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Gradient Methods

Gradient Methods. Yaron Lipman May 2003. Preview. Background Steepest Descent Conjugate Gradient. Preview. Background Steepest Descent Conjugate Gradient. Background. Motivation The gradient notion The Wolfe Theorems. Motivation. The min(max) problem:

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Gradient Methods

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  1. Gradient Methods Yaron Lipman May 2003

  2. Preview • Background • Steepest Descent • Conjugate Gradient

  3. Preview • Background • Steepest Descent • Conjugate Gradient

  4. Background • Motivation • The gradient notion • The Wolfe Theorems

  5. Motivation • The min(max) problem: • But we learned in calculus how to solve that kind of question!

  6. Motivation • Not exactly, • Functions: • High order polynomials: • What about function that don’t have an analytic presentation: “Black Box”

  7. Motivation • “real world” problem finding harmonic mapping • General problem: find global min(max) • This lecture will concentrate on finding localminimum.

  8. Background • Motivation • The gradient notion • The Wolfe Theorems

  9. Directional Derivatives: first, the one dimension derivative:

  10. Directional Derivatives : Along the Axes…

  11. Directional Derivatives : In general direction…

  12. Directional Derivatives

  13. The Gradient: Definition in In the plane

  14. The Gradient: Definition

  15. The Gradient Properties • The gradient defines (hyper) plane approximating the function infinitesimally

  16. The Gradient properties • By the chain rule: (important for later use)

  17. The Gradient properties • Proposition 1: is maximal choosing is minimal choosing (intuitive: the gradient point the greatest change direction)

  18. The Gradient properties Proof: (only for minimum case) Assign: by chain rule:

  19. The Gradient properties On the other hand for general v:

  20. The Gradient Properties • Proposition 2: let be a smooth function around P, if f has local minimum (maximum) at p then, (Intuitive: necessary for local min(max))

  21. The Gradient Properties Proof: Intuitive:

  22. The Gradient Properties Formally: for any We get:

  23. The Gradient Properties • We found the best INFINITESIMAL DIRECTIONat each point, • Looking for minimum: “blind man” procedure • How can we derive the way to the minimum using this knowledge?

  24. Background • Motivation • The gradient notion • The Wolfe Theorems

  25. The Wolfe Theorem • This is the link from the previous gradient properties to the constructive algorithm. • The problem:

  26. The Wolfe Theorem • We introduce a model for algorithm: Data: Step 0: set i=0 Step 1: if stop, else, compute search direction Step 2: compute the step-size Step 3: set go to step 1

  27. The Wolfe Theorem The Theorem: suppose C1 smooth, and exist continuous function: And, And, the search vectors constructed by the model algorithm satisfy:

  28. The Wolfe Theorem And Then if is the sequence constructed by the algorithm model, then any accumulation point y of this sequence satisfy:

  29. The Wolfe Theorem The theorem has very intuitive interpretation : Always go in decent direction.

  30. Preview • Background • Steepest Descent • Conjugate Gradient

  31. Steepest Descent • What it mean? • We now use what we have learned to implement the most basic minimization technique. • First we introduce the algorithm, which is a version of the model algorithm. • The problem:

  32. Steepest Descent • Steepest descent algorithm: Data: Step 0: set i=0 Step 1: if stop, else, compute search direction Step 2: compute the step-size Step 3: set go to step 1

  33. Steepest Descent • Theorem: if is a sequence constructed by the SD algorithm, then every accumulation point y of the sequence satisfy: Proof: from Wolfe theorem

  34. Steepest Descent • From the chain rule: • Therefore the method of steepest descent looks like this:

  35. Steepest Descent

  36. Steepest Descent • The steepest descent find critical point and local minimum. • Implicit step-size rule • Actually we reduced the problem to finding minimum: • There are extensions that gives the step size rule in discrete sense. (Armijo)

  37. Preview • Background • Steepest Descent • Conjugate Gradient

  38. Conjugate Gradient • Modern optimization methods : “conjugate direction” methods. • A method to solve quadratic function minimization: (H is symmetric and positive definite)

  39. Conjugate Gradient • Originally aimed to solve linear problems: • Later extended to general functions under rational of quadratic approximation to a function is quite accurate.

  40. Conjugate Gradient • The basic idea: decompose the n-dimensional quadratic problem into n problems of 1-dimension • This is done by exploring the function in “conjugate directions”. • Definition: H-conjugate vectors:

  41. Conjugate Gradient • If there is an H-conjugate basis then: • N problems in 1-dimension (simple smiling quadratic) • The global minimizer is calculated sequentially starting from x0:

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