7 logistic regression - Penn Engineeringcis520/lectures/7_logistic_regression.… · Logistic Regression uUse as the model for class k uGradient descent simultaneously updates all

Logistic RegressionLyle Ungar

Learning objectivesLogistic model & lossDecision boundaries as hyperplanesMulti-class regression

What do you do with a binary y?u Can you use linear regression?

l y = wTxu How about a different link function?

l y = f(wTx)u Or a different probability distribution

l P(y=1|x) = f(wTx)

Logistic function

h✓(x) =1

1 + e�✓Tx

Logistic Regression

Log odds

Log likelihood of data

y = 1 or -1

Decision Boundary

Representing Hyperplanes• How do we represent a line?

• In general, a hyperplane is defined by

The red vector (w) defines the green hyper plane that is orthogonal to it.

Why bother with this weird representation?

0 = wTx

[1,-1]

Projections

Now classification is easy!

h(x) = sgn(wTx)

Computing MLEu Use gradient ascent

Loss function = log-likelihood

Computing MAPu Prior

u So solve

u Again use gradient descent

gg

gg g

2g2

2g2

Multi-Class Classification

Disease diagnosis: healthy / cold / flu / pneumoniaObject classification: desk / chair / monitor / bookcase

x1

x2

x1

x2

Binary classification:

Multi-class classification:

Multi-Class Logistic Regressionu For 2 classes:

u For K classes:

l Called the softmax functionn maps a vector to a probability distribution

h✓(x) =1

1 + exp(�✓Tx)=

exp(✓Tx)

1 + exp(✓Tx)h✓(x) =

1

1 + exp(�✓Tx)=

exp(✓Tx)

1 + exp(✓Tx)

weight assigned to

y = -1

weight assigned to

y = 1

p(y = k | x; ✓1, . . . , ✓k) = exp(✓>k x)PK

k=1 exp(✓>k x)

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Multi-Class Logistic Regression

u Train a logistic regression classifier for each class k to predict the probability that y = k with

x1

x2

Split into One vs. Rest:

hk(x) =exp(✓>

k x)Pkk=1 exp(✓k>x)

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q1q3

q2

Implementing Multi-Class Logistic Regression

u P(y=k|x) estimated by:u Gradient descent simultaneously updates all

parameters for all modelsl Same derivative as before, just with the above hk(x)

u Predict class label as the most probable label

hk(x) =exp(✓>

k x)Pkk=1 exp(✓k>x)

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You should knowu Logistic model & loss

l Linear in log-odds

u Decision boundariesl hyperplane

u Softmaxl Maps vector to probability distribution