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: But we learned in calculus how to solve that kind of question!

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

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

Background Motivation The gradient notion The Wolfe Theorems

Directional Derivatives: first, the one dimension derivative:

Directional Derivatives : Along the Axes …

Directional Derivatives : In general direction …

Directional Derivatives

In the plane The Gradient: Definition in

The Gradient: Definition

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

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

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

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

The Gradient properties On the other hand for general v:

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))

The Gradient Properties Proof: Intuitive:

The Gradient Properties Formally: for any We get:

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

Background Motivation The gradient notion The Wolfe Theorems

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

The Wolfe Theorem We introduce a model for algorithm: Data: Step 0:set i=0 Step 1:ifstop, else, compute search direction Step 2: compute the step-size Step 3:setgo to step 1

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

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

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

Preview Background Steepest Descent Conjugate Gradient

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:

Steepest Descent Steepest descent algorithm: Data: Step 0:set i=0 Step 1:ifstop, else, compute search direction Step 2: compute the step-size Step 3:setgo to step 1

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

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

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)

Preview Background Steepest Descent Conjugate Gradient

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

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

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:

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 x 0 :