Performance Surfaces and Optimum Points

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1 CSC Neural Networks Performance Surfaces and Optimum Points 1

2 Entrance Performance learning is another important class of learning law. Network parameters are adjusted to optimize the performance of the network. Investigating the performance surface. Determine conditions for the existence of minima and maxima of the performance surface. 2

3 Performance Learning Several different learning laws that fall under the category of performance learning. During a training process network parameters are adjusted to optimize the performance of a network. Performance index a quantitative measure which measures the network performance. Small when the network performs well. Large when the network performs poorly. 3

4 Optimization Process Two steps: Define a performance index Search the parameter space in order to reduce the performance index. 4

5 Taylor Expansion Lets assume that F(x) is the performance index and x is the scalar parameter we are adjusting. Assume also F(x) is analytic (all of its derivatives exist) Then the Taylor s expansion about the point x * 5

6 Example 6

7 Plot of Approximation All three approximations are accurate if x is close to x * = 0. However, as x moves farther away from x * only the higher order approximations are accurate. 7

8 Vector Case The neural network performance index will not be a functions of a scalar x. It will be a function of all the network parameters (weights and biases) 8

9 Matrix Form The gradient and the Hessian are very important to understand the behaviour of a performance surface. 9

10 Directional Derivatives 10

11 Example The direction p is orthogonal to the gradient vector. 11

12 Plots Contour Plot (a series of curves along which the function value remains constant) The maximum derivative occurs in the direction of the gradient. The zero derivative is in the direction orthogonal to the gradient (tangent to the contour line). 12

13 Minima 13

14 Scalar Example 14

15 Necessary and Sufficient Conditions A necessary condition of a statement must be satisfied for the statement to be true. Formally, a statement P is a necessary condition of a statement Q if Q implies P. A sufficient condition is one that, if satisfied, assures the statement's truth. Formally, a statement P is a sufficient condition of a statement Q if P implies Q. E.g. Being a mammal (P) is necessary but not sufficient to being human (Q) 15

16 Stationary Points First order, necessary (but not sufficient) condition for x* to be a local minimum point. Any point that satisfies the above equation are called stationary points. 16

17 Positive Definite Matrix Positive Definite Matrix A is positive definite matrix if If all eigenvalues are positive, then the matrix is positive definite. Positive Semidefinite Matrix A is positive semidefinite matrix if If all eigenvalues are non-negative, then the matrix is positive semidefinite. 17

18 Necessary and Sufficient Conditions for a Minimum Sufficient condition (for a strong minimum) Hessian matrix must be positive definite. Necessary condition Hessian matrix must be positive semidefinite. E.g x * 0 = x F( x) = = = x 2x 0 2 F ( x) + x 2 F( x) 12x = = x = 0 2 x= 0 Eigen values λ = 0, 2 Hessian is positive semidefinite We cannot guarantee it is a minimum point, but we have not eliminated it as a possible minimum point. 18

19 Another Example 19

20 Quadratic Function h constant vector 20

21 Properties (1) If the eigenvalues of the Hessian matrix are all positive, the function will have a single strong minimum. If the eigenvalues of the Hessian matrix are all negative, the function will have a single strong maximum. If some eigenvalues are positive and others are negative, the function will have a single saddle point. 21

22 Properties (2) If the eigenvalues are all nonnegative, but some eigenvalues are zero, then the function will either have a weak minimum, or will have no stationary point. If the eigenvalues are all nonpositive, but some eigenvalues are zero, then the function will either have a weak maximum, or will have no stationary point. 22

23 Properties (3) If d is nonzero and A is invertible, then the stationary point of the function is x* = -A -1 d 23

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