04 logistic regression - colorado state universitycs545/fall18/lib/exe/fetch... · cross entropy...

Linear models: Logistic regression

Chapter 3.3

Predicting probabilities

Objective: learn to predict a probability P(y | x) for a binary classification problem using a linear classifier The target function: For positive examples P(y = +1 | x) = 1 whereas P(y = +1 | x) = 0 for negative examples.

The Target Function is Inherently Noisy

f(x) = P[y = +1 | x].

The data is generated from a noisy target function:

P (y | x) =

⎪⎨

⎪⎩

f(x) for y = +1;

1− f(x) for y = −1.

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 6 /23 When is h good? −→

Objective: learn to predict a probability P(y | x) for a binary classification problem using a linear classifier The target function: For positive examples P(y = +1 | x) = 1 whereas P(y = +1 | x) = 0 for negative examples. Can we assume that P(y = +1 | x) is linear?

The Target Function is Inherently Noisy

f(x) = P[y = +1 | x].

The data is generated from a noisy target function:

P (y | x) =

⎪⎨

⎪⎩

f(x) for y = +1;

1− f(x) for y = −1.

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 6 /23 When is h good? −→

Logistic regression

h(x) = (s) h(x) = s h(x) = θ(s)

h x( )

⃝ AML

θ(s) =es

1 + es

θ(s)1

⃝ AML

θ(s) =es

1 + es

θ(s)1

⃝ AML

The logistic function (aka squashing function):

The signal is the basis for several linear models:

Properties of the logistic function

Predicting a Probability

Will someone have a heart attack over the next year?

age 62 yearsgender maleblood sugar 120 mg/dL40,000HDL 50LDL 120Mass 190 lbsHeight 5′ 10′′

. . . . . .

Classification: Yes/No

Logistic Regression: Likelihood of heart attack logistic regression ≡ y ∈ [0, 1]

h(x) = θ

= θ(wtx)θ(s)

θ(s) =es

1 + es=

1 + e−s.

θ(−s) =e−s

1 + e−s=

1 + es= 1− θ(s).

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 4 /23 Data is binary ±1 −→

Predicting a Probability

Will someone have a heart attack over the next year?

age 62 yearsgender maleblood sugar 120 mg/dL40,000HDL 50LDL 120Mass 190 lbsHeight 5′ 10′′

. . . . . .

Classification: Yes/No

Logistic Regression: Likelihood of heart attack logistic regression ≡ y ∈ [0, 1]

h(x) = θ

= θ(wtx)θ(s)

θ(s) =es

1 + es=

1 + e−s.

θ(−s) =e−s

1 + e−s=

1 + es= 1− θ(s).

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 4 /23 Data is binary ±1 −→

Fitting the data means finding a good hypothesis h

What Makes an h Good?

‘fiting’ the data means finding a good h

h is good if:

h(xn) ≈ 1 whenever yn = +1;

h(xn) ≈ 0 whenever yn = −1.

A simple error measure that captures this:

Ein(h) =1

h(xn)−12(1 + yn)

Not very convenient (hard to minimize).

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 7 /23 Cross entropy error −→

The Probabilistic Interpretation

Suppose that h(x) = θ(wtx) closely captures P[+1|x]:

P (y | x) =

⎪⎨

⎪⎩

θ(wtx) for y = +1;

1− θ(wtx) for y = −1.

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 9 /23 1− θ(s) = θ(−s) −→

P (y | x) =

⎪⎨

⎪⎩

θ(wtx) for y = +1;

1− θ(wtx) for y = −1.

Fitting the data means finding a good hypothesis h

What Makes an h Good?

‘fiting’ the data means finding a good h

h is good if:

h(xn) ≈ 1 whenever yn = +1;

h(xn) ≈ 0 whenever yn = −1.

A simple error measure that captures this:

Ein(h) =1

h(xn)−12(1 + yn)

Not very convenient (hard to minimize).

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 7 /23 Cross entropy error −→

P (y | x) =

⎪⎨

⎪⎩

θ(wtx) for y = +1;

1− θ(wtx) for y = −1.

So, if h(x) = θ(wtx) closely captures P[+1|x]:

P (y | x) =

⎪⎨

⎪⎩

θ(wtx) for y = +1;

θ(−wtx) for y = −1.

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 10 /23 Simplify to one equation −→

So, if h(x) = θ(wtx) closely captures P[+1|x]:

P (y | x) =

⎪⎨

⎪⎩

θ(wtx) for y = +1;

θ(−wtx) for y = −1.

. . . or, more compactly,P (y | x) = θ(y ·wtx)

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 11 /23 The likelihood −→

More compactly:

Is logistic regression really linear?

To figure out how the decision boundary looks like set solving for x we get: i.e.

P (y = +1|x) = exp(w

|x) + 1

P (y = �1|x) = 1� P (y = +1|x) = 1

|x) + 1

P (y = +1|x) = P (y = �1|x)

|x) = 1

|x = 0

Maximum likelihood

We will find w using the principle of maximum likelihood.

The Likelihood

P (y | x) = θ(y ·wtx)

Recall: (x1, y1), . . . , (xN, yN) are independently generated

Likelihood:The probability of getting the y1, . . . , yN in D from the corresponding x1, . . . ,xN :

P (y1, . . . , yN | x1, . . . ,xn) =N!

P (yn | xn).

The likelihood measures the probability that the data were generated if f were h.

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 12 /23 Maximize the likelihood −→

The Likelihood

P (y | x) = θ(y ·wtx)

Recall: (x1, y1), . . . , (xN, yN) are independently generated

Likelihood:The probability of getting the y1, . . . , yN in D from the corresponding x1, . . . ,xN :

P (y1, . . . , yN | x1, . . . ,xn) =N!

P (yn | xn).

The likelihood measures the probability that the data were generated if f were h.

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 12 /23 Maximize the likelihood −→

Valid since

Maximizing the likelihood

Maximizing The Likelihood (why?)

n=1P (yn | xn)

⇔ max ln"!N

n=1P (yn | xn)#

≡ max$N

n=1 lnP (yn | xn)

⇔ min − 1N

$Nn=1 lnP (yn | xn)

≡ min 1N

$Nn=1 ln

1P (yn|xn)

≡ min 1N

$Nn=1 ln

1θ(yn·wtxn)

← we specialize to our “model” here

≡ min 1N

$Nn=1 ln(1 + e−yn·w

Ein(w) =1

ln(1 + e−yn·wtxn)

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 13 /23 How to minimize Ein(w) −→

Summary: maximizing the likelihood is equivalent to

θ(yn w xn)

θ(s) =1

1 + e−s

E (w) =1

1 + e−ynw xn

+ ,- .

(h(xn),yn)⃝ AML

minimize

Cross entropy error

Summary: maximizing the likelihood is equivalent to

θ(yn w xn)

θ(s) =1

1 + e−s

E (w) =1

1 + e−ynw xn

+ ,- .

(h(xn),yn)⃝ AML

minimize

Cross entropy error

Ein(w) =1

I(yn = +1) ln1

h(xn)+ I(yn = �1) ln

1� h(xn)

Exercise: check that this is equivalent to:

Digression: information theory

I am thinking of an integer between 0 and 1,023. You want to guess it using the fewest number of questions. Most of us would ask “is it between 0 and 512?” This is a good strategy because it provides the most information about the unknown number. It provides the first binary digit of the number. Initially you need to obtain log2(1024) = 10 bits of information. After the first question you only need log2(512) = 9 bits.

Information and Entropy

By halving the search space we obtained one bit. In general, the information associated with a probabilistic outcome: Why the logarithm? Assume we have two independent events x, and y. We would like the information they carry to be additive. Let’s check: Entropy:

I(p) = � log p

I(x, y) = � logP (x, y) = � logP (x)P (y)

= � logP (x)� logP (y) = I(x) + I(y)

H(P ) = �X

P (x) logP (x)

Entropy

For a Bernoulli random variable: Maximal when p = ½.

H(p) = �p log p� (1� p) log(1� p)

KL divergence

The KL divergence between distributions P and Q: Properties: ²  Non-negative, equal to 0 iff P = Q ²  It is not symmetric

(P ||Q) = �X

P (x) log

KL divergence

The KL divergence between distributions P and Q:

(P ||Q) = �X

P (x) log

(P ||Q) = �X

P (x) logQ(x) +

P (x) logP (x)

cross entropy - entropy

Cross entropy and logistic regression

The logistic regression cost function: It is the average cross entropy between the learned P(y | x) and the observed probabilities Cross entropy And for binary variables:

Ein(w) =1

I(yn = +1) ln1

h(xn)+ I(yn = �1) ln

1� h(xn)

H(y, y) = �y log y � (1� y) log(1� y)

H(P,Q) = �X

P (x) logQ(x)

In-sample error

The in-sample error for logistic regression

θ(yn w xn)

θ(s) =1

1 + e−s

E (w) =1

1 + e−ynw xn

+ ,- .

(h(xn),yn)⃝ AML

minimize

Cross entropy error t = 0 w(0)

t = 0, 1, 2, . . .

∇E = −1

1 + eynw (t)xn

w(t + 1) = w(t)− η∇E

⃝ AML

t = 0 w(0)t = 0, 1, 2, . . .

∇E = −1

1 + eynw (t)xn

w(t + 1) = w(t)− η∇E

⃝ AML

t = 0 w(0)t = 0, 1, 2, . . .

∇E = −1

1 + eynw (t)xn

w(t + 1) = w(t)− η∇E

⃝ AML

Digression: gradient ascent/descent

Topographical maps can give us intuition on how to optimize a cost function

http://www.csus.edu/indiv/s/slaymaker/archives/geol10l/shield1.jpg http://www.sir-ray.com/touro/IMG_0001_NEW.jpg

Digression: gradient descent

21 Images from http://en.wikipedia.org/wiki/Gradient_descent

Given a function E(w), the gradient is the direction of steepest ascent Therefore to minimize E(w), take a step in the direction of the negative of the gradient

Notice that the gradient is perpendicular to contours of equal E(w)

Gradient descent

Gradient descent is an iterative process How to pick ?

Our Ein Has Only One Valley

Weights, w

In-sam

pleError,E

. . . because Ein(w) is a convex function of w. (So, who care’s if it looks ugly!)

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 16 /23 How to roll down? −→

How to “Roll Down”?

Assume you are at weights w(t) and you take a step of size η in the direction v.

w(t + 1) = w(t) + ηv

We get to pick v ← what’s the best direction to take the step?

Pick v to make Ein(w(t + 1)) as small as possible.

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 17 /23 The gradient −→

w(t+ 1) = w(t) + ⌘v

Gradient descent

The gradient is the best direction to take to optimize Ein(w):

The Gradient is the Fastest Way to Roll Down

Approximating the change in Ein

∆Ein = Ein(w(t+ 1))− Ein(w(t))

= Ein(w(t) + ηv)− Ein(w(t))

= η∇Ein(w(t))tv

! "# $

minimized at v = − ∇Ein(w(t))||∇Ein(w(t)) ||

+O(η2) (Taylor’s Approximation)

>≈ −η||∇Ein(w(t)) || ←attained at v = − ∇Ein(w(t))

||∇Ein(w(t)) ||

The best (steepest) direction to move is the negative gradient:

v = −

∇Ein(w(t))

||∇Ein(w(t)) ||

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 18 /23 Iterate the gradient −→

The Gradient is the Fastest Way to Roll Down

Approximating the change in Ein

∆Ein = Ein(w(t+ 1))− Ein(w(t))

= Ein(w(t) + ηv)− Ein(w(t))

= η∇Ein(w(t))tv

! "# $

minimized at v = − ∇Ein(w(t))||∇Ein(w(t)) ||

+O(η2) (Taylor’s Approximation)

>≈ −η||∇Ein(w(t)) || ←attained at v = − ∇Ein(w(t))

||∇Ein(w(t)) ||

The best (steepest) direction to move is the negative gradient:

v = −

∇Ein(w(t))

||∇Ein(w(t)) ||

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 18 /23 Iterate the gradient −→

Choosing the step size

The choice of the step size affects the rate of convergence:

The ‘Goldilocks’ Step Size

η too small η too large variable ηt – just right

Weights, w

In-sam

pleError,E

Weights, w

In-sam

pleError,E

Weights, w

In-sam

pleError,E

large η

small η

η = 0.1; 75 steps η = 2; 10 steps variable ηt; 10 steps

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 20 /23 Fixed learning rate gradient descent −→

Let’s use a variable learning rate:

w(t+ 1) = w(t) + ⌘tv

⌘t = ⌘ · ||rEin(w(t))||

||rEin(w(t))|| ! 0When approaching the minimum:

Choosing the step size

Weights, w

In-sam

pleError,E

Weights, w

In-sam

pleError,E

Weights, w

In-sam

pleError,E

large η

small η

Let’s use a variable learning rate:

w(t+ 1) = w(t) + ⌘tv

⌘t = ⌘ · ||rEin(w(t))||

⌘tv = �⌘ · ||rEin(w(t))|| · rEin(w(t))

||rEin(w(t))|| = �⌘rEin(w(t))

The final form of gradient descent

Weights, w

In-sam

pleError,E

Weights, w

In-sam

pleError,E

Weights, w

In-sam

pleError,E

large η

small η

w(t+ 1) = w(t)� ⌘rEin(w(t))

Logistic regression using gradient descent

We will use gradient descent to minimize our error function. Fortunately, the logistic regression error function has a single global minimum:

Our Ein Has Only One Valley

Weights, w

In-sam

pleError,E

. . . because Ein(w) is a convex function of w. (So, who care’s if it looks ugly!)

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 16 /23 How to roll down? −→

Finding The Best Weights - Hill Descent

Ball on a complicated hilly terrain

— rolls down to a local valley↑

this is called a local minimum

Questions:

How to get to the bottom of the deepest valey?

How to do this when we don’t have gravity?

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 15 /23 Our Ein is convex −→

So we don’t need to worry about getting stuck in local minima

Logistic regression using gradient descent Putting it all together: This is called batch gradient descent

Fixed Learning Rate Gradient Descent

ηt = η · ||∇Ein(w(t)) ||

||∇Ein(w(t)) ||→ 0 when closer to the minimum.

v = −ηt ·∇Ein(w(t))

||∇Ein(w(t)) ||

= −η · ||∇Ein(w(t)) || ·∇Ein(w(t))

||∇Ein(w(t)) ||

v = −η ·∇Ein(w(t))

1: Initialize at step t = 0 to w(0).2: for t = 0, 1, 2, . . . do3: Compute the gradient

gt = ∇Ein(w(t)). ←− (Ex. 3.7 in LFD)

4: Move in the direction vt = −gt.5: Update the weights:

w(t+ 1) = w(t) + ηvt.

6: Iterate ‘until it is time to stop’.7: end for8: Return the final weights.

Gradient descent can minimize any smooth function, for example

Ein(w) =1

ln(1 + e−yn·wtx) ← logistic regression

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 21 /23 Stochastic gradient descent −→

t = 0 w(0)t = 0, 1, 2, . . .

∇E = −1

1 + eynw (t)xn

w(t + 1) = w(t)− η∇E

⃝ AML

θ(yn w xn)

θ(s) =1

1 + e−s

E (w) =1

1 + e−ynw xn

+ ,- .

(h(xn),yn)⃝ AML

Logistic regression

Comments: v  In practice logistic regression is solved by faster

methods than gradient descent v  There is an extension to multi-class classification

Stochastic gradient descent

Variation on gradient descent that considers the error for a single training example: Tends to converge faster than the batch version.

Stochastic Gradient Descent (SGD)

A variation of GD that considers only the error on one data point.

Ein(w) =1

ln(1 + e−yn·wtx) =

e(w,xn, yn)

• Pick a random data point (x∗, y∗)

• Run an iteration of GD on e(w,x∗, y∗)

w(t + 1)← w(t)− η∇we(w,x∗, y∗)

1. The ‘average’ move is the same as GD;

2. Computation: fraction 1N

cheaper per step;

3. Stochastic: helps escape local minima;

4. Simple;

5. Similar to PLA.

Logistic Regression:

w(t + 1)← w(t) + y∗x∗

1 + ey∗wtx∗

(Recall PLA: w(t+ 1)← w(t) + y∗x∗)

c⃝ AML Creator: Malik Magdon-Ismail Logistic Regression and Gradient Descent: 22 /23 GD versus SGD, a picture −→

Ein(w) =1

e(w,xn, yn)

w(t + 1)← w(t)− η∇we(w,x∗, y∗)

cheaper per step;

4. Simple;

5. Similar to PLA.

w(t + 1)← w(t) + y∗x∗

1 + ey∗wtx∗

Ein(w) =1

e(w,xn, yn)

w(t + 1)← w(t)− η∇we(w,x∗, y∗)

cheaper per step;

4. Simple;

5. Similar to PLA.

w(t + 1)← w(t) + y∗x∗

1 + ey∗wtx∗

Summary of linear models

Linear methods for classification and regression:

The Linear Signal

s = wtx−→

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

→ sign(wtx) {−1,+1}

→ wtx R

→ θ(wtx) [0, 1]

y = θ(s)

c⃝ AML Creator: Malik Magdon-Ismail Linear Classification and Regression: 6 /21 Classification and PLA −→

More to come!

04 logistic regression - colorado state universitycs545/fall18/lib/exe/fetch... · cross entropy...

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