Machine Learning

Week 1, Lecture 2

RecapSupervised Learning

Data Set

Learning Algorithm

Hypothesis hh(x) ≈ f(x)

Unknown Target f

Hypothesis Set5 0 4 1 9 2 1 3 1 4

Hyperplane

Halfspace >0

Halfspace < 0

w

Classification Regression

np-hard in general

Assume Data Is Linear Separable!!!Perception find separating hyperplane Convex

Today

• Convex Optimization – Convex sets– Convex functions

• Logistic Regression– Maximum Likelihood– Gradient Descent

• Maximum likelihood and Linear Regression

Convex Optimization

Optimization problem, in general very hard (if possible at all)!!!

For convex optimization problemstheoretical (polynomial time) and practical solutions exist(most of the time)

Example:

Convex Sets

Convex Set Non-convex Set

The “line” from x to y must also be in the set

Convex Sets

Union of convex setsmay not be convex

Intersection of convex sets

Convex Functions

x,f(x)

y,f(y)

f is concave if –f is convex

Concave?, Convex? Both

Differentiable Convex Functions

x,f(x)

y,f(y)

f(x)+f’(x)(y-x)

Example

Twice Differentiable Convex Functions

f is convex if the Hessian is positive semi-definite for all x.

Real symmetric matrix A is positive semidefinite if for all nonzero x

1D:

Simple 2D Example

More Examples

Quadratic Functions:

Convex if A is positive semidefinite

Affine Functions:

Convexity of Linear Regression

Quadratic Functions:

Convex if A is positive semidefinite

Real and Symmetric: Clearly

EpigraphConnection between convex sets and convex functions

f is convex if epi(f) is a convex set

Sublevel setsConvex function

Define α-Sublevel set:

Is Convex

Convex Optimization

f and g are convex, h is affine

Local Minima are Global Minima

Examples of Convex Optimization

• Linear Programming

• Quadratic Programming (P is positive semidefinite)

Summary

Rockafellar stated, in his 1993 SIAM Review survey paper “In fact the great watershed in optimization isn’t between linearity and nonlinearity, but convexity and nonconvexity”

Convex GOOD!!!!

Estimating Probabilities

• Probability of getting cancer given your situation.

• Probability that AGF wins against Viborg given the last 5 results.

• Probability that the loan is not payed back as a function of credit worthiness

• Probability of a student getting an A in Machine Learning given his grades.

Data is actual events not probabilities, e.g. some students that failed and some who did not…

Breast Cancerhttp://archive.ics.uci.edu/ml/datasets/Breast+Cancer+Wisconsin+%28Original%29

• 1. Sample code number: id number • 2. Clump Thickness: 1 - 10 • 3. Uniformity of Cell Size: 1 - 10 • 4. Uniformity of Cell Shape: 1 - 10 • 5. Marginal Adhesion: 1 - 10 • 6. Single Epithelial Cell Size: 1 - 10 • 7. Bare Nuclei: 1 - 10 • 8. Bland Chromatin: 1 - 10 • 9. Normal Nucleoli: 1 - 10 • 10. Mitoses: 1 - 10

Input Features• benign • malignant

Target Function

PREDICT PROBABILITY OF BENIGN AND MALIGNANT ON FUTURE PATIENTS

Maximum LikelihoodBiased Coin, (bias θ probability of heads)Flip it n times independently (Bernoulli trials), Count the number of heads k

Fix θ, What is the probability of seeing DTake Logs

After seeing the data what can we infer

Likelihood of the data

Maximum Likelihood

solve for 0

Compute Gradient

Negative Log Likelihood of the data (log is monotone)

Maximize

Minimize

Bayesian Perspective

Bayes Rule:

Want: Need: A Prior

Likelihood x Prior

Normalizing factor

Posterior

Bayesian Perspective

• Compute the probability of each hypotheses• Pick the most likely and use for predictions

(map = maximum a posteriori)• Compute Expected Values (Weighed average

over all hypotheses)

Logistic Regression

Assume Independent Data Points, Apply Maximum Likelihood (there is a Bayesian version to)

Hard Threshold Hard and Soft Threshold

Can and is used for classification. Predict most likely y

Maximum Likelihood Logistic Regression

Neg. Log likelihood is convex

Cannot solve for zero analytically

Descent Methods

Iteratively move toward a better solution

where f is twice continuously differentiable

• Pick start point x• Repeat Until Stopping Criterion Satisfied• Compute Descent Direction v• Line Search: Compute Step Size t• Update: x = x + t v

Gradient Descent

Line (Ray) Search• Pick start point x• Repeat Until Stopping Criterion Satisfied• Compute Descent Direction v• Line Search: Compute Step Size t• Update: x = x + t v

• Solve analytically (if possible)• Backwards Search start high and decrease until

improving distance found [SL 9.2]• Fix to a small constant• Use size of the gradient scaled with small constant.• Start with constant, let it decrease slowly or when to high

Stopping Criteria

• Gradient becomes very small• Max number of iterations used

Gradient Descent for Linear Reg.

GD For Linear RegressionMatlab style

function theta= GD(X,y,theta)LR = 0.1for i=1:50

cost = (1/length(y))* sum((X*theta-y).^2)grad = (1/length(y))*2.*X'*(X*theta-y)

theta = theta – LR * gradend

Note we do not scale gradient to unit vector

Learning Rate Learning Rate Learning Rate

Gradient Descent Jump Around

Use Exact Line Search Starting From (10,1)

Gradient Descent Running Time

• Number of iterations x Cost per iteration.• Cost Per Iteration is usually not a problem.• Number of iterations depends choice of line

search and stopping Criterion clearly.– Very Problem and Data Specific– Need a lot of math to give bounds. – We will not cover it in this course.

Gradient Descent For Logistic Regression

Handin 1!A long with multiclass extension

Stochastic Gradient Descent

Pick at random and use

Use K points chosen at randomMini-Batch Gradient Descent

Linear Classification with K classes

• Use Logistic regression All Vs one.– Train K classifiers one for each class– Input X is the same. Y is 1 for all elements from that

class and 0 otherwise (All vs. One)– Prediction, compute the probability for all K classifiers

output class with highest probability.• Use Softmax Regression– Extension of logistic function to K classes in some

sense– Covered in Handin 1.

Maximum Likelihood and Linear Regression (Time to spare slide)

Assume: Independently

Todays Summary• Convex Optimizations

– Many Definitions– Local Optimal is Global Optimal– Usually theoretical and practically feasible

• Maximum likelihood– Use as a proxy for– Assume Independent Data

• Gradient Descent– Minimize function – Iteratively finding better solution by local steps based on gradient– First order method (Uses gradient)– Other methods exist, e.g. Second order methods (use hessian)