1. METHOD OF STEEPEST DESCENT ELE 774 - Adaptive Signal Processing1 Week 5

2. Mean Square Error (Revisited) For a transversal filter (of length M), the output is written as and the error term wrt. a certain desired response is

3. Mean Square Error (Revisited) Following these terms, the MSE criterion is defined as Substituting e(n) and manupulating the expression, we get where Quadratic in w !

4. Mean Square Error (Revisited) For notational simplicity, express MSE in terms of vector/matrices where

5. Mean Square Error (Revisited) We found that the solution (optimum filter coef.s w o ) is given by the Wiener-Hopf eqn.s Inversion of R can be very costly. J(w) is quadratic in w → convex in w → for w o,  Surface has a single minimum and it is global, then Can we reach to w o, i.e. with a less demanding algorithm?

6. Basic Idea of the Method of Steepest Descent Can we find w o in an iterative manner?

7. Basic Idea of the Method of Steepest Descent Starting from w(0), generate a sequence {w(n)} with the property Many sequences can be found following different rules. Method of steepest descent generates points using the gradient  Gradient of J at point w, i.e. gives the direction at which the function increases most.  Then gives the direction at which the function decreases most.  Release a tiny ball on the surface of J → it follows negative gradient of the surface.

8. Basic Idea of the Method of Steepest Descent For notational simplicity, let, then going in the direction given by the negative gradient How far should we go in –g → defined by the step size param. μ  Optimum step size can be obtained by line search - difficult  Generally a constant step size is taken for simplicity. Then, at each step improvement in J is (from Taylor series expansion)

9. Application of SD to Wiener Filter For w(n) From the theory of Wiener Filter we know that Then the update eqn. Becomes which defines a feedback connection.

10. Convergence Analysis Feedback → may cause stability problems under certain conditions.  Depends on The step size, μ The autocorrelation matrix, R Does SD converge?  Under which conditions?  What is the rate of convergence? We may use the canonical representation. Let the weight-error vector be then the update eqn. becomes

11. Convergence Analysis Let be the eigendecomposition of R. Then Using QQ H =I Apply the change of coordinates Then, the update eqn. becomes

12. Convergence Analysis We know that Λ is diagonal, then the k-th natural mode is or, with the initial values v k (0), we have Note the geometric series

13. Convergence Analysis Obviously for stability or, simply Geometric series results in an exponentially decaying curve with time constant τ k, where letting or

14. Convergence Analysis We have but We know that Q is composed of the eigenvectors of R, then or Each filter coefficient decays exponentially. The overall rate of convergence is limited by the slowest and fastest modes then

15. Convergence Analysis For small step size What is v(0)? The initial value v(0) is For simplicity assume that w(0)=0, then

16. Convergence Analysis Transient behaviour:  From the canonical form we know that  then As long as the upper limit on the step size parameter μ is satisfied, regardless of the initial point

17. Convergence Analysis The progress of J(n) for n=0,1,... is called the learning curve. The learning curve of the steepest-descent algorithm consists of a sum of exponentials, each of which corresponds to a natural mode of the problem. # natural modes = # filter taps

18. Example A predictor with 2 taps (w 1 (n) and w 2 (n)) is used to find the params. of the AR process Examine the transient behaviour for  Fixed step size, varying eigenvalue spread  Fixed eigenvalue spread, varying step size. σ v 2 is adjusted so that σ u 2 =1.

19. Example The AR process: Two eigenmodes Condition number (Eigenvalue Spread):

20. Example (Experiment 1) Experiment 1: Keep the step size fixed at Change the eigenvalue spread

21. Example (Experiment 1)

23. Example (Experiment 2) Keep the eigenvalue spread fixed at Change the step size (μ max =1.1)

25. Example (Experiment 2) Depending on the value of μ, the learning curve can be  Overdamped, moves smoothly to the min. ((very) small μ)  Underdamped, oscillates towards the min. (large μ< μ max )  Critically damped  Generally rate of convergence is slow for the first two.

26. Observations SD is a ‘deterministic’ algorithm, i.e. we assume that  R and p are known exactly. In practice they can only be estimated  Sample average?  Can have high computational complexity. SD is a local search algorithm, but for Wiener filtering,  the cost surface is convex (quadratic)  convergence is guaranteed as long as μ< μ max is satisfied.

27. Observations The origin of SD comes from the Taylor series expansion (as many other local search optimization algorithms) Convergence can we very slow. To speed up the process, second term can also be included as in the Newton’s Method High computational complexity (inversion), numerical stability problems. Hessian