1. Math for CSLecture 51 Function Optimization

2. Math for CSLecture 52 There are three main reasons why most problems in robotics, vision, and arguably every other science or endeavor take on the form of optimization problems. One is that the desired goal may not be achievable, and so we try to get as close as possible to it. The second reason is that there may be more ways to achieve the goal, and so we can choose one by assigning a quality to all the solutions and selecting the best one. The third reason is that we may not know how to solve the system of equations f(x) = 0, so instead we minimize the norm If(x)I, which is a scalar function of the unknown vector x. Why Function Optimization ?

3. Math for CSLecture 53 Local Minimization and Steepest Descent k=0; while x k is not a minimum compute step direction P k with IIP k II = 1 compute step size a k x k+l = x k + a k P k ; k=k+1 end. Suppose that we want to find a local minimum for the scalar function f of the vector variable x, starting from an initial point x o. Picking an appropriate x o is crucial, but also very problem- dependent. We start from x o, and we go downhill. At every step of the way, we must make the following decisions : Whether to stop. In what direction to proceed. How long a step to take. The following algorithm reflects various ‘descent minimization’ procedures:

4. Math for CSLecture 54 The best direction of descent is not necessarily the direction of steepest descent. Consider a function: where Q is a symmetric, positive definite matrix. Positive definite means that for every nonzero x the quantity x T Qx is positive. In this case, the graph of f(x) - c is a plane a T x plus a paraboloid x T Qx. Of course, if f were this simple, no descent methods would be necessary. In fact the minimum of f can be found by setting its gradient to zero: so that the minimum x* is the solution to the linear system Qx=-a Since Q is positive definite, it is also invertible (why?), and the solution x* is unique. Minimization of Positive Definite functions 1 (1)

5. Math for CSLecture 55 Minimization of Positive Definite functions 2 In order to simplify the mathematics, we observe that if we let then we have so that e and f differ only by a constant. Since e is simpler, we consider that we are minimizing e rather than f. In addition, we can let what shifts the origin of the domain to x*, and study the function (2)

6. Math for CSLecture 56 The steepest descent direction where e in minimum reaches a value of zero : let our steepest descent algorithm find this minimum by starting from the initial point the algorithm chooses the direction of steepest descent: Which is opposite to the gradient of e evaluated at y k :

7. Math for CSLecture 57 The step size The most favorable step size will take us from y k to the lowest point in the direction of p k. This can be found by differentiating the function with respect to α, and setting the derivative to zero to obtain the optimal step α k and setting this to zero yields

8. Math for CSLecture 58 The step size 2 Thus, the basic step of our steepest descent can be written as follows: or (3)

9. Math for CSLecture 59 e(y) descend rate How much closer does one step bring us to the solution y* = 0? In other words, how much smaller is e(y k+1 ) relatively to e(y k )? From the definition (2) of e(y) and equation (3) for y k+1, we obtain: (4)

10. Math for CSLecture 510 e(y) descend rate 2 Since Q is invertible, we have: And,what allows to rewrite (4) as: or

11. Math for CSLecture 511 Kantorovich inequality. Kantorovich inequality: Let Q be a positive definite, symmetric, n x n matrix. Then, for any vector y there holds: This inequality allows to prove the Steepest Descent Rate theorem.

12. Math for CSLecture 512 Steepest Descent Rate theorem 1 Steepest Descent Rate theorem: Let be a quadratic function of x, with Q symmetric and positive definite. For any x o, the method of steepest descent,where

13. Math for CSLecture 513 Steepest Descent Rate theorem 2 Converges to the unique minimum point The difference at every step satisfies, where σ 1 and σ n are the respectively the largest and the smallest singular values of Q

14. Math for CSLecture 514 Proof From the definitions we obtain Here, the Kantorovich inequality was used.

15. Math for CSLecture 515 Analysis The ratio Is called a condition number of Q. The larger the condition number (ratio between the largest and the smallest singular values), the smaller the ratio And therefore the slower the convergence.

16. Math for CSLecture 516 Illustration Consider the two dimensional case, x R 2. The figure shows a trajectory x k, imposed in the isocontours of f(x). The greater the ratio between the singular values, of Q (which is the aspect ratio of the ellipses), the slower the convergence rate. If the isocontours are circular (k(Q)=1) or the trajectory started from the ellipses axis, the single step brings us to x*.

17. Math for CSLecture 517 Convergence rate To characterize the speed of convergence of different minimization algorithms, we introduce the order of convergence. It is defined as the largest value of q, for which the Is finite. If is the limit, then we can write (for large values of k) The distance from x* is reduced by the q-th power at every step, therefore thehigher the order of convergence, the better.

18. Math for CSLecture 518 Stop criteria We do not know the x* and therefore f(x*). Thus the stop criteria is not trivial. The criteria can be |f(x k )-f(x k-1 )| or |x k -x k-1 |. The second criteria is better, since it indicates proximity of x*.

19. Math for CSLecture 519 Line search The steepest descend can be applied to general cases of f, not necessarily quadratic and not defined via In these cases, Q is the matrix of the second derivatives of f with respect to x, called a Hessian of f. In this case, only n first derivatives are needed to calculate the direction p k. The step size requires calculation of Hessian of f(x), which requires computing second derivatives, and therefore is very expensive. Using the line search allows to reach the minimum of f(x) in the direction p k without the Hessian calculation. (1)

20. Math for CSLecture 520 Line search runs as following. Let Be the scalar function of α representing the possible values of f(x) in the direction of p k. 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: If b-a>c-b, u=(a+b)/2; If f(u)<f(b) (a,b,c)=(a,u,b) Else (a,b,c)=(u,b,c) Line search 2 a b c u If b-a<c-b, u=(b+c)/2; If f(u)<f(b) (a,b,c)=(b,u,c) Else (a,b,c)=(a,b,u)