Taylor Series in AI.
Source: tai_blog

P.S. Read thru this article a bit slowly  word by word; youll thank me later ;)         Lets see what the Taylor Series is and how it relates to its applications in AI & Processing.   The study of Taylor series is largely about taking non-polynomial functions and finding polynomials that approximate them near some input  3Blue1Brown.   Okay  lets try to rephrase that to understand better:   Imagine you have a really complicated function  like a curve on a graph  and you want to understand what it looks like near a certain point. The Taylor Series helps us do this by breaking the function into a bunch of smaller  easier pieces called polynomials.   It is a way to approximate a function using an infinite sum of simpler terms. These terms are calculated using the functions values and its derivatives (which tell us the slope and how the function changes!).   Consider this:If you have a function f(x)  and you want to approximate it near a point  say at x = a  then this is what the Taylor Series looks like:   f(x) = f(a) + f(a)(x-a) + f(a)/2! (x-a) + f(a)/3!(x-a)   Take a second to go thru that again.   Here    f(a) is the value of the function at x = af(a) is the slope at x = af(a) is how the slope is changing at x = aWe all know that n! stands for n factorial  which is the product of all positive integers up to n.   ex: 3! = 1 x 2 x 3 = 6         Lets look at a very simple example to understand this better: the exponential function of e^x.   For e^x around x = 0 is:   (try formulating it yourself first  referring to the formula above ;)   e^x = 1 + x + x/2! + x/3! + x/4!    ConceptualThink of the Taylor Series as a recipe for building a copy of a function near the point a  sort of like a stencil. The more terms  or in this case  ingredients you add  the closer you will get to the original function  and the closer your approximation will be.   So  if you want to estimate e^x for small values of x  you can just use the first few terms:   e^x = 1 + x + x/2 + x/6   This exercise should give you a good idea of how e^x looks like at x = 0.   Pro-Tip: Repeat this exercise a few times to better grasp the concept.   Okay  so what? How is this useful in the real world?Well  The Taylor series allows us to approximate complex functions with simpler polynomials  which makes calculations easier and faster!   Here are a few examples    PhysicsExample: Pendulum Motion   Imagine a pendulum  like a clock. Scientists use math to understand how it swings. The exact math is tricky  but for small swings  the Taylor Series helps simplify it  making it easier to predict the pendulums motion.   So that you can be late for school.   EngineeringExample: Control Systems   Think about a cars cruise control  which keeps the car at a steady speed. Engineers use the Taylor Series to simplify complex math so the system can react smoothly and keep the car at the right speed.   So that you can ignore the speed limit.   EconomicsExample: Interest Rates   When banks calculate interest on savings  they sometimes use complicated formulas. The Taylor series helps simplify these calculations so they can more easily determine how much money youll earn!   So that the government can take the right percentage of that in taxes.   Computer ScienceExample: Machine Learning   In ML  computers learn from data. The Taylor series helps simplify the math behind these learning algorithms so computers can learn faster and more effectively.   So that you become lazy and spend all day on them.   MedicineExample: Medical Imaging   When doctors take MRI or CT scans  they receive a lot of data. The Taylor Series helps turn this data into clear images of the inside of the body  making it easier for doctors to diagnose problems!   So that you ignore their advice and walk to McDonald's (cuz you dont run XD)   Everyday TechnologyExample: GPS Systems   When you use GPS on your phone  it calculates your location using satellites. The Taylor series helps make the math simpler so your GPS can quickly and accurately tell you where you are.   So that you can lie about where you are.   Weather ForecastingExample: Predicting Temperature   Meteorologists predict the weather using complicated math. The Taylor series helps simplify these equations  allowing them to make more accurate forecasts about temperature  rain  and wind.   So that you never open the weather app and always forget an umbrella.         So YOU might not use the Taylor Series in the real world  ever; but its used every day to make your life simpler!         Now  for the interesting bit:   How do we use the Taylor Series in AI? U+1F525Youve already taken a look into how this is used in ML above and how it helps simplify the math behind these learning algorithms so computers can learn faster and more effectively.   Lets dive deeper:   First  where can we even use this in AI?Forget the term AI for a while. Just think of where we use the Taylor Series in everyday mathematical and engineering problems. We can later extrapolate that into how we use it in AI and Machine Learning.   Weve already discussed how we use it in physics  engineering  economics  CS  medicine  GPS  and weather forecasting. I suggest you scroll back to that again; itll click more now and at the end of this article. U+1F5B1   In AI  we often deal with complex math problems. The Taylor series helps simplify these problems so our AI can learn and make better decisions.   Example:   For Training AI Models:When we train an AI model  like a neural network  we want to improve its prediction accuracy. We do this by adjusting its parameters (like weights in a neural network) to minimize errors. (w&b)   Taylor series helps here by letting us approximate how small changes in the parameters will affect the error. This approximation helps us find the best way to adjust the parameters to improve the models predictions.   Training Neural Networks:When training a neural network  we want to minimize a loss function  which is how we measure the difference between the predicted outputs and the actual targets. To achieve this  we adjust the networks parameters (weights and biases) to reduce the loss. This is usually done by using gradient-based optimization methods.   ExampleImagine youre on a big hill and you want to find the lowest point. To get there  you need to figure out which direction to walk.   The Hill: Think of the hill as the loss function  which shows how good or bad your predictions are. The steeper parts of the hill represent higher loss (bad predictions)  and the flatter parts represent lower loss (better predictions).Finding the Best Path: When youre on the hill  you cant see the whole thing  just the part right around you. To decide which way to walk  you use the slope (how steep it is) right where you are. This is like the gradient in ML  which tells you the direction that increases the loss the most.Using the Slope: If you want to get to the lowest point  you walk in the opposite direction of the slope (since you want to go downhill). You keep taking small steps in this direction to lower the loss.Where does the Taylor Series HelpThe Taylor series is like having a small map that shows you how the hill looks around you. It helps you understand the local slope better  so you can make better decisions about which way to walk.   Simple Map: The basic Taylor series is like a simple map that shows the hills slope around you.Detailed Map: If you want a more accurate map  you might also look at how the hill curves  which is like adding more details to your Taylor series.1. Training AI Models: Gradient DescentCost FunctionSame analogy again: Imagine the cost function as a hill we need to climb down to find the lowest point (the best solution). As stated  the lower the value  the better it is.   GradientThe gradient tells us the direction of the steepest slope.   Gradient Descent:The Taylor Series helps us approximate the cost function around a point  telling us how it changes when we adjust the parameters slightly. This approximation makes it easier to determine which direction to move in to reduce the cost.   Example:   Imagine youre trying to adjust the angle of a ramp to make a ball roll into a target. The cost function tells you how far the ball is from the target. The Taylor series helps you understand how changing the ramps angle (parameters) will affect the balls position (cost) so you can make better adjustments.   2. Making Calculations EasierNeural networks use something called activation functions to decide whether to activate a neuron (like a switch). One common activation function is the sigmoid function.   ExampleThink of the Sigmoid Function as a dimmer switch that adjusts light brightness. The Taylor series helps simplify the math behind how much the light should dim based on the input  making it easier for the neural network to process. It helps a neural network decide whether to activate a neuron. The Taylor series can approximate this function and speed up calculations.   3. Approximating Complex FunctionsIn Reinforcement Learning  an AI learns by trying different actions and getting rewards or penalties (trial and error). The value function estimates the expected rewards for actions.   How the Taylor Series HelpsThe Taylor series approximates the value function  which can be very complex. This approximation helps the AI predict rewards more easily  allowing it to choose better actions.   ExampleImagine youre playing a video game  and you want to predict which moves will earn you the most points. The value function helps with this prediction  and the Taylor series simplifies the calculations  making it easier to decide the best moves.   4. Handling Uncertainty: Bayesian InferenceSometimes  we need to understand how uncertain our AI model is about its predictions. The Taylor series helps us estimate this uncertainty  making our AI more reliable.   Example: Bayesian InferenceIn Bayesian inference  we update our beliefs about the AI models parameters based on new data. The Taylor series helps simplify these updates  making them easier to calculate.   5. Understanding Model BehaviorThe Taylor Series can also be employed to understand and interpret the behavior of machine learning models. By expanding the models function around a point  we can gain insights into how changes in input affect the output  which is crucial for tasks like feature importance analysis and debugging models.   Specific ApplicationsNeural Networks Training: In training neural networks  the backpropagation algorithm often uses the Taylor Series for calculating the gradients of weights.Regularization Techniques: Some regularization techniques in machine learning  like Tikhonov regularization  can be understood and derived using the Taylor Series expansion.Non-linear Models: For non-linear models  the Taylor Series provides a way to linearize the model around a point  which is useful for analysis and optimization.Algorithm Development: Advanced machine learning algorithms  like Gaussian processes and some ensemble methods  sometimes use the Taylor Series for development and refinement.The fundemental intuition to keep in mind is that they translate derivative information at a single point to approximation information around that point  3Blue1Brown         So  with the multiple examples and instances  weve discussed how the concept of the Taylor Series eases our lives  from real-world applications in Engineering & Computer Science to how it simplifies working with and building AI.   I think that the Taylor series is like a magic tool that turns complicated math into simpler math because it helps AI learn faster  make better decisions  and handle complex problems more efficiently. Thats the inference and understanding I got from the research Ive done and while drafting this article.   Now  as were approaching the end  I want you to reflect back: What exactly do we mean when we say Taylor Series  instances of using it irl  examples of Taylor series use  and finally  the cherry on top  how do we use Taylor series in AI.   Read through the entire article again  and compare it with the understanding you have now; youll notice the difference  as I did ;)   Thats it for this time; thanks for Reading and Happy Learning!   References: How I learned this concept Taylor series U+007C Chapter 11  Essence of calculus (youtube.com) (3Blue1Brown)   Exploring the Role of Taylor Series in Machine Learning: From Function Approximation to Model Optimization U+007C by Everton Gomede  PhD U+007C  . U+007C Medium   A Gentle Introduction to Taylor Series  MachineLearningMastery.com   How is Taylor series used in deep learning? (analyticsindiamag.com)