Extreme Value Analysis of Simulated Annealing, Simulated ... · Travelling salesman ! Knapsack ! Bin packing ! and many others… 4 NP-hard problems

||Damian Steiger 1

Extreme Value Analysis of Simulated Annealing, Simulated Quantum Annealing and a Mean-Field

modelDamian Steiger

||Damian Steiger 2

Ising spin glasses

Jij = {±1}

hi = 0

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1

HIsing = �X

i<j

Jij�zi �

zj �

X

i

hi�zi , �z

i = {±1}

||Damian Steiger 3

NP-hard

▪▪ ▪

▪ Travelling salesman !

▪ Knapsack !

▪ Bin packing !

▪ and many others…

4

NP-hard problems

||Damian Steiger 5

Heuristic Solvers inspired by Physics

Image credit ANFF NSW node, University of New South

Simulated Annealing Simulated Quantum AnnealingMean-Field Model

||Damian Steiger

▪ Single spin updates ▪ Linear schedule

6

Simulated Annealing

Kirkpatrick, S., Gelatt, C. D. & Vecchi, M. P. Optimization by Simulated Annealing. Science 220, 671–680. (1983).

Image credit ANFF NSW node, University of New South

||Damian Steiger 7

Simulated Quantum Annealing

H(t) = �A(t)X

i

�x

i

�B(t)X

i<j

Jij

�z

i

�z

j

0.0

0.2

0.4

0.6

0.8

1.0

steps

energy

a.u.

t

B t

A

||Damian Steiger 8

Mean-Field Model

Shin et al. How "Quantum" is the D-Wave Machine?. arXiv:1401.7087 . (2014)

H(t) = �A(t)X

i

�x

i

�B(t)X

i<j

Jij

�z

i

�z

j

H(t) = �A(t)X

i

sin ✓i �B(t)X

i<j

Jij cos ✓i cos ✓j

||Damian Steiger 9

Distribution of time to solution for these algorithms?

Damian Steiger

||Damian Steiger

1. Create 20’000 random instances for each system size

2. Optimize annealing time for each algorithm

3. Find single run success probability s for each instance

4. Calculate mean number of repetitions 𝜏 to find the ground state for each instance

5. Make a histogram10

Test with Random Problem Instances

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SA 288 spins

||Damian Steiger 11

Scaling

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions Τ

instances

SA 32 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SA 512 spins

?Scaling prediction would be easier if parametric model of distribution function would be known

Extreme Value Theory provides a parametric model for hard instances

||Damian Steiger 12

Central Limit Theorem

X1 X2 Xn X1 X2 Xn X1 X2 Xn. . . . . . . . .

Block of size n

Block maximum

u

threshold

excesses

+ + ... +

Sum Sn=X1+...+Xn

Central Limit Theorem

for n large enough:Sn follows approx.

a normal df

Limit law for block maxima

for n large enough:Block maximum follows approx. a

GEV df

Limit law for excesses

for threshold uhigh enough:

excesses followapprox. a GP df

exceedance

Let X be an iid random variable with distribution F.

For n large enough: Sn follows approximately a Normal distribution !

(if F satisfies some assumptions)

||Damian Steiger 13

Limit Law for Block Maxima

X1 X2 Xn X1 X2 Xn X1 X2 Xn. . . . . . . . .

Block of size n

Block maximum

u

threshold

excesses

+ + ... +

Sum Sn=X1+...+Xn

Central Limit Theorem

for n large enough:Sn follows approx.

a normal df

Limit law for block maxima

for n large enough:Block maximum follows approx. a

GEV df

Limit law for excesses

for threshold uhigh enough:

excesses followapprox. a GP df

exceedanceFor n large enough: Block maximum follows approximately a Generalized Extreme Value distribution !

(if F satisfies some assumptions)

Let X be an iid random variable with distribution F.

||Damian Steiger 14

Limit Law for Excesses

X1 X2 Xn X1 X2 Xn X1 X2 Xn. . . . . . . . .

Block of size n

Block maximum

u

threshold

excesses

+ + ... +

Sum Sn=X1+...+Xn

Central Limit Theorem

for n large enough:Sn follows approx.

a normal df

Limit law for block maxima

for n large enough:Block maximum follows approx. a

GEV df

Limit law for excesses

for threshold uhigh enough:

excesses followapprox. a GP df

exceedance

For u large enough: Excesses follow approximately a Generalized Pareto distribution !

(if F satisfies some assumptions)

Let X be an iid random variable with distribution F.

||Damian Steiger 15

Scaling

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions Τ

instances

SA 32 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SA 512 spins

?

||Damian Steiger 16

Extreme Value Theoryprobability

density

log(x)u

threshold

fitted GP probability density

observed excesses yi

range of observations xi range of extrapolation

||Damian Steiger 17

Balkema - de Haan - Pickands Theorem

Pickands III, J. Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131. (1975).

Balkema, A. A. & de Haan, L. Residual life time at great age. Ann. Probab. 2, 792–804. (1974).

Reiss, R.-D. & Thomas, M. Statistical Analysis of Extreme Values: with Applications to Insurance, Finance, Hydrology and Other Fields 3rd ed.

(Birkhäuser, 2007)

Let X be an iid random variable with distribution F. If F satisfies some assumptions and threshold u is high enough:

Pr {X � u y | X > u} ⇡ H⇠(y) := 1�✓1 +

⇠y

�̃

◆�1/⇠

generalized Pareto E

�Y k

�= 1 for k � 1/⇠

||Damian Steiger 18

Generalized Pareto Distribution

x

Log!Pro

babi

lity

Den

sity"

!x"!1#1#Ξ"exp!"x#Σ"

end"point

shape param.Ξ 0Ξ 0Ξ 0

Σ 0scale param.

at "Σ#Ξ

||Damian Steiger 19

Optimizing Algorithms

Damian Steiger

||Damian Steiger 20

Optimize algorithms for random Ising spin glasses

101 102 103 104

106

107

108

sweeps

numberofspinflips

200 spins quantiles0.9990.99

0.90.750.50.250.10.01

0.95

SA

||Damian Steiger 21

Bimodality changes

0.0 0.2 0.4 0.6 0.8 1.00

500

1000

1500

2000

success probability

instances

SQA 200 spins

0.0 0.2 0.4 0.6 0.8 1.00

500

1000

1500

2000

2500

3000

3500

success probability

instances

SQA 200 spins

10000 sweeps 150 sweeps

||Damian Steiger 22

Results

Damian Steiger

||Damian Steiger 23

Distribution of SQA

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SQA 32 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SQA 72 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SQA 128 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SQA 200 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions Τ

instances

SQA 288 spins

||Damian Steiger 24

Distribution of SA

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions Τ

instances

SA 32 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions Τinstances

SA 72 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SA 128 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions Τ

instances

SA 200 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SA 288 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SA 392 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SA 512 spins

||Damian Steiger 25

-

-

-- -

-

-

-- -

-

-- -

- --

-

-- -

- --

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions t

instances

SA 288 spins

100 101 102 103 104 105 106 107 108 109

100

101

102

103

104

mean number of repetitions Τ

instances

SQA 288 spins

x

Log!Pro

babi

lity

Den

sity"

!x"!1#1#Ξ"exp!"x#Σ"

end"point

shape param.Ξ 0Ξ 0Ξ 0

Σ 0scale param.

at "Σ#Ξ

||Damian Steiger 26

Tail Comparison

-

-

-- -

-

-

-- -

-

-- -

- --

-

-- -

- --

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

SQA (Low Temp, Slow)

SA (Fast)

||Damian Steiger

-

-

-- -

-

-

-- -

-

--

-

-

--

-

-

-

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

-

-

-- -

-

-

-- -

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

-

-

-- -

-

-

-- -

-

--

-

-

--

-

-

-

--

--

-

--

- --

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

27

Optimising helpsSQA (Low Temp, Slow)

SQA (Low Temp, Fast)

SQA (High Temp, Fast)

||Damian Steiger 28

Comparing the algorithms

-

--

-

-

--

-

-

-

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

-

--

-

-

--

-

-

-

-

-- -

- --

-

-- -

- --

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

-

--

-

-

--

-

-

-

--

--

- -

--

--

- -

-

-- -

- --

-

-- -

- --

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

-

--

-

-

--

-

-

-

--

--

-

--

- --

--

--

- -

--

--

- -

-

-- -

- --

-

-- -

- --

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

SQA (Low Temp, Fast)

SA (Fast)

MF (High Temp, Fast)

SQA (High Temp, Fast)

▪▪ ▪

||Damian Steiger

▪ SQA and MF have worse tail behaviour than SA !!

▪ SQA has a better tail behaviour if it is run faster and even better if it runs faster at a higher temperature

29

Summary

-

-

-- -

-

-

-- -

-

--

-

-

--

-

-

-

--

--

-

--

- --

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spinsshapeparameterx

-

--

-

-

--

-

-

-

--

--

-

--

- --

--

--

- -

--

--

- -

-

-- -

- --

-

-- -

- --

32 72 128 200 288 392 512-0.5

0.0

0.5

1.0

1.5

2.0

spins

shapeparameterx

||Damian Steiger 30

Collaborators

Matthias Troyer (ETH Zürich)

Troels Rønnow (ETH Zürich)

Ilia Zintchenko (ETH Zürich)

||Damian Steiger 31

Damian Steiger