Simplifying Mathematics Behind AI | Machine Learning | Community Webinar

Key Takeaways

okay awesome so the topic that we're going to be covering today is basically going to be simplifying mathematics behind AI now uh for a lot of people maths can seem a bit complex it can seem a bit daunting um and the reality is that behind a lot of uh your AI use cases uh and algorithms there is going to be some maths right so our goal of this webinar today is to basically break down those Concepts and try to understand them and simplify them so just to go over my introduction again so my name is Albert I am a senior data scientist along with an instructor here at data science Dojo so I have my academic Foundation as a mathematics graduate and I have also been participating in teaching various courses here at data science Tojo including the data science boot Cam and python for data science and while it comes to my hands-on experience I work on cloud-based AI machine learning projects uh and the goal of that is to of course turn complex Concepts into action insights for diverse audiences now in our session today uh this is what the agenta is going to be we will be covering pattern recognition in Ai and we'll be understanding what role mathematics plays in AI we will be learning how we can turn those patterns of AI into mathematical equations we'll also be understanding some mathematical Concepts such as regression and Bri and desense and we'll be touching upon calculus and linear algebra use cases in Ai and then we will be looking at word and sentence embeddings and similarity metrics so without further Ado let's go ahead and get started with our session for today now ai or artificial intelligence is mainly based on pattern recognition the goal of AI is to basically observe your data and identify any sort of underlying patterns or relationships are present over there for example if you were to give it a picture of a cat it's going to extract some information from that by recognizing those whiskers or those pointed ears so the AI is basically recognizing some sort of shapes edges or objects with that use case or if you were to give it some text it will try to find some sort of meaning and context from the words that are being used in that text right so simply AI is basically all about finding those patterns so you start off by giving it your raw data that data is is cleaned and pre-processed and from that some sort of information or features are extracted and that basically is used to figure out what pattern is present within your data once that pattern has been uncovered you can go ahead and start caring out your analytics or you can go ahead and start carrying out predictions and these can assist a business in making some informed decisions now here we have an example of a neuron Network which basically is given an image and the goal of that neural network is to take that image and extract some features from it so basically it goes through some layers and after that that image is simplified into a vector or a matrix so what's happening over there is that there is some sort of pattern recognition happening now our goal is to understand how we can take those p and represent it with a mathematical equation so the role of mathematics in AI now math is very critical because it helps us in training the models making predictions and understanding different behaviors of AI system and mainly math can help you bridge the patterns and equations together for example if you were to see any sort of pattern present than your data for example if I was collecting my data of how much I watch YouTube videos and I noticed some sort of pattern that at 900 p.m. I watch certain categories of videos now that's a pattern that I've observed and I want to turn it into an equation and use that equation to make predictions about the future and for doing that maths comes into place so there are different concepts of maths that actually help you out with AI and that includes linear algebra and calculus now linear algebra basically helps you convert those patterns into matrices and calculus helps you optimize equations to improve pattern accuracy so let's go ahead and dive into each of these in details and understand how does that actually happen so for starters in our journey of turning a pattern to an equation let's imagine a data set that consist information about different houses and each houses will have a respective price let's take out one data point from my data set this is basically a house that is 1500 square ft and it has a cost of $300,000 we are going to treat this data point as a DOT and we're going to try to plot it on a graph now in this graph what we are doing is we are assuming our xais to be the size of the house and our y AIS is the price of the house in $1,000 so if you were to take this data point which was, 1500 square ft and it cost $300,000 it's mathematical representation is basically going to be X is 1500 and Y is going to be 300 which is basically $300,000 so if I were to plot this on a graph you see this the x value of 1500 and this is the Y value of 300 it shows up as a singular dot on our graph so what we have done we have taken one data point and we have mapped it onto this graph now we can repeat that process for different data points so over here we have a bunch of different houses this house is 100 square ft and it cost $200,000 the next house 12200 ft cost $240,000 and so on each of these data points can be mapped onto this graph remembering that on our x-axis which is this axis we are plotting the house size and on our y-axis we are plotting the corresponding price so going over one each of these one by one this house is 1,000 square ft and it cost $200,000 so price in $1,000 would be 200 so this will be our first data point now repeating this for the remainder of the data points this house is 1200 ft cost $240,000 so 240 and 1200 this will be our next data point and we can keep repeating that for the remainder of the data points what we have done is basically we have taken our data and we have represented it into a graph now looking at this graph there is going to be some sort of pattern that you're seeing and that's a very simple pattern the pattern that you're seeing over here is that as you as the house size increases so does the price of the house that means a bigger house is going to cost more now the pattern that you're seeing over here can be represented with the help of a line and this line basically suggests that there is a linear relationship between the size of the house and the price of the house as the house size increases so does the price and we had all these different data points and what we did we basically created a line that passes through all those points this is basically a linear relationship that we're representing with the help of this line now we noticed this pattern in our data that as the house size was getting increasing the price of the house was increasing so what we did we basically represented that with the help of the line now going back to uh your high school pre-calculus you would remember that lines can be repres presented with the help of some equations so we notice this pattern and now our goal is going to be to represent it with the help of an equation so there are two things that we're going to be covering right now one is going to be the slope of a line and the second is going to be the intercept of a line a slope of a line is basically how inclin that line is think about climbing up a mountain if it's very steep it's going to have a greater slope and if not it's going to have a lesser slope so looking at all of these lines this line is very incline that means it will have a greater slope then a line like this will have a lesser slope and a line like this will have a lesser slope so this is incl very inclin this is inclin but less than this this one has no incline because it's just a flat line and this one isn't the opposite direction so this is negatively inclined now you can associate a number with all of these inclinations so just for simplification purpose is let's assume that the slope of this line is five the slope of this line is one and since this has no slope it's going to be zero and since this goes in the opposite direction it's going to be negative now that's the first thing the slope of a line the next thing is going to be the Y intercept which basically means the place where the line cuts the y- axis this is the y axis and that's the place the line cuts it this is going to be your Y intercept now the slope that we just talked about that can be calculated with this formula which basically is change in y divided by change in X Change in y means how much y value changes and change in X is basically how much your x value changes so let's assume with an example this was what your equation was like and you want to represent it with the help of the equation that we just talked about which was y = to MX plus C where m is the slope of a line and C is where the line cuts the Y AIS so m is calculated by basically measuring change in y / change in X and C is basically the Y intercept so if you were wanted to represent this equation it would end up looking something like this where you have this value as the change in y and this value as the change in X so if you're going from this point to this point your value of x changes by 500 and if you're going from this point to this point your value y changes by 100 right so basically just pick out any two random points from this graph and notice how X and Y are changing going from 1,00 to 1500 your X changes by 500 and going from 200 to 300 your y changes by 100 so if you wanted to calculate the value of M that's basically going to be your change in y divid by change in X which is 100 upon 500 which ends up giving you the value of 0.2 now that's the first part of your equation m y = to 0.2x the next thing is going to be the intercept which basically means where does this equation once you extrapolate it cuts the Y AIS and over here if you see it cuts it at yal to0 so that's basically what your intercept is going to be so putting all these values back into the equation y equals to your M value is 0.2 and your C value is is zero so you end up with this equation Y = 0.2x so as a recap you started off with all these different house prices along with their sizes and you've noticed that there was a pattern that as your house size increases your price increases and you ended up converting that into this simple equation and what this equation does is that it lets you make predictions if you were given the the value of x which basically means the size of the house you could predict the value of y which is going to be the price of the house so you had your on the x-axis you had the size of house and on the y- axis you had the price of house so for carrying this calculation out what you will do is assume that you were given the size 2,000 Square ft all you have to do is multiply it with the slope putting this value over here you end up with Y as 400 that basically means if the size of your house was 2,000 square ft your price of house will be $400,000 this multiplied by th000 now that that's a simple case right this is the case of linear regression where your equation can be simplified into a linear line whatever pattern that you're seeing is linear now that might not always be the case your equations can be polinomial and there are other cases that you can end up with but for a simple case that we're talking about right now what we did we found a pattern and we basically turned into this mathematical equation which basically means that your price of house is equals to 0.2 times the size of house and you can replicate this for other use cases as well now calculating the gradient of a uh linear line is very easy right it's Delta y upon Delta X but what about if the line was curved how do you calculate the gradient in that case what if I wanted to know what the slope was at this point or if I wanted to know what the slope was at this point looking at a linear line the slope remains constant throughout all you have to do is just do a Delta y upon Delta X or or you can do it over here as well or you can do it over here as well but what if the line was a slope for that what you do is you basically will build a line that's perpendicular to that point sorry uh parallel to that point and you will measure the gradient of that so if I were to build a line over here and I measure the gradient of that that would be the gradient at that point let's understand this better with diving more into details about how calculus in AI works so calculus is basically essential for understanding how AI model learns from data and it helps us calculate changes and optimize model through mathematical functions now previously we were using this notation where we were calling the change in the value of x as Delta X and the change in value of y as Delta Y and this was one coordinate X X1 y1 and this was the other coordinate X2 Y2 now what we're going to do is we're going to just change this notation a bit and instead of saying y we're going to replace it with f ofx which basically means f is a function of X so we're just renaming the notation everything else is the same so instead of Y I'm saying F ofx and X Remains the Same now for X initially we had X1 and X2 what I'm going to do now is that instead of saying X2 I'm going to call it x + Delta X let me explain what that means this was our value of x and the difference between X and X1 and X2 was Delta X so instead of saying X1 and X2 what I'm just doing it that this is X and since the distance between these two points is Delta X this value is going to be X+ Delta X so now this value is X Plus Delta X and this value is X so this value is going to be f of x and this value is going to be f of x plus Delta X which basically means that f is a function of X right this was clear so over here it's simple now the value of x is X Plus Delta X has increased slightly by this number so we're going to replace this with X Plus Delta X and this is the notation that we're going to use moving forward now if I wanted to measure the gradient of this line it's the concept is going to remain the same it was change in y divided by change in x what that means is that your Y is changing by this much and your X is changing by this much so this is your value of Y the second one and this is your value of y the first one so it's basically f ofx plus Delta X this is the second value minus F ofx which is the first value and then we're dividing it by the change in the value of x which is going to be Delta X and that's basically what the slope of this line is right it's just the same thing as Delta y upon Delta X we have just slightly changed that notation and represented it like this now this is going to help us with calcul ating derivatives for the cases where the line is not a straight line over here if you think about it we had this curve right we cannot simply use Delta y upon Delta X to calculate the gradient because the gradient at each point is different the line at this point is like this the line at this point is like this then it just keeps getting flat so the gradient is constantly changing and to calculate the gradient at each point we need to look at it a bit zoomed in if we had a curve if you were to zoom in for a brief period of time it will just look like a straight line now let's apply that logic that we had talked about before over here where the gradient was f of x + Delta xus f of x / Delta X now our goal is to make this distance as short as possible to the point it approaches zero what that means is that we are assuming that this change in X is getting very small so the limit of Delta X is basically approaching zero and using that we can calculate the gradient of the line at this point now it's important to understand gradients because they have many use cases in AI for starters um one second let me just go to the okay so for starters what we're going to be cing is the importance of gradients in gradient descent which is basically a way AI can use to improve itself if you have any sort of model and you want to make it better gradient descent can help you achieve that process so going back to our example of linear regression where we had this line which was representing some sort of data we had the house prices on the xaxis and prices on the Y axxis and over here what happens is that you have these data points and you're trying to make a line that passes through most of them or find the best fit line now in reality data is not very clean the pattern that you figure out it's not going to be 100% accurate there are going to be errors it could be due to um errors in measuring the data it could be that data does not follow an exact path so there is going to be some sort of error in your prediction and your real data so over here we have these data points and we also have this prediction that we're carrying out and there is going to be some sort of error between the real value and the predicted value now these errors can be represented with the help of some equations and over here we're going to be using something called mean square error which is basically what we can use to figure out how good is our model or how bad is our model we're basically calculating all these errors and we're summing them up together and that's what basically a cost function is a cause function is how AI measures how wrong it is think about it as a scorecard that basically tells you how far is the model's guesses from the actual answers for example if a model predicts three and the actual value is five there's this distance of two between the real value and the predicted value that is represented with the help of cost function and our goal is to basically minimize this cost function as as much as possible we want to make this value as small as possible minimize the error within our data and for achieving that gradient descent comes into place what it does is that it helps you figure out at what point is your cost the minimum so over here we have this cost function right that cost function can be presented in a graph and it will let's assume it looks something like this now in this C function we have this point which is called the Minima and at this point is your error the least and what you're trying to do is you're trying to reach this point you want to minimize your error as much as possible in your model there is going to be some error and you want to figure out how can we reach this point how can we minimize this cost function there are going to be some parameters that you want to play around with to end up reaching that point and for doing that Gan descent works so you have this CA function that's covered in orange color and you want to whichever point you start from you want to end up reaching this point which has the minimum cost function and for doing that you will be using something called the derivative of that line so at each point you will have a derivative of that line which is basically the gradient of that line so for example over here the gradient is let's start from the right hand side that's going much easier the gradient is a lot then the slope is decreasing decreasing decreasing till it gets flat which means the gradient over here is going to be zero over here it's the most let's say five then four 3 2 until it reaches zero while over here it's in the opposite direction so the gradient is going to be negative it's 1 -2 -34 andg 5 now whichever point you start from you want to go from that point up until this point now how do you figure that out how do you know should I go towards this side or should I go towards this side well the derivative helps you achieve that process basically if you if you we just talked about how we calculate those D derivatives of these lines right we basically are going to build a parallel line and we will calculate its slope or its gradient remembering that the gradient over here is negative and the gradient over here is positive so how does the model know should I should it take a step in the right hand direction or the left hand Direction well if it's positive if the gradient is positive it knows to move towards the left so if the slope of the line is positive go towards this direction and if the slope of the line is negative go towards this direction so if you were to start from this point as an example it will first figure out what the gradient of the line is at that point and over here since it's positive it knows to take a step in this direction and it will move further and it will keep doing that until the gradient becomes zero remembering that a slope of a line like this is zero so this is the part where your cost is minimum this means this is the cost where you are reaching the least uh error associated with your model so your goal is to reach that point and with the help of the gradient descent you can figure out what the optimal point is and using that that will give you your optimal weights for your machine learning model so think of it is like rolling a ball down the hill it finds the lowest point and yeah that's basically all about gradiant descent now moving on we're going to be covering the next topic which is going to be linear algebra and lar algebra in AI is basically the Foundation of many algorithms and data science especially those involving data transformation and optimization so there is a huge boom in the generative AI field uh llms and generative AI is at the very Peak right now chart GPT and Bard and all these other models at the back end they are making use of linear algebra which is why we're going to be taking the remainder of the session to understand how lar algebra works and how they work in different machine learning algorithms so for starter the first concept that we're going to be covering is going to be vectors and vectors basically represent data points as coordinates in multi-dimensional space so let's say that we have a vector a which is 2 1 and we have another Vector which is B52 so basically think about these as individual points0 and that's going to be your or vector a and then we have the second Vector B52 which is going that's your vector basically the goal of vectors is to your data points inial space now you have am restarting my screen share okay so we were talking about vectors and the next thing that we wanted to cover was going to be vector addition let's say you have this Vector a which has the points 225 689 and B which has the point 52 651 now if you want to add those two vectors together a which is something like this and then B which is let's assuming something like like this and you want to add those two vectors together to get a plus b the calculation goes something like this you have each of these individual values added together one by one so these first two values get added together then these next two values then these next two values then these next two values and then these final two values so if you want to add a plus b it's going to be 2 + 5 2 + 2 6 + 6 8 + 5 and 9 + 1 that is 7 4 12 13 and 10 so the sum of these two vectors is basically 7 4 12 13 10 now why do you uh why are we talking about vector addition why are we talking about vectors well that is to come in the later slides when we start diving into the import how embeding work and how are they used to create your generative AI models and so on before that let's talk about something else which is matrices so your AI relies heavily on vectors and also on your matrices your vectors were basically uh just your uh simple vectors now matrices it adds up into a grid of numbers previously only had a single list of numbers with matrices is going to be a grid and matrices can be used to represent your data and inputs and outputs like this for example assume that you had an image of this number five in black and white and you wanted the machine to be able to use it and interpret it and make use of any machine learning models how would you go ahead and do that how can the machine understand this image at this at the moment it's basically your raw data or your model that you're building whatever AI that you're using they will not be able to understand it until you transform it into a form of numbers and you can achieve that by converting it into a matrix representation and how does that work well think of this box as a 7x6 matrix and each pixel will represent one color right so let's build this box quickly and over here this box is completely wide this box is also completely wide this box is also completely wide all of these are completely wide so what we're going to do is we're going to use this representation where white is represented by zero and something that's completely black is represented by 255 so your numbers over here are going to be 0 0 0 0 0 and 0 this is also zero but over here this box is completely black so this will get the number 255 similarly this will also get the number 255 255 this box is slightly gray so it's it's more towards White and less towards black so it's be something like uh let's say 50 depending on how the intensity of the black color is and this is again white so that's zero and that will repeat for the entirety of the image so this is something like 50 this is 0 0 0 this might be something like 20 this is also something like 20 all of these blacks are to 55s and all these whites are zeros so what you have done is you have basically taken this image and you have now converted into this 7x6 Matrix where you have all these uh seven values over here and you have these six values over here now once you have your data converted into a matrix format you can then pass it to the machine and it will be able to interpreted and it can apply different processing and calculations on top of now we do the same thing for text Data as well and that's what goes be behind large language models which is taking your text data and converting it converting it into some sort of numerical format and this is achieved with the help of word edics so word edics are basically words being mapped to some sort of vector space based on their meaning and the goal of these embeddings is to basically capture semantic relationship between different words like words like king and queen they are very similar to each other and they're not going to be very similar to Apple so once you create those embeddings king and queen are going to have similar representation than Apple so in mathematics if you were to take an example something like cat would then be mapped to something like 0.12 0 negative 0.34 2.5 3.7 and 1.9 this is just a random example and you might be wondering where do these numbers come from well these numbers are extracted to the help of neural networks and what they do is they try to find what sort of pattern is present within your data so words that are appearing together such as dog and pet will have similar Vector representation and that's because in a document where somebody says I have a dog they will also say that I pet my dog or something like that similarly if a document is talking about King it might also talk about Queen so wor that occur together they end up having similar Vector representation that means that that they are closer to each other so if it's what what do you mean by similar well if it says 0 0 0.5 and one this will have similar values 0 0 maybe something like 0.55 and 0.91 these two values are very close to each other and these two values are very close to each other and these two values are the same so basically what it does is that it uses the information of occurring together to generate these embedding and the values are so that's what happens over here with your word embeddings now models like word to what you do is you feed them large amount of text documents and what they do do is they automatically extract those patterns present within those text documents or basically let's say if you feeded a lot of English documents it will try to figure out how are words in English language to each other they might be used interchangeably so they these words since they're occurring together they sorry uh I think my voice was breaking off so let me repeat so words that occur together and words are similar to each other they will end up having similar representation so that's how what we do we take those text words and we come up with this mathematical representation for it now why is this important well fors this sentence consists of six different words and each of these will end up having some sort of vector representation for simplification purposes let's assume that the has the number 0.1 0.2 0.3 cat has this number sat on the map right so this sentence each of the word has this respective representation now if you want to see how can we represent this entire sentence mathematically we would have to combine all these numbers together somehow and for doing that what you can do is you can take an average of the vectors for the sentence so you have this word sentence the C sat on the mat and you have all the individual uh numbers for those vectors so if you go back all of these are here and I want to see how can I represent this entire sentence with the help of a vector well I will take these unique representation add them up together so these all added together will give me one value these all added up together will give me one value and these all added up together will give me another value since I have a total of 1 2 3 4 five six six words I'm going to divide it by six so as an example let's do this 0.1 0.5 0.2 0.3 0.1 and 0.6 adding them all up together divided by six we will end up getting the average value so for example we have this cat sat on a map this will be averaged out into a value something like 0.338 0.336 and 0.3 so that's what we did right now we represented one sentence right so if we had a bunch of documents we can represent each of those documents into their respective representation now how does this help us converting documents into these embeddings it let lets us apply a lot of different machine learning algorithms for starters we can see which article is more similar to another article we took an article we calculated the individual word embeddings and then the combined embedding average and it looks something like this and now I want to see which other article is this this the most similar to so for doing that we have these similarity Matrix so let's talk about cosign similarity which basically tells us how similar is one document to the other so if I go back what I've done is I've taken my document and I've represented as a point another document again as a point and another document as another point and this distances between these points B basically tells me how similar are these documents to each other if it's very further apart that means is less similar and if it's closer together it means it's more similar so if if this distance is short that means the document is very similar and if the distance is large it means those two documents are not very similar now the distances can be calculated using different formulas one of them is ukan distance which looks something like this X2 - X1 + Y2 - y1 which basically means we subtract this from this their X values and then we subtract the Y values so as an example let's just do one quickly let's say that we have this point 3 comma 0 and we have this point 0 comma 1 this is X1 this is X2 this is y1 this is Y2 and if you were to measure the distance it's going to be 3 minus 0 2 + 0 - 1 2 and this becomes 9 - 9 + 1 that's the under root of 10 that's the distance between this now if this was under root 10 and this was under root 51 we know that this distance is much larger so this document is not very similar to this document but it is very similar to this document because this distance is much shorter now this distance can also be calculated using cosine similarity which basically tells you which is another Matrix to figure out how similar are documents to each other now uh the formula for that is basically written over here and it basically rather than seeing the distance it sees the angle between points so if the angle is shorter let's said that this was one vector this was the other vector and this was the third Vector this this angle is shorter that means A and B are more similar this angle is much bigger that means A and C are less similar now uh you can learn more about these similarity matrics by Googling them and how you can calculate them um since we are short of time I would like to uh allow some time to the questions that anyone might have so this would be a good point to wrap up our session and I hope this session was very helpful to all of you as a starting point for understanding how mathematics can be implemented into different domains of AI I will now open the question Forum to different questions that anyone might have so that we can um uh address them quickly all right so let me just go through the chat I'm just going to the chat just give me a minute okay so there was one question about uh the data is not that messy thus the cost function is not necessarily smooth but quite bumpy and chaotic okay so um so um it could be so the cost function is basically uh an equation that you create yourself right so um you figure that out with the help of the data that you had in our use case uh these were just sample visualizations and graphs they were not based on real life data so it might be that at one point it could be not matching the other one this was just for explanation purposes so yeah I hope that addresses your question let me just go through the chat from other questions okay so there is one question about how do we apply this in coding so um this is this was basically the goal of understanding how these different models work right what is happening at the backend of these models when it comes to coding um all the mathematics has already been done for you majority of the times the models that exist have all these uh things already written for you and sometimes it's pretty much as simple as just writing just one line of code that does your task for you however it's very important that you also understand what's happening at the back end uh because if you were to use a model without having any information about it you might end up piing the wrong model in the wrong scenario right so the the goal of this session is to understand what happens at the back end of AI um for implementing all of this in uh activities and coding uh there are numerous resources available out there for python or there are also drag and drop models available out there um so yeah this is this was what the goal was right and can you recommend any simple tools for experimenting with creating simple models um so if you want to create if you don't want to dive into much about coding um there are methods available which have a drag and drop interface where you're actually not writing the code all you're doing is just dragging and dropping and using the tools as is uh for example Azure uh machine Learning Studio it provides you the opportunity for carrying that out uh but for if you want to dive into more about coding I personal preferen is python Some people prefer using R uh there are again numerous resources available on the web for that uh for creating the models from scratch or using existing libraries in Python SK learn has all these different libraries that allow you to carry out different sort of um machine learning models building and all of that um so yeah hopefully that addresses your question um so there is the question so one needs to know calculus in algebra and Statics and probability to understand machine learning better is there a sequence one should follow to learn these math topic so I would say there is no exact sequence that you would need to follow uh calculus so these are covering some of the concepts and areas where uh these come into play in Ai and U machine learning right there it actually goes beyond that it has many other use cases I would say that we have only touched the tip of the iceberg it was a very introductory section um there are other areas as well where these mathematical Concepts come into place right a play sorry um so calculus linear algebra and statistics and probability these are the things that actually are used in Ai and data science throughout each and every single uh domain that you will come across in different or various ways so there is no exact sequence that you should follow it's recommended that you know all of these um so you can understand how exactly is your machine learning model working or your AI model working um it's good to know the theory behind what's exactly happening so in case if you run into any errors or issues you can debug it and get improve the model further and get to a better solution so yeah hopefully that addresses your question um so there is a question about explaining mini batch and sastic gradient descents in the context of epoch so that would be an entire webinar on its own it's a very topic but I would suggest that going to that you can make use of and understanding this better all right that sounds good thank you Sandra uh yeah so so I guess that is all the questions uh the