<!-- Abstract: CS 461 Introduction to Deep Learning Lecture 01: Introduction --> <!-- To left align everything <style type="text/css"> .reveal p { text-align: left; } .reveal ul { display: block; } .reveal ol { display: block; } </style> --> # CS 461 - Lecture 01 ## Machine Learning Principles Bernhard Firner 2025-09-08 --- ## Course Details * My email: `bfirner@cs.rutgers.edu` * Canvas * Not yet * Office: Hill 273 * Office hours: TBD --- ## Syllabus: Grading * Current plan * 20% Quizzes * 50% Assignments * 30% Final Exam --- ## Syllabus: Tools * Python * Numpy * Maybe Pandas or PyTorch for specific tasks * For example, to show how SVMs can be used with feature vectors from a neural network * In general, we want you to learn how to implement things on your own --- ## Academic Integrity * [CS Academic Integrity Policy](https://www.cs.rutgers.edu/academics/undergraduate/academic-integrity-policy) * Feel free to use any LLM in this course * Record all prompts and responses * We will use them as part of a discussion at the end of the course --- ## Absences & Late assignments * Use the self-reporting tool * [https://sims.rutgers.edu/ssra/](https://sims.rutgers.edu/ssra/) * Late assignments * 20% deduction for up to 2 days late * 50% deduction for up to 7 days late * Exceptions for major illnesses * Missed exams * You must have a valid excuse to schedule a make up exam --- ## Exceptions * Inform me ahead of time of special circumstances * Meaning non-emergency situations * If you have an emergency, please don't stop to email me * You can self-report after the fact --- ## Appropriate Conversations * I can help with problems related to this class * Can probably help with professional questions * I'm not a psychologist. I probably can't help with other issues. * I am also required to report some topics, so don't assume everything you tell me will remain private --- ## My Background * 8 years doing autonomous driving at NVIDIA * Hundreds of miles between highway failures after 4 years * Other industry experience in avionics, embedded systems, and autonomous drones * You may think that we stopped using "old" machine learning techniques * Not true; I saw them used constantly --- ## Course Goals * My goals * Teach you enough that you can use ML in real life * Give you a strong foundation for future courses * Your goals? --- ## Material * CS440 is AI * CS461 is ML, a subset of AI * CS462 is DL, a subset of ML --- ## Machine Learning * Nowadays we do this with computers * Origins are in hand calculations * Important distinctions from other approaches * Data-driven * Make assumptions about statistics of data * About noise, bias, and measurement errors * Techniques are formulaic --- ## Rough Course Outline 1. Least squares fitting, regression, classification, and statistical models 2. "Model free" machine learning 3. Introduction to deep learning --- ## Course Philosophy * Try to tell a story from early techniques through today * Neural networks may overshadow other techniques * But NNs are not always appropriate * And some techniques work well *with* NNs --- ## Early Motivations * Astronomers were the rock stars of their day * Early astronomers used movements of stellar object to predict the future * Later astronomers used math to predict future movements of objects * Many techniques were created in the 1700s * There is a lot to be learned from a brief history lesson --- ## Epicycles <style> .container { display: flex; } .col {flex: 1;} </style> <div class="container"> <div class="col"> * Kepler's [laws of planetary motion](https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion) were published in the early 1600s * Finally supplanted [epicycles](https://en.wikipedia.org/wiki/Deferent_and_epicycle) for good * Epicycles had planets moving in circular orbits and smaller cycles overlaid on top of the larger orbit * They matched observations from Earth * Other planets' relative speed isn't constant </div> <div class="col"> <img style="width: 85%" class="r-stretch" src="./figures/Cassini_apparent.jpg" alt="A picture of a geocentric model with spirograph-looking orbits of the planets. By James Ferguson (1710-1776), based on similar diagrams by Giovanni Cassini (1625-1712) and Dr Roger Long (1680-1770); engraved for the Encyclopaedia by Andrew Bell. - Encyclopaedia Britannica (1st Edition, 1771; facsimile reprint 1971), Volume 1, Fig. 2 of Plate XL facing page 449., Public Domain, https://commons.wikimedia.org/w/index.php?curid=10884763" /> </div> </div> --- ## Early Overfitting * It turns out that epicycles can be used to match *any* orbit * The model was too powerful! * This is a fundamental problem with many ML techniques * If you decide upon a model, how do you know it is the right one? * If you can't make predictions using a model, what is the point? --- ## Predictions * Kepler's laws are relatively simple * The equations governing motion are not * Especially not when combined with measurement error * This had practical impacts when attempting predictions * Ceres is a well known example --- ## Gauss * Gauss predicted Ceres' reemergence from behind the Sun's glare * Big result at the time * That name should be familiar; he discovered the Gaussian distribution * Happened during work on estimation --- ## Predicting a World * Gauss solved for Ceres' orbit by combining days worth of observations * The number of observations outnumbered the unknowns in his equations * However, each observation had some error * This is where statistics comes into play * As the number of samples grows, the mean of the errors approaches 0 * If the errors are unbiased and uncorrelated --- ## Morals * Machine learning is all about models and statistics * Is your model capable of solving the problem? * Do you have enough data to make your statistics favorable? * Do you understand the noise and errors in your data? --- ## Notation * This should be a review * Let's say we want to predict the value of a variable * $Y$ is the thing we want to predict * $X$ is the information we use * The thing we use to predict is the model * It is a function * Technically, $\hat{f}(X) = m$ * The hat means that $f$ is an estimator --- ## Talking About Error * How wrong is a model? * *Mean squared error* is our typical error function * Going to write that as *MSE* * The expected error of a model using MSE is * $ \mathbb{E}[(Y - m)^2] $ --- ## Bias and Variance * We can remove the variance term * $\mathbb{E}[(Y - m)^2] = (\mathbb{E}[Y-m])^2 + Var[Y-m]$ * The first term is now the squared bias of our estimating function * The second is the variance, and reduces to $Var[Y]$ * Since our estimate has no impact on $Var[Y]$ * $\mathbb{E}[(Y - m)^2] = (\mathbb{E}[Y] - m)^2 + Var[Y]$ --- ## Bias * Bias is $\mathbb{E}[\hat{f}(X) - Y(X)$ * An unbiased estimator has 0 expected error * Not 0 error! * The sum of all errors is 0! * E.g., $\hat{f}(X) = 0$ is an unbiased estimate of $sin$ * But a terrible one; the variance never goes to 0 --- ## Best Predictor * It should be obvious that the best predictor is an unbiased one * If you were to only make one guess for all values of $x$, $m(x)$, guess the mean of $Y$ * Regardless of the variance in $Y$ * If $Y$ has nonzero variance, then guess $m(x) = \mathbb{E}(Y|x)$ * How do we get the best estimate of the mean? --- ## Central Limit Theorem * If every dataset required new analytics, life would be hard * But we are lucky! * Let's say $Y = signal + noise$ * As $n \rightarrow \infty$, the mean estimate approaches the true mean * $\frac{1}{n}\Sigma_{i=1}^{n}X_{i} \rightarrow \mathcal{N}(\mathbb{E}[X], Var[X]/n)$ * True if samples of $Y$ are independent and have the same finite variance * This is `independent and identically distributed`, `IID` --- ## Reducing Variance * Sometimes we want to reduce the variance of our predictions * Even at the cost of increasing error * One example is $\frac{1}{(n+1)}\Sigma_{i=1}^{n}X_{i}$ * This is a biased estimator * Biased towards 0 * But still converges as $n \rightarrow \infty$ --- ## Challenges in ML * So more data means a better estimate * However, adding more data isn't always easy * Data is going to be contradictory * Data can grow stale * Distributions may actually be long-tailed --- ## Starting Example * Let's say we just want to guess a single number * Predict $y_i = \hat{f}(x_i)$ * Observe some $y_i$ given samples, $x_1, x_2, ..., x_n$ * Goal is to minimize $MSE$ --- ## Fitting a simple curve <style> .container { display: flex; } .col {flex: 1;} </style> <div class="container"> <div class="col"> * Let's fit a polynomial to the `sin` function * Range from $0$ to $\pi/4$ * We'll use a first degree polynomial to predict $y$ given $x$ * $y = \beta_0 + \beta_{1}x$ * This is our `model` </div> <div class="col"> <img style="width: 95%" class="r-stretch" src="./figures/ex1_sin.png" /> </div> </div> --- ## Without Noise * We'll have observations of the `sin` function from 0 to $\pi/4$ * This gives us `x`, `y` pairs * With no noise and a proper model we can easily minimize error * But what happens with noise? * $Y = \beta_0 + \beta_{1}x + \epsilon_i$ * That's next --- ## Estimating with Least Squares * Let's minimize the $MSE$ * $error = Y - \beta_0 + \beta_{1}X$ * $MSE = \frac{1}{n}\Sigma_{i=1}^{n}(Y - (\beta_0 + \beta_{1}X))^2$ * $ = \beta_{0}^2 + 2\beta_{0}\beta_{1}X - 2\beta_{0}Y + \beta_{1}^{2}X^2 - 2\beta_{1}XY + Y^2$ --- ## More Math... * Use the derivative with respect to $\beta_0$ and $\beta_1$ * Then set each to 0 * $0 = 2\beta_0 + 2\beta_{1}X - 2Y$ * $0 = 2\beta_{0}X + 2\beta_{1}X^2 - 2XY$ --- ## Samples * Let's say we have two samples: * $f(Y|x=0) = 0$ * $f(Y|x=0.5) \approx 0.48$ --- ## 2 Equations, 2 Unknowns * Plug in $f(X=0)=0$ * From equation 1 * $0 = 2\beta_{0}$ * $\beta_{0} = 0$ --- ## 1 Equation, 1 Unknown <style> .container { display: flex; } .col {flex: 1;} </style> <div class="container"> <div class="col"> * Plug in $f(X=0.5) \approx 0.48$ * $0 = 2\beta_{0}X + 2\beta_{1}X^2 - 2XY$ * $0 = 0.5\beta_{1} - 0.48$ * $\beta_{1} = 0.96$ </div> <div class="col"> <img style="width: 95%" class="r-stretch" src="./figures/ex1_solution.png" /> </div> </div> --- ## Observations * We just solved for the intercept and slope * But things will get complicated with more complex equations * Which is too bad, because our model is too weak * And what if there is noise? * How do we combine samples? * And how can we tell that our model is too weak when samples are noisy? --- ## Machine Learning * Matrix math the is the beating heart of ML * And I guess data is the air is breathes * Both made possible with computers * So we've tackled harder problems as a result * Next class: least squares with Gaussian noise * A solution that improve as we add data