Gradient descent method

Gradient descent method
2013.11.10
SanghyukChun
Many contents are from
Large Scale Optimization Lecture 4 & 5 by Caramanis& Sanghavi
Convex Optimization Lecture 10 by Boyd & Vandenberghe
Convex Optimization textbook Chapter 9 by Boyd & Vandenberghe 1

Contents
•Introduction
•Example code & Usage
•Convergence Conditions
•Methods & Examples
•Summary
2

Introduction
Unconstraint minimization problem, Description, Pros and Cons
3

Unconstrained minimization problems
•Recall: Constrained minimization problems
•From Lecture 1, the formation of a general constrained convex optimization problem is as follows
•min푓푥푠.푡.푥∈χ
•Where 푓:χ→Ris convex and smooth
•From Lecture 1, the formation of an unconstrained optimization problem is as follows
•min푓푥
•Where 푓:푅푛→푅is convex and smooth
•In this problem, the necessary and sufficient condition for optimal solution x0 is
•훻푓푥=0푎푡푥=푥0
4

Unconstrained minimization problems
•Minimize f(x)
•When f is differentiable and convex, a necessary and sufficient condition for a point 푥∗to be optimal is훻푓푥∗=0
•Minimize f(x) is the same as fining solution of 훻푓푥∗=0
•Min f(x): Analytically solving the optimality equation
•훻푓푥∗=0: Usually be solved by an iterative algorithm
5

Description of Gradient Descent Method
•The idea relies on the fact that −훻푓(푥(푘))is a descent direction
•푥(푘+1)=푥(푘)−η푘훻푓(푥(푘))푤푖푡ℎ푓푥푘+1<푓(푥푘)
•Δ푥(푘)is the step, or search direction
•η푘is the step size, or step length
•Too small η푘will cause slow convergence
•Too large η푘could cause overshoot the minima and diverge
6

Description of Gradient Descent Method
•Algorithm (Gradient Descent Method)
•given a starting point 푥∈푑표푚푓
•repeat
1.Δ푥≔−훻푓푥
2.Line search: Choose step size ηvia exact or backtracking line search
3.Update 푥≔푥+ηΔ푥
•untilstopping criterion is satisfied
•Stopping criterion usually 훻푓(푥)2≤휖
•Very simple, but often very slow; rarely used in practice
7

Pros and Cons
•Pros
•Can be applied to every dimension and space (even possible to infinite dimension)
•Easy to implement
•Cons
•Local optima problem
•Relatively slow close to minimum
•For non-differentiable functions, gradient methods are ill-defined
8

Example Code & Usage
Example Code, Usage, Questions
9

Gradient Descent Example Code
•http://mirlab.org/jang/matlab/toolbox/machineLearning/
10

Usage of Gradient Descent Method
•Linear Regression
•Find minimum loss function to choose best hypothesis
11
Example of Loss function:
푑푎푡푎푝푟푒푑푖푐푡−푑푎푡푎표푏푠푒푟푣푒푑 2
Find the hypothesis (function) which minimize the loss function

Usage of Gradient Descent Method
•Neural Network
•Back propagation
•SVM (Support Vector Machine)
•Graphical models
•Least Mean Squared Filter
…and many other applications!
12

Questions
•Does Gradient Descent Method always converge?
•If not, what is condition for convergence?
•How can make Gradient Descent Method faster?
•What is proper value for step size η푘
13

Convergence Conditions
L-Lipschitzfunction, Strong Convexity, Condition number
14

L-Lipschitzfunction
•Definition
•A function 푓:푅푛→푅is called L-Lipschitzif and only if 훻푓푥−훻푓푦2≤퐿푥−푦2,∀푥,푦∈푅푛
•We denote this condition by 푓∈퐶퐿, where 퐶퐿is class of L-Lipschitzfunctions
15

L-Lipschitzfunction
•Lemma 4.1
•퐼푓푓∈퐶퐿,푡ℎ푒푛푓푦−푓푥−훻푓푥,푦−푥≤ 퐿 2 푦−푥2
•Theorem 4.2
•퐼푓푓∈퐶퐿푎푛푑푓∗=min 푥 푓푥>−∞,푡ℎ푒푛푡ℎ푒푔푟푎푑푖푒푛푡푑푒푠푐푒푛푡 푎푙푔표푟푖푡ℎ푚푤푖푡ℎ푓푖푥푒푑푠푡푒푝푠푖푧푒푠푡푎푡푖푠푓푦푖푛푔η< 2 퐿 푤푖푙푙푐표푛푣푒푟푔푒푡표 푎푠푡푎푡푖표푛푎푟푦푝표푖푛푡
16

Strong Convexity and implications
•Definition
•If there exist a constant m > 0 such that 훻2푓≻=푚퐼푓표푟∀푥∈푆, then the function f(x) is strongly convex function on S
17

•Lemma 4.3
•If f is strongly convex on S, we have the following inequality:
•푓푦≥푓푥+<훻푓푥,푦−푥>+ 푚 2 푦−푥2푓표푟∀푥,푦∈푆
•Proof
18
( )
useful as stopping criterion (if you know m)

19
Proof

Upper Bound of 훻2푓(푥)
•Lemma 4.3 implies that the sublevel sets contained in S are bounded, so in particular, S is bounded. Therefore the maximum eigenvalue of 훻2푓푥is bounded above on S
•ThereexistsaconstantMsuchthat훻2푓푥=≺푀퐼푓표푟∀푥∈푆
•Lemma 4.4
•퐹표푟푎푛푦푥,푦∈푆,푖푓훻2푓푥=≺푀퐼푓표푟푎푙푙푥∈푆푡ℎ푒푛 푓푦≤푓푥+<훻푓푥,푦−푥>+ 푀 2 푦−푥2
20

Condition Number
•From Lemma 4.3 and 4.4 we have
푚퐼=≺훻2푓푥=≺푀퐼푓표푟∀푥∈푆,푚>0,푀>0
•The ratio k=M/m is thus an upper bound on the condition number of the matrix 훻2푓푥
•When the ratio is close to 1, we call it well-conditioned
•When the ratio is much larger than 1, we call it ill-conditioned
•When the ratio is exactly 1, it is the best case that only one step will lead to the optimal solution (there is no wrong direction)
21

Condition Number
•Theorem 4.5
•Gradient descent for a strongly convex function f and step sizeη= 1 푀 will converge as
•푓푥∗−푓∗≤푐푘푓푥0−푓∗,푤ℎ푒푟푒푐≤1− 푚 푀
•Rate of convergence c is known as linear convergence
•Since we usually do not know the value of M, we do line search
•For exact line search, 푐=1− 푚 푀
•For backtracking line search, 푐=1−min2푚훼,2훽훼푚 푀 <1
22

Methods & Examples
Exact Line Search, Backtracking Line Search, Coordinate Descent Method, Steepest Descent Method
23

Exact Line Search
• The optimal line search method in which η is chosen to
minimize f along the ray 푥 − η훻푓 푥 , as shown in below
• Exact line search is used when the cost of minimization
problem with one variable is low compared to the cost of
computing the search direction itself.
• It is not very practical
24

Exact Line Search
•Convergence Analysis
•
•푓푥푘−푓∗decreases by at least a constant factor in every iteration
•Converging to 0 geometric fast. (linear convergence)
25
1

Backtracking Line Search
•It depends on two constants 훼,훽푤푖푡ℎ0<훼<0.5,0<훽<1
•It starts with unit step size and then reduces it by the factor 훽until the stopping condition
푓(푥−η훻푓(푥))≤푓(푥)−훼η훻푓푥2
•Since −훻푓푥is a descent direction and −훻푓푥2<0, so for small enough step size η, we have
푓푥−η훻푓푥≈푓푥−η훻푓푥2<푓푥−훼η훻푓푥2
•It shows that the backtracking line search eventually terminates
•훼is typically chosen between 0.01 and 0.3
•훽is often chosen to be between 0.1 and 0.8
26

•
27

•Convergence Analysis
•Claim: η≤ 1 푀 always satisfies the stopping condition
•Proof
28

•Proof (cont)
29

Line search types
•Slide from Optimization Lecture 10 by Boyd 30

Line search example
•Slide from Optimization Lecture 10 by Boyd 31

Coordinate Descent Method
• Coordinate descent belongs to the class of several non
derivative methods used for minimizing differentiable
functions.
• Here, cost is minimized in one coordinate direction in each
iteration.
32

•Pros
•It is well suited for parallel computation
•Cons
•May not reach the local minimum even for convex function
33

Converge of Coordinate Descent
•Lemma 5.4
34

•Method of selecting the coordinate for next iteration
•Cyclic Coordinate Descent
•Greedy Coordinate Descent
•(Uniform) Random Coordinate Descent
35

Steepest Descent Method
•The gradient descent method takes many iterations
•Steepest Descent Method aims at choosing the best direction at each iteration
•Normalized steepest descent direction
•Δ푥푛푠푑=푎푟푔푚푖푛훻푓푥푇푣푣=1}
•Interpretation: for small 푣,푓푥+푣≈푓푥+훻푓푥푇푣direction Δ푥푛푠푑 is unit-norm step with most negative directional derivative
•Iteratively, the algorithm follows the following steps
•Calculate direction of descent Δ푥푛푠푑
•Calculate step size, t
•푥+=푥+푡Δ푥푛푠푑
36

Steepest Descent for various norms
•The choice of norm used the steepest descent direction can be have dramatic effect on converge rate
•푙2norm
•The steepest descent direction is as follows
•Δ푥푛푠푑= −훻푓(푥) 훻푓(푥)2
•푙1norm
•For 푥1= 푖푥푖,a descent direction is as follows,
•Δ푥푛푑푠=−푠푖푔푛 휕푓푥 휕푥푖 ∗푒푖 ∗
•푖∗=푎푟푔min 푖 휕푓 휕푥푖
•푙∞norm
•For 푥∞=argmin 푖 푥푖,a descent direction is as follows
•Δ푥푛푑푠=−푠푖푔푛−훻푓푥
37

38
Quadratic Norm
푙1-Norm

•Example
39

Steepest Descent Convergence Rate
•Fact: Any norm can be bounded by ∙2, i.e., ∃훾, 훾∈(0,1] such that, 푥≥훾푥2푎푛푑푥∗≥훾푥2
•Theorem 5.5
•If f is strongly convex with respect to m and M, and ∙2has 훾, 훾as above then steepest decent with backtracking line search has linear convergence with rate
•푐=1−2푚훼 훾2min1, 훽훾 푀
•Proof: Will be proved in the lecture 6
40

Summary
•Unconstrained Convex Optimization Problem
•Gradient Descent Method
•Step Size Trade-off between safety and speed
•Convergence Conditions
•L-LipschtizFunction
•Strong Convexity
•Condition Number
42

Summary
•Exact Line Search
•Backtracking Line Search
•Coordinate Descent Method
•Good for parallel computation but not always converge
•Steepest Descent Method
•The choice of norm is important
43

Gradient descent method

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