1. Lecture 04: Logistic Regression
CS480/680: Intro to ML
Lecture 04: Logistic Regression
9/20/18
Yao-Liang Yu

2. Outline Announcements Bernoulli model Logistic regression Computation
9/20/18
Yao-Liang Yu

3. Outline Announcements Bernoulli model Logistic regression Computation
9/20/18
Yao-Liang Yu

4. Announcements
Assignment 1 due next week
9/20/18
Yao-Liang Yu

5. Outline Announcements Bernoulli model Logistic regression Computation
9/20/18
Yao-Liang Yu

6. Classification revisited
ŷ = sign( xTw + b )
How confident we are about ŷ?
|xTw + b| seems a good indicator
real-valued; hard to interpret
ways to transform into [0,1]
Better(?) idea: learn confidence directly
9/20/18
Yao-Liang Yu

7. Conditional probability
P(Y=1 | X=x): conditional on seeing x, what is the chance of this instance being positive, i.e., Y=1?
obviously, value in [0,1]
P(Y=0 | X=x) = 1 – P(Y=1 | X=x), if two classes
more generally, sum to 1
Notation (Simplex). Δc-1 := { p in Rc: p ≥ 0, Σk pk = 1 }
9/20/18
Yao-Liang Yu

8. Reduction to a harder problem
P(Y=1 | X=x) = E(1Y=1 | X=x)
Let Z = 1Y=1, then regression function for (X, Z)
use linear regression for binary Z?
Exploit structure!
conditional probabilities are in a simplex
Never reduce to unnecessarily harder problem
9/20/18
Yao-Liang Yu

9. Bernoulli model Let P(Y=1 | X=x) = p(x; w), parameterized by w
Conditional likelihood on {(x1, y1), … (xn, yn)}:
simplifies if independence holds
Assuming yi is {0,1}-valued
9/20/18
Yao-Liang Yu

10. Naïve solution Find w to maximize conditional likelihood
What is the solution if p(x; w) does not depend on x?
What is the solution if p(x; w) does not depend on w?
9/20/18
Yao-Liang Yu

11. Generalized linear models (GLM)
y ~ Bernoulli(p); p = p(x; w) natural parameter
Logistic regression
y ~ Normal(μ, σ2); μ = μ(x; w)
(weighted) least-squares regression
GLM: y ~ exp( θ φ(y) – A(θ) )
log-partition function
sufficient statistics
9/20/18
Yao-Liang Yu

12. Outline Announcements Bernoulli model Logistic regression Computation
9/20/18
Yao-Liang Yu

13. Logit transform p(x; w) = wTx? p >=0 not guaranteed…
log p(x; w) = wTx? better!
LHS negative, RHS real-valued…
Logit transform
Or equivalently
odds ratio
9/20/18
Yao-Liang Yu

14. Prediction with confidence
ŷ = 1 if p = P(Y=1 | X=x) > ½ iff wTx > 0
Decision boundary wTx = 0
ŷ = sign(wTx) as before, but with confidence p(x; w)
9/20/18
Yao-Liang Yu

15. Not just a classification algorithm
Logistic regression does more than classification
it estimates conditional probabilities
under the logit transform assumption
Having confidence in prediction is nice
the price is an assumption that may or may not hold
If classification is the sole goal, then doing extra work
as shall see, SVM only estimates decision boundary
9/20/18
Yao-Liang Yu

16. More than logistic regression
F(p) transforms p from [0,1] to R
Then, equating F(p) to a linear function wTx
But, there are many other choices for F!
precisely the inverse of any distribution function!
9/20/18
Yao-Liang Yu

17. Logistic distribution
Cumulative Distribution Function
Mean mu, variance s2π2/3
9/20/18
Yao-Liang Yu

18. Outline Announcements Bernoulli model Logistic regression Computation
9/20/18
Yao-Liang Yu

19. Maximum likelihood Minimize negative log-likelihood 9/20/18
Yao-Liang Yu

20. Newton’s algorithm η = 1: iterative weighted least-squares PSD
Uncertain predictions get
bigger weight
η = 1: iterative weighted least-squares
9/20/18
Yao-Liang Yu

21. Comparison
9/20/18
Yao-Liang Yu

22. A word about implementation
Numerically computing exponential can be tricky
easily underflows or overflows
The usual trick
estimate the range of the exponents
shift the mean of the exponents to 0
9/20/18
Yao-Liang Yu

23. More than 2 classes Softmax Again, nonnegative and sum to 1
Negative log-likelihood (y is one-hot)
9/20/18
Yao-Liang Yu

24. Questions?
9/20/18
Yao-Liang Yu