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+WEBVTT Kind: captions; Language: en-US
+
+NOTE
+Created on 2024-02-07T20:52:49.1946189Z by ClassTranscribe
+
+00:01:20.650 --> 00:01:21.930
+Alright, good morning everybody.
+
+00:01:25.660 --> 00:01:27.860
+So I just wanted to start with a little
+
+00:01:27.860 --> 00:01:28.850
+Review.
+
+00:01:28.940 --> 00:01:29.540
+
+
+00:01:30.320 --> 00:01:32.885
+So first question, and don't yell out
+
+00:01:32.885 --> 00:01:34.190
+the answer I'll give you.
+
+00:01:34.190 --> 00:01:35.960
+I want to give everyone a couple a
+
+00:01:35.960 --> 00:01:37.150
+little bit to think about it.
+
+00:01:37.150 --> 00:01:39.420
+Which of these tend to be decreased as
+
+00:01:39.420 --> 00:01:40.790
+the number of training examples
+
+00:01:40.790 --> 00:01:41.287
+increase?
+
+00:01:41.287 --> 00:01:43.350
+The Training Error, test error
+
+00:01:43.350 --> 00:01:45.380
+Generalization could be more than one.
+
+00:01:46.750 --> 00:01:47.700
+I'll give you.
+
+00:01:47.850 --> 00:01:49.840
+A little bit to think about it.
+
+00:02:02.100 --> 00:02:04.545
+Alright, so well, would you expect the
+
+00:02:04.545 --> 00:02:06.540
+Training Error to decrease as the
+
+00:02:06.540 --> 00:02:08.530
+number of training examples increases?
+
+00:02:09.920 --> 00:02:11.190
+Raise your hand if so.
+
+00:02:13.140 --> 00:02:14.430
+And raise your hand if not.
+
+00:02:16.110 --> 00:02:20.580
+So is have it a lot of abstains, but if
+
+00:02:20.580 --> 00:02:21.440
+I don't count them.
+
+00:02:21.440 --> 00:02:25.980
+So yeah, actually the Training Error
+
+00:02:25.980 --> 00:02:28.125
+will actually increase as the number of
+
+00:02:28.125 --> 00:02:30.230
+training examples increases because the
+
+00:02:30.230 --> 00:02:31.330
+model gets harder to fit.
+
+00:02:32.030 --> 00:02:33.390
+So assuming the Training Error is
+
+00:02:33.390 --> 00:02:35.170
+nonzero, then it will increase or the
+
+00:02:35.170 --> 00:02:36.895
+loss that you're fitting is going to
+
+00:02:36.895 --> 00:02:38.620
+increase because as you get more
+
+00:02:38.620 --> 00:02:40.110
+Training examples then.
+
+00:02:42.710 --> 00:02:44.780
+Then, given a single model, you're
+
+00:02:44.780 --> 00:02:46.380
+Error is going to go up all right.
+
+00:02:46.380 --> 00:02:47.310
+What about test Error?
+
+00:02:47.310 --> 00:02:49.580
+Would you expect that to increase or
+
+00:02:49.580 --> 00:02:51.100
+decrease or stay the same?
+
+00:02:51.100 --> 00:02:53.820
+I guess first just do you expect it to
+
+00:02:53.820 --> 00:02:54.180
+decrease?
+
+00:02:55.920 --> 00:02:57.160
+Raise your hand for decreased.
+
+00:02:57.940 --> 00:02:59.030
+All right, raise your hand for
+
+00:02:59.030 --> 00:02:59.520
+increase.
+
+00:03:00.820 --> 00:03:02.370
+Everyone expects to test their to
+
+00:03:02.370 --> 00:03:02.870
+decrease.
+
+00:03:03.910 --> 00:03:05.930
+And Generalization Error, do you expect
+
+00:03:05.930 --> 00:03:08.056
+that to increase or I mean sorry, do
+
+00:03:08.056 --> 00:03:09.060
+you expect it to decrease?
+
+00:03:10.110 --> 00:03:12.220
+Raise your hand if Generalization Error
+
+00:03:12.220 --> 00:03:12.990
+should decrease.
+
+00:03:14.860 --> 00:03:16.380
+And raise your hand if it should
+
+00:03:16.380 --> 00:03:16.770
+increase.
+
+00:03:18.520 --> 00:03:19.508
+Right, so you expect.
+
+00:03:19.508 --> 00:03:21.960
+So the Generalization Error should also
+
+00:03:21.960 --> 00:03:22.650
+decrease.
+
+00:03:22.650 --> 00:03:25.172
+And remember that the Generalization
+
+00:03:25.172 --> 00:03:26.710
+error is the.
+
+00:03:27.920 --> 00:03:31.000
+Test Error minus the Training error, so
+
+00:03:31.000 --> 00:03:32.930
+the typical curve you see.
+
+00:03:35.010 --> 00:03:37.080
+The typical curve you would see if this
+
+00:03:37.080 --> 00:03:39.720
+is the number of train.
+
+00:03:41.550 --> 00:03:43.080
+And this is the Error.
+
+00:03:43.880 --> 00:03:45.220
+Is that Training Error?
+
+00:03:45.220 --> 00:03:47.600
+We'll go like something like that.
+
+00:03:47.600 --> 00:03:49.020
+So this is the train.
+
+00:03:49.670 --> 00:03:52.230
+And the test error will go something
+
+00:03:52.230 --> 00:03:52.960
+like this.
+
+00:03:55.050 --> 00:03:58.389
+And this is the generalization error is
+
+00:03:58.390 --> 00:04:01.010
+a gap between training and test error.
+
+00:04:01.010 --> 00:04:02.540
+So actually.
+
+00:04:02.610 --> 00:04:05.345
+The generalization error will decrease
+
+00:04:05.345 --> 00:04:08.600
+the fastest because that gap is closing
+
+00:04:08.600 --> 00:04:10.419
+faster than the test error is going
+
+00:04:10.420 --> 00:04:10.920
+down.
+
+00:04:10.920 --> 00:04:12.920
+That has to be the case because the
+
+00:04:12.920 --> 00:04:13.790
+Training Error is going up.
+
+00:04:14.750 --> 00:04:17.230
+And then the test error decreased the
+
+00:04:17.230 --> 00:04:18.840
+second fastest, and the Training error
+
+00:04:18.840 --> 00:04:20.205
+is actually going to increase, so the
+
+00:04:20.205 --> 00:04:21.350
+Training loss will increase.
+
+00:04:22.560 --> 00:04:26.330
+Alright, second question and these are
+
+00:04:26.330 --> 00:04:28.655
+just Review questions that I took from
+
+00:04:28.655 --> 00:04:31.480
+the thing that I linked but wanted to
+
+00:04:31.480 --> 00:04:32.290
+do them here.
+
+00:04:32.290 --> 00:04:35.780
+So Classify the X with the plus using
+
+00:04:35.780 --> 00:04:37.710
+one nearest neighbor and three Nearest
+
+00:04:37.710 --> 00:04:39.475
+neighbor where you've got 2 features on
+
+00:04:39.475 --> 00:04:40.270
+the axis there.
+
+00:04:42.110 --> 00:04:45.370
+Alright, 41 Nearest neighbor.
+
+00:04:45.370 --> 00:04:47.220
+How many people think it's an X?
+
+00:04:48.790 --> 00:04:50.540
+OK, how many people think it's an O?
+
+00:04:51.700 --> 00:04:52.700
+Everyone said 784x1.
+
+00:04:52.700 --> 00:04:53.840
+That's correct.
+
+00:04:53.840 --> 00:04:55.100
+For three Nearest neighbor.
+
+00:04:55.100 --> 00:04:56.750
+How many people think it's an X?
+
+00:04:58.010 --> 00:04:59.300
+How many people think it's to know?
+
+00:05:00.460 --> 00:05:01.040
+Right.
+
+00:05:01.040 --> 00:05:02.130
+Yeah, you guys got that.
+
+00:05:02.130 --> 00:05:04.100
+So 3 Nearest neighbor, it's a no.
+
+00:05:05.650 --> 00:05:06.700
+Right now these I think.
+
+00:05:08.910 --> 00:05:10.670
+Also, I have a couple of probability
+
+00:05:10.670 --> 00:05:11.780
+questions.
+
+00:05:13.330 --> 00:05:15.340
+Alright, so first, just what assumption
+
+00:05:15.340 --> 00:05:16.890
+does the Naive based model make if
+
+00:05:16.890 --> 00:05:19.860
+there are two features X1 and X2?
+
+00:05:19.860 --> 00:05:21.710
+Give you a second to think about it,
+
+00:05:21.710 --> 00:05:22.030
+there's.
+
+00:05:22.730 --> 00:05:23.980
+Really two options there.
+
+00:05:23.980 --> 00:05:26.180
+They either it's one of it's either A
+
+00:05:26.180 --> 00:05:27.880
+or B, neither or both.
+
+00:05:29.280 --> 00:05:30.140
+I'll give you a moment.
+
+00:05:49.590 --> 00:05:52.900
+Alright, so how many say that A is an
+
+00:05:52.900 --> 00:05:54.960
+assumption that Naive Bayes makes?
+
+00:05:57.940 --> 00:05:58.180
+Right.
+
+00:05:58.180 --> 00:05:59.910
+How many people say that B is an
+
+00:05:59.910 --> 00:06:01.430
+assumption that Naive Bayes makes?
+
+00:06:03.950 --> 00:06:06.480
+How many say that neither of those are
+
+00:06:06.480 --> 00:06:06.860
+true?
+
+00:06:09.740 --> 00:06:12.120
+And how many say that both of those are
+
+00:06:12.120 --> 00:06:13.582
+true, that they're the same thing and
+
+00:06:13.582 --> 00:06:14.130
+they're both true?
+
+00:06:16.390 --> 00:06:18.810
+So I think there are maybe at least one
+
+00:06:18.810 --> 00:06:19.780
+vote for each of them.
+
+00:06:19.780 --> 00:06:23.410
+But so the answer is B that Naive Bayes
+
+00:06:23.410 --> 00:06:25.070
+assumes that the features are
+
+00:06:25.070 --> 00:06:27.675
+independent of each other given the
+
+00:06:27.675 --> 00:06:30.626
+given the Prediction given the label.
+
+00:06:30.626 --> 00:06:32.676
+And I'll consistently use X for
+
+00:06:32.676 --> 00:06:33.826
+features and Y for label.
+
+00:06:33.826 --> 00:06:34.089
+So.
+
+00:06:34.810 --> 00:06:36.920
+Hopefully that part is clear.
+
+00:06:36.920 --> 00:06:39.270
+So A is not true because it's not
+
+00:06:39.270 --> 00:06:42.180
+assuming that the in fact A is just
+
+00:06:42.180 --> 00:06:44.390
+never true or?
+
+00:06:45.200 --> 00:06:46.420
+Is that ever true?
+
+00:06:46.420 --> 00:06:48.230
+I guess it could be true if Y is always
+
+00:06:48.230 --> 00:06:50.180
+one or under certain weird
+
+00:06:50.180 --> 00:06:52.930
+circumstances, but a is like a bad
+
+00:06:52.930 --> 00:06:54.370
+probability statement.
+
+00:06:55.080 --> 00:06:58.555
+And then B assumes that X1 and X2 are
+
+00:06:58.555 --> 00:07:00.370
+independent given Y because remember
+
+00:07:00.370 --> 00:07:01.979
+that if A&B are independent.
+
+00:07:02.930 --> 00:07:04.496
+Then probability of AB equals
+
+00:07:04.496 --> 00:07:06.150
+probability of a times probability B.
+
+00:07:06.850 --> 00:07:08.580
+And similarly, even if it's
+
+00:07:08.580 --> 00:07:10.440
+conditional, if X1 and X2 are
+
+00:07:10.440 --> 00:07:12.700
+independent, then probability of X1 and
+
+00:07:12.700 --> 00:07:14.832
+X2 given Y is equal to probability of
+
+00:07:14.832 --> 00:07:16.642
+X1 given Y times probability of X2
+
+00:07:16.642 --> 00:07:17.150
+given Y.
+
+00:07:18.450 --> 00:07:21.660
+And they're and they're not equivalent,
+
+00:07:21.660 --> 00:07:24.010
+they're different expressions.
+
+00:07:24.010 --> 00:07:26.090
+OK, so now this one is probably the.
+
+00:07:26.090 --> 00:07:27.190
+This one is the most.
+
+00:07:28.600 --> 00:07:30.040
+Complicated to work through I guess.
+
+00:07:30.900 --> 00:07:33.060
+So let's say X1 and X2 are binary
+
+00:07:33.060 --> 00:07:35.780
+features and Y is a binary label.
+
+00:07:36.410 --> 00:07:37.180
+And.
+
+00:07:38.100 --> 00:07:40.830
+And then all I've set the probabilities
+
+00:07:40.830 --> 00:07:44.794
+so we know what X 1 = 1 given y = 0, X
+
+00:07:44.794 --> 00:07:46.712
+2 = 1 given y = 0.
+
+00:07:46.712 --> 00:07:48.130
+So I didn't fill out the whole
+
+00:07:48.130 --> 00:07:49.820
+probability table, but I gave enough
+
+00:07:49.820 --> 00:07:51.710
+maybe to do the first part.
+
+00:07:52.920 --> 00:07:55.190
+So if we make an app as assumption.
+
+00:07:55.800 --> 00:07:57.760
+So that's the assumption under B there.
+
+00:07:58.810 --> 00:08:01.860
+What is probability of y = 1?
+
+00:08:02.900 --> 00:08:06.920
+Given X 1 = 1 and X 2 = 1.
+
+00:08:08.240 --> 00:08:09.890
+I'll give you a little bit of time to
+
+00:08:09.890 --> 00:08:12.086
+start thinking about it, but I won't
+
+00:08:12.086 --> 00:08:12.504
+ask.
+
+00:08:12.504 --> 00:08:14.650
+I won't ask anyone to call it the
+
+00:08:14.650 --> 00:08:15.275
+answer.
+
+00:08:15.275 --> 00:08:17.220
+I'll just start working through it.
+
+00:08:19.050 --> 00:08:21.810
+So think about how you would solve it.
+
+00:08:22.230 --> 00:08:22.840
+
+
+00:08:24.940 --> 00:08:26.030
+What things you have to multiply
+
+00:08:26.030 --> 00:08:26.860
+together, et cetera.
+
+00:08:35.800 --> 00:08:36.110
+Nice.
+
+00:08:45.670 --> 00:08:48.020
+Alright, so I'll start working it out.
+
+00:08:48.020 --> 00:08:51.390
+So probability of Y1 given X1 and X2.
+
+00:08:52.190 --> 00:08:53.580
+So let's see.
+
+00:08:53.580 --> 00:08:56.940
+So probability of y = 1.
+
+00:08:57.750 --> 00:09:02.530
+Given X 1 = 1 and X 2 = 1.
+
+00:09:05.000 --> 00:09:10.400
+That's the probability of y = 1 X.
+
+00:09:11.390 --> 00:09:12.160
+1.
+
+00:09:13.210 --> 00:09:14.550
+Equals one.
+
+00:09:15.350 --> 00:09:17.180
+X 2 = 1.
+
+00:09:18.690 --> 00:09:19.830
+Divided by.
+
+00:09:20.740 --> 00:09:21.870
+Probability.
+
+00:09:22.300 --> 00:09:22.650
+
+
+00:09:24.810 --> 00:09:26.830
+I'll just do sum over K to save myself
+
+00:09:26.830 --> 00:09:27.490
+some rating.
+
+00:09:27.490 --> 00:09:28.920
+I don't like writing by hand much.
+
+00:09:30.120 --> 00:09:33.220
+So sum K in the values of zero to 1
+
+00:09:33.220 --> 00:09:34.430
+probability of Y.
+
+00:09:35.310 --> 00:09:41.530
+Equals K&X 1 = 1 and X 2 = 1.
+
+00:09:42.810 --> 00:09:45.100
+So the reason for this, whoops, the
+
+00:09:45.100 --> 00:09:46.740
+reason for that is that.
+
+00:09:46.810 --> 00:09:47.420
+
+
+00:09:49.330 --> 00:09:51.910
+I'm marginalizing out the Y so that is
+
+00:09:51.910 --> 00:09:53.430
+just equal to probability.
+
+00:09:53.430 --> 00:09:54.916
+On the denominator I have probability
+
+00:09:54.916 --> 00:09:58.889
+of X 1 = 1 and probability of X 2 = 1.
+
+00:10:05.270 --> 00:10:08.570
+And then this guy is going to be.
+
+00:10:09.770 --> 00:10:10.690
+I can get there.
+
+00:10:11.450 --> 00:10:15.750
+By probability of Y given X1 and X2.
+
+00:10:16.690 --> 00:10:18.440
+Equals probability.
+
+00:10:19.300 --> 00:10:20.960
+Sorry, I meant to flip that.
+
+00:10:23.670 --> 00:10:26.130
+Probability of X1 and X2.
+
+00:10:28.920 --> 00:10:31.812
+Given Y is equal to probability of X1
+
+00:10:31.812 --> 00:10:35.919
+given Y times probability of X 2 = y.
+
+00:10:35.920 --> 00:10:37.490
+That's the Naive Bayes assumption part.
+
+00:10:38.520 --> 00:10:39.570
+So the numerator.
+
+00:10:40.540 --> 00:10:41.630
+Is.
+
+00:10:41.740 --> 00:10:43.250
+Let's see.
+
+00:10:43.250 --> 00:10:46.790
+So the numerator will be 1/4 * 1/2.
+
+00:10:47.900 --> 00:10:50.590
+And then probability of Y is 5.
+
+00:10:50.590 --> 00:10:53.240
+So on the numerator of this expression
+
+00:10:53.240 --> 00:10:56.730
+here I have 1/4 * 1/2 * 5.
+
+00:10:58.030 --> 00:11:01.966
+And on the denominator I have 1/4 * 1/2
+
+00:11:01.966 --> 00:11:03.230
+* 1.5.
+
+00:11:04.370 --> 00:11:05.140
+Plus.
+
+00:11:07.090 --> 00:11:11.180
+2/3 * 1/3 * .5, right?
+
+00:11:11.180 --> 00:11:13.775
+This is a probability of X = 1 given y
+
+00:11:13.775 --> 00:11:16.730
+= 0 times that, times that or times.
+
+00:11:16.730 --> 00:11:18.678
+And then it's times 5 because the
+
+00:11:18.678 --> 00:11:20.561
+probability of y = 1 is .5.
+
+00:11:20.561 --> 00:11:23.249
+Then probability of y = 0 is 1 -, .5,
+
+00:11:23.250 --> 00:11:24.370
+which is also 05.
+
+00:11:25.650 --> 00:11:27.210
+That's how I solve that first part.
+
+00:11:29.150 --> 00:11:31.520
+And then under Naive base assumption,
+
+00:11:31.520 --> 00:11:34.340
+is it possible to calculate this given
+
+00:11:34.340 --> 00:11:35.990
+the information I provided in those
+
+00:11:35.990 --> 00:11:36.630
+equations?
+
+00:11:43.020 --> 00:11:45.180
+So it's not.
+
+00:11:45.180 --> 00:11:47.010
+Under first glance it might look like
+
+00:11:47.010 --> 00:11:49.480
+it is, but it's not because I don't
+
+00:11:49.480 --> 00:11:52.106
+know what the probability of X = 0
+
+00:11:52.106 --> 00:11:53.091
+given Y is.
+
+00:11:53.091 --> 00:11:54.810
+I didn't give any information about
+
+00:11:54.810 --> 00:11:54.965
+that.
+
+00:11:54.965 --> 00:11:57.436
+I only said what the probability of X1
+
+00:11:57.436 --> 00:11:59.916
+given Y is, and I can't figure out the
+
+00:11:59.916 --> 00:12:02.683
+probability of X0 given Y from the
+
+00:12:02.683 --> 00:12:04.149
+probability of X1 given Y.
+
+00:12:05.960 --> 00:12:07.340
+Or as I at least.
+
+00:12:08.090 --> 00:12:09.870
+I haven't thought through it in great
+
+00:12:09.870 --> 00:12:11.245
+detail, but I don't think I can figure
+
+00:12:11.245 --> 00:12:11.500
+it out.
+
+00:12:13.090 --> 00:12:13.490
+Alright.
+
+00:12:13.490 --> 00:12:16.180
+So then with without the Naive's
+
+00:12:16.180 --> 00:12:18.580
+assumption, yeah, under the nibs,
+
+00:12:18.580 --> 00:12:18.970
+sorry.
+
+00:12:19.710 --> 00:12:21.020
+I made a I was.
+
+00:12:21.180 --> 00:12:21.400
+OK.
+
+00:12:22.240 --> 00:12:24.030
+Under the name's assumption.
+
+00:12:24.030 --> 00:12:26.560
+Is it possible to figure that out?
+
+00:12:26.560 --> 00:12:27.250
+Let me think.
+
+00:12:28.140 --> 00:12:29.570
+Probability of X1.
+
+00:12:38.260 --> 00:12:39.630
+Yeah, sorry about that.
+
+00:12:39.630 --> 00:12:42.530
+I was I switched these in my head.
+
+00:12:42.530 --> 00:12:44.620
+So under the knob is assumption.
+
+00:12:44.620 --> 00:12:47.310
+Actually I can figure this out because.
+
+00:12:47.400 --> 00:12:47.990
+
+
+00:12:49.270 --> 00:12:52.470
+If because if X, since X is binary,
+
+00:12:52.470 --> 00:12:56.020
+then if X probability of X 1 = 1 given
+
+00:12:56.020 --> 00:12:56.779
+y = 0.
+
+00:12:57.440 --> 00:13:00.891
+Is 2/3 then probability of X 1 = 0
+
+00:13:00.891 --> 00:13:04.403
+given y = 0 is 1/3 and probability of
+
+00:13:04.403 --> 00:13:08.306
+X2 given equals zero given y = 0 is 2/3
+
+00:13:08.306 --> 00:13:12.599
+and probability of X 1 = 0 given y = 1
+
+00:13:12.599 --> 00:13:13.379
+is 3/4.
+
+00:13:13.380 --> 00:13:17.979
+So I know probability of X = 0 given Y.
+
+00:13:18.940 --> 00:13:22.590
+Equals 0 or y = 1 so I can solve this
+
+00:13:22.590 --> 00:13:22.800
+one.
+
+00:13:23.520 --> 00:13:25.720
+And then I kind of gave it away, but
+
+00:13:25.720 --> 00:13:28.440
+without the nib is assumption is it
+
+00:13:28.440 --> 00:13:30.860
+possible to calculate the probability
+
+00:13:30.860 --> 00:13:34.179
+of y = 1 given X 1 = 1 and X 2 = 1?
+
+00:13:37.370 --> 00:13:39.600
+No, I mean I already I said it, but.
+
+00:13:40.670 --> 00:13:41.520
+But no, it's not.
+
+00:13:41.520 --> 00:13:43.090
+And the reason is because I don't have
+
+00:13:43.090 --> 00:13:44.520
+any of the joint probabilities here.
+
+00:13:44.520 --> 00:13:45.740
+For that I would need to know
+
+00:13:45.740 --> 00:13:48.785
+something, the probability of X1 and X2
+
+00:13:48.785 --> 00:13:51.440
+and Y the full probability table.
+
+00:13:51.440 --> 00:13:53.205
+Or I would need to be given the
+
+00:13:53.205 --> 00:13:55.359
+probability of Y given X1 and X2.
+
+00:14:04.200 --> 00:14:06.700
+Alright, so that was just a little
+
+00:14:06.700 --> 00:14:07.795
+Review and warm up.
+
+00:14:07.795 --> 00:14:10.410
+So today I'm going to mainly talk about
+
+00:14:10.410 --> 00:14:13.418
+Linear models and in particular I'll
+
+00:14:13.418 --> 00:14:15.938
+talk about Linear, Logistic Regression
+
+00:14:15.938 --> 00:14:17.522
+and Linear Regression.
+
+00:14:17.522 --> 00:14:19.240
+And then as part of that I'll talk
+
+00:14:19.240 --> 00:14:20.290
+about this concept called
+
+00:14:20.290 --> 00:14:21.180
+regularization.
+
+00:14:24.880 --> 00:14:27.179
+Right, So what is the Linear model?
+
+00:14:27.179 --> 00:14:31.925
+A Linear model is a model in a model is
+
+00:14:31.925 --> 00:14:36.949
+linear in X if it is a X plus some plus
+
+00:14:36.950 --> 00:14:38.360
+maybe some constant value.
+
+00:14:39.030 --> 00:14:41.940
+So I can write that as W transpose X +
+
+00:14:41.940 --> 00:14:44.850
+B and remember using your linear
+
+00:14:44.850 --> 00:14:46.510
+algebra that that's the same as the sum
+
+00:14:46.510 --> 00:14:50.113
+over I of wixi plus B.
+
+00:14:50.113 --> 00:14:53.920
+So for any values of X&B these are WI
+
+00:14:53.920 --> 00:14:55.260
+and B are scalars.
+
+00:14:55.260 --> 00:14:57.835
+XI would be a scalar, so X is a vector,
+
+00:14:57.835 --> 00:14:58.370
+W vector.
+
+00:14:59.290 --> 00:15:02.680
+So this is a Linear model no matter how
+
+00:15:02.680 --> 00:15:03.990
+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.
+