On the influence of non-perfect random numbers on probabilistic algorithms

Transcription

1 On the influence of non-perfect random numbers on probabilistic algorithms Markus Maucher Bioinformatics Group University of Ulm

2 Outline 1 Motivation and introduction 2 Theoretical results 3 Experimental results 4 Conclusion

3 Motivation Probabilistic algorithms are often faster than their deterministic counterparts (e.g. Miller-Rabin primality test). Their error probability is usually negligible. But: real random numbers are hard to produce. Pseudorandom numbers are easy to produce, so we use these. Problem: Analysis of algorithms is usually based on the usage of independent, uniformly distributed random numbers.

4 Questions What kind of pseudorandom numbers is suited to be used with certain probabilistic algorithms? What do we know about using non-perfect random numbers? Is uniformity important? Is independence important? Should we prefer more complex pseudorandom generators to simpler (but maybe faster) ones? Is the period length important? Is a high discrepancy an advantage?

5 Real random numbers Computers are deterministic machines. Real randomness can only be gained from outside the system User input (time between keystrokes, time between interrupts,...) Atmospheric noise (e.g. random.org) Physical devices [T. Jennewein et al.: A fast and compact quantum random number generator, Rev. Sci. Inst. 71, 1675, (2000)]

6 Pseudorandom numbers Simple methods to create pseudorandom numbers Linear congruential generator X n+1 = ax n + b mod m. Explicit polynomial generator X n = k a i n i mod m i=0 (produces k-wise independent numbers) Mersenne Twister Combination of linear recurrences Used in many mathematical software packages, like R or Matlab.

7 Quasi-random numbers (Low discrepancy point sets) Discrepancy: D (P ) := sup x [0,1) d ( {x i P j.x (j) i <x (j) } N d i=1 x(i) ) Examples: van der Corput sequence representation of in base. n k n k 1... n 1 n 0 n b φ b (n) := 0.n 0 n 1... n k quasi-random sequence φ b (1), φ b (2),... Halton sequence: Combination of van der Corput sequences with different bases. Diehard sequence Halton sequence Others: Niederreiter sequences, Hammersley point sets,...

8 What happens if we use non-perfect random numbers? Theoretical results

9 Probabilistic comparison of polynomials Comparison of two polynomials: q 1 (x) = k i=1 (x a i) q 2 (x) = k i=1 (x b i) Are and equal? Solution: Evaluate q 1 and q 2 at random position x 0 (mod p). q 1 (x 0 ) q 2 (x 0 ) q 1 (x 0 )=q 2 (x 0 ) : Polynomials are not equal : Polynomials are equal, error probability k p Repeating this approach t times reduces the error probability ( using uniformly distributed, independent numbers: k p err p using pseudorandom numbers from a PRNG with s possible internal states: ( ) p err k t+1 s ) t

10 SAT - The Boolean Satisfiability Problem Given: A Boolean formula F (usually in conjunctive normal form) Problem: Does a satisfying assignment for F exist? Example: Formula {{x 1, x 2,x 3 }, {x 1,x 2, x 3 }, {x 1, x 2, x 3 }, {x 1, x 2,x 4 }, {x 1,x 2,x 4 }, {x 1,x 3, x 4 }, {x 1, x 3, x 4 }, {x 2, x 3,x 4 }, {x 2,x 3,x 4 }} Satisfying assignment x 1 0 x 2 1 x 3 1 x 4 0 SAT is NP-complete: Efficient algorithms are not known (and probably do not exist)

11 SAT - The Boolean Satisfiability Problem Schöning s random walk algorithm: Guess an assignment a 3n times do if a satisfies F output satisfiable ; end else choose a clause C in F that is not satisfied by a flip one of the variables in C end end output not satisfiable Success probability: (clause size at most k, n variables) p succ ( k 2k 2 ) n

12 SAT - The Boolean Satisfiability Problem What if our sources or randomness are biased? Assume that each bit of the initial assignment is set to the wrong value with probability 0.5+ɛ, each step of the random walk chooses the right variable with probability 1/k δ. Success probability decreases to: ( k 2ɛ(k 2+2δk) p err = 2k 2+2δk ) n originally: ( k p err = 2k 2 ) n

13 SAT - The Boolean Satisfiability Problem An exponential-time algorithm can be severely influenced by a biased random source: Error probability of a single run: 1 p n After repeating 20p n times: (1 p n ) 20 p n e Error probability of a single run: 1 (p ɛ) n (biased) After repeating (1 (p ɛ) n ) 20 p n 20p n times: = (1 (p ɛ) n 20(p ɛ) n ) p n (p ɛ) n e 20 ( p ɛ p ) n error probability p=0.75 epsilon=0.05 epsilon=0.02 epsilon= n

14 Randomized QuickSort QuickSort(A) IF(A contains at most 1 element) return(a) ELSE x := a random element from A Divide A into two subsequences A1, A2 such that a) all elements in A1 are smaller than x b) all elements in A2 are greater than x return(quicksort(a1) x QuickSort(A2)) Advantages: very fast (average case) easy to implement in-place implementation is possible.

15 Randomized QuickSort Expected number of comparisons when using independent, uniformly distributed random numbers: T (n) =1.39 n log 2 n θ(n). This is reached even for a worst case input. Number of comparisons when using deterministic pivot element: T (n) =θ(n 2 ). What will happen in-between? I.e. if numbers are random, but not perfectly random.

16 Randomized QuickSort - bounds Given: sequence of probability distributions (P 1,P 2,...) P n =(p n1,p n2,..., p nn ) distribution on ranks, : probability to choose element of rank i (out of n ). p ni with Then QuickSort s number of comparisons cg(n)n n T (n) g(n)n log 2 n with ( n ( i ) ) 1 g(n) i=1 p ni H n H(x) = x log 2 x (1 x) log 2 (1 x) : binary entropy function [List, Maucher et al. 2005, List, Maucher et al. 2009]

17 Examples Example 1 Uniformly distributed random numbers, p ni =1/n ( n ( ) ) 1 ( 1 i 1 dx) 1 g(n) = H H(x) =1.38 n n i=1 0 Example 2 6(i 1)(n i) Median of three variant of QuickSort, p ni = n(n 1)(n 2) ( g(n) = ) 1 12 ln 2 x(1 x)h(x) dx =

18 What happens if we use non-perfect random numbers? Experimental results

19 Simulated Annealing Search heuristic, inspired by a physical phenomenon: Slow annealing of metal leads to states with lower energy S. Kirkpatrick et al., Optimization by Simulated Annealing, Science, 1983 Core algorithm: Metropolis algorithm (Metropolis 53)

20 Simulated Annealing Input: Function f : A R Output: x that minimizes f (or an approximation thereof) Initialize temperature T x random element from A opt x WHILE T > T 0 y random neighbor of x IF f(y) <f(x) THEN x y ELSE x y w. prob. e f(y) f(x) T IF f(x) <f(opt) THEN opt x Decrease T END output opt

21 Genetic Algorithms Inspired by natural evolution (survival of the fittest) J.H. Holland, Adaptation in Natural and Artificial Systems, MIT Press 1975

22 Models of non-perfect randomness How to simulate bad random numbers Explicit polynomial generator with different degrees k-wise independence High/low modulus period length, granularity Artificial reduction of the period length Bias Deviation from uniform distribution

23 Experimental Results Simulated Annealing on TSP with varying period lengths Generator: Mersenne Twister, artificially reduced period length Length of shortest tour found The Traveling Salesman Problem: Given n cities and the distances between them, find the shortest tour that visits each city once. r 1009 r 2003 r 4001 r 8009 r r r r r r Generator

24 Experimental Results Genetic Algorithm on TSP with varying period lengths Generators: linear congruential generators with different period lengths Length of shortest tour found l 1000 l 2000 l 4000 l 8000 l 8192 l l l l l l MT D Q Generator

25 Experimental Results Simulated Annealing with biased generator Generators: random 12 bit numbers with different probabilities for 0 and 1 bits Length of shortest tour found b!0.05 b!0.1 b!0.15 b!0.2 b!0.25 b!0.3 b!0.35 b!0.4 b!0.45 b!0.5 b!0.55 b!0.6 b!0.65 b!0.7 b!0.75 b!0.8 b!0.85 b!0.9 b!0.95 Generator

26 MT hal2 23 hal2 25 hal2 27 hal2 35 hal2 37 vdc2 vdc8p1 vdc8p2 vdc8p3 vdc8p Generator Length of shortest tour found Experimental Results Simulated Annealing with quasi-random sequences x 1 x x 1 x 2 Generators: Halton and van der Corput sequences 2-dim. Halton v.d.corput in 2 dim.

27 Experimental Results Genetic Algorithm on TSP with quasi-random sequences Generators: Halton and van der Corput sequences Length of shortest tour found MT Q hal2 23 hal2 25 hal2 27 hal2 35 hal2 37 vdc2 vdc8p1 vdc8p2 vdc8p3 vdc8p4 Generator

28 Experimental Results Simulated Annealing: Similar results for various sources (linear congruential generator, polynomial generator, Mersenne Twister) of equal period lengths Quasi-random sequences lead to good results (as long as they have high dimension) k-wise independence seems to have no influence when varying k period length has strong influence bias has an influence

29 Experimental Results Genetic Algorithms: Are only marginally influenced by the quality of the generator - simple lcg sufficed Rather robust versus short period lengths Seem to be more robust versus use of lowdimensional quasi-random sequences bias has an influence [Maucher et al. 2008]

30 Conclusion For some problems, we can show how non-perfect randomness influences probabilistic algorithms. (Equality test for polynomials, SAT, QuickSort) For Simulated Annealing and Genetic Algorithms, experiments show that some properties of the random numbers have a strong influence (period length, bias, low dimension) Genetic Algorithm more robust than Simulated Annealing when using non-perfect random numbers Mersenne Twister is on par with true random numbers

31 Thank you for your attention.

32 References Beatrice List, Markus Maucher, Uwe Schöning und Rainer Schuler. Randomized quicksort and the entropy of the random source. In Computing and Combinatorics - COCOON 2005, pp Springer, Markus Maucher, Uwe Schöning und Hans A. Kestler. An empirical assessment of local and population based search methods with different degrees of pseudorandomness. Technical Report, University Ulm, Markus Maucher, Uwe Schöning und Hans A. Kestler. On the different notions of pseudorandomness. Technical Report, University Ulm, Beatrice List, Markus Maucher, Uwe Schöning und Rainer Schuler. Quicksort under an information theoretic view. In Wolfgang Arendt and Wolfgang Schleich (Eds.) Mathematical Analysis of Evolution, Information and Complexity. Wiley, 2008.