1 00:00:11,680 --> 00:00:16,840 In this lecture we are going to look at a slightly more advanced topic in PI talk which is creating 2 00:00:16,870 --> 00:00:19,000 custom lost functions. 3 00:00:19,030 --> 00:00:24,730 In addition we're going to apply this new knowledge to build a new kind of neural network one that not 4 00:00:24,730 --> 00:00:32,270 only estimates a function but also estimates the uncertainty in the function prediction previously. 5 00:00:32,370 --> 00:00:37,860 We spent the entire course only looking at neural networks that act as function approximations for each 6 00:00:37,890 --> 00:00:38,360 input. 7 00:00:38,400 --> 00:00:43,590 We get some output prediction but you may have noticed something curious. 8 00:00:43,590 --> 00:00:48,750 There is in fact a discrepancy between a regression and classification with regression. 9 00:00:48,750 --> 00:00:50,390 We only get one answer. 10 00:00:50,490 --> 00:00:54,740 The model's output is our best guess for the target given that input. 11 00:00:55,020 --> 00:00:58,810 But with classification we get a probability distribution. 12 00:00:58,980 --> 00:01:04,120 This probability distribution gives us some sense of the confidence in that prediction. 13 00:01:04,140 --> 00:01:09,720 For example if we're doing binary classification and the output is zero point nine nine then we might 14 00:01:09,720 --> 00:01:13,560 conclude that our model is very confident in that prediction. 15 00:01:13,560 --> 00:01:18,030 If the model outputs zero point five five then it would still make the same prediction. 16 00:01:18,330 --> 00:01:22,930 But intuitively we know that we are less confident in that prediction. 17 00:01:23,000 --> 00:01:26,480 You can apply the same reasoning to soft Max outputs. 18 00:01:26,480 --> 00:01:31,730 So given this knowledge you might wonder why don't we output probability distributions when we're doing 19 00:01:31,730 --> 00:01:32,390 a regression 20 00:01:37,470 --> 00:01:42,900 so if you've studied regression and the mean squared error more in-depth then you know that doing regression 21 00:01:42,900 --> 00:01:50,090 with the squared error is nothing more than maximum likelihood estimation with maximum likelihood estimation. 22 00:01:50,100 --> 00:01:51,490 We start with the likelihood. 23 00:01:51,630 --> 00:01:56,970 Take the log of the likelihood and then negate the likelihood to make it a loss to minimize rather than 24 00:01:56,970 --> 00:02:01,450 a utility to maximize in this form. 25 00:02:01,460 --> 00:02:07,920 We say that T which is the target is the random variable which we assume comes from a Gaza and distribution. 26 00:02:07,940 --> 00:02:14,090 Why is the model prediction which is the mean of the distribution and X is the input to the model we 27 00:02:14,090 --> 00:02:14,680 call theta. 28 00:02:14,690 --> 00:02:16,390 The collection of model parameters. 29 00:02:16,760 --> 00:02:21,110 So this can represent a neuron that work linear regression or any other regression model. 30 00:02:21,110 --> 00:02:21,680 Doesn't matter 31 00:02:26,810 --> 00:02:33,160 as you've seen the usual first step is to take the log of this function at this point your gut instinct 32 00:02:33,160 --> 00:02:34,260 should be this. 33 00:02:34,570 --> 00:02:39,880 We know that we want to end up with the T minus Y squared term and all the other terms should drop off. 34 00:02:39,880 --> 00:02:44,740 That's what gives us the sum of squared error or the mean square error and we know that we can do this 35 00:02:44,770 --> 00:02:50,910 because the log of 2 pi sigma squared is constant constants don't affect the gradient of the loss. 36 00:02:50,920 --> 00:02:56,920 So whatever model parameters maximize this log likelihood would also maximize the log likelihood minus 37 00:02:56,920 --> 00:02:58,360 or plus any constants 38 00:03:03,360 --> 00:03:06,850 but here's the major difference in this lecture in this lecture. 39 00:03:06,850 --> 00:03:09,670 We are not going to make Sigma a constant. 40 00:03:09,670 --> 00:03:14,350 Instead we can ask what if sigma is also a function of x. 41 00:03:14,530 --> 00:03:19,060 So now this term can't be dropped from the objective function because it is not constant. 42 00:03:19,840 --> 00:03:24,910 What this means is that in this scenario this whole thing will be our last function instead of just 43 00:03:24,910 --> 00:03:25,660 the squared error. 44 00:03:26,290 --> 00:03:31,570 And as usual we can take the mean instead of the sum which is equivalent to having a Monte Carlo estimate 45 00:03:31,840 --> 00:03:38,150 of the expected value. 46 00:03:38,150 --> 00:03:43,550 The next question you should have is OK now we're trying to predict two things the target and its corresponding 47 00:03:43,550 --> 00:03:44,570 variance. 48 00:03:44,600 --> 00:03:46,070 How can we do something like that. 49 00:03:47,270 --> 00:03:50,830 Well why not just have a neuron that work with two output nodes. 50 00:03:50,840 --> 00:03:55,370 Let's remember that it doesn't matter what the meaning of these two output nodes is. 51 00:03:55,400 --> 00:04:00,220 We've seen that a neuron that we can end with a linear layer with any number of output nodes. 52 00:04:00,290 --> 00:04:06,020 That's the case whether we're doing regression binary classification or multi class classification. 53 00:04:06,020 --> 00:04:09,260 So then what's the difference between these three cases. 54 00:04:09,320 --> 00:04:15,080 The difference is that the meaning is assigned by us the designer of the program and we assign that 55 00:04:15,080 --> 00:04:17,180 meaning through the loss function. 56 00:04:17,180 --> 00:04:21,980 In other words it's up to us to come up with the appropriate lost function such that the meaning of 57 00:04:21,980 --> 00:04:25,160 these two outputs is the Gaussian mean and variance 58 00:04:30,200 --> 00:04:32,840 but you should be suspicious at this point. 59 00:04:32,840 --> 00:04:38,770 A neural network layer can output any value but don't we have a constraint on the value of a variance. 60 00:04:38,960 --> 00:04:45,260 In fact we require that a variance is positive in order to enforce the variance to be positive. 61 00:04:45,320 --> 00:04:49,940 We can make the assumption that the none that work doesn't output the variance directly but rather the 62 00:04:49,940 --> 00:04:56,510 log of the variance if you think hard this is just like the classification case. 63 00:04:56,770 --> 00:04:59,530 The notion that work doesn't output probabilities directly. 64 00:04:59,530 --> 00:05:04,390 Instead it outputs logics which are related to log probabilities. 65 00:05:04,390 --> 00:05:09,520 In fact that has more constraints because not only do probability distributions have to have positive 66 00:05:09,520 --> 00:05:11,590 values they also have to sum to one 67 00:05:16,690 --> 00:05:21,580 using this information we can determine what the laws function should look like with respect to the 68 00:05:21,580 --> 00:05:23,770 neuron that works outputs. 69 00:05:23,950 --> 00:05:29,380 First let's assume that instead of having a neuron that work with two output nodes we simply have two 70 00:05:29,380 --> 00:05:34,660 neural networks in our model and then the fourth function returns the output from these individual and 71 00:05:34,660 --> 00:05:41,110 their own that works then the output of our model from the forward function will be a tuple with the 72 00:05:41,110 --> 00:05:51,110 first element is the mean predictions and the second element is the log variance predictions. 73 00:05:51,120 --> 00:05:57,220 Now we get back to our original question which was how do we create a custom lost function and Pi talk. 74 00:05:57,360 --> 00:06:02,250 As always it's easiest to start with what we're familiar with the mean squared error. 75 00:06:02,250 --> 00:06:07,170 If we had a regular road regression neural network that only outputs the mean prediction then here's 76 00:06:07,170 --> 00:06:08,750 how you would do it. 77 00:06:08,760 --> 00:06:10,760 The trick is there is no trick. 78 00:06:10,950 --> 00:06:16,180 The lost function is merely a regular Python function that takes in any arguments you want. 79 00:06:16,190 --> 00:06:22,440 Obviously some of them have to be pi to which variables for us this means the predictions and the targets 80 00:06:24,800 --> 00:06:29,900 then inside this function we find the element y's difference between the predictions and the targets 81 00:06:30,200 --> 00:06:32,140 square them and then take the mean. 82 00:06:32,300 --> 00:06:39,060 Pretty straightforward so a custom lost function is just a regular function that does mathematical operations 83 00:06:39,270 --> 00:06:45,400 using pi to urge. 84 00:06:45,440 --> 00:06:46,610 So how do we do this. 85 00:06:46,610 --> 00:06:52,820 When it comes to the full negative log likelihood First we still take in the same arguments the predictions 86 00:06:52,820 --> 00:06:54,500 and the targets. 87 00:06:54,500 --> 00:06:57,840 This time the predictions are tuples so we can split them up. 88 00:06:58,010 --> 00:07:03,750 The first element becomes the mean which I'll come to and the second element becomes the log variance. 89 00:07:03,950 --> 00:07:10,070 Since that XP function is the inverse of the log function I can take that e XP of the log to get the 90 00:07:10,070 --> 00:07:18,340 variance itself which I'll call V then we simply need to plug and chug into our formula for the negative 91 00:07:18,340 --> 00:07:19,870 log likelihood. 92 00:07:19,870 --> 00:07:24,730 This consists of the coefficient term and the exponent term since we're in log space. 93 00:07:24,730 --> 00:07:30,040 We can just add these two together and this gives us the element y's negative log likelihood after which 94 00:07:30,040 --> 00:07:31,000 we can take the mean 95 00:07:36,120 --> 00:07:40,350 so at this point you may be wondering how is a model like this useful. 96 00:07:40,350 --> 00:07:46,530 And naturally you can apply this technique to any problem that contains uncertainty in the target function 97 00:07:47,930 --> 00:07:50,300 if we recall the bias variance tradeoff. 98 00:07:50,300 --> 00:07:56,480 This is the statement that the error of your model is the bias squared plus the variance plus an irreducible 99 00:07:56,510 --> 00:08:02,960 error the irreducible error is called that because it's a characteristic of the data itself and your 100 00:08:02,960 --> 00:08:06,440 model can't make a perfect prediction because of it. 101 00:08:06,500 --> 00:08:08,240 This arises naturally. 102 00:08:08,390 --> 00:08:13,640 For example suppose you're measuring the relationship between how long students study on a test and 103 00:08:13,640 --> 00:08:20,650 the grade let's say you have one student who study 10 hours and got a ninety five percent. 104 00:08:20,670 --> 00:08:24,830 And you have another student who study 10 hours and got an 85 percent. 105 00:08:24,960 --> 00:08:29,010 In this case the input is the same but the target is different. 106 00:08:29,010 --> 00:08:34,110 And since your model is only a function of the inputs it can't make a perfect prediction. 107 00:08:34,110 --> 00:08:40,500 If there are two different possible target values for that single input as you can imagine this kind 108 00:08:40,500 --> 00:08:42,660 of noise can appear everywhere. 109 00:08:42,660 --> 00:08:48,390 So instead of ending with a generic statement like my model has a lower square you can quantify the 110 00:08:48,390 --> 00:08:49,980 uncertainty of your model instead 111 00:08:55,150 --> 00:08:56,740 when the variability of the target. 112 00:08:56,740 --> 00:08:59,470 With respect to the input is not constant. 113 00:08:59,470 --> 00:09:05,410 We say that it's hetero scholastic we can think of other situations where hetero skit activity city 114 00:09:05,410 --> 00:09:06,720 can arise. 115 00:09:06,730 --> 00:09:11,110 One popular application is with stock returns and stock prices. 116 00:09:11,110 --> 00:09:15,890 These are well known to be hetero scholastic in finance. 117 00:09:15,880 --> 00:09:20,360 We often want to avoid high variance because we think of variance as risk. 118 00:09:20,360 --> 00:09:24,150 And so in order to minimize risk we should minimize variance. 119 00:09:24,380 --> 00:09:29,810 And in order to minimize variance it would be useful to model not just the prices of the returns but 120 00:09:29,810 --> 00:09:30,980 the variance itself.