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Stochastic Differentiation. Lecture 3. Leonidas Sakalauskas Institute of Mathematics and Informatics Vilnius, Lithuania EURO Working Group on Continuous Optimization. Content. Concept of stochastic gradient Analytical differentiation of expectation

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stochastic differentiation

Stochastic Differentiation

Lecture 3

Leonidas Sakalauskas

Institute of Mathematics and Informatics

Vilnius, Lithuania

EURO Working Group on Continuous Optimization

content
Content
  • Concept of stochastic gradient
  • Analytical differentiation of expectation
  • Differentiation of the objective function of two-stage SLP
  • Finite difference approach
  • Stochastic perturbation approximation
  • Likelihood approach
  • Differentiation of integrals given by inclusion
  • Simulation of stochastic gradient
  • Projection of Stochastic Gradient
expected objective function
Expected objective function

The stochastic programming deals with the objective and/or constraint functions defined as expectation of random function:

  • elementary event in the probabilityspace:

- the measure, defined by probability density function:

concept of stochastic gradient
Concept of stochastic gradient

The methods of nonlinear stochastic programming are built using the concept of stochastic gradient.

The stochastic gradient of the function

is the random vector such that:

methods of stochasti c differentiation
Methods of stochastic differentiation
  • Several estimators examined for stochastic gradient:
    • Analytical approach (AA);
    • Finite difference approach (FD);
    • Likelihood ratio approach (LR)
    • Simulated perturbation approximation.
analytical approach aa
Analytical approach (AA)

Assume, density of random variable doesn’t depends on the decision variable.

Thus, the analytical stochastic gradient coincides with the gradient of random integrated function:

analytical approach aa8
Analytical approach (AA)

Let consider the two-stage SLP:

vectors q, h, and matrices W, T

can be random in general

analytical approach aa9
Analytical approach (AA)

The stochastic analytical gradient is defined as

by the a set of solutions of the dual problem

f inite difference fd approach
Finite difference (FD) approach

Let us approximate the gradient of the random function by finite differences.

Thus, the each ith component of the stochastic gradient is computed as:

is the vector with zero components except ith one, equal to 1, is some small value.

simulated p erturbation s tochastic a pproximation spsa
Simulated perturbation stochastic approximation (SPSA)

where is the random vector obtaining values 1 or -1 with probabilities p=0.5, is some small value

(Spall 1992).

stochastic differentiation of integrals given by inclusion
Stochastic differentiation of integrals given by inclusion

Let consider the integral on the set given by inclusion

stochastic differentiation of integrals given by inclusion14
Stochastic differentiation of integrals given by inclusion

The gradient of this function is defined as

where is defined through derivatives of p and f(see, Uryasev (1994), (2002))

simulation of stochastic gradient
Simulation of stochastic gradient

We assume here that the Monte-Carlo sample of a certain size N are provided for any

are independent random copies of

i.e., distributed according to the density

sampling estimators of the objective function
Sampling estimators of the objective function

the sampling estimator of the objective function:

and the sampling variance are computed

sampling estimator of the gradient
Sampling estimator of the gradient

The gradient is evaluated using the same random sample:

sampling estimator of the gradient18
Sampling estimator of the gradient

The sampling covariance matrix is applied

later on for normalising of the gradient estimator.

Say, the Hotelling statistics can be used for testing the value of the gradient:

computer simulation
Computer simulation
  • Two-stage stochastic linear optimisation problem.
  • Dimensions of the task are as follows:
    • the first stage has 10 rows and 20 variables;
    • the second stage has 20 rows and 30 variables.

http://www.math.bme.hu/~deak/twostage/ l1/20x20.1/

(2006-01-20).

wrap up and conclusions
Wrap-Up and conclusions
  • The methods of nonlinear stochastic programming are built using the concept of stochastic gradient
  • Several methods exist to obtain the stochastic gradient by evaluating the objective function and stochastic gradient by the same random sample.