1 00:00:11,570 --> 00:00:16,510 In this lecture we are going to start looking at machine learning from a new perspective. 2 00:00:16,580 --> 00:00:21,650 While this example might seem overly simplistic you might be surprised to learn that every example that 3 00:00:21,650 --> 00:00:26,810 we do in this course will rely on these basic steps which you are about to learn. 4 00:00:26,810 --> 00:00:29,880 First let's start by describing the basic task. 5 00:00:30,020 --> 00:00:34,940 I'm sure you're all familiar with this example from your grade school of science and math classes. 6 00:00:34,970 --> 00:00:38,260 Let's suppose we've done some kind of scientific experiment. 7 00:00:38,390 --> 00:00:44,120 For example we measured the voltage in a circuit with different resistors or we asked everyone in our 8 00:00:44,120 --> 00:00:51,230 class what their height and weight is. 9 00:00:51,240 --> 00:00:56,610 This gives us a dataset with two columns of numbers one that we can call X and the other that we can 10 00:00:56,610 --> 00:00:57,980 call y. 11 00:00:57,990 --> 00:01:03,150 Now imagine that we plot these points on a grid and we get something that looks pretty close to a line. 12 00:01:03,300 --> 00:01:09,450 We would like to draw a line on this grid that closely passes through these points in your grade school 13 00:01:09,450 --> 00:01:09,870 class. 14 00:01:09,870 --> 00:01:14,820 Maybe you did something like took a ruler and moved it around a little bit until you found the best 15 00:01:14,820 --> 00:01:15,670 possible line. 16 00:01:16,830 --> 00:01:21,150 But now that we're doing machine learning we need more sophisticated methods than just drawing dots 17 00:01:21,150 --> 00:01:25,440 on paper and using a ruler to estimate what the best line should be. 18 00:01:25,440 --> 00:01:29,340 The problem is what happens when we have thousands or millions of dots. 19 00:01:29,340 --> 00:01:34,680 What if our data is more than 2 dimensional then this approach is no longer possible. 20 00:01:34,770 --> 00:01:37,710 So perhaps you might say aha I know what to do. 21 00:01:37,740 --> 00:01:39,870 Let's use our trusty tool socket learn 22 00:01:45,170 --> 00:01:46,330 as you know by now. 23 00:01:46,430 --> 00:01:51,700 This is what a basic regression script might look like if we were to use psychic learning. 24 00:01:51,770 --> 00:01:54,890 First we load in some data call that x and y. 25 00:01:54,890 --> 00:01:58,790 These would just be the two columns of data that we were looking at earlier. 26 00:01:58,790 --> 00:02:03,630 Next we create a model for the coming lectures we'll be interested in linear regression. 27 00:02:03,680 --> 00:02:06,430 So let's create a linear regression model. 28 00:02:06,440 --> 00:02:10,280 Next we call the fit function to fit our model to the data. 29 00:02:10,310 --> 00:02:15,500 Next we can do multiple things with our fitted model such as making predictions with models that predicts 30 00:02:17,210 --> 00:02:23,120 the problem with this is this is just using an API but it doesn't tell us how any of this actually works 31 00:02:23,600 --> 00:02:29,050 what actually happens inside these functions and why is that important to know at this stage. 32 00:02:29,060 --> 00:02:32,900 What I want to highlight on this screen is what we should expect it to be different. 33 00:02:32,930 --> 00:02:38,210 When we look at Pi talk first model creation is going to be different. 34 00:02:38,300 --> 00:02:43,220 As I mentioned earlier pi torture is going to allow us to build all kinds of different models using 35 00:02:43,220 --> 00:02:45,710 compostable building blocks. 36 00:02:45,710 --> 00:02:50,430 So it's not going to be some predefined model as inside you learn next. 37 00:02:50,570 --> 00:02:57,290 I want to emphasize in PI talk there is no such thing as a fit function or a predict function. 38 00:02:57,290 --> 00:03:01,550 This is unlike tensor flowing carrots which do provide you with those functions. 39 00:03:01,940 --> 00:03:08,480 So if you're looking for something simpler than tensor flow or KRS maybe more your pace in PI talk however 40 00:03:08,510 --> 00:03:10,250 that will not be the case. 41 00:03:10,550 --> 00:03:15,530 Therefore we're going to have to find out how to do all these steps in PI talk using our knowledge of 42 00:03:15,530 --> 00:03:16,970 the concepts. 43 00:03:16,970 --> 00:03:23,390 Importantly however this lecture will focus only on the concepts and not on any PI talk in the next 44 00:03:23,390 --> 00:03:23,770 lecture. 45 00:03:23,770 --> 00:03:27,530 We'll look at how to actually implement these concepts in PI towards syntax 46 00:03:32,800 --> 00:03:35,120 so let's first look at what these concepts are. 47 00:03:35,230 --> 00:03:40,270 Before we discuss the details I'm going to go out of order in some sense but you'll see why that has 48 00:03:40,270 --> 00:03:41,600 to be the case. 49 00:03:41,680 --> 00:03:47,320 The three main concepts we want to discuss are number one what is the model architecture and how do 50 00:03:47,320 --> 00:03:48,760 we actually build a model. 51 00:03:48,760 --> 00:03:52,770 If it's not defined for us like it is inside get learn. 52 00:03:52,900 --> 00:03:55,960 Number two how do we make predictions with the model. 53 00:03:55,960 --> 00:04:00,290 In fact this is closely tied to number one and number three. 54 00:04:00,370 --> 00:04:06,310 How do we fit the model that is how do we find the slope and intercept parameters as you'll see. 55 00:04:06,310 --> 00:04:08,400 This is the most complicated step. 56 00:04:08,410 --> 00:04:12,790 It also depends on number one and number two which is why we have to discuss this last 57 00:04:18,150 --> 00:04:19,120 so number one. 58 00:04:19,160 --> 00:04:21,030 What is the model architecture. 59 00:04:21,050 --> 00:04:22,330 Well you already know this. 60 00:04:22,340 --> 00:04:24,940 It's why hat equals M X plus B. 61 00:04:25,100 --> 00:04:27,590 Notice our use of the hat symbol here. 62 00:04:27,590 --> 00:04:33,250 Typically we'll use Y for the true value also known as the target which is given in the dataset and 63 00:04:33,290 --> 00:04:36,250 y hat for the corresponding prediction. 64 00:04:36,260 --> 00:04:40,940 Fortunately for us this actually solves both of the first two problems for us. 65 00:04:40,970 --> 00:04:47,060 This not only defines the model architecture which is a line but it also tells us how to make predictions 66 00:04:47,540 --> 00:04:48,760 given some input x. 67 00:04:48,800 --> 00:04:51,660 I can see exactly how to calculate y hat. 68 00:04:51,830 --> 00:04:52,970 Pretty simple. 69 00:04:53,060 --> 00:04:55,760 So this is our model y hat equals M X plus b 70 00:05:01,030 --> 00:05:02,770 the hard part is number three. 71 00:05:02,890 --> 00:05:07,470 How do we actually find good values of the slope M and the intercept B. 72 00:05:07,810 --> 00:05:11,140 The key is to define what is called a lost function. 73 00:05:11,140 --> 00:05:15,310 I warn you that there are many synonyms for the lost function and you'll see them all. 74 00:05:15,370 --> 00:05:19,940 So don't be afraid if you see these different terms and think one of them might be wrong. 75 00:05:19,990 --> 00:05:21,040 That's not the case. 76 00:05:21,070 --> 00:05:26,770 These words are all used equivalently so you might see this referred to as a cause function or an objective 77 00:05:26,770 --> 00:05:28,630 function or an error function. 78 00:05:28,630 --> 00:05:29,950 These all mean the same thing. 79 00:05:32,950 --> 00:05:40,330 The basic idea is this if we pass in our data set X and Y along with some setting of MLB the lost function 80 00:05:40,360 --> 00:05:44,110 will tell us how good that setting of MLB is. 81 00:05:44,110 --> 00:05:50,230 Let's see how. 82 00:05:50,260 --> 00:05:56,770 First let's assume that we're given a set of pairs of data points X1 y 1 x 2 y 2 all the way up to X 83 00:05:56,770 --> 00:06:03,550 and Y N make a note to yourself now that it will be convention for us and many others to use the capital 84 00:06:03,550 --> 00:06:07,600 letter N to represent the number of samples in your dataset. 85 00:06:07,600 --> 00:06:12,730 This is not always the case but it's at the very least the convention I use and the convention and many 86 00:06:12,730 --> 00:06:13,880 others use. 87 00:06:13,900 --> 00:06:18,130 So if you see something different in the future or you've seen something different in the past don't 88 00:06:18,130 --> 00:06:19,720 freak out believe it or not. 89 00:06:19,720 --> 00:06:22,020 This happens sometimes anyway. 90 00:06:22,030 --> 00:06:27,400 The idea behind linear regression is we will not find a line that perfectly passes through all the points 91 00:06:29,030 --> 00:06:34,250 that's impossible since as you can see there is no such line that can pass to all the points. 92 00:06:34,250 --> 00:06:37,490 All we can do is try to find a line that is a good fit. 93 00:06:37,580 --> 00:06:39,940 So how do we define a good fit. 94 00:06:40,100 --> 00:06:43,280 As I mentioned this is called a lost function. 95 00:06:43,280 --> 00:06:48,680 I'm going to start by telling you what it is and then describing to you why it makes sense for some 96 00:06:48,680 --> 00:06:51,940 of you it'll make sense immediately and for others it might not. 97 00:06:52,400 --> 00:06:59,810 But hopefully by the end of this I will have convinced you that it does make sense so the last function 98 00:06:59,810 --> 00:07:04,820 that we use and regression is called the mean squared error or MSE for short. 99 00:07:04,820 --> 00:07:10,430 Basically you take all the y I's and you find the squared difference between them and the line itself. 100 00:07:10,430 --> 00:07:17,840 The Y had eyes you can think of this as the squared vertical distance from each y eye to the line each 101 00:07:17,840 --> 00:07:22,580 corresponding point on the line is a Y had I a prediction that given x y. 102 00:07:23,030 --> 00:07:29,030 Once we have all the squared differences we take their average you will know how to find averages so 103 00:07:29,030 --> 00:07:33,920 you know that is just the sum of the squared differences divided by the total number of points and 104 00:07:39,100 --> 00:07:42,350 the question is why does this make sense. 105 00:07:42,370 --> 00:07:47,100 I think this is a great exercise to try for yourself if you haven't done so before. 106 00:07:47,200 --> 00:07:52,690 Take a piece of graph paper and draw some dots and a line just like what you see here and calculate 107 00:07:52,690 --> 00:07:54,680 the squared error by hand. 108 00:07:54,760 --> 00:08:00,160 What you should discover is that if the line is not a good fit then the error will be larger. 109 00:08:00,160 --> 00:08:03,550 If the line is a good fit then the arrow will be smaller. 110 00:08:03,550 --> 00:08:05,650 I think that's pretty intuitive. 111 00:08:05,650 --> 00:08:11,410 In addition you should recognize that if the data points perfectly lie on the line then the mean squared 112 00:08:11,410 --> 00:08:17,620 error will be 0. 113 00:08:17,650 --> 00:08:24,180 Next question how do we use this lost function and our model to actually find the values of m and b. 114 00:08:24,820 --> 00:08:29,650 Let's see what happens if we plug in our expression for y hat into the loss. 115 00:08:29,650 --> 00:08:32,260 This is clearly the equation we would get. 116 00:08:32,260 --> 00:08:34,330 Now I have a quiz question for you. 117 00:08:34,330 --> 00:08:37,020 What are the variables in this equation. 118 00:08:37,030 --> 00:08:38,590 I'll give you a second to think about it. 119 00:08:38,770 --> 00:08:43,030 So please take a moment to come up with the answer before moving on to the next slide. 120 00:08:48,360 --> 00:08:53,660 Ok so I hope you thought about the answer to what the variables are in this equation. 121 00:08:53,760 --> 00:08:55,620 If you said x and y. 122 00:08:55,680 --> 00:08:57,380 This is not correct. 123 00:08:57,420 --> 00:09:03,120 Many beginners make this mistake because in a typical math course the variable is X and you're always 124 00:09:03,120 --> 00:09:05,760 trying to do something like solve for x. 125 00:09:05,760 --> 00:09:12,540 However it's important to recognize that in machine learning X is normally not a variable X Y and Y 126 00:09:12,620 --> 00:09:14,630 are just values from my dataset. 127 00:09:14,760 --> 00:09:18,090 You can think of them as numbers from your Excel spreadsheet. 128 00:09:18,150 --> 00:09:21,960 In fact in our case the variables are M and B. 129 00:09:21,960 --> 00:09:25,270 Remember that these are the values that we are currently trying to find. 130 00:09:25,350 --> 00:09:28,290 They are unknown and therefore they are variables 131 00:09:33,480 --> 00:09:34,170 at this point. 132 00:09:34,170 --> 00:09:40,020 We know that our goal is to minimize the loss with respect to M and B this is how we would write that 133 00:09:40,050 --> 00:09:41,700 mathematically. 134 00:09:41,700 --> 00:09:45,690 If you're afraid of math don't worry about this equation too much because it doesn't give us anything 135 00:09:45,690 --> 00:09:46,450 new. 136 00:09:46,500 --> 00:09:51,840 It's just a way of saying what we already said but using math we would like to find the best values 137 00:09:51,840 --> 00:09:58,530 of m and be called M star and B star where these values of m and b are the ones that minimize the loss 138 00:09:58,560 --> 00:10:04,290 l out of all possible values of m and b how do we solve such a problem. 139 00:10:09,470 --> 00:10:15,640 It might help to consider a similar but simply problem say we have a function f of x equals X squared. 140 00:10:15,680 --> 00:10:17,030 A quadratic. 141 00:10:17,030 --> 00:10:18,890 We all know this from high school. 142 00:10:19,010 --> 00:10:25,410 You might say well I can just look at the graph and see that the minimum value is zero when x is zero. 143 00:10:25,760 --> 00:10:30,770 But remember we don't do that in machine learning because in machine learning we work in multiple dimensions. 144 00:10:30,770 --> 00:10:36,230 You can't just look at it at this point we have to introduce our old friend calculus. 145 00:10:36,230 --> 00:10:42,740 If you recall the way that we solve this is you find the derivative D F by the X set it equal to zero 146 00:10:42,830 --> 00:10:44,510 and solve for x. 147 00:10:44,510 --> 00:10:51,450 This is because as you can see the derivative at the minimum is equal to zero so if you draw a tangent 148 00:10:51,450 --> 00:10:55,650 line at the minimum point it should be a horizontal line. 149 00:10:55,680 --> 00:10:59,850 There are of course a lot more details that I'm not including here but you should at least remember 150 00:10:59,850 --> 00:11:01,650 this from your high school math studies. 151 00:11:06,870 --> 00:11:14,060 So how do we find the solution in our case since we have two variables m and b we have to take partial 152 00:11:14,060 --> 00:11:20,990 derivatives but the idea is the same find the partial derivative of L with respect to M and set that 153 00:11:20,990 --> 00:11:22,040 to zero. 154 00:11:22,100 --> 00:11:26,690 Find the derivative of L with respect to B and set that to zero. 155 00:11:26,690 --> 00:11:33,960 This will give us two equations with two unknowns which are MLB at which point we can solve for MLB. 156 00:11:34,100 --> 00:11:39,170 In fact if you've never done this before I would encourage you to do this calculation and solve for 157 00:11:39,230 --> 00:11:41,360 MLB as an exercise. 158 00:11:41,360 --> 00:11:48,200 This can be done for any arbitrary data set X1 y one up to X and Y N So your solution should be in terms 159 00:11:48,200 --> 00:11:54,380 of those values. 160 00:11:54,440 --> 00:12:00,270 There is a problem with this approach however unfortunately finding the derivatives and setting them 161 00:12:00,270 --> 00:12:03,450 to zero only works for linear regression. 162 00:12:03,510 --> 00:12:07,570 It does not work for any other model we will discuss in this course. 163 00:12:07,590 --> 00:12:13,740 In fact the very next model we discuss for linear classification cannot be solved in this way. 164 00:12:13,740 --> 00:12:17,550 Instead we need to use a method called gradient descent. 165 00:12:17,550 --> 00:12:22,890 Very briefly the gradient is the multi-dimensional analogue of the derivative. 166 00:12:22,890 --> 00:12:28,830 So basically when you're working in multiple dimensions you have gradients in place of scalar derivatives 167 00:12:30,780 --> 00:12:34,980 the idea behind gradient descent can be understood with this simple picture. 168 00:12:35,100 --> 00:12:36,590 It's an iterative algorithm. 169 00:12:36,620 --> 00:12:40,480 So we're going to go in a loop on each iteration of the loop. 170 00:12:40,570 --> 00:12:44,410 We're going to take a small step in the direction of the gradient. 171 00:12:44,490 --> 00:12:50,010 It's been mathematically proven that if we keep doing this we will eventually end up at the minimum 172 00:12:50,010 --> 00:12:52,930 point in this lecture. 173 00:12:52,960 --> 00:12:55,330 I don't want to get into detail about gradient descent. 174 00:12:55,600 --> 00:13:00,610 However you are encouraged to check out the lectures in the ends up section of this course. 175 00:13:00,670 --> 00:13:02,310 If you are interested. 176 00:13:02,620 --> 00:13:06,780 Our goal is to get to how we do this in PI to work as quickly as possible. 177 00:13:07,090 --> 00:13:11,760 Gradient descent is a really cool algorithm because it works in pretty much all cases. 178 00:13:11,770 --> 00:13:17,770 That's why libraries that have automatic differentiation such as the ANO tensor flow and Pi torque have 179 00:13:17,770 --> 00:13:22,220 been so popular for machine learning aside from deep learning. 180 00:13:22,260 --> 00:13:26,270 You can use the exact same techniques for things like Hidden Markov models. 181 00:13:26,340 --> 00:13:29,340 K Means clustering Matrix Factorization and more 182 00:13:34,480 --> 00:13:34,830 all right. 183 00:13:34,860 --> 00:13:38,510 So since this has been a pretty long lecture let's recap the steps. 184 00:13:38,640 --> 00:13:44,640 Our goal in this lecture was to take you from the three steps of the cyclone API to what actually happens 185 00:13:44,640 --> 00:13:51,510 conceptually as you recall the three steps are creating the model fitting the model and making predictions 186 00:13:51,510 --> 00:13:52,910 with the model. 187 00:13:52,980 --> 00:13:57,900 After this lecture you now understand at a high level what actually happens during these three function 188 00:13:57,900 --> 00:13:58,520 calls. 189 00:13:59,100 --> 00:14:05,440 However the concepts don't provide the full picture because we still need to implement this in code. 190 00:14:05,460 --> 00:14:10,770 In fact I have a course that goes very in-depth into all these concepts and how you can take them and 191 00:14:10,770 --> 00:14:16,680 put them into a computer program using basic libraries like num pi in the Python programming language. 192 00:14:17,130 --> 00:14:19,400 But remember this course is about pi talk. 193 00:14:19,740 --> 00:14:24,270 So the next question we have to ask is how is this done in PI talk. 194 00:14:24,360 --> 00:14:27,530 Concepts are one thing but syntax is another. 195 00:14:27,570 --> 00:14:32,580 For example the syntax for Pi talk will be different from the syntax for tensor flow. 196 00:14:32,580 --> 00:14:34,650 Even if we are doing the exact same thing.