Tutorial 5-6. Function Optimization. Line Search. Taylor Series for R n Steepest Descent Newton’s Method Conjugate Gradients Method. a. c. u. b. Line search runs as following. Let
Taylor Series for Rn
Conjugate Gradients Method
Line search runs as following. Let
Be the scalar function of α representing the possible values of f(x) in the direction of pk. Let (a,b,c) be the three points of α, such, that the point of (constrained) minimum x’, is between a and c: a<x’<c.
Then the following algorithm allows to
approach x’ arbitrarily close:
The Taylor series for f(x) is
For the function of m variables, the expression is
Consider the elliptic function: f(x,y)=(x1-1)2+(2x2-2)2and find the first three terms of Taylor expansion.
2D Taylor Series: Example
Consider the elliptic function: f(x,y)=(x1-1)2+4(x2-2)2and find the first three terms of Taylor expansion.
In Lecture 5 we have seen that the steepest descent method can suffer from slow convergence. Newton’s method fixes this problem for cases, where the function f(x) near x* can be approximated by a paraboloid:
Here can suffer from slow convergence. Newton’s method fixes this problem for cases, where the function gk is the gradient and Qk is the Hessian of the function f, evaluated at xk. They appear in the 2nd and 3rd terms of the Taylor expansion of f(xk). Minimum of the function should require:
The solution of this equation gives the step direction and the step size towards the minimum of (2), which is, presumably, close to the minimum of f(x). The minimization algorithm in which xk+1=y(xk)=xk+∆, with ∆ defined by (2) is called a Newton’s method.
Newton’s Method 2
-f’(0) can suffer from slow convergence. Newton’s method fixes this problem for cases, where the function
Consider the same elliptic function: f(x,y)=(x1-1)2+4(x2-2)2and find the first step for Newton’s Method.
Newton’s Method: Example
Suppose that we want to minimize the quadratic function can suffer from slow convergence. Newton’s method fixes this problem for cases, where the function
where Q is a symmetric, positive definite matrix, and x has n components. As we saw in explanation of steepest descent, the minimum x* is the solution to the linear system
The explicit solution of this system requires about O(n3) operations and O(n2) memory, what is very expensive.
We now consider an alternative solution method that does not need Q, but only the gradient of f(xk)
evaluated at n different points x1 , . . ., xn.
Conjugate Gradients 2
Consider the case need n = 3, in which the variable x in f(x) is a three-dimensional vector . Then the quadratic function f(x) is constant over ellipsoids, called isosurfaces, centered at the minimum x* . How can we start from a point xo on one of these ellipsoids and reach x* by a finite sequence of one-dimensional searches? In the steepest descent, for the poorly conditioned Hessians orthogonal directions lead to many small steps, that is, to slow convergence.
Conjugate Gradients 3
When the ellipsoids are spheres, on the other hand, the convergence is much faster: first step takes from xo to x1 , and the line between xo and x1 is tangent to an isosurface at x1 . The next step is in the direction of the gradient, takes us to x* right away. Suppose however that we cannot afford to compute this special direction p1 orthogonal to po, but that we can only compute some direction p1 orthogonal to po (there is an n-1 -dimensional space of such directions!) and reach the minimum of f(x) in this direction.
In that case n steps will take us to x* of the sphere, since coordinate of the minimum in each on the n directions is independent of others.
Conjugate Gradients: Spherical Case
Any set of orthogonal directions, with a line search in each direction, will lead to the minimum for spherical isosurfaces. Given an arbitrary set of ellipsoidal isosurfaces, there is a one-to-one mapping with a spherical system: if Q = UEUT is the SVD of the symmetric, positive definite matrix Q, then we can write
Conjugate Gradients: Elliptical Case
Consequently, there must be a condition for the original problem (in terms of Q) that is equivalent to orthogonality for the spherical problem. If two directions qi and qj are orthogonal in the spherical context, that is, if
what does this translate into in terms of the directions pi and pj for the ellipsoidal problem? We have
Elliptical Case 2
Consequently, problem (in terms of
This condition is called Q-conjugacy, or Q-orthogonality : if equation (7) holds, then pi and pj are said to be Q-conjugate or Q-orthogonal to each other. Or simply say "conjugate".
Elliptical Case 3
In summary, if we can find n directions problem (in terms of po, . . .,pn_1 that are mutually conjugate, i.e. comply with (7), and if we do line minimization along each direction pk, we reach the minimum in at most n steps. Of course, we cannot use the transformation (5) in the algorithm, because E and especially UT are too large. So we need to find a method for generating n conjugate directions without using either Q or its SVD .
Elliptical Case 4
Hestenes Stiefel Procedure problem (in terms of
It is simple to see that p problem (in terms of k and pk+1 are conjugate. In fact,
Hestenes Stiefel Procedure 2
The proof that pi and pk+1 for i = 0, . . . , k are also conjugate can be
done by induction, based on the observation that the vectors pk are found by a generalization of Gram-Schmidt to produce conjugate rather than orthogonal vectors.
In the described algorithm the expression for problem (in terms of yk contains the Hessian Q, which is too large. We now show that yk can be rewritten in terms of the gradient values gk and gk+1 only. To this end, we notice
Removing the Hessian
We can therefore write problem (in terms of
and Q has disappeared .
This expression for yk can be further simplified by noticing that
because the line along pk is tangent to an isosurface at xk+l , while the gradient gk+l is orthogonal to the isosurface at xk +l.
Removing the Hessian 2
Similarly, problem (in terms of
Then, the denominator of yk becomes
In conclusion, we obtain the Polak-Ribiere formula