Lesson 07 – A softer perceptron, part III: gradient descent

So welcome to class. We made it. So we have today lesson seven. All right. So where did we left off last time? We were introducing at the end of the last lesson a loss function. A loss function. It was this curly L for my weights W given a specific data set. This was given to me by the average percent loss which is we can write like this average this is a percent loss of w for a specific x and y pair. What does the loss tell me? The loss tell me how bad my weights are for a given data set in this case. And then the curly L is just on average how bad my weight is for each X and Y of my data set. This is just the average. Then we said that my loss in our soft perceptron was going to be we ch make a choice in this case X and Y is going to be the binary cross entropy. So this is the cross entropy between y and then my prediction y given that y t is given to me by the sigmoid of my weight multiplied by my feature vector. This thing here is going to be equal to y log of y minus 1 - y * log of 1 - y t. So this is the binary cross entropy. Okay. And this is the the form the shape that I I usually remember y 1 - y 1 - there are the minuses such that the logs gives me a positive value. But then the loss is a function of my weights. And so let's rewrite this one in terms of weights x and y. So we can also write this one. The loss is going to be equal and I just use this identity for the Y t. And so I have that the loss is going to be Y times the solve plus of the negative inner product between my weight and the feature vector for my specific X. And then I have class 1 minus y and I have soft class of my weight transpose my feature vector of x. As you can see now I have both of these are equivalent but the second expression is using weight the x and then the y. So this is function of x wx and y in y entropy measure the distance is a cost and measure the distance between y and y t. So that's why I have y and y t in the expression. Remember how how is this soft working? So in the case of y= one we only have this part surviving. And then if you have y equals z then you have only this part here surviving. And so the cross entropy is you can think about it as the sum. I don't like it. You can think about this as having that curly like the curly bracket and then you have the two options for y= 1 and for y equals 0. You understand right? It's a binary option here. And so how does it work? So if I write on my horizontal axis this thing here. So first of all what is this item here? It's a matrix. It's a vector. It's a scalar. What is it? >> Scal this guy here. Where am I pointing the finger? >> It's a vector right? So these two things are maybe very large vectors like the vector with many many many components. But then as soon as I do the projection here I'm projecting the feature vector on the way. I just get a single scaler. And so here I I show you with a single axis the projection between weight and then my feature. And so according to whether I have the label equal one or zero I will have two different shapes right for this cross entropy. So here I'm going to be plotting the cross entropy y. So how does the soft be? So plus is going to be basically positive like like the bis sector for the positive things and then basically zero on the other side right where does it cross the vertical axis >> 7 and so this thing here is going to be something like comes down this way and then it's going to be zero here and then it's going to be going going up. This is for y equals z. And then the other case we had that it's going to be just what happens if you put a negative sign. You just you just flip it on the vertical axis, right? We I think we know this. So this one still comes down this way. It's going to be zero here and then it goes down crosses 7 and it goes basically down to zero. So what happens here? Let's think about it. So let's consider the case where y equal one. So if my inner product is positive I don't pay penalty right this is how much you pay I don't it's a cost right cost means how much you money you spend so if I if it's very positive my inner product if y equ= 1 my feature is align with a weight vector and it's very align then I pay nothing similarly if y equals 0 and my product is very negative meaning my f is opposite with W then I also pay nothing which is basically what we we're having in the perceptronum case right if it's positive if you're falling on the correct side of decision boundary you don't pay anything so if you're falling down here you don't pay anything how about the other case if my y equal one but my inner product is negative I pay a price that is proportional to how negative my inner product is right you can see if it's minus one the projection I pay one. If it's minus2, I pay + two, right? And therefore, you pay for having W and F with a not acute angle. Similarly, for the class Y equals zero, you're going to start paying whenever your feature and the weight are aligned. You pay you pay proportional to how much this inner product is. All right, this is kind of review from last class. Then we finish with a very nice uh plot which I'm going to be trying to reproduce here in a quick manner because I think I failed to show you something. So here we were trying to what we were trying to do what why we were trying to choose different weights last time again either maximize the likelihood that is correct but then in this class we've been using minimizing the the loss right the loss is telling us how bad the weights are so we'd like to get good weights and so there is like a scalar value which is telling us about the goodness or badness or spec a specific thing. Let me bring back a friend of ours from 1948. Who am I bringing back? >> What? >> Norbert. >> Norbert. Yes. Let me bring back Norbert. So you know you have readings right on the on the Google Drive. Let's bring back winner. So like let's say now I supposed to pick up my chalk, right? I I pick up a lead pencil. To do this, I have moved some muscles in my arm and whatever fingers to grab the thing. However, for all of us, but some a few expert anatomist, we don't know which what these muscles are. On the contrary, what we will is to pick up the pencil, right? I can pick up my chalk without any problem without knowing what muscles I'm using. So once we have determined on this our motion proceeds in such a way that we may say roughly that the amount by which the pensy is not yet picked up is decreased at each stage and so this somehow what would like I'd like to connect this to what we are trying to do here we have a value that is telling us how much we have to yet travel to get a very good weight and then we try to decrease this amount which is telling me oh you haven't not yet got there. So I try to decrease decrease decrease until I get the lowmost loss which is going to be basically telling me okay this is the best weight you can get right so there is some sort of analogy between this guy here and what we are trying to do so that's the first recall we we have that that's why I we are trying to figure out what's possible weight are going to be good weights for different what what did I show you last time so this stuff how do you pick a you need to determine what data set you're interested in. So different data sets will have different optimal weights. And so what I show you last time was here I was going to show you the different uh losses or how much you pay. Here I'm showing you just the feature space. Be careful. This is the inner product here. Since I cannot draw in whatever number of space I consider to have a single feature right. So considering my feature vector is simply one and then f okay single scalar feature this is from last time and this is going to be the loss I pay for specific choice choices of weights and then here let's say I have different data sets. So I have data set one, data set one, then I have data set two. Data set one was pretty nice. It had the red points all on one side and then the blue guys on the other side. And so we were going to pay. So if these are going to be the y= 1 and these are the y equals 0 let's say this is zero in this case we don't want to care too much so if these are the ones and I have positive f how much should be the we be my weight should be positive or negative if my if this is the zero posit because of that right we said that the inner product has to be positive If my f are positive, then I have to pick a positive weight in this case, right? And then we saw so if I pick a positive weight, I'm going to be ending up with something like this. So here we were looking at two possible choices. One is going to be this one where my W1 my my weight is going to be equal one. So I get exactly that one. And then in the second case, so this is W the weight equal one. And in the second case, I use my weight equal two. What happens if you put a y W equal two? You just speed up this thing, right? We already we saw that. I'm not explaining this. I'm just reviewing the content from last lesson. And so if I speed up this thing here, I go twice as fast. And so if you go twice as fast, sure the slow one was crossing slowly. The fast one is crossing fast, right? So the slow one will make you pay for the red guys, right? because it is a non zero non zero non zero lo or loss for these points. The more you make it steep and the more this guy will go down on the other side. Right? This is what we saw last time. Sweet. So where are we using W? W is used for coming up with my this guy. And this is what does this represent? This is representing what the predicted. >> Yes. Okay, that's correct. Ya is a prediction but is a number between 0 and one which we interpret as >> the probability of y >> y right? So this thing here we interpret as the probability of y= 1 for that specific x. Right? And so let's draw this case here. So let's draw I'm going to draw here the probability on this line for y = 1. How do I draw this one? Well, I just multiply w * f. And so if I check I choose this w2, I'm going to be getting something that is likely going to be one over here, zero over here, and then in the middle is going to be going down and this is my probability this is what we have seen last time I didn't draw this last part so today new content is the fact that I drew the probability which is using this specific choice of way right this is with with wal in the second case if I change my data set there will be obviously a different weight which is going to be optimal for the different data set. So let's now up the system here and I will get some of the samples that are going to be put on the wrong side. Right? So I will have here a red guy and I will have here a blue guy and then the rest is going to be the same. So we are still having the positive class on the positive side. So which should be the sign of my weight to minimize the cost? >> Still positive, right? We haven't changed the the positive the the sign of the weight. But in this case, what happens if I pick W equal to this guy over here? If I will check how much this guy pays, this one pays a lot. And this red one pay nothing. But what happens if I choose W equal one? Well, I pay much less. I pay this amount over here. Okay. And yes, the other one on the right hand side will pay a bit more than nothing. But the amount I decrease this one is much larger than the sum of all the amounts that this one are going to be paying more, right? And so how do we evaluate this curly in this case over here? Here I just show you the capital L. >> How do I evaluate the curly L? >> I take the average. >> I take the average. Right? So I take the the penalty the cost each of them pay and then I compute the average cost. Right? So there is a cost associated to each of them. But the cost associated to this one is so large that when I compute the average, this one is make me pay so much much so much more than I would pay if I choose a smaller weight which is going to be me making me pay now for the other red guys but much less overall than how much I would have pay for the single bad case in the case I would have chosen the W equal to. Okay, are we follow? All right. And so how do I draw now the probability for again the same stuff right? So probability y = 1 given x with w = 1 how how is going to change with respect to the previous soft right you can see that right so it's going to be likely one over here it's going to be likely zero over here it's going to be likely 0.5 over here and then This stuff will be like like that, right? And so here you can see that this one is much sharper. Right? If this is the zero I draw in a diagonal but you can see the probability that we come up with it's much smoother and so if there is some confusion in my points then the overall probability transition is going to be a smoother one that is reflecting the confusion around the decision boundary. If the data is completely linear I will just recover basically a vertical edge. All right. So that was concluding the lesson from last week. Now are there better way to do this? How do we call this? Okay, this is a joke. This is called GSD. You know what is GSD? Yes. Graduate student descent. But you're undergraduate. So it will be you GSD. >> Yeah. Whatever. Right. Okay. And you figure out by the end of today's lesson why this is a joke, right? for the one that don't know. Okay, don't worry. Okay, so is there a better way to figure out which weight to choose to minimize this average of badnesses? Yes. >> So what is this function over here? So let's let's get some uh some definitions. So assuming a given data set. So we are just looking at one of these data sets. Let's take this like let's let's let's stop it like let's fix it. So assuming a given data set this function it's mapping RN to R plus what what is the name of this thing? No not the the loss. What is the the symbol here? Do you know what is this notation? >> Like when you find distance usually. No. >> Right. Are you familiar with the column sum set arrow set? >> It's a function. >> This is a function. A function notation. I'm defining the domain and the image of a specific. Are you writing down notes? No. Okay. I don't know how you're going to be learning anything without writing down. Anyway, so this is going to be a non negative scalar field. What is a scalar field? It's basically a like an ark, right? This is a scalar field. This is this is like a scalar field. It's defined over a specific domain, but then there is only one height. So this like a it's a blanket on a specific domain underneath. And this blanket is actually only positive. We don't care the rate of change. of L at W along direction is called the directional derivative. Okay. So we have a specific so this guy here where does you live? So u it's a cursor is a unit vector. U is a unit vector that lives where >> yeah. All right. So it's just a direction. And so the directional derivative it tells me how much a scalar field changes in that specific direction. Okay. This is definition. Okay, the directional derivative d u of l e w0 is given to me by the inner product between this weird symbol here and u okay and this thing here let's find out what it is this item over here which is called the gradient of l in W0 is going to be the Jacobian of L in W which is simply the collection of the partial derivative with respect to all the parameters. Okay. So this guy over here is called the gradient of L in Z. This thing here is called the Jacobian. And this is just the this is a partial derivative, right? This is a partial derivative. You already seen this. Who hasn't seen this before? One, two, three, four. Okay. Have you seen partial derivatives? No. Have you seen derivatives? Yes. Okay. So, how many? This is going to be a function of multiple variables. Usually in this case so far, we only had a single W. Here we have multiple W's. We have W1, W2, W3. So the loss when I write loss of W this means I'm talking about loss with respect to the first component second component and n component there are multiple component and so each of these item here is the derivative of this function considering it as a function of the whatever component I show here. So this is here is the der the derivative of my loss function with respect to the second value. All the other I consider just constant like numbers. You don't you don't care. So just like w should be a vector right w is yes is a location which I call it w0. So let me put a zero on top so we don't get confused. >> I just mean like for the you so above to the right. Yes, yes, yes, yes. Let me put a zero on top so I don't get >> so this is a vector I call it w0 and these are all the partial derivative with respect to each component and then it's going to be evaluated at the specific location. This is a specific location. Sorry guys. So this is a specific location. We are looking at this napkin, right? This kind of cloth. And then if I'm at a specific point, which I call it W0, then what is the rate of change in this direction? Well, I can just check a specific direction. U is going to be one of the directions I'm looking at. And I see how much this function goes up and down. Okay. What w is >> a specific location. So but how do we choose? >> No, it's arbitrary location. So let's say I have in this case only two dimensional. So n equals 2. So I have w1 here, w2 here. This location over here I will call it my W0 which is a specific W 1 and W 2 0. It's a specific point. >> Okay. >> These are gradient is computed at just one point. >> Correct. So the gradient if I wouldn't put a zero will be the function >> is a gradient function. If I put the actual zero is the gradient evaluated at that point. It's a number. It's a it's a it's a vector. Okay, it's a numerical vector. This is going to be my my point here. I may have my something like that. This is my surface. And then I go up to the location on the surface. And at this I in this location I gonna check for example if this is going to be my u hat. This is the direction I'm looking at. And here I like to see in this direction how is this what is the the rate of change of the function. It's going to be up going down. So I can just check every possible direction. I'm on some surface. I can go in every direction and see oh it's going to be going up or down. Okay. I check this direction. It's going to be going up and down. Let me show you a picture so you can actually I mean I draw the thing but let me show you this going to be my whatever surface and all along these red lines the height doesn't change. Okay. So either you play plot things in 3D or you can actually plot them down on the bottom side. You can see all the locations where the height of this function is constant. But we were talking about these gradients. Let me show you the gradients in one sec. So this is going to be my surface I'm talking about. This is going to be my initial location, the W0. And then I look at a specific direction. That's my U direction. So in that U direction, I go up on my on my surface and I going to be checking what is the rate of change in that specific direction. This is the d directional derivative of my surface my my scalar field at this w location. Okay. So here we are looking at this. So what can I do now? I can change you and I will be able to look at the rate of change in possible different direction. My question for you, what is the direction where I have the largest rate of change in the direction of gradient? So what is the meaning of this thing here? So this is the inner product, right, between two vectors. >> This is how much is this stuff, right? This is a scalar, right? This is scalar value, right? It's the inner product of two different things. So this is going to be equal the magnitude of the first times the magnitude of the second one times of the angle between the two. Okay. How much is this guy here? The magnitude of the unit vector. >> Okay. Very good. How much is the magnitude of this thing? I don't know. is the magnitude of this vector over here. Now I can change u I can change the direction which is the direction which gives me the largest overall product of items. The only thing I can change is gonna be the angle here five right and so when is when is the cosine maximum >> zero >> when the phi is zero right and therefore whenever my direction u is aligned to the direction of this item over here I will have the maximum increase of this function. Okay. And so we have that this horrible vector over here tells me two different things. The first one tells me what what is the direction of maximum >> maximum increase of the function. The second thing that tells me which is the magnitude is the actual maximum rate of change. Right? So it tells you two different things. The direction of maximal maximum change and the amount of maximum change. So let's write it down. Okay. The gradient of L in W0 points in the direction of maximum increase its magnitude its length. tells us what >> rate of change. >> The maximum rate of change that function has the maximum rate of change. Well, the maximum directional derivative, right? Okay, whatever we can write it down. Maximum range of change of L in so this is not definition but we kind of understand right who's not understanding okay don't be shy let me back up to the picture I show you before so these are again these red lines are the locations where the height doesn't change so what is the rate of change along the red lines >> the directional derivative if I point if I am on one of these red circle circle let's say I am at specific location right can you see my mouse yeah I am at specific location here the purple one where okay where the the arrow is let's let's assume we are at the arrow and let's consider the directional derivative with the direction being tangent to the level curve. What is the derivative? What is the directional derivative for the direction tangent of the level curve? >> 0 >> zero. Why? >> Because we're not changing >> because along the level curve the height doesn't change by definition. Maybe I didn't show you the definition. Do I have the definition? Yes, I have the definition. So the definition of level curves that you haven't seen before is of a function f of in this case two variables are the curves of equations whatever the height is constant where a is a constant right so the the curves where the height is constant that's called a level curve okay therefore if you compute the directional derivative in that in the t for a direction that is tangent to the level curve is going to be zero now a question for What is going to be the direction of steepest ascent? Perpendicular to the level curve. Why? See, I don't want you to answer because you memorize something. This is not a class where you're memorizing things. Here we are trying to build some understanding. Don't get it wrong. Right? I'm trying to to want to make you reason about this. Why is this blue curve the direction of steepest ascent along the red line? We said the derivative is >> zero. Then I turn, I turn, I turn, I turn, I turn, I get orthogonal to the level curve. Then you have the maximum in this case. I go I go I go I go again when I am once again on the zero right and then I go again again again again again I get the direction of steepest descent oh what is the title of today's class >> oh so now we figure out how to find the direction to go downhill. Why do we want to go why do we want to go downhill? because we want to minimize this damn loss right of my weight. So, we're going to be using the direction of steepest descent. What is the steepest descent? Well, if the gradient points in the direction of maximum increase, this is direction of maximum increase. I just have it backward. >> Are we all on board? Are we lost? No, we understand. Yes. Question. Will the graph always look like this? Is any other shape possible? >> That's the next slide. We have so much time. Oh, okay. No, we don't have much time. I want to just You don't know what's coming, right? No, you don't know. Okay. I don't know. Okay. I'm improvising. There we go. This is going to be our exercise for now. We have to figure out which goes with which, right? This is like primary school thing. Okay. Which curve go? Which letter curves go with which one? Right. So A, where does A go? In first, second, third, fourth, fifth, and sixth. Show me with your Okay. You have fingers, right? Your in your hands. Both all of you have at least six fingers, right? You don't Okay. Show me with your hand with where A goes, right? Yeah, that's correct. I don't know that this those are correct. Those are not correct. Okay, some of you are mostly correct. Let's try number uh B, right? Where is B going? Okay, most of you are wrong. How about C? Where is C? Let me see. C. Yeah, C. Yeah, now you're correct. You're correct as well. You're correct. You're not correct. You're incorrect. What about you? Where's he go? See? >> Yeah, I know. You're supposed to show them. my fingers. Anyway, okay. Let's go here. If you know this is six, right? You know how to use, right? E R S. All right. How about D? Where is D? D. Yeah. Okay, very good. High five. How about E? Okay. And F. Well, it's the last one, right? Okay. Very good. So, we understood the the assignment. Amazing. Yeah. So, if you are not really familiar with these things, you haven't seen it them before, there is the chapter 14 of this book. You can find it wherever you want to find it. Right. All right. One more. This one. What happened here? What am I showing you here? As you can see, these are my gradients. So, where is the maximum of this function here? >> Think about what is maximum. Yeah. What does the what does this what does this symbol mean here? Look what what how do we call this in English or length? >> Pink. Pink. Pink. Just think. Yeah. Yeah. Sure. Okay. One more one more bonus thing. See, today we have a lot of things. >> Today also we have Wikipedia the end. So what's happening here? So as you see the black curves are representing the level curves. This one is going to be a maximum. The yellow means very high value. The blue means it's a minimum. We are starting at specific random well some locations and then all these blue curves are always always perpendicular to the level curves. Okay. So at given locations we are performing one step in what direction? >> Negative gradient. >> The negative gradient. Right. So how do we call this one? Let me actually write it down because we didn't even call it. Therefore, we will be mice add one sample at a time such that we're going to have my weight is going to be my weight minus EA gradient weight. Okay. is this procedure in the so okay additionally for now So ita is called learning and for now equal one. Same with beta where we don't whenever we don't care about beta or ea or whatever we just set it to one we're going to forget about ea for and so just pay attention the gradient is a vector. Meanwhile, here we were looking at the transpose gradient which is the Jacobian, right? The Jacobian is the the transpose of the of the gradient. So this is the gradient is a column vector. I subtract from my weight a proportion like something proportional to the negative gradient of my percent loss. Okay, L is the percent of those B per although we are trying to minimize the average of the Bness. It turns out that the procedure that we're going to be using for now is going to be just taking each sample each like one sample I compute one gradient I perform one step. I take another sample, I comput another gradient, I perform one other step and so on and so on and so on. So this is like iterative procedure. I do one step at a time because otherwise the alternative it would be to take all the points, compute all the gradients, average out all those gradients, then one do one step. It takes forever because every single time you compute all the gradients to do one step and it turns out that if you have duplicate in your data set, you're just wasting computation. If I have the same email twice and the same label, I'm just computing the same gradient twice. And is this reasonable? Yes, it is very reasonable because if all these emails weren't going to be kind of similar, it would be hopeless for your system to be able to detect spam and so if you pick one email, one spam, two spam, 100,000 spam, most of them will be looking like spam and so you're just getting gradients that are basically identical. And so if you just want to compute the gradient for this guy, you're just wasting computations. Meanwhile, what we are going to be doing is going to be for each email, I compute my gradient, I update my weight. I take the second email, I compute the gradient, I apply the weight. Third email, I compute the gradient, up the yada yada. Okay. Yeah. >> So, we're saying this approximates that. So uh if we use for example L it will be approximately equal to >> this gradient here with respect like the gradient of the capital L is a stochastic approximation of the gradient of the curly guy. The curly guy is the average of all these gradients. But if I have a data set of 100,000 emails I have to comput 100,000 gradients to just compute one gradient for the curve and do one step. It takes forever. If your data set is infinite, you're never stepping in like let's say you're training CHGPT on the internet. If you want to read the whole internet before doing a single step, you don't go anywhere, right? Meanwhile, you just should read one thing step one thing step one thing step right more or less. Other questions? >> Yeah. Where do we start from? >> Zero zero. >> Yes. Don't worry. I I I make sure in the next part we have >> do we start next part? Of course I'm going to start a new part but Okay. Do you have other questions? No. Right. Okay. So brace yourself for what's coming today. Should have been this the second part should have been the main part but then I had the different things with the with the screens. I didn't plan we we didn't have them before. Let's take a picture. All right. What was the last line I just erased? >> Stoastic gradient. >> Okay. So, where are we starting? >> We are going to compute a gradient of our capital L. What was the first line here? What is our L? You forgot >> average. >> Capital L. >> Average. >> No, no, no, no. That's curly L. >> Sure. Which per sample loss are we using? >> Entent entropy. So let's talk about let's compute gradient of the percent. We need the fact that y is defined as my sigma of omega f and which is going to be 1 + x of negative omega f omega sorry w minus one. Okay. And then we're going to be computing now well we're going to be reviewing what is the cross entropy for y = 1 and my y. This is remember I just use the inner product. I have simply the soft plus which one? the one right of the negative w okay which is simply correct right sweet so what is the gradient for this guy so instead of gradient let's go to jacobians again it's all the same right cares So I'm going to be computing the partial of this guy here of log. >> Okay. >> Derivative of function of functions is a function of a function. Right? has a multiplication inside right. Okay. So what multiplied by? >> Wip. >> Yeah. I I just write what I think you said. >> And then what? >> What's coming out? One more thing. Right. Okay, now I just do some cleaning. What is this? Very good. >> Okay. What is this thing here? This is one minus this thing, right? So I can also write down here. >> Wait a ask where can we learn to take derivatives of matrices and vectors? Is there like a book? >> You don't care. try just use it as a don't worry just just pretend there is it's not a it's not a vector you don't you don't care I know like there is a book there is Wikipedia there are many sources there's a book the book I I use for today's lesson but you don't care okay so this guy here is the what >> this thing Here is >> this thing here is >> yeah which is >> this is the what what is this stuff giving us? This is >> so what is this thing here yals? >> Yeah exactly. Thank you. Okay. So for the case where y = 1, we have that my weight will be the weight minus the gradient. So minus the gradient. The gradient will be the transpose of one thing transpose advance one >> this is the Jacobian this is the gra >> no but like it's there it's minus one minus y till the frpose >> correct >> but very you wrote just as f >> that is correct from the previous blackboard this is jacobian this is transpose algebra just notation just forget about the the transpose doesn't matter so what happened in this case we can also write yes any question yeah >> each row the joy matrix is a gradient right that's why we're doing the transpose thing right here >> yeah so the the definition from previous from previous board here we had that the transpose so this guy here the Transpose of this one is the Jacobian of L which is the row vector of all the partial okay so this is a column vector so they are one oh I didn't write the another notation for this one here is the following there you go my bad so this thing here is k another notation for this jacobian here which is going to be this thing over my bad I didn't define that this notation over here what's happening if I perform a prediction so this is going to be again the probability of y= z if x so you make a prediction You have you assign the some belongness of this specific X to the red class and to the blue class. The email was a red red one was plus one. So I will sum F to my W to my weight proportionally to how much I think it should be the blue class. If I think it's not the blue class, I don't sum anything. If I think it's just a little bit the blue class, then I sum just a little bit of F. If I think it's really the blue class, then I sum a lot my F. So this one allows me to sum between zero to like is a multiplier that goes from 0 to one, which is allowing me to sum to my current weight, the feature vector scaled by this proportional value. Okay, this this multiplier. Okay, let's do the second one. Second part is going to be basically identical. I have the fire cell of my cross entropy for the other class with respect to the weight is going to be what? Well, we had to actually write down this on the right. So, I didn't write the definition. Let's write down here. So, we have the cross entropy for this this case. This is going to be what? Soft plus of the positive thing. Right? This is soft plus of the positive inner product which is going to be 1 log 1 + x of this thing here. Right? So we take the derivative. It's going to be identical but there is the minus not without a minus minus. So we had x w transpose f / 1 + x w f* frppose I divide it and I get 1 / 1 + x of negative w F * F transpose which is going to be this is our sigmoid right this is my sigmoid of W of F times my left transpose and we know this one is again my Y tild this guy here is my Y right going to be again the probability the same right as the probability of y = 1 given my x and so for the second case what happens if I have that with y = 0 we are going to be updating my weight as my weight minus this thing here right minus my weight times my So in this case we were summing the feature proportional to how much we are giving the attribute to the other class. In this class we were supposed to get zero and so we're going to be subtracting the feature. Right? If I want to have zero I have to lower the inner product. How much do I lower proportional to the this item over here? I have y equals z. I have to lower my inner product. So I subtract. I have y= 1. I want to add to increment the inner product. Right? It's very similar to what we were doing before. Let me add some more things. What is this thing over here? Cool. Probability y - y tilda. Yeah. Same thing there. It will be y - y tilda. Okay, for both cases, if I if I summarize, we have that we sum to my wave factor the difference between y and y tilda. Finally, if I put together, we have the given data set and a zero vector. I pick one email label Y compute the feature and use the following This is known as this regression. Although a better name logistic classification and the logistic comes from the sigma, right? Okay, so we kind of put all together there is a few derivatives there which we computed but then overall what is this rule? This looks the same as our >> perception the same thing, right? But the only difference well the only it's a major difference. This thing is a number between zero and one. Okay, but this you do exactly the same thing. You just have always this is no longer error correcting algorithm because you're not just correcting errors. You always have a nonzero difference here. So you always change in this way. That was today's lesson. Questions? No questions.