| 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 | |
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| 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 | |
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| the answer I'll give you. | |
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| I want to give everyone a couple a | |
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| little bit to think about it. | |
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| Which of these tend to be decreased as | |
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| the number of training examples | |
| 00:01:40.790 --> 00:01:41.287 | |
| increase? | |
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| The Training Error, test error | |
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| Generalization could be more than one. | |
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| I'll give you. | |
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| A little bit to think about it. | |
| 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 | |
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| 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 | |
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| 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 | |
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| Training examples then. | |
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| Then, given a single model, you're | |
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| Error is going to go up all right. | |
| 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 | |
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| 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 | |
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| 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 | |
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| should decrease. | |
| 00:03:14.860 --> 00:03:16.380 | |
| And raise your hand if it should | |
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| increase. | |
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| 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 | |
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| error is the. | |
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| Test Error minus the Training error, so | |
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| the typical curve you see. | |
| 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. | |
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| 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 | |
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| 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 | |
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| down. | |
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| 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 | |
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| 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 | |
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| just Review questions that I took from | |
| 00:04:28.655 --> 00:04:31.480 | |
| the thing that I linked but wanted to | |
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| do them here. | |
| 00:04:32.290 --> 00:04:35.780 | |
| So Classify the X with the plus using | |
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| one nearest neighbor and three Nearest | |
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| neighbor where you've got 2 features on | |
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| the axis there. | |
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| Alright, 41 Nearest neighbor. | |
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| How many people think it's an X? | |
| 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. | |
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| 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 | |
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| 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 | |
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| there are two features X1 and X2? | |
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| Give you a second to think about it, | |
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| there's. | |
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| Really two options there. | |
| 00:05:23.980 --> 00:05:26.180 | |
| They either it's one of it's either A | |
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| 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. | |
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| How many people say that B is an | |
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| 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 | |
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| independent of each other given the | |
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| given the Prediction given the label. | |
| 00:06:30.626 --> 00:06:32.676 | |
| And I'll consistently use X for | |
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| features and Y for label. | |
| 00:06:33.826 --> 00:06:34.089 | |
| So. | |
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| Hopefully that part is clear. | |
| 00:06:36.920 --> 00:06:39.270 | |
| So A is not true because it's not | |
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| assuming that the in fact A is just | |
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| never true or? | |
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| Is that ever true? | |
| 00:06:46.420 --> 00:06:48.230 | |
| I guess it could be true if Y is always | |
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| one or under certain weird | |
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| circumstances, but a is like a bad | |
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| probability statement. | |
| 00:06:55.080 --> 00:06:58.555 | |
| And then B assumes that X1 and X2 are | |
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| 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. | |
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| Complicated to work through I guess. | |
| 00:07:30.900 --> 00:07:33.060 | |
| So let's say X1 and X2 are binary | |
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| features and Y is a binary label. | |
| 00:07:36.410 --> 00:07:37.180 | |
| And. | |
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| 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. | |