Black Box Search By Unbiased Variation

Black Box Search By Unbiased Variation Per Kristian Lehre and Carsten Witt CERCIA, University of Birmingham, UK DTU Informatics, Copenhagen, Denmark ThRaSH - March 24th 2010

State of the Art in Runtime Analysis of RSHs OneMax (1+1) EA O(n log n) [Mühlenbein, 1992] (1+λ) EA O(λn + n log n) [Jansen et al., 2005] (µ+1) EA O(µn + n log n) [Witt, 2006] 1-ANT O(n 2 ) w.h.p. [Neumann and Witt, 2006] (µ+1) IA O(µn + n log n) [Zarges, 2009] Linear Functions (1+1) EA Θ(n log n) [Droste et al., 2002] and [He and Yao, 2003] cga Θ(n 2+ε ), ε > 0 const. [Droste, 2006] Max. Matching (1+1) EA e Ω(n), PRAS [Giel and Wegener, 2003] Sorting (1+1) EA Θ(n 2 log n) [Scharnow et al., 2002] SS Shortest Path (1+1) EA O(n 3 log(nw max)) [Baswana et al., 2009] MO (1+1) EA O(n 3 ) [Scharnow et al., 2002] MST (1+1) EA Θ(m 2 log(nw max)) [Neumann and Wegener, 2007] (1+λ) EA O(nλ log(nw max)), λ = m2 [Neumann and Wegener, 2007] n 1-ANT O(mn log(nw max)) [Neumann and Witt, 2008] Max. Clique (1+1) EA Θ(n 5 ) [Storch, 2006] (rand. planar) (16n+1) RLS Θ(n 5/3 ) [Storch, 2006] Eulerian Cycle (1+1) EA Θ(m 2 log m) [Doerr et al., 2007] Partition (1+1) EA PRAS, avg. [Witt, 2005] Vertex Cover (1+1) EA e Ω(n), arb. bad approx. [Friedrich et al., 2007] and [Oliveto et al., 2007a] Set Cover (1+1) EA e Ω(n), arb. bad approx. [Friedrich et al., 2007] SEMO Pol. O(log n)-approx. [Friedrich et al., 2007] Intersection of (1+1) EA 1/p-approximation in [Reichel and Skutella, 2008] p 3 matroids O( E p+2 log( E w max)) UIO/FSM conf. (1+1) EA e Ω(n) [Lehre and Yao, 2007] See survey [Oliveto et al., 2007b].

Motivation - A Theory of Randomised Search Heuristics Computational Complexity Classification of problems according to inherent difficulty. Common limits on the efficiency of all algorithms. Assuming a particular model of computation. Computational Complexity of Search Problems Polynomial-time Local Search [Johnson et al., 1988]. Black-Box Complexity [Droste et al., 2006].

Black Box Complexity A f Function class F Photo: E. Gerhard (1846). [Droste et al., 2006]

Black Box Complexity f(x 1 ), f(x 2 ), f(x 3 ),... x 1, x 2, x 3,... A f Function class F Photo: E. Gerhard (1846). [Droste et al., 2006]

Black Box Complexity f(x 1 ), f(x 2 ), f(x 3 ),..., f(x t ) x 1, x 2, x 3,..., x t A f Function class F Photo: E. Gerhard (1846). Black box complexity on function class F T F := min max T A,f A f F [Droste et al., 2006]

Results with old Model Very general model with few restrictions on resources. Example: Needle has BB complexity (2 n + 1)/2. Some NP-hard problems have polynomial BB complexity. Artificially low BB complexity on example functions, e.g. n/ log(2n + 1) 1 on OneMax n/2 o(n) on LeadingOnes

Refined Black Box Model A f Function class F Photo: E. Gerhard (1846).

Refined Black Box Model f(x 0 ) 0 f(x 0 ) x 0 A f Function class F Photo: E. Gerhard (1846).

Refined Black Box Model f(x 0 ), f(x 1 ) 0, 0 f A f(x 0 ) f(x 1 ) x 0 x 1 Function class F Photo: E. Gerhard (1846).

Refined Black Box Model f(x 0 ), f(x 1 ), f(x 2 ) 0, 0, 2 f(x 0 ) x 0 A f(x 1 ) x 1 Function class F f f(x 2 ) x 2 Photo: E. Gerhard (1846).

Refined Black Box Model f(x 0 ), f(x 1 ), f(x 2 ), f(x 3 ) 0, 0, 2, 3 f(x 0 ) x 0 A f(x 1 ) x 1 Function class F f f(x 2 ) f(x 3 ) x 2 x 3 Photo: E. Gerhard (1846).

Refined Black Box Model f(x 0 ), f(x 1 ), f(x 2 ), f(x 3 ), f(x 4 ), f(x 5 ), f(x 6 ) 0, 0, 2, 3, 0, 2 f(x 0 ) x 0 A f(x 1 ) x 1 Function class F f f(x 2 ) f(x 3 ) x 2 x 3 f(x 4 ) x 4 Photo: E. Gerhard (1846). f(x 5 ) x 5 f(x 6 ) x 6

Refined Black Box Model f(x 0 ) x 0 A f(x 1 ) x 1 Function class F f f(x 2 ) f(x 3 ) x 2 x 3 f(x 4 ) x 4 Photo: E. Gerhard (1846). f(x 5 ) x 5 f(x 6 ) x 6 Unbiased black box complexity on function class F T F := min max T A,f A f F

Unbiased Variation Operators 1 Encoding of solution by bitstring x = x 1 x 2 x 3 x 4 x 5 x 1 x 5 x 2 x 4 x 3 1 Figure by Dake, available under a Creative Commons Attribution-Share Alike 2.5 Generic license.

Unbiased Variation Operators 1 Encoding of solution by bitstring x = x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 5 1 Figure by Dake, available under a Creative Commons Attribution-Share Alike 2.5 Generic license.

Unbiased Variation Operators 1 Encoding of solution by bitstring x = x 1 x 2 x 3 x 4 x 5 x 1 x2 = 1 = blue in! x 3 x 4 = 1 = orange in! x 5 1 Figure by Dake, available under a Creative Commons Attribution-Share Alike 2.5 Generic license.

Unbiased Variation Operators 1 Encoding of solution by bitstring x = x 1 x 2 x 3 x 4 x 5 x 1 x2 = 1 = blue in! x 3 x 4 = 1 = orange out! x 5 1 Figure by Dake, available under a Creative Commons Attribution-Share Alike 2.5 Generic license.

Unbiased Variation Operators p(y x) For any bitstrings x, y, z and permutation σ, we require 1) p(y x) = p(y z x z) 2) p(y x) = p(y σ(1) y σ(2) y σ(n) x σ(1) x σ(2) x σ(n) ) We consider unary operators, but higher arities possible. [Droste and Wiesmann, 2000, Rowe et al., 2007]

Unbiased Variation Operators x x r y Condition 1) and 2) imply Hamming-invariance.

Unbiased Black-Box Algorithm Scheme 1: t 0. 2: Choose x(t) uniformly at random from {0, 1} n. 3: repeat 4: t t + 1. 5: Compute f(x(t 1)). 6: I(t) (f(x(0)),..., f(x(t 1))). 7: Depending on I(t), choose a prob. distr. p s on {0,..., t 1}. 8: Randomly choose an index j according to p s. 9: Depending on I(t), choose an unbiased variation op. p v ( x(j)). 10: Randomly choose a bitstring x(t) according to p v. 11: until termination condition met. (µ +, λ) EA, simulated annealing, metropolis, RLS, any population size, any selection mechanism, steady state EAs, cellular EAs, ranked based mutation...

Simple Unimodal Functions Algorithm LeadingOnes (1+1) EA Θ(n 2 ) (1+λ) EA Θ(n 2 + λn) (µ+1) EA Θ(n 2 + µn log n) BB Ω(n)

Simple Unimodal Functions Algorithm LeadingOnes (1+1) EA Θ(n 2 ) (1+λ) EA Θ(n 2 + λn) (µ+1) EA Θ(n 2 + µn log n) BB Ω(n) Theorem The expected runtime of any black box algorithm with unary, unbiased variation on LeadingOnes is Ω(n 2 ).

Simple Unimodal Functions Algorithm LeadingOnes (1+1) EA Θ(n 2 ) (1+λ) EA Θ(n 2 + λn) (µ+1) EA Θ(n 2 + µn log n) BB Ω(n) Theorem The expected runtime of any black box algorithm with unary, unbiased variation on LeadingOnes is Ω(n 2 ). Proof idea Potential between n/2 and 3n/4. # 0-bits flipped hypergeometrically distributed. Lower bound by polynomial drift.

Escaping from Local Optima Jump(x) x m

Escaping from Local Optima Jump(x) x m Theorem For any m n(1 ε)/2 with 0 < ε < 1, the expected runtime of any black box algorithm with unary, unbiased variation is at least 2 cm with probability 1 2 Ω(m). ( n rm) cm with probability 1 2 Ω(m ln(n/(rm))). These bounds are lower than the Θ(n m ) bound for (1+1) EA!

Escaping from Local Optima Jump(x) x m Proof idea Simplified drift in gaps 1. Expectation of hypergeometric distribution. 2. Chvátal s bound.

General Pseudo-boolean Functions Algorithm OneMax (1+1) EA Θ(n log n) (1+λ) EA O(λn + n log n) (µ+1) EA O(µn + n log n) BB Ω(n/ log n)

General Pseudo-boolean Functions Algorithm OneMax (1+1) EA Θ(n log n) (1+λ) EA O(λn + n log n) (µ+1) EA O(µn + n log n) BB Ω(n/ log n) Theorem The expected runtime of any black box search algorithm with unbiased, unary variation on any pseudo-boolean function with a single global optimum is Ω(n log n).

General Pseudo-boolean Functions Algorithm OneMax (1+1) EA Θ(n log n) (1+λ) EA O(λn + n log n) (µ+1) EA O(µn + n log n) BB Ω(n/ log n) Theorem The expected runtime of any black box search algorithm with unbiased, unary variation on any pseudo-boolean function with a single global optimum is Ω(n log n). Proof idea Expected multiplicative weight decrease. Chvátal s bound.

Summary and Conclusion Refined black box model. Proofs are (relatively) easy! Comprises EAs never previously analysed. Ω(n log n) on general functions. Some bounds coincide with the runtime of (1+1) EA. Future work: k-ary variation operators for k > 1.

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