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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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