(x) = f(x v) f(x) = S4(x) = S1(x) = f(x 1) f(x) x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) We don’t need to store S1,4(x) since can be computed from others If none of 1 and 4 are improving moves, 1,4 will not be an improving move Pseudo-Boolean functions Scores Update Decomposition
  • 12. 12 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Example: 1,2 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) f(1) f(2) f(3) x1 x2 x3 x4 f(1) f(2) f(3) x1 x2 x3 x4 f(1) x1 x2 Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) Sv(x) = f(x v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x) S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) x1 and x2 “interact” Pseudo-Boolean functions Scores Update Decomposition
  • 13. 13 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Decomposition rule for scores • When can we decompose a score as the sum of lower order scores? • … when the variables in the move can be partitioned in subsets of variables that DON’T interact • Let us define the Co-occurrence Graph (a.k.a. Variable Interaction Graph, VIG) f(1) f(2) f(3) f(4) x1 x2 x3 x4 There is an edge between two variables if there exists a function that depends on both variables (they “interact”) x4 x3 x1 x2 Pseudo-Boolean functions Scores Update Decomposition
  • 14. 14 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work What is an improving move in MO? • An improving move is one that provides a solution dominating the current one • We define strong and weak improving moves. • We are interested in strong improving moves Taking improving moves Algorithm f2 f1 Assuming maximization Can we safely take only strong improving moves?
  • 15. 15 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work We need to take weak improving moves • We could miss some higher order strong improving moves if we don’t take weak improving moves S2 S1 stored stored not stored current solution Sv(x) = f(x v) f(x) Sv1[v2 (x) = Sv1 (x) + Sv2 (x) Taking improving moves Algorithm
  • 16. 16 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Cycling • We can make the hill climber to cycle if we take weak improving moves S2 S1 stored current solution S2 S1 stored current solution Taking improving moves Algorithm
  • 17. 17 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Solution: weights for the scores • Given a weight vector w with wi > 0 S2 S1 w 1st strong improving 2nd w-improving 2nd w-improving A w-improving move is one with w · Sv(x) > 0 Taking improving moves Algorithm
  • 18. 18 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work MNK Landscape Problem Results Source Code Why NKq and not NK? Floating point precision • An MNK Landscape is a multi-objective pseudo-Boolean problem where each objective is an NK Landscape (Aguirre, Tanaka, CEC 2004: 196-203) • An NK-landscape is a pseudo-Boolean optimization problem with objective function: where each subfunction f(l) depends on variable xl and K other variables •The subfunctions are randomly generated and the values are taken in the range [0,1] • In NKq-landscapes the subfunctions take integer values in the range [0,q-1] • We use NKq-landscapes in the experiments f(1) (x) + f(2) (x 2) f(2) (x) + f(3) (x 2) f(3) (x) 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f(3) (x 1, 2) f(3) (x) S1,2(x) 6= S1(x) + S2(x) f(x) = NX l=1 f(l) (x) 1 • In the adjacent model the variables are consecutive f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4
  • 19. 19 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Runtime 0 100 200 300 400 500 600 700 800 900 1000 1100 10 20 30 40 50 60 70 80 90 100 Averagetimepermove(microseconds) N (number of variables in thousands) r=1, d=2 r=1, d=3 r=2, d=2 r=2, d=3 r=3, d=2 r=3, d=3 Fig. 2. Average time per move in µs for the Multi-Start Hill Climber based on Algo- Neighborhood size: 166 trillion Problem Results Source Code
  • 20. 20 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Quality 600000 610000 620000 630000 640000 650000 660000 670000 610000 630000 650000 f2 f1 r=1 r=2 r=3 • 50% Empirical Attainment Functions (Knowles, ISDA 2005: 552-557) Problem Results Source Code
  • 21. 21 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Source Code https://github.com/jfrchicanog/EfficientHillClimbers Problem Results Source Code
  • 22. 22 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Conclusions and Future Work Conclusions & Future Work • We can efficiently identify improving moves in a ball of radius r around a solution in constant time (independent of n) • The space required to store the information (scores) is linear in the size of the problem n Conclusions • We can also deal with constrained problems (GECCO 2016) • Generalize to other search spaces • Combine with high-level algorithms Future Work
  • 23. 23 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Acknowledgements Efficient Hill Climber for Multi-Objective Pseudo-Boolean Optimization
  • 24. 24 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work • Whitley and Chen proposed an O(1) approximated steepest descent for MAX-kSAT and NK-landscapes based on Walsh decomposition • For k-bounded pseudo-Boolean functions its complexity is O(k2 2k) • Chen, Whitley, Hains and Howe reduced the time required to identify improving moves to O(k3) using partial derivatives • Szeider proved that the exploration of a ball of radius r in MAX-kSAT and kSAT can be done in O(n) if each variable appears in a bounded number of clauses Previous work Ball of radius r Improving moves Previous work Research Question D. Whitley and W. Chen. Constant time steepest descent local search with lookahead for NK-landscapes and MAX-kSAT. GECCO 2012: 1357–1364 W. Chen, D. Whitley, D. Hains, and A. Howe. Second order partial derivatives for NK-landscapes. GECCO 2013: 503–510 S. Szeider. The parameterized complexity of k-flip local search for SAT and MAX SAT. Discrete Optimization, 8(1):139–145, 2011
  • 25. 25 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores to store • In terms of the VIG a score can be decomposed if the subgraph containing the variables in the move is NOT connected • The number of these scores (up to radius r) is O((3kc)r n) • Details of the proof in the paper • With a linear amount of information we can explore a ball of radius r containing O(nr) solutions x4 x3 x1 x2 x4 x3 x1 x2 S2(x) = f(1) (x 2) f(1) (x) + f(2) (x 2) f(2) (x) + f S1,2(x) = f(1) (x 1, 2) f(1) (x)+f(2) (x 1, 2) f(2) (x)+f S1,2(x) 6= S1(x) + S2(x) l=1 l=1 S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) We need to store the scores of moves whose variables form a connected subgraph of the VIG Pseudo-Boolean functions Scores Update Decomposition
  • 26. 26 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores to update • Let us assume that x4 is flipped • Which scores do we need to update? • Those that need to evaluate f(3) and f(4) f(1) f(2) f(3) f(4) x1 x2 x3 x4 x4 x3 x1 x2 • The scores of moves containing variables adjacent or equal to x4 in the VIG Main idea Decomposition of scores Constant time update
  • 27. 27 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores to update and time required • The number of neighbors of a variable in the VIG is bounded by c k • The number of stored scores in which a variable appears is the number of spanning trees of size less than or equal to r with the variable at the root • This number is constant • The update of each score implies evaluating a constant number of functions that depend on at most k variables (constant), so it requires constant time x4 x3 x1 x2 f(1) f(2) f(3) f(4) x1 x2 x3 x4 O( b(k) (3kc)r |v| ) b(k) is a bound for the time to evaluate any subfunction Main idea Decomposition of scores Constant time update
  • 28. 28 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Results: checking the time in the random model • Random model: the number of subfunctions in which a variable appears, c, is not bounded by a constant NKq-landscapes • Random model • N=1,000 to 12,000 • K=1 to 4 • q=2K+1 • r=1 to 4 • 30 instances per conf. K=3 r=1 r=2 r=3 0 2000 4000 6000 8000 10000 12000 0 50 100 150 200 n TimeHsL r=1 r=2 r=3 0 2000 4000 6000 8000 10000 12000 0 100000 200000 300000 400000 N Scoresstoredinmemory NKq-landscapes Sanity check Random model Next improvement
  • 29. 29 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores Problem Formulation Landscape Theory Decomposition SAT Transf. Results f(1) f(2) f(3) x1 x2 x3 x4 f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4 f = + + +f(1)(x) f(3)(x)f(2)(x) f(4)(x) x1 x2 x3 x4
  • 30. 30 / 23EvoCOP 2016, Porto, Portugal, March-April 2016 Introduction Background MO Hill Climber Experiments Conclusions & Future Work Scores Problem Formulation Landscape Theory Decomposition SAT Transf. Results f(1) x1 x2 f(1) f(2) f(3) x1 x2 x3 x4 S1(x) = f(x 1) f(x) v) f(x) = mX l=1 (f(l) (x v) f(l) (x)) = mX l=1 S(l) (x) S4(x) = f(x 4) f(x) S1,4(x) = f(x 1, 4) f(x) S1,4(x) = S1(x) + S4(x) S1(x) = f(1) (x 1) f(1) (x)