1 00:00:11,690 --> 00:00:16,350 In this lecture we are going to answer the question how does a model learn. 2 00:00:16,370 --> 00:00:21,950 Let's start with linear regression and its most basic form linear regression just means line of best 3 00:00:21,950 --> 00:00:22,370 fit. 4 00:00:23,120 --> 00:00:30,080 As you may recall from your high school math studies a line has the equation y equals M X plus b here 5 00:00:30,080 --> 00:00:35,420 X is the input variable M is the slope and B is the y intercept. 6 00:00:35,420 --> 00:00:41,690 When we put these together we get the equation for a line and our job of course is to find a line that 7 00:00:41,690 --> 00:00:43,010 best fits our input data 8 00:00:48,250 --> 00:00:53,470 so you can imagine that our input data is a bunch of data points that approximately create the shape 9 00:00:53,470 --> 00:00:54,670 of a line. 10 00:00:54,700 --> 00:00:56,310 This plot is called a scatter plot. 11 00:00:57,130 --> 00:01:03,010 So for each point x and y in our dataset we're going to draw a dot on this chart. 12 00:01:03,010 --> 00:01:07,360 Our goal is to find the line that best passes through all these data points. 13 00:01:07,420 --> 00:01:13,420 In practice this means finding the slope and y intercept one method. 14 00:01:13,420 --> 00:01:19,900 You may have used in high school is to take a ruler and to try and draw the best line on paper using 15 00:01:19,900 --> 00:01:21,620 visual inspection. 16 00:01:21,730 --> 00:01:25,840 Of course since this is data science we no longer use such methods. 17 00:01:25,840 --> 00:01:28,360 We have to be more systematic. 18 00:01:28,360 --> 00:01:32,000 The line that you draw might not be the same as the line that I draw. 19 00:01:32,050 --> 00:01:36,430 And furthermore if we are in multiple dimensions this won't be a line at all. 20 00:01:36,430 --> 00:01:44,430 So it's clear that we need something better and more quantitative. 21 00:01:44,460 --> 00:01:50,490 The idea is we're going to define an error function for each data point which is a pair made up of the 22 00:01:50,490 --> 00:01:53,110 input x eye and the target y. 23 00:01:53,200 --> 00:02:00,660 We're going to calculate a prediction y had I equals to m exile plus B then we are going to take the 24 00:02:00,660 --> 00:02:07,860 squared difference between Y and Y had I we're going to do this for each of our data points for I equals 25 00:02:07,860 --> 00:02:13,200 one up to n once we've done this we can add them all up and divide by n. 26 00:02:13,230 --> 00:02:16,520 This is called the mean squared error. 27 00:02:16,770 --> 00:02:21,470 It's the average squared deviation between our predictions and our targets. 28 00:02:21,480 --> 00:02:24,900 Basically this is going to tell us how accurate our model is. 29 00:02:25,800 --> 00:02:31,410 If our predictions are equal to the targets then this area will be zero because why I would be equal 30 00:02:31,410 --> 00:02:32,430 to I had. 31 00:02:33,090 --> 00:02:37,290 Then the more wrong we are the larger this error becomes. 32 00:02:37,290 --> 00:02:39,600 So that's generally how an error function should work 33 00:02:44,890 --> 00:02:45,770 as a side note. 34 00:02:45,790 --> 00:02:51,580 Since some students get confused by this at first we have multiple names for this error and we usually 35 00:02:51,580 --> 00:02:53,500 use them interchangeably. 36 00:02:53,650 --> 00:02:55,370 Sometimes we call it a loss. 37 00:02:55,390 --> 00:02:59,100 Sometimes we call it an objective or sometimes we call it a cost. 38 00:02:59,110 --> 00:03:01,360 However these all mean the same thing. 39 00:03:02,110 --> 00:03:07,840 I like the term cost because it makes intuitive sense for business people as any good businessmen would 40 00:03:07,840 --> 00:03:08,650 do. 41 00:03:08,650 --> 00:03:20,380 Your job is to minimize your cost and in fact this statement alone tells us how to find the ways W. 42 00:03:20,460 --> 00:03:25,290 So how do we minimize this cost or in other words make it as small as possible. 43 00:03:25,380 --> 00:03:31,410 Well it's time to turn to our old friend calculus if you recall we can find the minimum or a maximum 44 00:03:31,410 --> 00:03:36,190 of a function by finding where its derivative is equal to zero. 45 00:03:36,240 --> 00:03:42,120 If you're not convinced by this try to draw a curve which has a minimum or maximum and draw the slope 46 00:03:42,180 --> 00:03:44,360 at the minimum or maximum point. 47 00:03:44,640 --> 00:03:49,530 If you drew your line correctly it should be a horizontal line meaning that the slope at this point 48 00:03:49,530 --> 00:03:50,830 is zero. 49 00:03:51,000 --> 00:03:54,780 And remember the slope at each point is just the derivative. 50 00:03:54,840 --> 00:04:00,240 So that's why we want to find the derivative set it to zero and then solve for the parameter in question 51 00:04:05,450 --> 00:04:10,400 now usually we have more than one parameter even in one dimensional linear regression. 52 00:04:10,400 --> 00:04:14,990 This is the case because we have both the slope and the y intercept. 53 00:04:14,990 --> 00:04:19,780 So what happens when we have a function of more than one variable in this case. 54 00:04:19,790 --> 00:04:23,090 The derivative is actually called the gradient. 55 00:04:23,090 --> 00:04:29,540 If you recall from your calculus studies the gradient is just a vector or tensor of partial derivatives 56 00:04:29,870 --> 00:04:36,650 for each of the variables we're taking the gradient with respect to so for example if you take the square 57 00:04:36,650 --> 00:04:41,900 function in one dimension that's called a parabola but if you have two input dimensions it's called 58 00:04:41,900 --> 00:04:43,520 a parabola void. 59 00:04:43,520 --> 00:04:50,150 Beyond that we can't picture it but the basic idea is still the same find the gradient set it to 0 solve 60 00:04:50,150 --> 00:04:54,920 for the parameters by the way the term parameter is just another name for the weights. 61 00:04:54,980 --> 00:04:58,340 So that's the W. and b we've been referring to throughout this section 62 00:05:03,610 --> 00:05:10,060 luckily in this course we won't be calculating any derivatives manually since I have many other courses 63 00:05:10,060 --> 00:05:11,430 which do that a lot. 64 00:05:11,710 --> 00:05:13,870 And by a lot I really mean a lot. 65 00:05:14,590 --> 00:05:19,930 So in this course we're kind of excused from doing that because pi torch already does that for us using 66 00:05:19,930 --> 00:05:27,170 a process called automatic differentiation luckily you don't have to know how that works either because 67 00:05:27,230 --> 00:05:30,410 as its name suggests it's automatic. 68 00:05:30,410 --> 00:05:36,260 So the only thing you really do have to know is that a PI torch is going to automatically find the gradient 69 00:05:36,290 --> 00:05:41,030 of all your weights and be pi which uses these gradients to train your model 70 00:05:46,810 --> 00:05:52,290 at this point you might be wondering if all I have to do is find the gradient and set it to zero. 71 00:05:52,390 --> 00:05:57,980 Then why in our previous code did it involve an iterative training process. 72 00:05:58,120 --> 00:06:03,670 Recall that we had to specify the number of epochs to train for and this resulted in a plot of loss 73 00:06:03,670 --> 00:06:09,020 per iteration which we could check to confirm that the training algorithm converged nicely. 74 00:06:09,310 --> 00:06:14,800 Well in actuality it's not possible to actually solve for the equation that you get when you set the 75 00:06:14,800 --> 00:06:16,030 gradient to zero. 76 00:06:16,030 --> 00:06:22,330 Most of the time the one exception to this in this course at least is linear regression where we can 77 00:06:22,330 --> 00:06:28,200 solve it we call the solution an analytical solution or a closed form solution. 78 00:06:28,210 --> 00:06:35,140 Basically this means that we can express the optimal value of W and B using an equation for logistic 79 00:06:35,140 --> 00:06:38,050 regression and the rest of the models we'll be discussing. 80 00:06:38,050 --> 00:06:40,390 It's not possible to do this. 81 00:06:40,600 --> 00:06:43,100 Therefore we need another approach. 82 00:06:43,120 --> 00:06:45,450 That approach is called gradient descent. 83 00:06:50,490 --> 00:06:53,860 Now there's a lot more to gradient descent than what's in this lecture. 84 00:06:54,000 --> 00:06:58,290 And you can check out the in-depth section of this course for that if you're interested. 85 00:06:58,290 --> 00:07:01,150 For now let's just talk about the basics. 86 00:07:01,200 --> 00:07:07,090 Essentially what we do is we start at a randomly initialize point for both WMD. 87 00:07:07,530 --> 00:07:11,850 Since this is random these probably don't lead to a small cost. 88 00:07:12,000 --> 00:07:19,440 So then we find the gradient of our loss with respect to W and b we then take small steps in this direction 89 00:07:19,770 --> 00:07:23,310 to update W and B on each iteration. 90 00:07:23,310 --> 00:07:27,600 Remember these iterations are called epochs mathematically. 91 00:07:27,660 --> 00:07:33,210 You can prove that this leads to a decrease in cost although we won't discuss that in this lecture. 92 00:07:33,210 --> 00:07:38,580 You can imagine that this is exactly what goes on inside the optimizer that step function. 93 00:07:38,580 --> 00:07:45,520 This is really all that's happening for epoch in range epochs and then set w equal to W minus eight 94 00:07:45,810 --> 00:07:47,100 times the gradient of J. 95 00:07:47,100 --> 00:07:57,070 With respect to W and set B to be minus eight times the gradient of J with respect to B. 96 00:07:57,090 --> 00:08:03,060 So again all we're doing is taking small steps in the direction of the gradient with respect to W and 97 00:08:03,060 --> 00:08:11,290 B and one obvious question that arises from this is how small should these small steps actually be. 98 00:08:11,310 --> 00:08:17,640 Well the step size is specified by this Greek letter ADA which we call the learning rate that specifies 99 00:08:17,640 --> 00:08:20,950 how fast or slow we want to train our model. 100 00:08:20,970 --> 00:08:26,880 It's important to set this value right because if you don't then your model won't get good results even 101 00:08:26,880 --> 00:08:29,030 if your model is actually a good model 102 00:08:34,180 --> 00:08:38,750 unfortunately there is no direct method of choosing a good learning rate. 103 00:08:38,770 --> 00:08:42,720 Generally speaking the learning rate is something we call a hyper parameter. 104 00:08:42,790 --> 00:08:48,370 This is to differentiate it from W and B which are just regular old parameters. 105 00:08:48,370 --> 00:08:53,500 It's called a hyper parameter because it's still a parameter but it's not a parameter of the model itself. 106 00:08:53,560 --> 00:08:59,890 It's sort of like a meta parameter and in fact no hyper parameters are really chosen directly. 107 00:08:59,890 --> 00:09:06,010 It's more of a process of trial and error along with intuition that you gain from practicing a lot and 108 00:09:06,010 --> 00:09:07,780 seeing a lot of different examples 109 00:09:12,900 --> 00:09:14,430 as mentioned before. 110 00:09:14,430 --> 00:09:19,680 One way to know if your learning rate is too high or too low is to check the last per iteration after 111 00:09:19,680 --> 00:09:21,060 training. 112 00:09:21,060 --> 00:09:26,550 This is somewhat unfortunate because training can take a long time and you only know the result once 113 00:09:26,550 --> 00:09:27,930 it's done. 114 00:09:27,930 --> 00:09:33,540 If you use a bad learning rate and got bad results while you still had to wait for them I really like 115 00:09:33,540 --> 00:09:38,610 this visualization because it encapsulates how to choose the lending rate quite well. 116 00:09:38,760 --> 00:09:43,410 If you're learning rate is too high then your loss will just shoot off to infinity. 117 00:09:43,410 --> 00:09:48,450 Intuitively this is because you're overshooting the minimum and just ending up on the other side of 118 00:09:48,450 --> 00:09:54,660 the canyon that means your steps are too large and you need to make them smaller. 119 00:09:54,830 --> 00:10:00,290 Sometimes your lending rate might be a bit too high so you'll see some convergence but it'll converge 120 00:10:00,290 --> 00:10:07,710 to a sub optimal value if you're lending rate is too low then you'll see a really shallow curve. 121 00:10:07,720 --> 00:10:11,440 This is also not good because as I mentioned in training takes time. 122 00:10:11,470 --> 00:10:14,940 So if your lending rate is too low you'll have to wait longer. 123 00:10:15,190 --> 00:10:20,050 But not only that sometimes you can just get stuck at a suboptimal point. 124 00:10:20,050 --> 00:10:21,750 That's also not good. 125 00:10:22,030 --> 00:10:28,030 A good learning rate is in between these two extremes and so what you'll probably end up doing is finding 126 00:10:28,030 --> 00:10:33,520 the limits of your particular dataset what's too high what's too low and then trying different numbers 127 00:10:33,520 --> 00:10:36,890 in between until you find something good. 128 00:10:37,170 --> 00:10:43,350 Normally we try numbers that are powers of 10 for example zero point one zero point zero one zero point 129 00:10:43,350 --> 00:10:44,990 zero zero one and so forth.