Stochastic Approximation and Simulated Annealing

Lecture 8

Leonidas SakalauskasInstitute of Mathematics and InformaticsVilnius, Lithuania EURO Working Group on Continuous Optimization

Introduction.

Stochastic Approximation: SPSA with Lipschitz perturbation operator; SPSA with Uniform perturbation operator; Standard Finite Difference Approximation

algorithm.

Simulated Annealing

Implementation and Applications

Wrap-Up and Conclusions

Content

In many practical problems of technical design some of the data may be subject to significant uncertainty which is reduced to probabilistic-statistical models.

The performance of such problems can be viewed like constrained stochastic optimization programming tasks.

Stochastic Approximation can be considered as alternative to traditional optimization methods, especially when objective functions are no differentiable or computed with noise.

Introduction

Application of Stochastic Approximation to solving of optimization problems, while the objective function is non-differentiable or nonsmooth and computed with noise is a topical theoretical and practical problem.

The known methods of Stochastic Approximation for solving of these problems use the idea of stochastic gradient and certain rules of changing of step length for ensuring the convergence.

Stochastic Approximation

The optimization problem is (minimization)

as follows:

where is a bounded from below Lipshitz function.

nx

minxf

n:f

Formulation of the optimization problem

Let be generalized gradient of this function.

Assume to be a set of stationary points:

and to be a set of function values:

*X

,xfxX * 0

*F

.Xx,xfzzF **

)x(f

Formulation of the optimization problem

We consider a function smoothed by perturbation operator:

where is the value of the perturbation parameter.

The functions smoothed by this operator are twice continuously differentiable (Rubinstein & Shapiro (1993), Bartkute & Sakalauskas (2004)), that offers certain opportunities creating optimization algorithms.

0

, , ~f x Ef x p

At last time the interesting research was focussed on

Stochastic Perturbation Stochastic Approximation (SPSA)

It is enough to calculate values of the function only in one or some points for the estimation of the stochastic gradient in SPSA algorithms, that promises for us to reduce numerical complexity of optimization.

Advantages of SPSA

1. SPSA with Lipschitz perturbation operator.

2. SPSA with Uniform perturbation operator.

3. Standard Finite Difference Approximation algorithm.

SA algorithms

... 2, ,1 k ,gxx kk

k1k

.0,,,,,, xgxgxgExg

General Stochastic Approximation scheme

where stochastic gradient

and

kkkk xgg ,,

This scheme is the same for different Stochastic Approximation algorithms whose distinguish only by approach for stochastic gradient estimation.

xfxf,,xg

SPSA with Lipschitz perturbation operator

Gradient estimator of the SPSA with Lipschitz perturbation operator is expressed as:

where -is the value of the perturbation parameter,

-is uniformly distributed in the unit ball vector

.yif,

,yif,Vy n

10

11

nV -is the volume of the n-dimensional ball (Bartkute & Sakalauskas (2007))

2

xfxf,,xg

SPSA with Uniform perturbation operator

Gradient estimator of the SPSA with Uniform perturbation operator is expressed as:

where -is the value of the perturbation parameter,

n ,....,, 21 -is a vector consisting of variables uniformly distributed from the interval [-1;1] (Mikhalevitch et al (1987)).

Standard Finite Difference Approximation algorithm

Gradient estimator of the Standard Finite Difference Approximation algorithm is expressed as:

where -is the value of the perturbation parameter,

-is uniformly distributed in the unit ball; vector

xfxf,,,xg i

i

0,.....,1,....,0,0,0i -is the vector with zero components except ith one, which is equal to 1. (Mikhalevitch et al (1987)).

Let consider that the function f(x) has a sharp

minimum in the point , in which the algorithm converges

when

Then

where A>0, H>0, K>0 are certain constants, is minimum point of the smoothed function.

*x

,1121

22*1

1

b

aHk

k

okH

baKAxxE

k

,k

ak ,0a ,

k

bk ,10,0 b .

2

1

Hb

a

1

*

kx

Rate of convergence

The proposed methods were tested with following

functions:

where is a set of real numbers randomly and uniformly

generated in the interval ,

ka

Computer simulation

n

kkk Mxaf

1

K, .K 0

The samples of T=500 test functions were generated, when .K, 52

Empirical and theoretical rates of convergence by SA methods

Theoretical rates

1.5 1.75 1.9

Empirical rates

SPSA (Lipshitz perturbation)

n = 2 1.45509 1.72013 1.892668

n = 4 1.41801 1.74426 1.958998

SPSA ( Uniform perturbation)

n = 2 1.605244 1.938319 1.988265

n = 4 1.551486 1.784519 1.998132

Stochastic Difference Approximation method

n = 2 1.52799 1.76399 1.90479

n = 4 1.50236 1.75057 1.90621

50. 750. 90.

The rate of convergence (n = 2)2*xxE k

The rate of convergence (n = 10)

2*xxE k

Let us consider the application of SA to the minimization of the mean absolute pricing error for the parameter calibration in the Heston Stochastic Volatility model [Heston S. L.(1993)].

We consider the mean absolute pricing error (MAE) defined as :

Volatility estimation by Stochastic Approximation algorithm

N

ii

Hi CC

NMAE

1

v,, , , ,1

v,, , , ,

where N is the total number of options, iC and HiC

the realized market price and the implied the theoretical model price, respectively, while (n=6) are the parameters of the Heston model to be estimated.

v,, , , ,

represent

To compute option prices by the Heston model, one needs input parameters that can hardly be found from the market data. We need to estimate the above parameters by an appropriate calibration procedure. The estimates of the Heston model

parameters are obtained by minimizing MAE:

Let consider the Heston model for the Call option on SPX (29 May 2002).

min v,, , , , MAE

Minimization of the mean absolute pricing error by SPSA and SFDA methods

In cargo oil tankers design, it is necessary to

choose such sizes for bulkheads, that the weight of

bulkheads would be minimal.

Optimal Design of Cargo Oil Tankers

min885.5

22

231

314

xxx

xxxxf

subject to 094.8

6

14.0 2

223131421

xxxxxxxxg

094.82.212

12.0 3

422

231314

222

xxxxxxxxg

015.00156.0 143 xxxg

015.00156.0 344 xxxg

005.145 xxg

0236 xxxg

The minimization of weight of bulkheads for the cargo oil tank we can formulate like nonlinear programing task (Reklaitis et al (1986)):

where 1x - width, 2x -debt, 3x - lenght, 4x - thikness.

SPSA with Lipschitz perturbation for the cargo oil target design

6.5

6.6

6.7

6.8

6.9

7

7.1

7.2

7.3

7.4

7.5

100 1000 1900 2800 3700 4600 5500 6400 7300 8200 9100 10000

Number of iterations

Confidence bounds of the minimum (A=6.84241, T=100, N=1000)

6.4

6.5

6.6

6.7

6.8

6.9

7

7.1

2 102 202 302 402 502 602 702 802 902 Number of iterations

Upper bound

Lower bound

Minimum of theobjective function

Simulated AnnealingGlobal optimization methods

Global algorithms (bounds and branch algorithms, dynamic programming, full selection, etc)

Greedy optimization (local search) Heuristic optimization

Metaheuristics Simulated Annealing Genetic Algorithms Swarm Intelligence Ant Colony Taboo search Scatter search Variable neighborhood Neural Networks Etc.

Simulated Annealing algorithm

Simulated Annealing algorithm is developed by modeling steel annealing process (Metropolis et al. (1953))

A lot of applications in Operational Research and Data Analysis, etc.

Simulated Annealing

Main idea: to simulate drift of current solution with

probability distribution

to improve solution updating - temperature function - neighborhood function

kTk

),( kTxP

Simulated Annealing algorithm

Step 1. Choose , , , set . Step 2. Generate drift with probability distribution Step 3. If and (Metropolis rule)

then accept: ; otherwise Step 2

0T00x 0k1kZ

),( kTxPkkZ 1

k

kkk

T

Zxfxf

e)()( 1

11 kkk Zxx

)1,0(U

Improvement of SA by Pareto Type models

The theoretical investigation of SA convergence shows, that in these algorithms Pareto type models can be applied to form search sequence (Yang (2000)).

Class of Pareto models, main feature and parameter:

Pareto model’s distributions have "heavy tails“.

α - the main parameter of these models, which impacts the heaviness of the tail

α –stable distributions are Pareto (follows to C.L.T.)

Pareto type (Heavy-tailed) distributions

Main features:

infinite variance, infinite mean

Introduced by Pareto in the 1920’s

Mandelbrot established the use of heavy-tailed distributions to model real-world fractal phenomena.

There are a lot of other applications (financial market, traffic in computer and telecommunication networks, etc.).

Pareto type (Heavy-tailed) distributions

Decay of DistributionsHeavy-Tailed - Power Law (polynomial) Decay (e.g. Pareto-Levy):

0,~}{ xCxxXP where 0 < α < 2 and C > 0 are constants

- stable distributions

Comparison of tail probabilities for standard normal, Cauchy and Levy distributions

In this table were compared the tail probabilities for the three distributions. It is clear that the tail probability for the normal quickly becomes negligible, whereas the other two distributions have a significant probability mass in the tail.

Improvement of SAS by Pareto type models

The convergence conditions (Yang (2000)) indicate that, under suitable conditions, an appropriate choice of the temperature and neighborhood size updating functions ensures the convergence of the SA algorithm to the global minimum of the objective function over the domain of interest.

The following corollaries give different forms of temperature and neighborhood size updating functions corresponding to different kinds of generation probability density functions to guarantee the global convergence of the SA algorithm.

Convergence of Simulated Annealing

Improvement of SA in continuous optimization

The above corollaries indicate that a different form of temperature updating function has to be used with respect to a different kind of generation probability density function in order to ensure the global convergence of the corresponding SA algorithm.

Convergence of Simulated Annealing

Some Pareto-type models were explored. See the Table 1.

Convergence of Simulated Annealing

Testing of SA for continuous optimization

In global and combinatorial optimization problems, when optimization algorithms are used, the reliability and efficiency of these algorithms is needed to be tested. Special testing functions, known in literature, are used for this.Some of these functions have one or more global minimum, some of them have global and local minimums. With the help of these functions it can be ensured, that the methods are efficient enough, thus, it is possible to test and prevent algorithms from being trapped in local minimum, as well as the speed and accuracy of convergence and other parameters can be watched.

Testing criteriaBy modeling SA algorithm with some testing functions with two different distributions, and changing some optional parameters, there were some questions:

which of these distributions guarantees the faster convergence to global minimum by value of objective function;

what are probabilities of finding global minimum, how can impact these probabilities the changing of some parameters;

what the proper number of iterations, which guarantees the finding global minimum with desirable probability.

Testing criteria

value of minimized objective function; probability to find global minimum after some

number of iterations.

These characteristics were computed by Monte-Carlo method - N realizations (N=100, 500, 1000) with K iterations each (K=100, 500, 1000, 3000, 10000, 30000).

Characteristics evaluated by Monte-Carlo simulation:

Testing functions

An example of test function:Branin’s RCOS (RC) function (2 variables):

RC(x1,x2)=(x2-(5/(42))x12+(5/)x1-6)2+10(1-(1/(8)))cos(x1)+10; Search domain: 5 < x1 < 10, 0 < x2 < 15;

3 minima: (x1 , x2)*=(- , 12.275), ( , 2.275), (9.42478 , 2.475);RC((x1 , x2)*)=0.397887.

Simulation resulsts

Simulation results

Simulation results

Simulation results

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

Iteracijų skaičius

Fig. 1. Probability to find global minimum by SA for Rastrigin function

1. The SA methods have been considered for comparison SPSA with Lipschitz perturbation operator; SPSA with Uniform perturbation operator and SFDA method as well Simulated Annealing;

2. Computer simulation by Monte-Carlo method has shown that the empirical estimates of the rate of convergence of SA for nondifferentiable functions corroborate the theoretical rates

Wrap-Up and Conclusions

21,1

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