CS_441_2023_Spring_January_26,_2023.vtt

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NOTE
Created on 2024-02-07T20:52:49.1946189Z by ClassTranscribe

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Alright, good morning everybody.

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So I just wanted to start with a little

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Review.

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So first question, and don't yell out

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the answer I'll give you.

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I want to give everyone a couple a

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little bit to think about it.

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Which of these tend to be decreased as

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the number of training examples

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increase?

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The Training Error, test error

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Generalization could be more than one.

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I'll give you.

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A little bit to think about it.

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Alright, so well, would you expect the

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Training Error to decrease as the

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number of training examples increases?

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Raise your hand if so.

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And raise your hand if not.

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So is have it a lot of abstains, but if

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I don't count them.

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So yeah, actually the Training Error

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will actually increase as the number of

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training examples increases because the

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model gets harder to fit.

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So assuming the Training Error is

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nonzero, then it will increase or the

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loss that you're fitting is going to

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increase because as you get more

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Training examples then.

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Then, given a single model, you're

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Error is going to go up all right.

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What about test Error?

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Would you expect that to increase or

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decrease or stay the same?

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I guess first just do you expect it to

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decrease?

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Raise your hand for decreased.

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All right, raise your hand for

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increase.

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Everyone expects to test their to

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decrease.

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And Generalization Error, do you expect

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that to increase or I mean sorry, do

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you expect it to decrease?

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Raise your hand if Generalization Error

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should decrease.

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And raise your hand if it should

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increase.

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Right, so you expect.

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So the Generalization Error should also

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decrease.

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And remember that the Generalization

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error is the.

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Test Error minus the Training error, so

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the typical curve you see.

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The typical curve you would see if this

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is the number of train.

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And this is the Error.

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Is that Training Error?

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We'll go like something like that.

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So this is the train.

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And the test error will go something

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like this.

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And this is the generalization error is

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a gap between training and test error.

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So actually.

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The generalization error will decrease

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the fastest because that gap is closing

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faster than the test error is going

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down.

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That has to be the case because the

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Training Error is going up.

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And then the test error decreased the

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second fastest, and the Training error

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is actually going to increase, so the

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Training loss will increase.

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Alright, second question and these are

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just Review questions that I took from

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the thing that I linked but wanted to

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do them here.

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So Classify the X with the plus using

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one nearest neighbor and three Nearest

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neighbor where you've got 2 features on

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the axis there.

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Alright, 41 Nearest neighbor.

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How many people think it's an X?

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OK, how many people think it's an O?

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Everyone said 784x1.

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That's correct.

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For three Nearest neighbor.

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How many people think it's an X?

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How many people think it's to know?

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Right.

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Yeah, you guys got that.

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So 3 Nearest neighbor, it's a no.

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Right now these I think.

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Also, I have a couple of probability

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questions.

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Alright, so first, just what assumption

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does the Naive based model make if

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there are two features X1 and X2?

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Give you a second to think about it,

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there's.

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Really two options there.

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They either it's one of it's either A

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or B, neither or both.

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I'll give you a moment.

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Alright, so how many say that A is an

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assumption that Naive Bayes makes?

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Right.

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How many people say that B is an

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assumption that Naive Bayes makes?

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How many say that neither of those are

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true?

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And how many say that both of those are

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true, that they're the same thing and

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they're both true?

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So I think there are maybe at least one

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vote for each of them.

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But so the answer is B that Naive Bayes

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assumes that the features are

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independent of each other given the

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given the Prediction given the label.

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And I'll consistently use X for

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features and Y for label.

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So.

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Hopefully that part is clear.

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So A is not true because it's not

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assuming that the in fact A is just

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never true or?

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Is that ever true?

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I guess it could be true if Y is always

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one or under certain weird

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circumstances, but a is like a bad

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probability statement.

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And then B assumes that X1 and X2 are

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independent given Y because remember

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that if A&B are independent.

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Then probability of AB equals

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probability of a times probability B.

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And similarly, even if it's

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conditional, if X1 and X2 are

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independent, then probability of X1 and

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X2 given Y is equal to probability of

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X1 given Y times probability of X2

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given Y.

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And they're and they're not equivalent,

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they're different expressions.

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OK, so now this one is probably the.

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This one is the most.

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Complicated to work through I guess.

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So let's say X1 and X2 are binary

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features and Y is a binary label.

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And.

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And then all I've set the probabilities

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so we know what X 1 = 1 given y = 0, X

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2 = 1 given y = 0.

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So I didn't fill out the whole

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probability table, but I gave enough

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maybe to do the first part.

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So if we make an app as assumption.

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So that's the assumption under B there.

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What is probability of y = 1?

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Given X 1 = 1 and X 2 = 1.

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I'll give you a little bit of time to

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start thinking about it, but I won't

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ask.

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I won't ask anyone to call it the

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answer.

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I'll just start working through it.

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So think about how you would solve it.

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What things you have to multiply

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together, et cetera.

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Nice.

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Alright, so I'll start working it out.

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So probability of Y1 given X1 and X2.

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So let's see.

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So probability of y = 1.

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Given X 1 = 1 and X 2 = 1.

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That's the probability of y = 1 X.

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1.

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Equals one.

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X 2 = 1.

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Divided by.

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Probability.

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I'll just do sum over K to save myself

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some rating.

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I don't like writing by hand much.

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So sum K in the values of zero to 1

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probability of Y.

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Equals K&X 1 = 1 and X 2 = 1.

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So the reason for this, whoops, the

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reason for that is that.

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I'm marginalizing out the Y so that is

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just equal to probability.

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On the denominator I have probability

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of X 1 = 1 and probability of X 2 = 1.

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And then this guy is going to be.

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I can get there.

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By probability of Y given X1 and X2.

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Equals probability.

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Sorry, I meant to flip that.

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Probability of X1 and X2.

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Given Y is equal to probability of X1

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given Y times probability of X 2 = y.

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That's the Naive Bayes assumption part.

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So the numerator.

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Is.

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Let's see.

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So the numerator will be 1/4 * 1/2.

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And then probability of Y is 5.

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So on the numerator of this expression

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here I have 1/4 * 1/2 * 5.

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And on the denominator I have 1/4 * 1/2

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* 1.5.

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Plus.

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2/3 * 1/3 * .5, right?

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This is a probability of X = 1 given y

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= 0 times that, times that or times.

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And then it's times 5 because the

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probability of y = 1 is .5.

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Then probability of y = 0 is 1 -, .5,

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which is also 05.

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That's how I solve that first part.

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And then under Naive base assumption,

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is it possible to calculate this given

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the information I provided in those

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equations?

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So it's not.

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Under first glance it might look like

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it is, but it's not because I don't

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know what the probability of X = 0

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given Y is.

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I didn't give any information about

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that.

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I only said what the probability of X1

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given Y is, and I can't figure out the

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probability of X0 given Y from the

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probability of X1 given Y.

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Or as I at least.

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I haven't thought through it in great

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detail, but I don't think I can figure

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it out.

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Alright.

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So then with without the Naive's

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assumption, yeah, under the nibs,

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sorry.

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I made a I was.

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OK.

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Under the name's assumption.

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Is it possible to figure that out?

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Let me think.

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Probability of X1.

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Yeah, sorry about that.

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I was I switched these in my head.

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So under the knob is assumption.

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Actually I can figure this out because.

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If because if X, since X is binary,

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then if X probability of X 1 = 1 given

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y = 0.

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Is 2/3 then probability of X 1 = 0

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given y = 0 is 1/3 and probability of

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X2 given equals zero given y = 0 is 2/3

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and probability of X 1 = 0 given y = 1

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is 3/4.

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So I know probability of X = 0 given Y.

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Equals 0 or y = 1 so I can solve this

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one.

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And then I kind of gave it away, but

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without the nib is assumption is it

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possible to calculate the probability

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of y = 1 given X 1 = 1 and X 2 = 1?

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No, I mean I already I said it, but.

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But no, it's not.

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And the reason is because I don't have

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any of the joint probabilities here.

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For that I would need to know

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something, the probability of X1 and X2

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and Y the full probability table.

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Or I would need to be given the

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probability of Y given X1 and X2.

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Alright, so that was just a little

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Review and warm up.

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So today I'm going to mainly talk about

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Linear models and in particular I'll

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talk about Linear, Logistic Regression

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and Linear Regression.

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And then as part of that I'll talk

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about this concept called

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regularization.

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Right, So what is the Linear model?

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A Linear model is a model in a model is

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linear in X if it is a X plus some plus

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maybe some constant value.

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So I can write that as W transpose X +

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B and remember using your linear

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algebra that that's the same as the sum

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over I of wixi plus B.

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So for any values of X&B these are WI

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and B are scalars.

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XI would be a scalar, so X is a vector,

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W vector.

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So this is a Linear model no matter how

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I choose those coefficients.

00:15:05.750 --> 00:15:07.730
And there's two main kinds of Linear

00:15:07.730 --> 00:15:08.330
models.

00:15:08.330 --> 00:15:10.603
There's a Linear classifier and a

00:15:10.603 --> 00:15:11.570
Linear regressor.

00:15:12.370 --> 00:15:15.210
So in a Linear classifier.

00:15:16.180 --> 00:15:19.450
This W transpose X + B is giving you a

00:15:19.450 --> 00:15:21.490
score for how likely.

00:15:22.190 --> 00:15:27.400
A feature vector is to belong to one

00:15:27.400 --> 00:15:28.790
class or the other class.

00:15:30.020 --> 00:15:31.300
So that's shown down here.

00:15:31.300 --> 00:15:33.854
We have like some O's and some

00:15:33.854 --> 00:15:34.320
triangles.

00:15:34.320 --> 00:15:36.240
I've got a Linear model here.

00:15:36.240 --> 00:15:40.555
This is the West transpose X + B and

00:15:40.555 --> 00:15:44.270
that gives me a score that say that say

00:15:44.270 --> 00:15:46.220
that class is equal to 1.

00:15:46.220 --> 00:15:48.170
Maybe I'm saying the triangles are ones

00:15:48.170 --> 00:15:49.370
are y = 1.

00:15:50.220 --> 00:15:54.692
So if I this line will project all of

00:15:54.692 --> 00:15:57.242
these different points onto the line.

00:15:57.242 --> 00:15:59.847
The West transpose X + B projects all

00:15:59.847 --> 00:16:01.640
of these points onto this line.

00:16:02.650 --> 00:16:05.146
And then we tend to look at when you

00:16:05.146 --> 00:16:07.140
when you see like diagrams of Linear

00:16:07.140 --> 00:16:07.990
Classifiers.

00:16:07.990 --> 00:16:09.480
Often what people are showing is the

00:16:09.480 --> 00:16:10.150
boundary.

00:16:10.890 --> 00:16:13.550
Which is where W transpose X + b is

00:16:13.550 --> 00:16:14.400
equal to 0.

00:16:16.460 --> 00:16:18.580
So all the points that project on one

00:16:18.580 --> 00:16:20.699
side of the boundary will be one class

00:16:20.700 --> 00:16:22.264
and all the ones that project on the

00:16:22.264 --> 00:16:24.132
other side of the boundary or the other

00:16:24.132 --> 00:16:24.366
class.

00:16:24.366 --> 00:16:26.670
Or in other words, if W transpose X + B

00:16:26.670 --> 00:16:27.830
is greater than 0.

00:16:28.470 --> 00:16:29.270
It's one class.

00:16:29.270 --> 00:16:30.675
If it's less than zero, it's the other

00:16:30.675 --> 00:16:30.960
class.

00:16:32.640 --> 00:16:34.020
A Linear regressor.

00:16:34.020 --> 00:16:36.720
You're directly fitting the data

00:16:36.720 --> 00:16:40.790
points, and you're solving for a line

00:16:40.790 --> 00:16:42.430
that passes through.

00:16:43.630 --> 00:16:45.130
The target and features.

00:16:46.030 --> 00:16:48.495
So that you're more directly so that

00:16:48.495 --> 00:16:50.880
you're able to predict the target

00:16:50.880 --> 00:16:54.310
value, the Y given your features and so

00:16:54.310 --> 00:16:57.740
in 2D I can plot that as a 2D line, but

00:16:57.740 --> 00:16:59.740
it can be ND it could be a high

00:16:59.740 --> 00:17:00.600
dimensional line.

00:17:01.300 --> 00:17:04.890
And you have y = W transpose X + B.

00:17:06.290 --> 00:17:09.150
So in Classification, typically it's

00:17:09.150 --> 00:17:11.550
not Y equals W transpose X + B, it's

00:17:11.550 --> 00:17:15.210
some kind of score for how it's a score

00:17:15.210 --> 00:17:17.340
for Y, and in Regression you're

00:17:17.340 --> 00:17:20.290
directly fitting Y with that line.

00:17:21.820 --> 00:17:22.200
Question.

00:17:27.440 --> 00:17:33.335
I almost all situations so at the so at

00:17:33.335 --> 00:17:35.740
the end of the day, like for example if

00:17:35.740 --> 00:17:36.890
you're doing deep learning.

00:17:37.520 --> 00:17:40.510
All of the different layers of the most

00:17:40.510 --> 00:17:42.503
of the layers of the feature, I mean of

00:17:42.503 --> 00:17:43.960
the network, you can think of as

00:17:43.960 --> 00:17:46.260
learning a feature representation and

00:17:46.260 --> 00:17:47.510
at the end of it you have a Linear

00:17:47.510 --> 00:17:50.190
classifier that maps from the features

00:17:50.190 --> 00:17:51.420
into the target label.

00:18:06.090 --> 00:18:06.560


00:18:14.220 --> 00:18:16.592
So the so the question is if you were

00:18:16.592 --> 00:18:18.170
if you were trying to predict whether

00:18:18.170 --> 00:18:20.400
or not somebody is caught based on a

00:18:20.400 --> 00:18:21.150
bunch of features.

00:18:22.260 --> 00:18:23.979
You could use the Linear classifier for

00:18:23.980 --> 00:18:24.250
that.

00:18:24.250 --> 00:18:26.840
So a Linear classifier is always a

00:18:26.840 --> 00:18:28.843
binary classifier, but you can also use

00:18:28.843 --> 00:18:30.530
it in Multiclass cases.

00:18:30.530 --> 00:18:33.777
So for example if you want to Classify

00:18:33.777 --> 00:18:35.860
if you have a picture of some animal

00:18:35.860 --> 00:18:37.366
and you want to Classify what kind of

00:18:37.366 --> 00:18:37.900
animal it is.

00:18:38.890 --> 00:18:40.634
And you have a bunch of features.

00:18:40.634 --> 00:18:43.470
Features could be like image Pixels, or

00:18:43.470 --> 00:18:45.950
it could be more complicated features

00:18:45.950 --> 00:18:48.420
than you would have a Linear model for

00:18:48.420 --> 00:18:50.860
each of the possible kinds of animals,

00:18:50.860 --> 00:18:54.040
and you would score each of the classes

00:18:54.040 --> 00:18:55.665
according to that model, and then you

00:18:55.665 --> 00:18:56.930
would choose the one with the highest

00:18:56.930 --> 00:18:57.280
score.

00:18:58.790 --> 00:19:00.930
So there's so some examples of Linear

00:19:00.930 --> 00:19:03.720
models are support vector.

00:19:03.720 --> 00:19:06.570
The only the two main examples I would

00:19:06.570 --> 00:19:08.936
say are support vector machines and

00:19:08.936 --> 00:19:10.009
Logistic Regression.

00:19:10.009 --> 00:19:11.619
Linear Logistic Regression.

00:19:12.630 --> 00:19:14.750
Naive Bayes is also a Linear model,

00:19:14.750 --> 00:19:15.220
but.

00:19:16.230 --> 00:19:18.080
And many other kinds of Classifiers.

00:19:18.080 --> 00:19:19.670
If you like, do the math, you can show

00:19:19.670 --> 00:19:21.150
that it's also a Linear model at the

00:19:21.150 --> 00:19:23.565
end of the day, but it's less thought

00:19:23.565 --> 00:19:24.510
that way.

00:19:31.680 --> 00:19:35.210
Cannon is not a Linear model.

00:19:35.210 --> 00:19:37.390
It has a non linear decision boundary.

00:19:38.070 --> 00:19:40.948
And boosted decision trees you can

00:19:40.948 --> 00:19:42.677
think of it as.

00:19:42.677 --> 00:19:45.180
So first like I will talk about trees

00:19:45.180 --> 00:19:47.750
and Bruce the decision trees next week.

00:19:47.750 --> 00:19:50.020
So I'm not going to fill in the details

00:19:50.020 --> 00:19:51.140
for those who don't know what they are.

00:19:51.140 --> 00:19:53.109
But basically you can think of it as

00:19:53.110 --> 00:19:55.010
that the tree is creating a

00:19:55.010 --> 00:19:56.280
partitioning of the features.

00:19:57.470 --> 00:19:59.115
Given that partitioning, you then have

00:19:59.115 --> 00:20:02.160
a Linear model on top of it, so you can

00:20:02.160 --> 00:20:03.570
think of it as an encoding of the

00:20:03.570 --> 00:20:04.850
features plus a Linear model.

00:20:06.030 --> 00:20:06.300
Yeah.

00:20:24.510 --> 00:20:26.290
How many like different models you need

00:20:26.290 --> 00:20:26.610
or.

00:20:26.610 --> 00:20:28.350
So it's the.

00:20:28.350 --> 00:20:29.020
It depends.

00:20:29.020 --> 00:20:30.800
It's kind of given by the problem

00:20:30.800 --> 00:20:32.440
setup, so if you're told.

00:20:33.930 --> 00:20:35.890
If you for example.

00:20:36.970 --> 00:20:37.750


00:20:38.900 --> 00:20:39.590


00:20:40.930 --> 00:20:42.670
OK, I'll just choose an image example

00:20:42.670 --> 00:20:44.010
because this pop into my head most

00:20:44.010 --> 00:20:44.680
easily.

00:20:44.680 --> 00:20:45.970
So if you're trying to Classify

00:20:45.970 --> 00:20:47.720
something between male or female,

00:20:47.720 --> 00:20:49.670
Classify an image between is it a male

00:20:49.670 --> 00:20:50.210
or female?

00:20:50.210 --> 00:20:51.678
Then you know you have two classes so

00:20:51.678 --> 00:20:54.164
you need to fit two models, one or need

00:20:54.164 --> 00:20:54.820
to fit.

00:20:55.640 --> 00:20:57.200
And the two class model you only have

00:20:57.200 --> 00:20:58.670
to fit one model because either it's

00:20:58.670 --> 00:20:59.270
one or the other.

00:20:59.980 --> 00:21:03.445
If you have, if you're trying to

00:21:03.445 --> 00:21:05.153
Classify, let's say you're trying to

00:21:05.153 --> 00:21:06.845
Classify a face into different age

00:21:06.845 --> 00:21:07.160
groups.

00:21:07.160 --> 00:21:09.460
Is it somebody that's under 10, between

00:21:09.460 --> 00:21:11.832
10 and 2020 and 30 and so on, then you

00:21:11.832 --> 00:21:13.660
would need like one model for each of

00:21:13.660 --> 00:21:14.930
those age groups.

00:21:14.930 --> 00:21:17.566
So usually as a problem set up you say

00:21:17.566 --> 00:21:19.880
I have these like features available to

00:21:19.880 --> 00:21:22.194
make my Prediction, and I have these

00:21:22.194 --> 00:21:23.890
things that I want to Predict.

00:21:23.890 --> 00:21:28.460
And if the things are a like a set of

00:21:28.460 --> 00:21:30.560
categories, then you would need one

00:21:30.560 --> 00:21:32.060
Linear model per category.

00:21:33.040 --> 00:21:35.390
And if the thing that you're trying to

00:21:35.390 --> 00:21:38.990
Predict is a set of continuous values,

00:21:38.990 --> 00:21:40.940
then you would need one Linear model

00:21:40.940 --> 00:21:42.030
per continuous value.

00:21:42.710 --> 00:21:45.160
If you're using like Linear models.

00:21:45.860 --> 00:21:46.670
Does that make sense?

00:21:47.400 --> 00:21:49.980
And then you mentioned like.

00:21:50.790 --> 00:21:52.850
You mentioned hidden hidden layers or

00:21:52.850 --> 00:21:54.230
something, but that would be part of

00:21:54.230 --> 00:21:56.650
neural networks and that would be like.

00:21:57.450 --> 00:22:00.790
A design choice for the network that we

00:22:00.790 --> 00:22:02.320
can talk about when we get to network.

00:22:05.500 --> 00:22:05.810
OK.

00:22:09.100 --> 00:22:12.500
A Linear classifier, you would say that

00:22:12.500 --> 00:22:16.150
the label is 1 if W transpose X + B is

00:22:16.150 --> 00:22:16.950
greater than 0.

00:22:17.960 --> 00:22:19.410
And then there's this important concept

00:22:19.410 --> 00:22:21.040
called linearly separable.

00:22:21.040 --> 00:22:22.200
So that just means that you can

00:22:22.200 --> 00:22:24.425
separate the points, the features of

00:22:24.425 --> 00:22:25.530
the two classes.

00:22:26.450 --> 00:22:27.750
Cleanly so.

00:22:30.220 --> 00:22:33.780
So for example, which of these is

00:22:33.780 --> 00:22:36.000
linearly separable, the left or the

00:22:36.000 --> 00:22:36.460
right?

00:22:38.250 --> 00:22:41.130
Right the left is linearly separable

00:22:41.130 --> 00:22:42.950
because I can put a line between them

00:22:42.950 --> 00:22:44.680
and all the triangles will be on one

00:22:44.680 --> 00:22:46.290
side and the circles will be on the

00:22:46.290 --> 00:22:46.520
other.

00:22:47.210 --> 00:22:48.660
But the right side is not linearly

00:22:48.660 --> 00:22:49.190
separable.

00:22:49.190 --> 00:22:51.910
I can't put any line to separate those

00:22:51.910 --> 00:22:53.780
from the triangles.

00:22:55.970 --> 00:22:58.595
So it's important to note that.

00:22:58.595 --> 00:23:01.150
So sometimes, like the fact that I have

00:23:01.150 --> 00:23:03.230
to draw everything in 2D on slides can

00:23:03.230 --> 00:23:04.240
be a little misleading.

00:23:04.860 --> 00:23:07.220
It may make you think that Linear

00:23:07.220 --> 00:23:09.200
Classifiers are not very powerful.

00:23:10.080 --> 00:23:11.930
Because in two dimensions they're not

00:23:11.930 --> 00:23:13.860
very powerful, I can create lots of

00:23:13.860 --> 00:23:16.070
combinations of points where I just

00:23:16.070 --> 00:23:17.580
can't get very good Classification

00:23:17.580 --> 00:23:18.170
accuracy.

00:23:19.410 --> 00:23:21.410
But as you get into higher dimensions,

00:23:21.410 --> 00:23:23.340
the Linear Classifiers become more and

00:23:23.340 --> 00:23:24.140
more powerful.

00:23:25.210 --> 00:23:28.434
And in fact, if you have D dimensions,

00:23:28.434 --> 00:23:31.460
if you have D features, that's what I

00:23:31.460 --> 00:23:32.419
mean by D dimensions.

00:23:33.050 --> 00:23:35.850
Then you can separate D + 1 points with

00:23:35.850 --> 00:23:37.700
any arbitrary labeling.

00:23:37.700 --> 00:23:40.370
So as an example, if I have one

00:23:40.370 --> 00:23:42.300
dimension, I only have one feature

00:23:42.300 --> 00:23:42.880
value.

00:23:43.610 --> 00:23:45.350
I can separate two points whether I

00:23:45.350 --> 00:23:47.740
label this as X and this is O or

00:23:47.740 --> 00:23:49.400
reverse I can separate them.

00:23:50.930 --> 00:23:52.430
But I can't separate these three

00:23:52.430 --> 00:23:53.100
points.

00:23:53.100 --> 00:23:55.730
So if it were like XI could separate

00:23:55.730 --> 00:23:58.519
it, but when it's Oxo I can't separate

00:23:58.520 --> 00:24:02.050
that with A1 dimensional 1 dimensional

00:24:02.050 --> 00:24:02.880
linear separator.

00:24:04.770 --> 00:24:07.090
In 2 dimensions, I can separate these

00:24:07.090 --> 00:24:08.730
three points no matter how I label

00:24:08.730 --> 00:24:11.570
them, whether it's ox or ox, no matter

00:24:11.570 --> 00:24:13.365
how I do it, I can put a line between

00:24:13.365 --> 00:24:13.590
them.

00:24:14.240 --> 00:24:16.220
But I can't separate four points.

00:24:16.220 --> 00:24:18.030
So that's a concept called shattering

00:24:18.030 --> 00:24:20.926
and an idea and Generalization theory

00:24:20.926 --> 00:24:22.017
called the VC dimension.

00:24:22.017 --> 00:24:24.120
The more points you can shatter, like

00:24:24.120 --> 00:24:26.175
the more powerful your classifier, but

00:24:26.175 --> 00:24:27.630
more importantly.

00:24:28.430 --> 00:24:30.910
The If you think about if you have 1000

00:24:30.910 --> 00:24:31.590
features.

00:24:32.320 --> 00:24:34.630
That means that if you have 1000 data

00:24:34.630 --> 00:24:38.130
points, random feature points, and you

00:24:38.130 --> 00:24:40.386
label them arbitrarily, there's two to

00:24:40.386 --> 00:24:41.274
the one.

00:24:41.274 --> 00:24:43.939
There's two to the 1000 different

00:24:43.940 --> 00:24:45.960
labels that you could assign different

00:24:45.960 --> 00:24:47.478
like label sets that you could assign

00:24:47.478 --> 00:24:49.965
to those 1000 points because either one

00:24:49.965 --> 00:24:50.440
could be.

00:24:50.440 --> 00:24:51.650
Every point can be positive or

00:24:51.650 --> 00:24:51.940
negative.

00:24:53.320 --> 00:24:55.150
For all of those two to the 1000

00:24:55.150 --> 00:24:57.110
different labelings, you can linearly

00:24:57.110 --> 00:24:59.110
separate it perfectly with 1000

00:24:59.110 --> 00:24:59.560
features.

00:25:00.500 --> 00:25:02.490
So that's pretty crazy.

00:25:02.490 --> 00:25:04.080
So this Linear classifier.

00:25:04.720 --> 00:25:07.480
Can deal with these two to the 1000

00:25:07.480 --> 00:25:09.370
different cases perfectly.

00:25:11.010 --> 00:25:12.420
So as you get into very high

00:25:12.420 --> 00:25:14.315
dimensions, Linear classifier gets very

00:25:14.315 --> 00:25:15.100
very powerful.

00:25:22.530 --> 00:25:23.060


00:25:23.940 --> 00:25:26.850
So the question is, more dimensions

00:25:26.850 --> 00:25:28.110
mean more storage?

00:25:28.110 --> 00:25:30.970
Yes, but it's only Linear, so.

00:25:31.040 --> 00:25:33.710
So that's not usually too much of a

00:25:33.710 --> 00:25:34.290
concern.

00:25:37.990 --> 00:25:38.230
Yes.

00:26:14.610 --> 00:26:16.100
So the question is like how do you

00:26:16.100 --> 00:26:18.160
visualize 1000 features?

00:26:18.830 --> 00:26:20.260
And.

00:26:20.400 --> 00:26:23.870
And so I will talk about essentially

00:26:23.870 --> 00:26:25.180
you have to map it down into 2

00:26:25.180 --> 00:26:26.750
dimensions or one dimension in

00:26:26.750 --> 00:26:29.390
different ways and I'll talk about that

00:26:29.390 --> 00:26:30.945
later in this semester.

00:26:30.945 --> 00:26:33.890
So there's the simplest methods are

00:26:33.890 --> 00:26:36.720
Linear Linear projections, principal

00:26:36.720 --> 00:26:38.490
component analysis, where you'd project

00:26:38.490 --> 00:26:40.230
it down under the dominant directions.

00:26:41.180 --> 00:26:43.220
There's also like nonlinear local

00:26:43.220 --> 00:26:46.640
embeddings that will create a better

00:26:46.640 --> 00:26:48.100
mapping out of all the features.

00:26:49.700 --> 00:26:51.880
You can also do things like analyze

00:26:51.880 --> 00:26:53.490
each feature by itself to see how

00:26:53.490 --> 00:26:54.380
predictive it is.

00:26:55.260 --> 00:26:56.750
And.

00:26:56.860 --> 00:26:57.750
But like.

00:26:58.850 --> 00:27:00.807
Ultimately you kind of need to do a

00:27:00.807 --> 00:27:01.010
test.

00:27:01.010 --> 00:27:03.120
So you what you would do is you do some

00:27:03.120 --> 00:27:04.936
kind of validation test where you would

00:27:04.936 --> 00:27:08.640
train a train a Linear model on say

00:27:08.640 --> 00:27:10.600
like 80% of the data and test it on the

00:27:10.600 --> 00:27:12.860
other 20% to see if you're able to

00:27:12.860 --> 00:27:15.200
predict the remaining 20% or if you

00:27:15.200 --> 00:27:16.439
want to just see if it's linearly

00:27:16.439 --> 00:27:16.646
separable.

00:27:16.646 --> 00:27:18.678
Then if you train it on all the data,

00:27:18.678 --> 00:27:20.633
if you get perfect Training Error then

00:27:20.633 --> 00:27:21.471
it's linearly separable.

00:27:21.471 --> 00:27:23.317
And if you don't get perfect Training

00:27:23.317 --> 00:27:25.180
Error then it's then it's not.

00:27:25.180 --> 00:27:27.830
Unless you like if you didn't apply a

00:27:27.830 --> 00:27:29.070
very strong regularization.

00:27:30.640 --> 00:27:31.060
You're welcome.

00:27:31.930 --> 00:27:33.380
Yeah, but you can't really visualize

00:27:33.380 --> 00:27:34.310
more than two dimensions.

00:27:34.310 --> 00:27:36.870
That's always a challenge, and it leads

00:27:36.870 --> 00:27:38.820
sometimes to bad intuitions.

00:27:40.520 --> 00:27:41.370
So.

00:27:42.610 --> 00:27:44.100
The thing is though that there is still

00:27:44.100 --> 00:27:45.970
like there might be many different ways

00:27:45.970 --> 00:27:48.560
that I can separate the points, so all

00:27:48.560 --> 00:27:50.500
of these will achieve 0 training error.

00:27:50.500 --> 00:27:53.000
So the different Classifiers, the

00:27:53.000 --> 00:27:54.860
different Linear Classifiers just have

00:27:54.860 --> 00:27:56.680
different ways of choosing the line

00:27:56.680 --> 00:27:58.600
essentially that make different

00:27:58.600 --> 00:27:59.200
assumptions.

00:28:00.850 --> 00:28:02.360
The.

00:28:02.420 --> 00:28:04.450
Common principles are that you want to

00:28:04.450 --> 00:28:06.670
get everything correct if you can, so

00:28:06.670 --> 00:28:08.295
it's kind of obvious like ideally you

00:28:08.295 --> 00:28:10.190
want to separate the positive from

00:28:10.190 --> 00:28:11.700
negative examples with your Linear

00:28:11.700 --> 00:28:12.210
classifier.

00:28:13.030 --> 00:28:14.860
Or you want the scores to predict the

00:28:14.860 --> 00:28:15.460
correct label?

00:28:17.150 --> 00:28:18.820
But you also want to have some high

00:28:18.820 --> 00:28:22.160
margin, so I would generally prefer

00:28:22.160 --> 00:28:25.110
this separating boundary than this one.

00:28:26.090 --> 00:28:28.465
Because this one, like everything, has

00:28:28.465 --> 00:28:30.340
like at least this distance away from

00:28:30.340 --> 00:28:32.860
the line, where with this boundary some

00:28:32.860 --> 00:28:34.415
of the points come pretty close to the

00:28:34.415 --> 00:28:34.630
line.

00:28:35.230 --> 00:28:37.420
And there's theory that shows that the

00:28:37.420 --> 00:28:40.340
bigger your margin for the same like

00:28:40.340 --> 00:28:41.320
weight size.

00:28:41.950 --> 00:28:44.590
The more likely you're classifier is to

00:28:44.590 --> 00:28:45.360
generalize.

00:28:45.360 --> 00:28:46.820
It kind of makes sense if you think of

00:28:46.820 --> 00:28:48.055
this as a random sample.

00:28:48.055 --> 00:28:50.346
If I were to Generate like more

00:28:50.346 --> 00:28:52.400
triangles from the sample, you could

00:28:52.400 --> 00:28:54.118
imagine that maybe one of the triangles

00:28:54.118 --> 00:28:55.595
would fall on the wrong side of the

00:28:55.595 --> 00:28:56.690
line and then this would make a

00:28:56.690 --> 00:28:58.800
Classification Error, while that seems

00:28:58.800 --> 00:29:00.270
less likely given this line.

00:29:05.420 --> 00:29:07.760
So that brings us to Linear Logistic

00:29:07.760 --> 00:29:08.390
Regression.

00:29:09.230 --> 00:29:12.440
And in Linear Logistic Regression, we

00:29:12.440 --> 00:29:14.390
want to maximize the probability of the

00:29:14.390 --> 00:29:15.560
labels given the data.

00:29:17.530 --> 00:29:19.747
And the probability of the label equals

00:29:19.747 --> 00:29:21.950
one given the data is given by this

00:29:21.950 --> 00:29:24.210
expression, here 1 / 1 + e to the

00:29:24.210 --> 00:29:25.710
negative my Linear model.

00:29:26.730 --> 00:29:29.620
This function 1 / 1 / 1 + E to the

00:29:29.620 --> 00:29:32.023
negative whatever is a Logistic

00:29:32.023 --> 00:29:34.056
function, that's called the Logistic

00:29:34.056 --> 00:29:34.449
function.

00:29:34.450 --> 00:29:37.132
So that's why this is Logistic Linear

00:29:37.132 --> 00:29:39.270
Logistic Regression because I've got a

00:29:39.270 --> 00:29:41.020
Linear model inside my Logistic

00:29:41.020 --> 00:29:41.500
function.

00:29:42.170 --> 00:29:44.060
So I'm regressing the Logistic function

00:29:44.060 --> 00:29:44.900
with a Linear model.

00:29:46.860 --> 00:29:48.240
This is called a logic.

00:29:48.240 --> 00:29:51.270
So this statement up here the second

00:29:51.270 --> 00:29:53.410
line implies that my Linear model.

00:29:54.200 --> 00:29:56.225
Is fitting the.

00:29:56.225 --> 00:29:59.210
It's called the odds log ratio.

00:29:59.210 --> 00:30:01.469
So it's the log or log odds ratio.

00:30:02.210 --> 00:30:04.673
It's the log of the probability of y =

00:30:04.673 --> 00:30:06.962
1 given X over the probability of y = 0

00:30:06.962 --> 00:30:07.450
given X.

00:30:08.360 --> 00:30:10.390
So if this is greater than zero, it

00:30:10.390 --> 00:30:13.373
means that probability of y = 1 given X

00:30:13.373 --> 00:30:16.216
is more likely than probability of y =

00:30:16.216 --> 00:30:18.480
0 given X, and if it's less than zero

00:30:18.480 --> 00:30:19.590
then the reverse is true.

00:30:20.780 --> 00:30:24.042
This ratio is always 2 alternatives, so

00:30:24.042 --> 00:30:24.807
it's one.

00:30:24.807 --> 00:30:26.350
It's either going to be one class or

00:30:26.350 --> 00:30:27.980
the other class, and this is the ratio

00:30:27.980 --> 00:30:29.060
of those probabilities.

00:30:34.620 --> 00:30:37.640
So if we think about Linear Logistic

00:30:37.640 --> 00:30:39.900
Regression versus Naive Bayes.

00:30:41.460 --> 00:30:43.350
They actually both have this Linear

00:30:43.350 --> 00:30:45.620
model for at least Naive Bayes does for

00:30:45.620 --> 00:30:47.420
many different probability functions.

00:30:48.070 --> 00:30:49.810
For all the probability functions and

00:30:49.810 --> 00:30:52.710
exponential family, which includes

00:30:52.710 --> 00:30:55.000
Bernoulli, multinomial, Gaussian,

00:30:55.000 --> 00:30:57.790
Laplacian, and many others, they're the

00:30:57.790 --> 00:31:00.010
favorite favorite probability family of

00:31:00.010 --> 00:31:00.970
statisticians.

00:31:02.600 --> 00:31:04.970
The Naive Bayes predictor is also

00:31:04.970 --> 00:31:07.610
Linear in X, but the difference is that

00:31:07.610 --> 00:31:09.580
in Logistic Regression you're free to

00:31:09.580 --> 00:31:11.460
independently tune these weights in

00:31:11.460 --> 00:31:14.580
order to achieve your overall label

00:31:14.580 --> 00:31:15.250
likelihood.

00:31:16.110 --> 00:31:17.835
While in Naive Bayes you're restricted

00:31:17.835 --> 00:31:19.650
to solve for each coefficient

00:31:19.650 --> 00:31:22.260
independently in order to maximize the

00:31:22.260 --> 00:31:24.580
probability of each feature given the

00:31:24.580 --> 00:31:24.940
label.

00:31:25.980 --> 00:31:27.620
So for that reason, I would say

00:31:27.620 --> 00:31:29.430
Logistic Regression model is typically

00:31:29.430 --> 00:31:31.060
more expressive than IBS.

00:31:31.870 --> 00:31:33.736
It's possible for your data to be

00:31:33.736 --> 00:31:35.610
linearly separable, but Naive Bayes

00:31:35.610 --> 00:31:37.980
does not achieve 0 training error while

00:31:37.980 --> 00:31:39.080
four Logistic Regression.

00:31:39.080 --> 00:31:40.637
You could always achieve 0 training

00:31:40.637 --> 00:31:42.335
error if your data is linearly

00:31:42.335 --> 00:31:42.830
separable.

00:31:45.160 --> 00:31:47.470
And then finally, it's important to

00:31:47.470 --> 00:31:48.930
note that Logistic Regression is

00:31:48.930 --> 00:31:50.810
directly fitting this discriminative

00:31:50.810 --> 00:31:52.500
function, so it's mapping from the

00:31:52.500 --> 00:31:54.826
features to a label and solving for

00:31:54.826 --> 00:31:55.339
that mapping.

00:31:56.050 --> 00:31:58.364
While many bees is trying to model the

00:31:58.364 --> 00:32:00.773
probability of the features given the

00:32:00.773 --> 00:32:02.405
data, so Logistic Regression doesn't

00:32:02.405 --> 00:32:02.840
model that.

00:32:02.840 --> 00:32:04.541
It just cares about the probability of

00:32:04.541 --> 00:32:06.383
the label given the data, not the

00:32:06.383 --> 00:32:07.486
probability of the data given the

00:32:07.486 --> 00:32:07.670
label.

00:32:09.020 --> 00:32:10.190
That probably features.

00:32:12.600 --> 00:32:13.050
Question.

00:32:14.990 --> 00:32:18.900
So Logistic Regression, sometimes

00:32:18.900 --> 00:32:20.520
people will say it's a discriminative

00:32:20.520 --> 00:32:22.529
function because you're trying to

00:32:22.530 --> 00:32:23.980
discriminate between the different

00:32:23.980 --> 00:32:25.330
things you're trying to Predict,

00:32:25.330 --> 00:32:28.130
meaning that you're trying to fit the

00:32:28.130 --> 00:32:29.560
probability of the thing that you're

00:32:29.560 --> 00:32:30.190
trying to Predict.

00:32:30.860 --> 00:32:33.170
Given the features or given the data.

00:32:34.120 --> 00:32:36.870
Where sometimes people say that.

00:32:36.940 --> 00:32:40.000
That, like Naive Bayes model is a

00:32:40.000 --> 00:32:42.490
generative model and they mean that

00:32:42.490 --> 00:32:45.270
you're trying to fit the probability of

00:32:45.270 --> 00:32:47.706
the data or the features given the

00:32:47.706 --> 00:32:48.100
label.

00:32:48.100 --> 00:32:49.719
So with Naive Bayes you end up with a

00:32:49.720 --> 00:32:52.008
joint distribution of all the data and

00:32:52.008 --> 00:32:52.384
features.

00:32:52.384 --> 00:32:54.500
With Logistic Regression you would just

00:32:54.500 --> 00:32:56.222
have the probability of the label given

00:32:56.222 --> 00:32:56.730
the features.

00:33:02.750 --> 00:33:03.200
So.

00:33:03.960 --> 00:33:06.140
With Linear Logistic Regression, the

00:33:06.140 --> 00:33:07.510
further you are from the lion, the

00:33:07.510 --> 00:33:08.700
higher the confidence.

00:33:08.700 --> 00:33:10.875
So if you're like way over here, then

00:33:10.875 --> 00:33:11.990
you're really confident you're a

00:33:11.990 --> 00:33:12.360
triangle.

00:33:12.360 --> 00:33:14.086
If you're just like right over here,

00:33:14.086 --> 00:33:15.076
then you're not very confident.

00:33:15.076 --> 00:33:16.595
And if you're right on the line, then

00:33:16.595 --> 00:33:18.165
you have equal confidence in triangle

00:33:18.165 --> 00:33:18.820
and circle.

00:33:21.820 --> 00:33:23.626
So the Logistic Regression algorithm

00:33:23.626 --> 00:33:25.300
there's always, as always, there's a

00:33:25.300 --> 00:33:26.710
Training and a Prediction phase.

00:33:27.790 --> 00:33:30.690
So in Training, you're trying to find

00:33:30.690 --> 00:33:31.810
the weights.

00:33:32.420 --> 00:33:35.450
That minimize this expression here

00:33:35.450 --> 00:33:36.635
which has two parts.

00:33:36.635 --> 00:33:39.750
The first part is a negative sum of log

00:33:39.750 --> 00:33:42.030
probability of Y given X and the

00:33:42.030 --> 00:33:42.400
weights.

00:33:43.370 --> 00:33:46.160
So breaking this down, South the reason

00:33:46.160 --> 00:33:47.022
for negative.

00:33:47.022 --> 00:33:49.177
So this is the negative.

00:33:49.177 --> 00:33:52.400
This is the same as.

00:33:52.470 --> 00:33:57.010
Maximizing the total probability of the

00:33:57.010 --> 00:33:58.100
labels given the data.

00:34:00.030 --> 00:34:01.670
The reason for the negative is just so

00:34:01.670 --> 00:34:03.960
I can write argument instead of argmax,

00:34:03.960 --> 00:34:05.830
because generally we tend to minimize

00:34:05.830 --> 00:34:07.320
things in machine learning, not

00:34:07.320 --> 00:34:07.960
maximize them.

00:34:08.680 --> 00:34:13.630
But the log is making it so that I turn

00:34:13.630 --> 00:34:14.220
my.

00:34:14.220 --> 00:34:15.820
Normally if I want to model a joint

00:34:15.820 --> 00:34:18.210
distribution, I have to take a product

00:34:18.210 --> 00:34:19.630
over all the different.

00:34:20.340 --> 00:34:21.760
Over all the different likelihood

00:34:21.760 --> 00:34:22.150
terms.

00:34:23.020 --> 00:34:24.570
But when I take the log of the product,

00:34:24.570 --> 00:34:25.940
it becomes the sum of the logs.

00:34:26.840 --> 00:34:29.360
And now another thing is that I'm

00:34:29.360 --> 00:34:31.940
assuming here that all of that each

00:34:31.940 --> 00:34:34.419
label only depends on its own features.

00:34:34.420 --> 00:34:36.764
So if I have 1000 data points, then

00:34:36.764 --> 00:34:38.938
each of the thousand labels only

00:34:38.938 --> 00:34:40.483
depends on the features for its own

00:34:40.483 --> 00:34:41.677
data point, it doesn't depend on all

00:34:41.677 --> 00:34:42.160
the others.

00:34:43.610 --> 00:34:45.700
And then I'm assuming that they all

00:34:45.700 --> 00:34:47.110
come from the same distribution.

00:34:47.110 --> 00:34:50.470
So I'm assuming IID independent and

00:34:50.470 --> 00:34:52.520
identically distributed data, which is

00:34:52.520 --> 00:34:55.120
always an almost always an unspoken

00:34:55.120 --> 00:34:56.390
assumption in machine learning.

00:34:58.540 --> 00:35:00.360
Alright, so the first term is saying I

00:35:00.360 --> 00:35:02.370
want to maximize the likelihood of my

00:35:02.370 --> 00:35:04.040
labels given the features over the

00:35:04.040 --> 00:35:04.610
Training set.

00:35:05.220 --> 00:35:06.460
So that's reasonable.

00:35:07.200 --> 00:35:08.880
And then the second term is a

00:35:08.880 --> 00:35:11.000
regularization term that says I prefer

00:35:11.000 --> 00:35:12.246
some models over others.

00:35:12.246 --> 00:35:14.280
I prefer models that have smaller

00:35:14.280 --> 00:35:16.280
weights, and I'll get into that a

00:35:16.280 --> 00:35:17.660
little bit more in a later slide.

00:35:20.460 --> 00:35:22.170
So that Prediction is straightforward,

00:35:22.170 --> 00:35:23.910
it's just I kind of already went

00:35:23.910 --> 00:35:24.680
through it.

00:35:24.680 --> 00:35:26.360
Once you have the weights, all you have

00:35:26.360 --> 00:35:28.160
to do is multiply your weights by your

00:35:28.160 --> 00:35:30.330
features, and that gives you the score

00:35:30.330 --> 00:35:31.180
question.

00:35:38.860 --> 00:35:40.590
Yeah, so I should explain the notation.

00:35:40.590 --> 00:35:42.090
There's different ways of denoting

00:35:42.090 --> 00:35:42.960
this, so.

00:35:44.230 --> 00:35:48.050
Usually when somebody puts a bar, they

00:35:48.050 --> 00:35:50.680
mean that it's given some features,

00:35:50.680 --> 00:35:52.440
given some data points or whatever.

00:35:53.130 --> 00:35:55.156
And then when somebody puts like a semi

00:35:55.156 --> 00:35:56.330
colon, or at least when I do it.

00:35:56.330 --> 00:35:58.450
But I see this a lot, if somebody puts

00:35:58.450 --> 00:36:00.660
like a semi colon here, then they're

00:36:00.660 --> 00:36:02.580
saying that these are the parameters.

00:36:02.580 --> 00:36:04.070
So what we're saying is that this

00:36:04.070 --> 00:36:05.380
probability function.

00:36:06.360 --> 00:36:08.830
Is like parameterized by W.

00:36:09.640 --> 00:36:13.536
And the input to that function is X and

00:36:13.536 --> 00:36:15.030
the output of the function.

00:36:15.810 --> 00:36:18.590
Is that probability of Y?

00:36:23.080 --> 00:36:24.688
The other way that you can write it

00:36:24.688 --> 00:36:26.676
that you it sometimes, and I first had

00:36:26.676 --> 00:36:28.443
it this way and then I switched it, is

00:36:28.443 --> 00:36:30.890
you might write like a subscript, so it

00:36:30.890 --> 00:36:33.635
might be P under score West.

00:36:33.635 --> 00:36:35.590
And part of the reason why you put this

00:36:35.590 --> 00:36:37.480
in here is just because otherwise it's

00:36:37.480 --> 00:36:39.776
not obvious that this term depends on

00:36:39.776 --> 00:36:40.480
West at all.

00:36:40.480 --> 00:36:43.405
And if you were like if you looked at

00:36:43.405 --> 00:36:45.170
it quickly and you were like trying to

00:36:45.170 --> 00:36:46.440
solve, you just be like, I don't care

00:36:46.440 --> 00:36:47.620
about that term, I'm just doing

00:36:47.620 --> 00:36:48.380
regularization.

00:36:49.600 --> 00:36:50.260
Question.

00:36:57.930 --> 00:37:00.370
So I forgot to say this out loud.

00:37:04.110 --> 00:37:06.070
So it is simplify the notation.

00:37:06.070 --> 00:37:08.980
I may omit the B which can be avoided

00:37:08.980 --> 00:37:10.971
by putting A1 at the end of the feature

00:37:10.971 --> 00:37:11.225
vector.

00:37:11.225 --> 00:37:12.702
So basically you can always take your

00:37:12.702 --> 00:37:14.326
feature vector and add a one to the end

00:37:14.326 --> 00:37:16.763
of all your features and then the B

00:37:16.763 --> 00:37:19.230
just becomes one of the W's and so I'm

00:37:19.230 --> 00:37:20.830
going to leave out the BA lot of times

00:37:20.830 --> 00:37:21.950
because otherwise it just kind of

00:37:21.950 --> 00:37:23.060
clutters up the equations.

00:37:27.540 --> 00:37:28.080
Thanks for.

00:37:28.970 --> 00:37:30.430
Pointing out though.

00:37:32.040 --> 00:37:34.090
Alright, so as I said before, one

00:37:34.090 --> 00:37:34.390
second.

00:37:34.390 --> 00:37:36.430
As I said before the this is the

00:37:36.430 --> 00:37:38.370
probability function that Logistic

00:37:38.370 --> 00:37:39.390
Regression assumes.

00:37:39.390 --> 00:37:41.691
If I multiply the top and the bottom by

00:37:41.691 --> 00:37:44.115
east to the West transpose X, then it's

00:37:44.115 --> 00:37:46.478
this because east to the West transpose

00:37:46.478 --> 00:37:47.630
X times that is 1.

00:37:48.540 --> 00:37:50.370
And then this generalizes.

00:37:50.370 --> 00:37:53.020
If I have multiple classes, then I

00:37:53.020 --> 00:37:54.740
would have a different weight vector

00:37:54.740 --> 00:37:55.640
for each class.

00:37:55.640 --> 00:37:57.435
So this is summing over all the classes

00:37:57.435 --> 00:37:59.545
and the final probability is given by

00:37:59.545 --> 00:38:02.120
this expression, so it's east to the

00:38:02.120 --> 00:38:02.980
Linear model.

00:38:04.170 --> 00:38:06.028
Divided by E to the sum of all the

00:38:06.028 --> 00:38:06.830
other Linear models.

00:38:06.830 --> 00:38:08.780
So it's basically your score for one

00:38:08.780 --> 00:38:10.646
model, divided by the score for all the

00:38:10.646 --> 00:38:12.513
other models, sum of score for all the

00:38:12.513 --> 00:38:12.979
other models.

00:38:14.140 --> 00:38:15.060
Was there a question?

00:38:15.060 --> 00:38:16.859
I thought somebody had a question,

00:38:16.860 --> 00:38:17.010
yeah.

00:38:25.670 --> 00:38:26.490
Yeah, good question.

00:38:26.490 --> 00:38:28.010
It's just the log of the probability.

00:38:28.820 --> 00:38:31.700
And the sum over N is just the

00:38:31.700 --> 00:38:33.690
probability term, it's not summing

00:38:33.690 --> 00:38:36.080
over, it's not the regularization times

00:38:36.080 --> 00:38:36.370
north.

00:38:39.350 --> 00:38:39.700
Question.

00:38:46.280 --> 00:38:50.170
If you're doing back prop, it depends

00:38:50.170 --> 00:38:51.770
on your activation functions, so.

00:38:52.600 --> 00:38:55.500
We will get into neural networks, but

00:38:55.500 --> 00:38:59.120
so you would if all your if at the end

00:38:59.120 --> 00:39:01.250
you have a Linear Logistic regressor.

00:39:01.880 --> 00:39:03.580
Then you would basically calculate the

00:39:03.580 --> 00:39:06.170
error due to your predictions in the

00:39:06.170 --> 00:39:08.170
last layer and then you would like

00:39:08.170 --> 00:39:10.234
accumulate those into the previous

00:39:10.234 --> 00:39:11.684
features and the previous features in

00:39:11.684 --> 00:39:12.409
the previous features.

00:39:13.980 --> 00:39:15.900
But sometimes people use like Velu or

00:39:15.900 --> 00:39:17.580
other activation functions, so then it

00:39:17.580 --> 00:39:18.100
would be different.

00:39:22.890 --> 00:39:24.900
So how do we train this thing?

00:39:24.900 --> 00:39:26.210
How do we optimize West?

00:39:27.330 --> 00:39:28.880
First, I want to explain the

00:39:28.880 --> 00:39:29.790
regularization term.

00:39:30.510 --> 00:39:31.710
There's two main kinds of

00:39:31.710 --> 00:39:32.610
regularization.

00:39:32.610 --> 00:39:35.740
There's L2 2 regularization and L1

00:39:35.740 --> 00:39:36.420
regularization.

00:39:37.080 --> 00:39:39.280
So L2 2 regularization is that you're

00:39:39.280 --> 00:39:41.756
minimizing the sum of the square values

00:39:41.756 --> 00:39:42.680
of the weights.

00:39:43.330 --> 00:39:45.908
I can write that as an L2 norm squared.

00:39:45.908 --> 00:39:48.985
That double bar thing is means like

00:39:48.985 --> 00:39:52.635
norm and the two under it means it's an

00:39:52.635 --> 00:39:55.132
L2 and the two above it means it's

00:39:55.132 --> 00:39:55.340
squared.

00:39:56.380 --> 00:39:58.500
Or I can write or I can do A1

00:39:58.500 --> 00:40:00.210
regularization, which is a sum of the

00:40:00.210 --> 00:40:01.660
absolute values of the weights.

00:40:02.920 --> 00:40:03.570
And.

00:40:05.220 --> 00:40:07.540
And I can write that as the norm like

00:40:07.540 --> 00:40:08.210
subscript 1.

00:40:09.350 --> 00:40:11.700
And then those are weighted by some

00:40:11.700 --> 00:40:13.670
Lambda which is a parameter that has to

00:40:13.670 --> 00:40:15.910
be set by the algorithm designer.

00:40:17.180 --> 00:40:20.100
Or based on some data like validation

00:40:20.100 --> 00:40:20.710
optimization.

00:40:21.820 --> 00:40:23.910
So these may look really similar

00:40:23.910 --> 00:40:25.650
squared absolute value.

00:40:25.650 --> 00:40:28.140
What's the difference as W goes higher?

00:40:28.140 --> 00:40:30.580
It means that you get a bigger penalty

00:40:30.580 --> 00:40:31.180
in either case.

00:40:31.890 --> 00:40:33.420
But they behave actually like quite

00:40:33.420 --> 00:40:33.960
differently.

00:40:34.830 --> 00:40:37.710
So if you look at this plot of L2

00:40:37.710 --> 00:40:39.990
versus L1, when the weight is 0,

00:40:39.990 --> 00:40:40.822
there's no penalty.

00:40:40.822 --> 00:40:43.090
When the weight is 1, the penalties are

00:40:43.090 --> 00:40:43.700
equal.

00:40:43.700 --> 00:40:45.760
When the weight is less than one, then

00:40:45.760 --> 00:40:48.207
the L2 penalty is smaller than the L1

00:40:48.207 --> 00:40:48.490
penalty.

00:40:48.490 --> 00:40:50.080
It has this like little basin where

00:40:50.080 --> 00:40:51.820
basically the penalty is almost 0.

00:40:52.760 --> 00:40:54.880
And but when the weight gets far from

00:40:54.880 --> 00:40:56.960
one, the L2 penalty shoots up.

00:40:57.870 --> 00:41:00.820
So L2 2 regularization hates really

00:41:00.820 --> 00:41:03.060
large weights, and they're perfectly

00:41:03.060 --> 00:41:05.030
fine with like lots of tiny little

00:41:05.030 --> 00:41:05.360
weights.

00:41:06.560 --> 00:41:08.490
L1 regularization doesn't like any

00:41:08.490 --> 00:41:10.600
weights, but it kind of doesn't like

00:41:10.600 --> 00:41:11.760
the mall roughly equally.

00:41:11.760 --> 00:41:14.170
So it doesn't like weights of three,

00:41:14.170 --> 00:41:16.699
but it's not as bad as it doesn't

00:41:16.700 --> 00:41:18.250
dislike them as much as L2 2.

00:41:19.130 --> 00:41:21.410
It also doesn't even a weight of 1.

00:41:21.410 --> 00:41:23.150
It's going to try just as hard to push

00:41:23.150 --> 00:41:24.722
that down as it does to push a weight

00:41:24.722 --> 00:41:25.200
of three.

00:41:27.020 --> 00:41:28.990
So when you think about when you when

00:41:28.990 --> 00:41:30.870
you think about optimization, you

00:41:30.870 --> 00:41:32.099
always want to think about the

00:41:32.100 --> 00:41:35.010
derivative as well as the.

00:41:35.390 --> 00:41:37.510
Like pure function, because you're

00:41:37.510 --> 00:41:38.830
always Minimizing, you're always

00:41:38.830 --> 00:41:40.310
setting a derivative equal to 0, and

00:41:40.310 --> 00:41:42.100
the derivative is what is like guiding

00:41:42.100 --> 00:41:45.400
your function optimization towards some

00:41:45.400 --> 00:41:46.270
optimal value.

00:41:47.590 --> 00:41:49.040
So if you're doing.

00:41:49.150 --> 00:41:49.800


00:41:51.230 --> 00:41:52.550
If you're doing L2.

00:41:54.530 --> 00:41:56.360
L2 2 minimization.

00:41:57.120 --> 00:41:59.965
And I plot the derivative, then the

00:41:59.965 --> 00:42:01.890
derivative is just going to be Linear,

00:42:01.890 --> 00:42:02.780
right?

00:42:02.780 --> 00:42:03.950
It's going to be.

00:42:04.820 --> 00:42:06.510
2/2 times.

00:42:06.590 --> 00:42:07.140


00:42:07.990 --> 00:42:10.420
It's going to be Lambda 2 WI and

00:42:10.420 --> 00:42:12.110
sometimes people put a 1/2 in front of

00:42:12.110 --> 00:42:13.800
Lambda just so that the two and the 1/2

00:42:13.800 --> 00:42:14.850
cancel out Mainly.

00:42:16.560 --> 00:42:17.850
Don't feel like it's necessary.

00:42:17.850 --> 00:42:21.350
If you do L2 one, then the derivatives

00:42:21.350 --> 00:42:26.830
are -, 1 if it's greater than zero, and

00:42:26.830 --> 00:42:29.310
positive one if it's less than 0.

00:42:30.270 --> 00:42:33.200
So basically, if it's L1 minimization,

00:42:33.200 --> 00:42:35.570
the regularization is like he's forcing

00:42:35.570 --> 00:42:38.080
things in towards zero with equal

00:42:38.080 --> 00:42:39.600
pressure no matter where it is.

00:42:40.240 --> 00:42:42.815
Wherewith L2 2 minimization, if you

00:42:42.815 --> 00:42:44.503
have a high value then it's like

00:42:44.503 --> 00:42:46.830
forcing it down, like really hard, and

00:42:46.830 --> 00:42:48.839
if you have a low low value then it's

00:42:48.840 --> 00:42:50.190
not forcing it very hard at all.

00:42:50.900 --> 00:42:52.500
And that's regularization is always

00:42:52.500 --> 00:42:53.960
struggling against the other term.

00:42:53.960 --> 00:42:55.640
These are like counterbalancing terms.

00:42:56.510 --> 00:42:58.000
So the regularization is trying to say

00:42:58.000 --> 00:42:58.790
your weights are small.

00:42:59.580 --> 00:43:02.400
But the log log likelihood term is

00:43:02.400 --> 00:43:04.750
trying to do whatever it can to solve

00:43:04.750 --> 00:43:07.710
that likelihood Prediction and so

00:43:07.710 --> 00:43:10.410
sometimes there sometimes there are

00:43:10.410 --> 00:43:11.080
odds with each other.

00:43:12.530 --> 00:43:14.700
Alright, so based on that, can anyone

00:43:14.700 --> 00:43:18.540
explain why it is that L2 1 tends to

00:43:18.540 --> 00:43:20.140
lead to sparse weights, meaning that

00:43:20.140 --> 00:43:21.890
you get a lot of 0 values for your

00:43:21.890 --> 00:43:22.250
weights?

00:43:25.980 --> 00:43:26.140
Yeah.

00:43:47.140 --> 00:43:48.630
Yeah, that's right.

00:43:48.630 --> 00:43:49.556
So L2.

00:43:49.556 --> 00:43:52.030
So the answer was that L2 1 prefers

00:43:52.030 --> 00:43:53.984
like a small number of features that

00:43:53.984 --> 00:43:56.300
have a lot of weight that have a lot of

00:43:56.300 --> 00:43:57.970
representational value or predictive

00:43:57.970 --> 00:43:58.370
value.

00:43:59.140 --> 00:44:01.370
Where I'll two really wants everything

00:44:01.370 --> 00:44:02.700
to have a little bit of predictive

00:44:02.700 --> 00:44:03.140
value.

00:44:03.770 --> 00:44:05.970
And you can see that by looking at the

00:44:05.970 --> 00:44:07.740
derivatives or just by thinking about

00:44:07.740 --> 00:44:08.500
this function.

00:44:09.140 --> 00:44:12.380
That L2 one just continually forces

00:44:12.380 --> 00:44:14.335
everything down until it hits exactly

00:44:14.335 --> 00:44:16.970
0, and while there's not necessarily a

00:44:16.970 --> 00:44:19.380
big penalty for some weight, so if you

00:44:19.380 --> 00:44:20.730
have a few features that are really

00:44:20.730 --> 00:44:22.558
predictive, it's going to allow those

00:44:22.558 --> 00:44:24.040
features to have a lot of weights,

00:44:24.040 --> 00:44:26.314
while if the other features are not

00:44:26.314 --> 00:44:27.579
predictive, given those few features,

00:44:27.579 --> 00:44:29.450
it's going to force them down to 0.

00:44:30.760 --> 00:44:33.132
With L2 2, if you have a lot of, if you

00:44:33.132 --> 00:44:34.440
have some features that are really

00:44:34.440 --> 00:44:35.870
predictive and others that are less

00:44:35.870 --> 00:44:38.040
predictive, it's still going to want

00:44:38.040 --> 00:44:40.260
those very predictive features to have

00:44:40.260 --> 00:44:41.790
like a bit smaller weight.

00:44:42.440 --> 00:44:44.520
And it's going to like try to make that

00:44:44.520 --> 00:44:46.530
up by having the other features will

00:44:46.530 --> 00:44:47.810
have just like a little bit of weight

00:44:47.810 --> 00:44:48.430
as well.

00:44:54.130 --> 00:44:56.360
So in consequence, we can use L2 1

00:44:56.360 --> 00:44:58.340
regularization to select the best

00:44:58.340 --> 00:45:01.260
features if we have if we have a bunch

00:45:01.260 --> 00:45:01.880
of features.

00:45:02.750 --> 00:45:04.610
And we want to instead have a model

00:45:04.610 --> 00:45:05.890
that's based on a smaller number of

00:45:05.890 --> 00:45:07.080
features.

00:45:07.080 --> 00:45:09.950
You can do solve for L1 Logistic

00:45:09.950 --> 00:45:11.790
Regression or L1 Linear Regression.

00:45:12.400 --> 00:45:14.160
And then choose the features that are

00:45:14.160 --> 00:45:17.000
non zero or greater than some epsilon

00:45:17.000 --> 00:45:20.470
and then just use those for your model.

00:45:22.810 --> 00:45:24.840
OK, I will answer this question for you

00:45:24.840 --> 00:45:26.430
to save a little bit of time.

00:45:27.540 --> 00:45:29.500
When is regularization absolutely

00:45:29.500 --> 00:45:30.110
essential?

00:45:30.110 --> 00:45:31.450
It's if your data is linearly

00:45:31.450 --> 00:45:31.970
separable.

00:45:33.390 --> 00:45:35.190
Because if your data is linearly

00:45:35.190 --> 00:45:37.445
separable then you just boost.

00:45:37.445 --> 00:45:38.820
You could boost your weights to

00:45:38.820 --> 00:45:41.083
Infinity and keep on separating it more

00:45:41.083 --> 00:45:41.789
and more and more.

00:45:42.530 --> 00:45:45.360
So if you have like 2.

00:45:46.270 --> 00:45:49.600
If you have two feature points here and

00:45:49.600 --> 00:45:50.020
here.

00:45:50.970 --> 00:45:54.160
Then you create this line.

00:45:55.260 --> 00:45:56.030
WX.

00:45:56.690 --> 00:45:59.088
If it's just one-dimensional and like

00:45:59.088 --> 00:46:02.220
if W is equal to 1, then maybe I have a

00:46:02.220 --> 00:46:04.900
score of 1 or -, 1 for each of these.

00:46:04.900 --> 00:46:08.215
But if test equals like 10,000, now my

00:46:08.215 --> 00:46:09.985
score is 10,000 and -, 10,000.

00:46:09.985 --> 00:46:11.355
So that's like even better, they're

00:46:11.355 --> 00:46:13.494
even further from zero and so there's

00:46:13.494 --> 00:46:15.130
no like there's no end to it.

00:46:15.130 --> 00:46:17.090
You're W would just go totally out of

00:46:17.090 --> 00:46:19.420
control and you would get an error

00:46:19.420 --> 00:46:21.500
probably that you're like that your

00:46:21.500 --> 00:46:22.830
optimization didn't converge.

00:46:23.730 --> 00:46:26.020
So you pretty much always want some

00:46:26.020 --> 00:46:27.610
kind of regularization weight, even if

00:46:27.610 --> 00:46:31.940
it's really small, to avoid this case

00:46:31.940 --> 00:46:34.760
where you don't have a unique solution

00:46:34.760 --> 00:46:35.990
to the optimization problem.

00:46:39.580 --> 00:46:41.240
There's a lot of different ways to

00:46:41.240 --> 00:46:43.890
optimize this and it's not that simple.

00:46:43.890 --> 00:46:47.440
So you can do various like gradient

00:46:47.440 --> 00:46:50.650
descents or things based on 2nd order

00:46:50.650 --> 00:46:54.868
terms, or lasso Regression for L1 or

00:46:54.868 --> 00:46:57.110
lasso lasso optimization.

00:46:57.110 --> 00:46:59.319
So there's a lot of different

00:46:59.320 --> 00:46:59.850
optimizers.

00:46:59.850 --> 00:47:01.540
I linked to this paper by Tom Minka

00:47:01.540 --> 00:47:03.490
that like explains like several

00:47:03.490 --> 00:47:05.290
different choices and their tradeoffs.

00:47:06.390 --> 00:47:07.760
At the end of the day, you're going to

00:47:07.760 --> 00:47:10.399
use a library, and so it's not really

00:47:10.400 --> 00:47:12.177
worth quoting this because it's a

00:47:12.177 --> 00:47:13.703
really explored problem and you're not

00:47:13.703 --> 00:47:15.040
going to make something better than

00:47:15.040 --> 00:47:15.840
somebody else did.

00:47:17.110 --> 00:47:19.000
So you want to use the library.

00:47:19.000 --> 00:47:20.810
It's worth like it's worth

00:47:20.810 --> 00:47:21.830
understanding the different

00:47:21.830 --> 00:47:25.540
optimization options a little bit, but

00:47:25.540 --> 00:47:26.800
I'm not going to talk about it.

00:47:30.030 --> 00:47:30.390
All right.

00:47:31.040 --> 00:47:31.550
So.

00:47:33.150 --> 00:47:35.760
Here I did an example where I visualize

00:47:35.760 --> 00:47:38.006
the weights that are learned using L2

00:47:38.006 --> 00:47:39.850
regularization and L1 regularization

00:47:39.850 --> 00:47:41.050
for some digits.

00:47:41.050 --> 00:47:42.820
So these are the average Pixels of

00:47:42.820 --> 00:47:43.940
digits zero to 4.

00:47:44.810 --> 00:47:47.308
These are the L2 2 weights and you can

00:47:47.308 --> 00:47:49.340
see like you can sort of see the

00:47:49.340 --> 00:47:51.125
numbers in it a little bit like you can

00:47:51.125 --> 00:47:52.820
sort of see the three in these weights

00:47:52.820 --> 00:47:53.020
that.

00:47:53.730 --> 00:47:56.437
And the zero, it wants these weights to

00:47:56.437 --> 00:47:58.428
be white, and it wants these weights to

00:47:58.428 --> 00:47:59.030
be dark.

00:47:59.690 --> 00:48:01.320
I mean these features to be dark,

00:48:01.320 --> 00:48:03.262
meaning that if you have a lit pixel

00:48:03.262 --> 00:48:05.099
here, it's less likely to be a 0.

00:48:05.099 --> 00:48:07.100
If you have a lit pixel here, it's more

00:48:07.100 --> 00:48:08.390
likely to be a 0.

00:48:10.300 --> 00:48:13.390
But for the L2 one, it's a lot sparser,

00:48:13.390 --> 00:48:15.590
so if it's like that blank Gray color,

00:48:15.590 --> 00:48:17.060
it means that the weights are zero.

00:48:18.220 --> 00:48:19.402
And if it's brighter or darker?

00:48:19.402 --> 00:48:20.670
If it's brighter, it means that the

00:48:20.670 --> 00:48:21.550
weight is positive.

00:48:22.260 --> 00:48:26.480
If it's darker than this uniform Gray,

00:48:26.480 --> 00:48:27.960
it means the weight is negative.

00:48:27.960 --> 00:48:30.430
So you can see that for L2 one, it's

00:48:30.430 --> 00:48:32.952
going to have like some subset of the

00:48:32.952 --> 00:48:35.123
L2 features are going to get all the

00:48:35.123 --> 00:48:36.900
weight, and most of the weights are

00:48:36.900 --> 00:48:38.069
very close to 0.

00:48:40.120 --> 00:48:42.000
So for one, it's only going to look at

00:48:42.000 --> 00:48:44.026
this small number of pixel, small

00:48:44.026 --> 00:48:45.990
number of pixels, and if any of these

00:48:45.990 --> 00:48:46.640
guys are.

00:48:47.400 --> 00:48:49.010
Are.

00:48:49.070 --> 00:48:51.130
Let then it's going to get a big

00:48:51.130 --> 00:48:52.500
penalty to being a 0.

00:48:53.150 --> 00:48:55.560
If any of these guys are, it gets a big

00:48:55.560 --> 00:48:56.939
boost to being a 0.

00:48:59.420 --> 00:48:59.780
Question.

00:49:36.370 --> 00:49:38.230
OK, let me explain a little bit more

00:49:38.230 --> 00:49:38.730
how I get this.

00:49:39.410 --> 00:49:42.470
1st So first this is up here is just

00:49:42.470 --> 00:49:45.510
simply averaging all the images in a

00:49:45.510 --> 00:49:46.370
particular class.

00:49:47.210 --> 00:49:49.550
And then I train 2 Logistic Regression

00:49:49.550 --> 00:49:50.240
models.

00:49:50.240 --> 00:49:52.780
One is trained using the same data that

00:49:52.780 --> 00:49:55.096
was used to Average, but to maximize

00:49:55.096 --> 00:49:57.480
the train, to maximize the probability

00:49:57.480 --> 00:49:59.670
of the labels given the data but under

00:49:59.670 --> 00:50:02.290
the L2 regularization penalty.

00:50:03.040 --> 00:50:05.090
And the other was trained to maximize

00:50:05.090 --> 00:50:06.320
the probability of the label is given

00:50:06.320 --> 00:50:08.450
the data under the L1 regularization

00:50:08.450 --> 00:50:08.920
penalty.

00:50:10.410 --> 00:50:12.355
The way that once you have these

00:50:12.355 --> 00:50:12.630
weights.

00:50:12.630 --> 00:50:14.512
So these weights are the W's.

00:50:14.512 --> 00:50:16.750
These are the coefficients that were

00:50:16.750 --> 00:50:19.220
learned as part of as your Linear

00:50:19.220 --> 00:50:19.560
model.

00:50:20.460 --> 00:50:22.310
In order to apply these weights to do

00:50:22.310 --> 00:50:23.320
Classification.

00:50:24.010 --> 00:50:26.000
You would multiply each of these

00:50:26.000 --> 00:50:27.760
weights with the corresponding pixel.

00:50:28.490 --> 00:50:31.280
So given a new test sample, you would

00:50:31.280 --> 00:50:34.510
take the sum over all the pixels of the

00:50:34.510 --> 00:50:36.900
pixel value times this weight.

00:50:37.720 --> 00:50:40.257
So if the way here is bright, it means

00:50:40.257 --> 00:50:41.755
that if the pixel value is bright, then

00:50:41.755 --> 00:50:43.170
the score is going to go up.

00:50:43.170 --> 00:50:45.805
And if the weight here is dark, that

00:50:45.805 --> 00:50:46.910
means it's negative.

00:50:46.910 --> 00:50:50.190
Then when you if the pixel value is on,

00:50:50.190 --> 00:50:52.169
then this is going, then the score is

00:50:52.169 --> 00:50:53.130
going to go down.

00:50:53.130 --> 00:50:55.330
So that's how to interpret.

00:50:56.370 --> 00:50:57.930
How to interpret the weights and?

00:50:57.930 --> 00:50:59.570
Normally it's just a vector, but I've

00:50:59.570 --> 00:51:01.340
reshaped it into the size of the image

00:51:01.340 --> 00:51:03.290
so you could see how it corresponds to

00:51:03.290 --> 00:51:04.160
the Pixels.

00:51:07.190 --> 00:51:08.740
Where Minimizing 2 things.

00:51:08.740 --> 00:51:10.540
One is that we're minimizing the

00:51:10.540 --> 00:51:11.900
negative log likelihood of the labels

00:51:11.900 --> 00:51:12.700
given the data.

00:51:12.700 --> 00:51:16.170
So in other words, we're maximizing the

00:51:16.170 --> 00:51:17.020
label likelihood.

00:51:17.930 --> 00:51:19.740
And the other is that we're minimizing

00:51:19.740 --> 00:51:21.237
the sum of the weights or the sum of

00:51:21.237 --> 00:51:21.920
the squared weights.

00:51:43.810 --> 00:51:44.290
Right.

00:51:44.290 --> 00:51:44.580
Yeah.

00:51:44.580 --> 00:51:45.385
So I Prediction time.

00:51:45.385 --> 00:51:47.530
So at Training time you have that

00:51:47.530 --> 00:51:48.388
regularization term.

00:51:48.388 --> 00:51:49.700
At Prediction time you don't.

00:51:49.700 --> 00:51:52.630
So at Prediction time, it's just the

00:51:52.630 --> 00:51:55.510
score for zero is the sum of all these

00:51:55.510 --> 00:51:57.340
coefficients times the corresponding

00:51:57.340 --> 00:51:58.100
pixel values.

00:51:58.760 --> 00:52:00.940
And the score for one is the sum of all

00:52:00.940 --> 00:52:02.960
these coefficient values times the

00:52:02.960 --> 00:52:04.947
corresponding pixel values, and so on

00:52:04.947 --> 00:52:05.830
for all the digits.

00:52:06.570 --> 00:52:08.210
And then at the end you choose.

00:52:08.210 --> 00:52:09.752
If you're just assigning a label, you

00:52:09.752 --> 00:52:11.240
choose the label with the highest

00:52:11.240 --> 00:52:11.510
score.

00:52:12.230 --> 00:52:12.410
Yeah.

00:52:13.580 --> 00:52:14.400
That did that help?

00:52:15.100 --> 00:52:15.360
OK.

00:52:17.880 --> 00:52:18.570
Alright.

00:52:24.020 --> 00:52:25.080
So.

00:52:26.630 --> 00:52:28.980
Alright, so then there's a question of

00:52:28.980 --> 00:52:29.990
how do we choose the Lambda?

00:52:31.260 --> 00:52:34.685
So selecting Lambda is often called a

00:52:34.685 --> 00:52:35.098
hyperparameter.

00:52:35.098 --> 00:52:37.574
A hyperparameter is it's a parameter

00:52:37.574 --> 00:52:40.366
that the algorithm designer sets that

00:52:40.366 --> 00:52:42.520
is not optimized directly by the

00:52:42.520 --> 00:52:43.120
Training data.

00:52:43.120 --> 00:52:45.530
So the weights are like Parameters of

00:52:45.530 --> 00:52:46.780
the Linear model.

00:52:46.780 --> 00:52:48.660
But the Lambda is a hyperparameter

00:52:48.660 --> 00:52:50.030
because it's a parameter of your

00:52:50.030 --> 00:52:51.714
objective function, not a parameter of

00:52:51.714 --> 00:52:52.219
your model.

00:52:56.490 --> 00:52:59.610
So when you're selecting values for

00:52:59.610 --> 00:53:02.660
your hyperparameters, the you can do it

00:53:02.660 --> 00:53:05.260
based on intuition, but more commonly

00:53:05.260 --> 00:53:07.780
you would do some kind of validation.

00:53:08.970 --> 00:53:11.210
So for example, you might say that

00:53:11.210 --> 00:53:14.000
Lambda is in this range, one of these

00:53:14.000 --> 00:53:16.125
values, 1/8, one quarter, one half one.

00:53:16.125 --> 00:53:18.350
It's usually not super sensitive, so

00:53:18.350 --> 00:53:21.440
there's no point going into like really

00:53:21.440 --> 00:53:22.840
tiny differences.

00:53:22.840 --> 00:53:24.919
And it also tends to be like

00:53:24.920 --> 00:53:27.010
exponential in its range.

00:53:27.010 --> 00:53:28.910
So for example, you don't want to

00:53:28.910 --> 00:53:32.650
search from 1/8 to 8 in steps of 1/8

00:53:32.650 --> 00:53:34.016
because that will be like a ton of

00:53:34.016 --> 00:53:36.080
values to check and like a difference

00:53:36.080 --> 00:53:39.090
between 7:00 and 7/8 and eight is like

00:53:39.090 --> 00:53:39.610
nothing.

00:53:39.680 --> 00:53:40.790
It won't make any difference.

00:53:41.830 --> 00:53:43.450
So usually you want to keep doubling it

00:53:43.450 --> 00:53:45.770
or multiplying it by a factor of 10 for

00:53:45.770 --> 00:53:46.400
every step.

00:53:47.690 --> 00:53:49.540
You train the model using a given

00:53:49.540 --> 00:53:51.489
Lambda from the training set, and you

00:53:51.490 --> 00:53:52.857
measure and record the performance from

00:53:52.857 --> 00:53:55.320
the validation set, and then you choose

00:53:55.320 --> 00:53:57.053
the Lambda and the model that gave you

00:53:57.053 --> 00:53:58.090
the best performance.

00:53:58.090 --> 00:53:59.540
So it's pretty straightforward.

00:54:00.500 --> 00:54:03.290
And you can optionally then retrain on

00:54:03.290 --> 00:54:05.330
the training and the validation set so

00:54:05.330 --> 00:54:07.150
that you didn't like only use your

00:54:07.150 --> 00:54:09.510
validation parameters for selecting

00:54:09.510 --> 00:54:11.992
that Lambda, and then test on the test

00:54:11.992 --> 00:54:12.299
set.

00:54:12.300 --> 00:54:13.653
But I'll note that you don't have to do

00:54:13.653 --> 00:54:14.866
that for the homework, you should, and

00:54:14.866 --> 00:54:16.350
the homework you should generally just.

00:54:17.480 --> 00:54:20.280
Use your validation for like measuring

00:54:20.280 --> 00:54:22.660
performance and selection and then just

00:54:22.660 --> 00:54:24.070
leave your Training.

00:54:24.070 --> 00:54:25.700
Leave the models trained on your

00:54:25.700 --> 00:54:25.960
Training set.

00:54:28.300 --> 00:54:30.010
And then once you've got your final

00:54:30.010 --> 00:54:32.170
model, you just test it on the test set

00:54:32.170 --> 00:54:33.680
and then that's the measure of the

00:54:33.680 --> 00:54:34.539
performance of your model.

00:54:36.890 --> 00:54:38.525
So you can start.

00:54:38.525 --> 00:54:41.020
So as I said, you typically will keep

00:54:41.020 --> 00:54:42.080
on like multiplying your

00:54:42.080 --> 00:54:44.190
hyperparameters by some factor rather

00:54:44.190 --> 00:54:45.380
than doing a Linear search.

00:54:46.390 --> 00:54:48.510
You can also start broad and narrow.

00:54:48.510 --> 00:54:51.405
So for example, if I found that 1/4 and

00:54:51.405 --> 00:54:54.320
1/2 were the best two values, but it

00:54:54.320 --> 00:54:55.570
seemed like there was actually like a

00:54:55.570 --> 00:54:56.960
pretty big difference between

00:54:56.960 --> 00:54:58.560
neighboring values, then I could then

00:54:58.560 --> 00:55:01.640
try like 3/8 and keep on subdividing it

00:55:01.640 --> 00:55:04.270
until I feel like I've gotten squeezed

00:55:04.270 --> 00:55:05.790
what I can out of that hyperparameter.

00:55:07.080 --> 00:55:09.750
Also, if you're searching over many

00:55:09.750 --> 00:55:13.450
Parameters simultaneously, the natural

00:55:13.450 --> 00:55:14.679
thing that you would do is you would do

00:55:14.680 --> 00:55:16.420
a grid search where you do for each

00:55:16.420 --> 00:55:19.380
Lambda and for each alpha, and for each

00:55:19.380 --> 00:55:21.510
beta you search over some range and try

00:55:21.510 --> 00:55:23.520
all combinations of things.

00:55:23.520 --> 00:55:25.145
That's actually really inefficient.

00:55:25.145 --> 00:55:28.377
The best thing to do is to randomly

00:55:28.377 --> 00:55:30.720
select your alpha, beta, gamma, or

00:55:30.720 --> 00:55:32.790
whatever things you're searching over,

00:55:32.790 --> 00:55:34.440
randomly select them within the

00:55:34.440 --> 00:55:35.410
candidate range.

00:55:36.790 --> 00:55:42.020
By probabilistic sampling and then try

00:55:42.020 --> 00:55:44.286
like 100 different variations and then

00:55:44.286 --> 00:55:46.173
and then choose the best combination.

00:55:46.173 --> 00:55:48.880
And the reason for that is that often

00:55:48.880 --> 00:55:50.530
the Parameters don't depend that

00:55:50.530 --> 00:55:51.550
strongly on each other.

00:55:52.140 --> 00:55:54.450
And that way in some Parameters will be

00:55:54.450 --> 00:55:55.920
much more important than others.

00:55:56.730 --> 00:55:58.620
And so if you randomly sample in the

00:55:58.620 --> 00:56:00.440
range, if you have multiple Parameters,

00:56:00.440 --> 00:56:02.270
then you get to try a lot more

00:56:02.270 --> 00:56:04.315
different values of each parameter than

00:56:04.315 --> 00:56:05.540
if you're doing a grid search.

00:56:09.500 --> 00:56:11.270
So validation.

00:56:11.390 --> 00:56:11.980


00:56:13.230 --> 00:56:14.870
You can also do cross validation.

00:56:14.870 --> 00:56:16.520
That's just if you split your Training,

00:56:16.520 --> 00:56:19.173
split your data set into multiple parts

00:56:19.173 --> 00:56:22.330
and each time you train on North minus

00:56:22.330 --> 00:56:24.642
one parts and then test on the north

00:56:24.642 --> 00:56:27.420
part and then you cycle through which

00:56:27.420 --> 00:56:28.840
part you use for validation.

00:56:29.650 --> 00:56:30.860
And then you Average all your

00:56:30.860 --> 00:56:31.775
validation performance.

00:56:31.775 --> 00:56:33.960
So you might do this if you have a very

00:56:33.960 --> 00:56:36.280
limited Training set, so that it's

00:56:36.280 --> 00:56:38.270
really hard to get both Training

00:56:38.270 --> 00:56:39.740
Parameters and get a measure of the

00:56:39.740 --> 00:56:41.770
performance with that one Training set,

00:56:41.770 --> 00:56:43.620
and so you can.

00:56:44.820 --> 00:56:47.600
You can then make more efficient use of

00:56:47.600 --> 00:56:48.840
your Training data this way.

00:56:48.840 --> 00:56:49.870
Sample efficient use.

00:56:50.650 --> 00:56:52.110
And the extreme you can do leave one

00:56:52.110 --> 00:56:53.780
out cross validation where you train

00:56:53.780 --> 00:56:55.777
with all your data except for one and

00:56:55.777 --> 00:56:58.050
then test on that one and then you

00:56:58.050 --> 00:57:00.965
cycle which point is used for

00:57:00.965 --> 00:57:03.749
validation through all the data

00:57:03.750 --> 00:57:04.300
samples.

00:57:06.440 --> 00:57:09.770
This is only practical if you if you're

00:57:09.770 --> 00:57:11.229
doing like Nearest neighbor for example

00:57:11.230 --> 00:57:12.890
where Training takes no time, then

00:57:12.890 --> 00:57:14.259
that's easy to do.

00:57:14.260 --> 00:57:16.859
Or if you're able to adjust your model

00:57:16.860 --> 00:57:19.657
by adjust it for the influence of 1

00:57:19.657 --> 00:57:19.885
sample.

00:57:19.885 --> 00:57:21.550
If you can like take out one sample

00:57:21.550 --> 00:57:23.518
really easily and adjust your model

00:57:23.518 --> 00:57:24.740
then you might be able to do this,

00:57:24.740 --> 00:57:26.455
which you could do with Naive Bayes for

00:57:26.455 --> 00:57:27.060
example as well.

00:57:32.060 --> 00:57:33.460
Right, so Summary of Logistic

00:57:33.460 --> 00:57:35.180
Regression.

00:57:35.180 --> 00:57:37.790
Key assumptions are that this log odds

00:57:37.790 --> 00:57:40.460
ratio can be expressed as a linear

00:57:40.460 --> 00:57:41.560
combination of features.

00:57:42.470 --> 00:57:44.589
So this probability of y = K given X

00:57:44.590 --> 00:57:46.710
over probability of Y not equal to K

00:57:46.710 --> 00:57:47.730
given X the log of that.

00:57:48.470 --> 00:57:51.770
Is just a Linear model W transpose X.

00:57:53.350 --> 00:57:55.990
I've got one coefficient per feature

00:57:55.990 --> 00:57:57.700
that's my model Parameters, plus maybe

00:57:57.700 --> 00:57:59.950
a bias term which the bias is modeling

00:57:59.950 --> 00:58:00.850
like the class prior.

00:58:02.320 --> 00:58:04.690
I can Choose L1 or L2 or both.

00:58:06.110 --> 00:58:08.110
Regularization in some weight on those.

00:58:09.810 --> 00:58:11.070
So this is really.

00:58:11.070 --> 00:58:13.090
This works well if you've got a lot of

00:58:13.090 --> 00:58:14.470
features, because again, it's much more

00:58:14.470 --> 00:58:16.100
powerful in a high dimensional space.

00:58:16.840 --> 00:58:18.740
And it's OK if some of those features

00:58:18.740 --> 00:58:20.520
are irrelevant or redundant, where

00:58:20.520 --> 00:58:22.110
things like Naive Bayes will get

00:58:22.110 --> 00:58:24.010
tripped up by irrelevant or redundant

00:58:24.010 --> 00:58:24.360
features.

00:58:25.480 --> 00:58:28.210
And it provides a good estimate of the

00:58:28.210 --> 00:58:29.380
label likelihood.

00:58:29.380 --> 00:58:32.290
So it tends to give you a well

00:58:32.290 --> 00:58:34.233
calibrated classifier, which means that

00:58:34.233 --> 00:58:36.425
if you look at its confidence, if the

00:58:36.425 --> 00:58:39.520
confidence is 8, then like 80% of the

00:58:39.520 --> 00:58:41.279
times that the confidence is .8, it

00:58:41.280 --> 00:58:41.960
will be correct.

00:58:42.710 --> 00:58:43.300
Roughly.

00:58:44.800 --> 00:58:46.150
Not to use and Weaknesses.

00:58:46.150 --> 00:58:47.689
If the features are low dimensional,

00:58:47.690 --> 00:58:49.410
then the Linear function is not likely

00:58:49.410 --> 00:58:50.600
to be expressive enough.

00:58:50.600 --> 00:58:52.824
So usually if your features are low

00:58:52.824 --> 00:58:54.395
dimensional to start with, you actually

00:58:54.395 --> 00:58:56.055
like turn them into high dimensional

00:58:56.055 --> 00:58:59.480
features first, like by doing trees or

00:58:59.480 --> 00:59:01.820
other ways of like turning continuous

00:59:01.820 --> 00:59:03.690
values into a lot of discrete values.

00:59:04.310 --> 00:59:05.900
And then you apply your Linear

00:59:05.900 --> 00:59:06.450
classifier.

00:59:10.310 --> 00:59:11.890
Right, so I was going to do like a

00:59:11.890 --> 00:59:13.600
Pause thing here, but since we only

00:59:13.600 --> 00:59:16.490
have 15 minutes left, I will use this

00:59:16.490 --> 00:59:18.470
as a Review question for the start of

00:59:18.470 --> 00:59:20.850
the next lecture.

00:59:20.850 --> 00:59:22.830
And I want to I do want to get into

00:59:22.830 --> 00:59:25.820
Linear Regression so apologies for.

00:59:26.860 --> 00:59:28.010
Fairly heavy.

00:59:29.390 --> 00:59:30.620
75 minutes.

00:59:33.310 --> 00:59:34.229
Yeah, there's a lot of math.

00:59:34.230 --> 00:59:37.080
There will be a lot of math every

00:59:37.080 --> 00:59:38.755
Lecture, pretty much.

00:59:38.755 --> 00:59:40.120
There's never not.

00:59:40.970 --> 00:59:42.075
There's always Linear.

00:59:42.075 --> 00:59:43.920
There's always Linear linear algebra,

00:59:43.920 --> 00:59:45.060
calculus, probability.

00:59:45.060 --> 00:59:47.920
It's part of every part of machine

00:59:47.920 --> 00:59:48.210
learning.

00:59:49.250 --> 00:59:50.380
So.

00:59:50.700 --> 00:59:52.002
Alright, so Linear Regression.

00:59:52.002 --> 00:59:53.470
Linear Regression is actually a little

00:59:53.470 --> 00:59:55.790
bit more intuitive I think than Linear

00:59:55.790 --> 00:59:57.645
Logistic Regression because you're just

00:59:57.645 --> 00:59:59.600
your Linear function is just like a

00:59:59.600 --> 01:00:01.440
lion, you're just fitting the data and

01:00:01.440 --> 01:00:02.570
we see this all the time.

01:00:02.570 --> 01:00:04.236
Like if you use Excel you can do a

01:00:04.236 --> 01:00:05.380
Linear fit to your plot.

01:00:06.120 --> 01:00:08.420
And there's a lot of reasons that you

01:00:08.420 --> 01:00:09.850
want to use Linear Regression.

01:00:09.850 --> 01:00:11.940
You might want to just like explain a

01:00:11.940 --> 01:00:12.580
trend.

01:00:12.580 --> 01:00:15.010
You might want to extrapolate the data

01:00:15.010 --> 01:00:18.330
to say if my Frequency were like 25 for

01:00:18.330 --> 01:00:21.530
chirps, then what is my likely cricket

01:00:21.530 --> 01:00:21.970
Temperature?

01:00:23.780 --> 01:00:25.265
You may want to do.

01:00:25.265 --> 01:00:26.950
You may actually want to do Prediction

01:00:26.950 --> 01:00:28.159
if you have a lot of features and

01:00:28.160 --> 01:00:29.580
you're trying to predict a single

01:00:29.580 --> 01:00:30.740
variable.

01:00:30.740 --> 01:00:32.650
Again, here I'm only showing 2D plots,

01:00:32.650 --> 01:00:34.500
but you can, like in your Temperature

01:00:34.500 --> 01:00:36.110
Regression problem, you can't have lots

01:00:36.110 --> 01:00:37.600
of features and use the Linear model

01:00:37.600 --> 01:00:37.800
on.

01:00:39.630 --> 01:00:41.046
The Linear Regression, you're trying to

01:00:41.046 --> 01:00:42.750
fit Linear coefficients to features to

01:00:42.750 --> 01:00:44.920
predicted continuous variable, and if

01:00:44.920 --> 01:00:46.545
you're trying to fit multiple

01:00:46.545 --> 01:00:48.560
continuous variables, then you do, then

01:00:48.560 --> 01:00:49.920
you have multiple Linear models.

01:00:52.450 --> 01:00:55.900
So this is evaluated by like root mean

01:00:55.900 --> 01:00:57.940
squared error, the sum of squared

01:00:57.940 --> 01:00:59.570
differences between the points.

01:01:01.560 --> 01:01:02.930
Square root of that.

01:01:02.930 --> 01:01:04.942
Or it could be like the median absolute

01:01:04.942 --> 01:01:06.890
error, which is the absolute difference

01:01:06.890 --> 01:01:08.858
between the points and the median of

01:01:08.858 --> 01:01:10.907
that, various combinations of that.

01:01:10.907 --> 01:01:13.079
And then here I'm showing the R2

01:01:13.080 --> 01:01:15.680
residual which is essentially the

01:01:15.680 --> 01:01:19.460
variance or the sum of squared error of

01:01:19.460 --> 01:01:20.490
the points.

01:01:21.110 --> 01:01:24.550
From the predicted line divided by the

01:01:24.550 --> 01:01:27.897
sum of squared difference between the

01:01:27.897 --> 01:01:29.771
points and the average of the points,

01:01:29.771 --> 01:01:31.378
the predicted values and the target

01:01:31.378 --> 01:01:33.252
values, and the average of the target

01:01:33.252 --> 01:01:33.519
values.

01:01:35.360 --> 01:01:37.750
It's 1 minus that thing, and so this is

01:01:37.750 --> 01:01:39.825
essentially the amount of variance that

01:01:39.825 --> 01:01:42.810
is explained by your Linear model.

01:01:43.550 --> 01:01:44.690
That's the R2.

01:01:45.960 --> 01:01:48.460
And if R2 is close to zero, then it

01:01:48.460 --> 01:01:50.810
means that the Linear model that you

01:01:50.810 --> 01:01:52.680
can't really linearly explain your

01:01:52.680 --> 01:01:54.880
target variable very well from the

01:01:54.880 --> 01:01:55.440
features.

01:01:56.470 --> 01:01:58.390
If it's close to one, it means that you

01:01:58.390 --> 01:02:00.060
can explain it almost perfectly.

01:02:00.060 --> 01:02:01.310
In other words, you can get an almost

01:02:01.310 --> 01:02:03.440
perfect Prediction compared to the

01:02:03.440 --> 01:02:04.230
original variance.

01:02:05.570 --> 01:02:08.330
So you can see here that this isn't

01:02:08.330 --> 01:02:09.060
really.

01:02:09.060 --> 01:02:10.500
If you look at the points, there's

01:02:10.500 --> 01:02:12.060
actually a curve to it, so there's

01:02:12.060 --> 01:02:14.203
probably a better fit than this Linear

01:02:14.203 --> 01:02:14.649
model.

01:02:14.650 --> 01:02:16.220
But the Linear model still isn't too

01:02:16.220 --> 01:02:16.670
bad.

01:02:16.670 --> 01:02:18.789
We have an R sqrt 87.

01:02:20.350 --> 01:02:23.330
Here the Linear model seems pretty

01:02:23.330 --> 01:02:25.410
decent, but there's a lot of as a

01:02:25.410 --> 01:02:25.920
choice.

01:02:25.920 --> 01:02:28.200
But there's a lot of variance to the

01:02:28.200 --> 01:02:28.570
data.

01:02:28.570 --> 01:02:30.632
Even for this exact same data, exact

01:02:30.632 --> 01:02:32.210
same Frequency, there's many different

01:02:32.210 --> 01:02:32.660
temperatures.

01:02:33.430 --> 01:02:35.400
And so here the amount of variance that

01:02:35.400 --> 01:02:37.010
can be explained is 68%.

01:02:42.160 --> 01:02:43.010
The Linear.

01:02:44.090 --> 01:02:44.630
Whoops.

01:02:45.760 --> 01:02:48.400
This should actually Linear Regression

01:02:48.400 --> 01:02:49.670
algorithm, not Logistic.

01:02:52.200 --> 01:02:54.090
So the Linear Regression algorithm.

01:02:54.090 --> 01:02:55.520
It's an easy mistake to make because

01:02:55.520 --> 01:02:56.570
they look almost the same.

01:02:57.300 --> 01:02:59.800
Is just that I'm Minimizing.

01:02:59.800 --> 01:03:01.440
Now I'm just minimizing the squared

01:03:01.440 --> 01:03:03.580
difference between the Linear model and

01:03:03.580 --> 01:03:04.630
the.

01:03:05.480 --> 01:03:08.640
And the target value over all of the.

01:03:09.380 --> 01:03:11.050
XNS so also.

01:03:11.970 --> 01:03:13.280
Let me fix.

01:03:17.040 --> 01:03:19.170
So this should be X.

01:03:21.580 --> 01:03:21.900
OK.

01:03:23.800 --> 01:03:25.740
Right, so I'm minimizing the sum of

01:03:25.740 --> 01:03:27.820
squared error here between the

01:03:27.820 --> 01:03:29.718
predicted value and the true value, and

01:03:29.718 --> 01:03:32.280
you could have different variations on

01:03:32.280 --> 01:03:32.482
that.

01:03:32.482 --> 01:03:34.140
You could minimize the sum of absolute

01:03:34.140 --> 01:03:35.825
error, which is a harder thing to

01:03:35.825 --> 01:03:38.030
minimize but more robust to outliers.

01:03:38.030 --> 01:03:39.340
And then I also have this

01:03:39.340 --> 01:03:41.520
regularization term that Prediction is

01:03:41.520 --> 01:03:43.340
just the sum of weights times the

01:03:43.340 --> 01:03:45.950
features or W transpose X.

01:03:45.950 --> 01:03:47.500
So straightforward.

01:03:50.060 --> 01:03:52.780
In terms of the optimization, it's just

01:03:52.780 --> 01:03:55.070
if you have L2 2 regularization, then

01:03:55.070 --> 01:03:55.920
it's just a.

01:03:57.260 --> 01:03:59.130
At least squares optimization.

01:03:59.810 --> 01:04:00.320
So.

01:04:01.360 --> 01:04:03.050
I did like a sort of Brief.

01:04:03.620 --> 01:04:06.760
Brief derivation, just Minimizing that

01:04:06.760 --> 01:04:07.970
function, taking the derivative,

01:04:07.970 --> 01:04:08.790
setting it equal to 0.

01:04:09.640 --> 01:04:12.180
At the end you will skip most of the

01:04:12.180 --> 01:04:13.770
steps because it's just a.

01:04:14.830 --> 01:04:15.905
It's the least squares problem.

01:04:15.905 --> 01:04:17.520
It shows up in a lot of cases and I

01:04:17.520 --> 01:04:19.020
didn't want to focus on it.

01:04:19.700 --> 01:04:21.079
At the end you will get this thing.

01:04:21.080 --> 01:04:24.000
So you'll say that A is the thing that

01:04:24.000 --> 01:04:25.810
minimizes this squared term.

01:04:27.340 --> 01:04:28.810
Or this is just a different way of

01:04:28.810 --> 01:04:31.508
writing that problem and so this is an

01:04:31.508 --> 01:04:32.970
N by M matrix.

01:04:32.970 --> 01:04:36.506
So these are your N examples and M

01:04:36.506 --> 01:04:36.984
features.

01:04:36.984 --> 01:04:38.690
This is the thing that we're

01:04:38.690 --> 01:04:39.420
optimizing.

01:04:39.420 --> 01:04:41.590
It's an M by 1 vector if I have M

01:04:41.590 --> 01:04:41.890
features.

01:04:42.630 --> 01:04:44.900
These are my values that I want to

01:04:44.900 --> 01:04:45.540
Predict.

01:04:45.540 --> 01:04:47.200
This is an north by 1 vector.

01:04:47.200 --> 01:04:49.420
That's my Different labels for the

01:04:49.420 --> 01:04:50.370
North examples.

01:04:50.950 --> 01:04:53.550
And then I'm squaring that term in

01:04:53.550 --> 01:04:54.700
matrix wise.

01:04:55.570 --> 01:04:58.577
And the solution this is just that a is

01:04:58.577 --> 01:05:01.125
the pseudo inverse of X * Y which

01:05:01.125 --> 01:05:02.920
pseudo inverse is given here.

01:05:05.640 --> 01:05:08.470
And again if you have.

01:05:09.510 --> 01:05:10.400
So.

01:05:11.060 --> 01:05:13.180
The regularization is exactly the same.

01:05:13.180 --> 01:05:15.455
It's usually used L2 or L1

01:05:15.455 --> 01:05:16.900
regularization and they do the same

01:05:16.900 --> 01:05:18.050
things that they did in Logistic

01:05:18.050 --> 01:05:18.335
Regression.

01:05:18.335 --> 01:05:19.890
They want the weights to be small, but

01:05:19.890 --> 01:05:23.280
L2 one wants is OK with some sparse

01:05:23.280 --> 01:05:25.186
higher values where L2 2 wants all the

01:05:25.186 --> 01:05:25.850
weights to be small.

01:05:27.820 --> 01:05:30.020
So L2 2 Linear Regression is pretty

01:05:30.020 --> 01:05:31.540
easy to implement, it's just going to

01:05:31.540 --> 01:05:37.020
be like in pseudocode or roughly exact

01:05:37.020 --> 01:05:37.290
code.

01:05:37.970 --> 01:05:41.530
It would just be inverse X * Y.

01:05:41.530 --> 01:05:42.190
That's it.

01:05:42.190 --> 01:05:44.360
So W equals inverse X * Y.

01:05:45.070 --> 01:05:47.700
And if you add some regularization

01:05:47.700 --> 01:05:50.080
term, you just have to add to XA little

01:05:50.080 --> 01:05:51.830
bit and add on to that.

01:05:51.830 --> 01:05:53.330
The target for West is 0.

01:05:55.330 --> 01:05:55.940
And.

01:05:56.740 --> 01:05:58.610
L1 regularization is actually a pretty

01:05:58.610 --> 01:06:00.850
tricky optimization problem, but I

01:06:00.850 --> 01:06:02.920
would just say you can also use the

01:06:02.920 --> 01:06:04.620
library for either of these.

01:06:04.620 --> 01:06:07.260
So similar to 1 Logistic Regression,

01:06:07.260 --> 01:06:08.890
Linear Regression is ubiquitous.

01:06:08.890 --> 01:06:10.470
No matter what program language you're

01:06:10.470 --> 01:06:12.190
using, there's going to be a library

01:06:12.190 --> 01:06:14.310
that you can use to solve this problem.

01:06:15.410 --> 01:06:18.517
So when I decide whether you should

01:06:18.517 --> 01:06:20.400
implement something by hand, or know

01:06:20.400 --> 01:06:22.202
how to implement it by hand, or whether

01:06:22.202 --> 01:06:24.240
you should just use a model, it's kind

01:06:24.240 --> 01:06:25.353
of a function of like.

01:06:25.353 --> 01:06:27.360
How complicated is that optimization

01:06:27.360 --> 01:06:30.200
problem also, are there?

01:06:30.200 --> 01:06:32.350
Is it like a really standard problem

01:06:32.350 --> 01:06:34.320
where you're pretty much guaranteed

01:06:34.320 --> 01:06:35.350
that for your own?

01:06:36.270 --> 01:06:37.260
Custom problem.

01:06:37.260 --> 01:06:39.530
You'll be able to just use a library to

01:06:39.530 --> 01:06:40.410
solve it.

01:06:40.410 --> 01:06:41.920
Or is it something where there's a lot

01:06:41.920 --> 01:06:43.380
of customization that's typically

01:06:43.380 --> 01:06:45.170
involved, like for a Naive Bayes for

01:06:45.170 --> 01:06:45.620
example.

01:06:47.590 --> 01:06:48.560
And.

01:06:49.670 --> 01:06:51.250
And that's basically it.

01:06:51.250 --> 01:06:53.750
So in cases where the optimization is

01:06:53.750 --> 01:06:55.750
hard and there's not much customization

01:06:55.750 --> 01:06:57.680
to be done and it's a really well

01:06:57.680 --> 01:07:00.140
established problem, then you might as

01:07:00.140 --> 01:07:01.536
well just use a model that's out there

01:07:01.536 --> 01:07:02.900
and not worry about the.

01:07:03.800 --> 01:07:05.050
Details of optimization.

01:07:07.130 --> 01:07:08.520
The one thing that's important to know

01:07:08.520 --> 01:07:11.150
is that sometimes you have, sometimes

01:07:11.150 --> 01:07:12.480
it's helpful to transform the

01:07:12.480 --> 01:07:13.050
variables.

01:07:13.920 --> 01:07:15.520
So it might be that originally your

01:07:15.520 --> 01:07:18.460
model is not very linearly predictive,

01:07:18.460 --> 01:07:19.250
so.

01:07:20.660 --> 01:07:24.330
Here I have a frequency of word usage

01:07:24.330 --> 01:07:25.160
in Shakespeare.

01:07:26.220 --> 01:07:29.270
And on the X axis is the rank of how

01:07:29.270 --> 01:07:31.360
common that word is.

01:07:31.360 --> 01:07:34.537
So the most common word occurs 14,000

01:07:34.537 --> 01:07:37.062
times, the second most common word

01:07:37.062 --> 01:07:39.290
occurs 4000 times, the third most

01:07:39.290 --> 01:07:41.190
common word occurs 2000 times.

01:07:41.960 --> 01:07:42.732
And so on.

01:07:42.732 --> 01:07:45.300
So it keeps on dropping by a big

01:07:45.300 --> 01:07:46.490
fraction every time.

01:07:47.420 --> 01:07:49.020
Most common word might be thy or

01:07:49.020 --> 01:07:49.500
something.

01:07:50.570 --> 01:07:53.864
So if I try to do a Linear fit to that,

01:07:53.864 --> 01:07:55.620
it's not really a good fit.

01:07:55.620 --> 01:07:57.670
It's obviously like not really lying

01:07:57.670 --> 01:07:59.085
along those points at all.

01:07:59.085 --> 01:08:01.220
It's way underestimating for the small

01:08:01.220 --> 01:08:03.140
values and weight overestimating where

01:08:03.140 --> 01:08:06.230
the rank is high, or reverse that

01:08:06.230 --> 01:08:06.990
weight, underestimating.

01:08:07.990 --> 01:08:09.810
It's underestimating both of those.

01:08:09.810 --> 01:08:11.680
It's only overestimating this range.

01:08:12.470 --> 01:08:13.010
And.

01:08:13.880 --> 01:08:17.030
But if I like think about it, I can see

01:08:17.030 --> 01:08:18.450
that there's some kind of logarithmic

01:08:18.450 --> 01:08:20.350
behavior here, where it's always

01:08:20.350 --> 01:08:22.840
decreasing by some fraction rather than

01:08:22.840 --> 01:08:24.540
decreasing by a constant amount.

01:08:25.830 --> 01:08:28.809
And so if I replot this as a log log

01:08:28.810 --> 01:08:31.100
plot where I have the log rank on the X

01:08:31.100 --> 01:08:33.940
axis and the log number of appearances.

01:08:34.610 --> 01:08:36.000
On the Y axis.

01:08:36.000 --> 01:08:39.680
Then I have this nice Linear behavior

01:08:39.680 --> 01:08:42.030
and so now I can fit a linear model to

01:08:42.030 --> 01:08:43.000
my log log plot.

01:08:43.860 --> 01:08:47.040
And then I can in order to do that, I

01:08:47.040 --> 01:08:49.380
would just then have essentially.

01:08:52.910 --> 01:08:56.150
I would say like let's say X hat.

01:08:57.550 --> 01:09:01.610
Equals log of X where X is the rank.

01:09:03.380 --> 01:09:06.800
And then Y hat equals.

01:09:07.650 --> 01:09:10.690
W transpose or here there's only One X,

01:09:10.690 --> 01:09:13.000
but leave it in vector format anyway.

01:09:13.000 --> 01:09:14.770
W transpose X hat.

01:09:17.320 --> 01:09:19.950
And then Y, which is the original thing

01:09:19.950 --> 01:09:22.060
that I wanted to Predict, is just the

01:09:22.060 --> 01:09:23.910
exponent of Y hat.

01:09:25.030 --> 01:09:28.070
Since Y was the.

01:09:29.110 --> 01:09:31.750
Since Y hat is the log Frequency.

01:09:33.680 --> 01:09:35.970
So I can just learn this Linear model,

01:09:35.970 --> 01:09:37.870
but then I can easily transform the

01:09:37.870 --> 01:09:38.620
variables.

01:09:39.290 --> 01:09:42.406
Get my prediction of the log number of

01:09:42.406 --> 01:09:43.870
appearances and then transform that

01:09:43.870 --> 01:09:47.350
back into the like regular number of

01:09:47.350 --> 01:09:47.760
appearances.

01:09:53.160 --> 01:09:55.890
It's also worth noting that if you are

01:09:55.890 --> 01:09:58.460
Minimizing a ^2 loss.

01:09:59.120 --> 01:10:01.760
Then you're then you're going to be

01:10:01.760 --> 01:10:04.860
sensitive to outliers, so as this

01:10:04.860 --> 01:10:07.240
example from the textbook and some a

01:10:07.240 --> 01:10:08.820
lot of these plots are examples from

01:10:08.820 --> 01:10:09.960
the Forsyth textbook.

01:10:12.120 --> 01:10:13.286
I've got these points here.

01:10:13.286 --> 01:10:15.379
I've got the exact same points here,

01:10:15.380 --> 01:10:18.290
but added one outlying .1 point that's

01:10:18.290 --> 01:10:19.050
way off the line.

01:10:19.890 --> 01:10:22.360
And you can see that totally messed up

01:10:22.360 --> 01:10:23.206
my fit.

01:10:23.206 --> 01:10:24.990
Like, now that fit hardly goes through

01:10:24.990 --> 01:10:28.040
anything, just from that one point.

01:10:28.040 --> 01:10:29.020
That's way off base.

01:10:30.070 --> 01:10:32.763
And so that's really a problem with the

01:10:32.763 --> 01:10:33.149
optimization.

01:10:33.149 --> 01:10:35.930
With the optimization objective, if I

01:10:35.930 --> 01:10:38.362
have a squared error, then I really,

01:10:38.362 --> 01:10:40.150
really, really hate points that are far

01:10:40.150 --> 01:10:42.670
from the line, so that one point is

01:10:42.670 --> 01:10:44.620
able to pull this whole line towards

01:10:44.620 --> 01:10:46.630
it, because this squared penalty is

01:10:46.630 --> 01:10:48.750
just so big if it's that far away.

01:10:49.950 --> 01:10:51.980
But if I have an L1, if I'm Minimizing

01:10:51.980 --> 01:10:55.380
the L2 one difference, then this will

01:10:55.380 --> 01:10:55.920
not happen.

01:10:55.920 --> 01:10:57.900
I would end up with roughly the same

01:10:57.900 --> 01:10:58.680
plot.

01:10:59.330 --> 01:11:02.380
Or the other way of dealing with it is

01:11:02.380 --> 01:11:05.960
to do something like me estimation,

01:11:05.960 --> 01:11:08.670
where I'm also estimating a weight for

01:11:08.670 --> 01:11:10.310
each point of how well it fits into the

01:11:10.310 --> 01:11:12.270
model, and then at the end of that

01:11:12.270 --> 01:11:13.730
estimation this will get very little

01:11:13.730 --> 01:11:15.250
weight and then I'll also end up with

01:11:15.250 --> 01:11:16.120
the original line.

01:11:17.220 --> 01:11:19.270
So I will talk more about or I plan

01:11:19.270 --> 01:11:21.880
anyway to talk more about like robust

01:11:21.880 --> 01:11:24.480
fitting later in the semester, but I

01:11:24.480 --> 01:11:25.790
just wanted to make you aware of this

01:11:25.790 --> 01:11:26.180
issue.

01:11:32.600 --> 01:11:34.260
Linear.

01:11:34.260 --> 01:11:34.630
OK.

01:11:34.630 --> 01:11:37.170
So just comparing these algorithms

01:11:37.170 --> 01:11:37.700
we've seen.

01:11:38.480 --> 01:11:41.635
So K&N between Linear Regression K&N

01:11:41.635 --> 01:11:42.770
and IBS.

01:11:42.770 --> 01:11:45.660
K&N is the most nonlinear of them, so

01:11:45.660 --> 01:11:47.530
you can fit nonlinear functions with

01:11:47.530 --> 01:11:47.850
K&N.

01:11:49.240 --> 01:11:50.880
Linear Regression is the only one that

01:11:50.880 --> 01:11:51.665
can extrapolate.

01:11:51.665 --> 01:11:54.250
So for a function like this like K&N

01:11:54.250 --> 01:11:56.290
and Naive Bayes will still give me some

01:11:56.290 --> 01:11:58.230
value that's within the range of values

01:11:58.230 --> 01:11:59.350
that I have observed.

01:11:59.350 --> 01:12:02.330
So if I have a frequency of like 5 or

01:12:02.330 --> 01:12:03.090
25.

01:12:04.000 --> 01:12:06.620
K&N is still going to give me like a

01:12:06.620 --> 01:12:08.716
Temperature that's in this range or in

01:12:08.716 --> 01:12:09.209
this range.

01:12:10.260 --> 01:12:11.960
Where Linear Regression can

01:12:11.960 --> 01:12:13.863
extrapolate, it can actually make a

01:12:13.863 --> 01:12:15.730
better like, assuming that it continues

01:12:15.730 --> 01:12:17.320
to be a Linear relationship, a better

01:12:17.320 --> 01:12:19.230
prediction for the extreme values that

01:12:19.230 --> 01:12:20.380
were not observed in Training.

01:12:22.370 --> 01:12:26.670
Linear Regression is compared to.

01:12:27.970 --> 01:12:31.460
Compared to K&N, Linear Regression is

01:12:31.460 --> 01:12:33.225
higher, higher bias and lower variance.

01:12:33.225 --> 01:12:35.140
It's a more constrained model than K&N

01:12:35.140 --> 01:12:37.816
because it's constrained to this Linear

01:12:37.816 --> 01:12:39.680
model where K&N is nonlinear.

01:12:41.140 --> 01:12:43.040
Linear Regression is more useful to

01:12:43.040 --> 01:12:46.439
explain a relationship than K&N or

01:12:46.440 --> 01:12:47.220
Naive Bayes.

01:12:47.220 --> 01:12:49.530
You can see things like well as the

01:12:49.530 --> 01:12:51.550
frequency increases by one then my

01:12:51.550 --> 01:12:53.280
Temperature tends to increase by three

01:12:53.280 --> 01:12:54.325
or whatever it is.

01:12:54.325 --> 01:12:56.420
So you get like a very simple

01:12:56.420 --> 01:12:57.960
explanation that relates to your

01:12:57.960 --> 01:12:59.030
features to your data.

01:12:59.030 --> 01:13:00.770
So that's why you do like a trend fit

01:13:00.770 --> 01:13:01.650
in your Excel plot.

01:13:04.020 --> 01:13:05.930
Linear compared to Gaussian I Bayes,

01:13:05.930 --> 01:13:08.485
Linear Regression is more powerful in

01:13:08.485 --> 01:13:10.700
the sense that it should always fit the

01:13:10.700 --> 01:13:12.350
Training data better because it has

01:13:12.350 --> 01:13:13.990
more freedom to adjust its

01:13:13.990 --> 01:13:14.700
coefficients.

01:13:16.340 --> 01:13:17.820
But it doesn't necessarily mean that

01:13:17.820 --> 01:13:19.030
will fit the test data better.

01:13:19.030 --> 01:13:20.980
So if your data is really Gaussian,

01:13:20.980 --> 01:13:22.830
then Gaussian nibs would be the best

01:13:22.830 --> 01:13:23.510
thing you could do.

01:13:28.290 --> 01:13:34.480
So the key it's basically that Y can be

01:13:34.480 --> 01:13:35.980
predicted by your Linear combination of

01:13:35.980 --> 01:13:36.590
features.

01:13:37.570 --> 01:13:38.354
You can.

01:13:38.354 --> 01:13:40.450
You want to use it if you want to

01:13:40.450 --> 01:13:42.380
extrapolate or visualize or quantify

01:13:42.380 --> 01:13:44.903
correlations or relationships, or if

01:13:44.903 --> 01:13:46.710
you have Many features that can be very

01:13:46.710 --> 01:13:47.620
powerful predictor.

01:13:48.580 --> 01:13:50.410
And you don't want to use it obviously

01:13:50.410 --> 01:13:51.860
if the relationships are very nonlinear

01:13:51.860 --> 01:13:53.540
and that or you need to apply a

01:13:53.540 --> 01:13:54.700
transformation first.

01:13:56.520 --> 01:13:58.850
I'll be done in just one second.

01:13:59.270 --> 01:14:02.490
And so these are used so widely that I

01:14:02.490 --> 01:14:03.420
couldn't think of.

01:14:03.420 --> 01:14:05.480
I felt like coming up with an example

01:14:05.480 --> 01:14:07.230
of when they're used would not give

01:14:07.230 --> 01:14:10.010
you, would not be the right thing to do

01:14:10.010 --> 01:14:11.940
because they're used millions of times,

01:14:11.940 --> 01:14:14.360
like almost all the time you're doing

01:14:14.360 --> 01:14:16.970
Linear Regression or Linear or Logistic

01:14:16.970 --> 01:14:17.550
Regression.

01:14:18.510 --> 01:14:20.300
If you have a neural network, the last

01:14:20.300 --> 01:14:22.130
layer is a Logistic regressor.

01:14:22.130 --> 01:14:24.240
So they use like really, really widely.

01:14:24.240 --> 01:14:24.735
They're the.

01:14:24.735 --> 01:14:26.080
They're the bread and butter of machine

01:14:26.080 --> 01:14:26.410
learning.

01:14:28.310 --> 01:14:29.010
I'm going to.

01:14:29.010 --> 01:14:30.480
I'll Recap this at the start of the

01:14:30.480 --> 01:14:31.040
next class.

01:14:31.820 --> 01:14:34.715
And I'll talk about, I'll go through

01:14:34.715 --> 01:14:36.110
the review at the start of the next

01:14:36.110 --> 01:14:37.530
class of homework one as well.

01:14:37.530 --> 01:14:39.840
This is just basically information,

01:14:39.840 --> 01:14:41.560
summary of information that's already

01:14:41.560 --> 01:14:42.539
given to you in the homework

01:14:42.540 --> 01:14:42.880
assignment.

01:14:44.960 --> 01:14:45.315
Alright.

01:14:45.315 --> 01:14:47.160
So next week I'll just go through that

01:14:47.160 --> 01:14:49.610
review and then I'll talk about trees

01:14:49.610 --> 01:14:51.390
and I'll talk about Ensembles.

01:14:51.390 --> 01:14:54.580
And remember that your homework one is

01:14:54.580 --> 01:14:56.620
due on February 6, so a week from

01:14:56.620 --> 01:14:57.500
Monday.

01:14:57.500 --> 01:14:58.160
Thank you.

01:15:03.740 --> 01:15:04.530
Question about.

01:15:06.630 --> 01:15:10.140
I observed the Training data and I

01:15:10.140 --> 01:15:13.110
think this occurrence is not simple one

01:15:13.110 --> 01:15:13.770
or zero.

01:15:13.770 --> 01:15:16.570
So how should we count the occurrence

01:15:16.570 --> 01:15:17.940
on each of the?

01:15:20.610 --> 01:15:24.257
So first you have to you threshold it

01:15:24.257 --> 01:15:28.690
so first you say like X train equals.

01:15:29.340 --> 01:15:30.810
784X1 train.

01:15:31.780 --> 01:15:33.580
Greater than 0.5.

01:15:34.750 --> 01:15:35.896
So that's what I mean by thresholding

01:15:35.896 --> 01:15:38.450
and now this will be zeros or zeros and

01:15:38.450 --> 01:15:40.820
ones and so now you can count.

01:15:42.360 --> 01:15:44.530
So that's how we.

01:15:46.270 --> 01:15:48.550
Now you can count it, yeah?

01:15:50.090 --> 01:15:51.270
Hi, I'm not sure if.

01:16:01.130 --> 01:16:01.790
So.

01:16:03.040 --> 01:16:05.420
In terms of so if you think it's the

01:16:05.420 --> 01:16:07.347
case that there's like a lot of.

01:16:07.347 --> 01:16:09.089
So first, if you think there's a lot of

01:16:09.090 --> 01:16:11.500
noisy features that aren't very useful

01:16:11.500 --> 01:16:13.200
and you have limited data, then L2 one

01:16:13.200 --> 01:16:15.400
might be better because it will be

01:16:15.400 --> 01:16:17.480
focused more on a few Useful features.

01:16:18.780 --> 01:16:21.150
The other is that if you have.

01:16:23.080 --> 01:16:24.960
If you want to select what are the most

01:16:24.960 --> 01:16:26.820
important features, then L2 one is

01:16:26.820 --> 01:16:27.450
better.

01:16:27.450 --> 01:16:28.750
It can do it in L2 2 can't.

01:16:30.170 --> 01:16:32.650
Otherwise, you often want to use L2

01:16:32.650 --> 01:16:34.370
just because the optimization is a lot

01:16:34.370 --> 01:16:34.940
faster.

01:16:34.940 --> 01:16:37.580
So one is a harder optimization problem

01:16:37.580 --> 01:16:39.440
and it will take a lot longer.

01:16:40.190 --> 01:16:41.840
From what I'm understanding, L2 one is

01:16:41.840 --> 01:16:43.210
only better when there are limited

01:16:43.210 --> 01:16:44.150
features and limited.

01:16:45.210 --> 01:16:48.160
If you think that some features are

01:16:48.160 --> 01:16:49.850
very valuable and there's a lot of

01:16:49.850 --> 01:16:51.396
other weak features, then it can give

01:16:51.396 --> 01:16:52.630
you a better result.

01:16:53.350 --> 01:16:53.870


01:16:54.490 --> 01:16:56.260
Or if you want to do feature selection.

01:16:56.260 --> 01:16:59.300
But in most practical cases you will

01:16:59.300 --> 01:17:01.450
get fairly similar accuracy from the

01:17:01.450 --> 01:17:01.800
two.

01:17:05.690 --> 01:17:07.740
Y is equal to 1 in this case would be.

01:17:14.630 --> 01:17:15.660
If it's binary.

01:17:17.460 --> 01:17:20.820
So if it's binary, then the score of Y,

01:17:20.820 --> 01:17:24.030
this Y the score for 0.

01:17:24.700 --> 01:17:28.010
Is the negative of the score, for one.

01:17:29.240 --> 01:17:31.730
So if it's binary then these relate

01:17:31.730 --> 01:17:34.080
because this would be east to the West

01:17:34.080 --> 01:17:34.690
transpose.

01:17:36.590 --> 01:17:40.100
784X1 over east to the West transpose X

01:17:40.100 --> 01:17:41.360
Plus wait.

01:17:41.360 --> 01:17:42.130
Am I doing that right?

01:17:49.990 --> 01:17:51.077
Sorry, I forgot.

01:17:51.077 --> 01:17:52.046
I can't explain.

01:17:52.046 --> 01:17:54.050
I forgot how to explain like why this

01:17:54.050 --> 01:17:56.059
is the same under the binary case.

01:17:56.060 --> 01:17:58.633
OK, so but there would be the same

01:17:58.633 --> 01:17:59.678
under the binary case.

01:17:59.678 --> 01:18:01.010
Yeah, they're still there.

01:18:01.010 --> 01:18:02.440
It ends up working out to be the same

01:18:02.440 --> 01:18:02.990
equation.

01:18:03.420 --> 01:18:04.580
You're welcome.

01:18:17.130 --> 01:18:17.650
Convert this.

01:18:38.230 --> 01:18:39.650
So you.

01:18:40.770 --> 01:18:41.750
I'm not sure if I understood.

01:18:41.750 --> 01:18:43.950
You said from audio you want to do

01:18:43.950 --> 01:18:44.360
what?

01:18:45.560 --> 01:18:48.660
I'm sitting on a beach this sentence.

01:18:49.440 --> 01:18:51.700
Or you are sitting OK.

01:18:52.980 --> 01:18:53.450
OK.

01:18:54.820 --> 01:18:57.130
My model or app should convert it as a.

01:19:00.490 --> 01:19:01.280
So that person.

01:19:05.870 --> 01:19:08.090
You want to generate a video from a

01:19:08.090 --> 01:19:08.840
speech.

01:19:12.670 --> 01:19:12.920
Right.

01:19:12.920 --> 01:19:14.760
That's like really, really complicated.

01:19:16.390 --> 01:19:17.070
So.