Bloomberg ML EDU presents:

Foundations of Machine Learning

Instructor , Office of the CTO at Bloomberg

Understand the Concepts, Techniques and Mathematical Frameworks Used by Experts in Machine Learning

About This Course

Bloomberg presents "Foundations of Machine Learning," a training course that was initially delivered internally to the company's software engineers as part of its "Machine Learning EDU" initiative. This course covers a wide variety of topics in machine learning and statistical modeling. The primary goal of the class is to help participants gain a deep understanding of the concepts, techniques and mathematical frameworks used by experts in machine learning. It is designed to make valuable machine learning skills more accessible to individuals with a strong math background, including software developers, experimental scientists, engineers and financial professionals.

The 30 lectures in the course are embedded below, but may also be viewed in this YouTube playlist . The course includes a complete set of homework assignments, each containing a theoretical element and implementation challenge with support code in Python, which is rapidly becoming the prevailing programming language for data science and machine learning in both academia and industry. This course also serves as a foundation on which more specialized courses and further independent study can build.

Please fill out this short online form to register for access to our course's Piazza discussion board. Applications are processed manually, so please be patient. You should receive an email directly from Piazza when you are registered. Common questions from this and previous editions of the course are posted in our FAQ .

The first lecture, Black Box Machine Learning , gives a quick start introduction to practical machine learning and only requires familiarity with basic programming concepts.

Highlights and Distinctive Features of the Course Lectures, Notes, and Assignments

  • Geometric explanation for what happens with ridge, lasso, and elastic net regression in the case of correlated random variables.
  • Investigation of when the penalty (Tikhonov) and constraint (Ivanov) forms of regularization are equivalent.
  • Concise summary of what we really learn about SVMs from Lagrangian duality.
  • Proof of representer theorem with simple linear algebra, emphasizing it as a way to reparametrize certain objective functions.
  • Guided derivation of the math behind the classic diamond/circle/ellipsoids picture that "explains" why L1 regularization gives sparsity (Homework 2, Problem 5)
  • From scrach (in numpy) implementation of almost all major ML algorithms we discuss: ridge regression with SGD and GD (Homework 1, Problems 2.5, 2.6 page 4), lasso regression with the shooting algorithm (Homework 2, Problem 3, page 4), kernel ridge regression (Homework 4, Problem 3, page 2), kernelized SVM with Kernelized Pegasos (Homework 4, 6.4, page 9), L2-regularized logistic regression (Homework 5, Problem 3.3, page 4),Bayesian Linear Regession (Homework 5, problem 5, page 6), multiclass SVM (Homework 6, Problem 4.2, p. 3), classification and regression trees (without pruning) (Homework 6, Problem 6), gradient boosting with trees for classification and regression (Homework 6, Problem 8), multilayer perceptron for regression (Homework 7, Problem 4, page 3)
  • Repeated use of a simple 1-dimensional regression dataset, so it's easy to visualize the effect of various hypothesis spaces and regularizations that we investigate throughout the course.
  • Investigation of how to derive a conditional probability estimate from a predicted score for various loss functions, and why it's not so straightforward for the hinge loss (i.e. the SVM) (Homework 5, Problem 2, page 1)
  • Discussion of numerical overflow issues and the log-sum-exp trick (Homework 5, Problem 3.2)
  • Self-contained introduction to the expectation maximization (EM) algorithm for latent variable models.
  • Develop a general computation graph framework from scratch, using numpy, and implement your neural networks in it.

Prerequisites

The quickest way to see if the mathematics level of the course is for you is to take a look at this mathematics assessment , which is a preview of some of the math concepts that show up in the first part of the course.

  • Solid mathematical background , equivalent to a 1-semester undergraduate course in each of the following: linear algebra, multivariate differential calculus, probability theory, and statistics. The content of NYU's DS-GA-1002: Statistical and Mathematical Methods would be more than sufficient, for example.
  • Python programming required for most homework assignments.
  • Recommended: At least one advanced, proof-based mathematics course
  • Recommended: Computer science background up to a "data structures and algorithms" course
  • (HTF) refers to Hastie, Tibshirani, and Friedman's book The Elements of Statistical Learning
  • (SSBD) refers to Shalev-Shwartz and Ben-David's book Understanding Machine Learning: From Theory to Algorithms
  • (JWHT) refers to James, Witten, Hastie, and Tibshirani's book An Introduction to Statistical Learning

With the abundance of well-documented machine learning (ML) libraries, programmers can now "do" some ML, without any understanding of how things are working. And we'll encourage such "black box" machine learning... just so long as you follow the procedures described in this lecture. To make proper use of ML libraries, you need to be conversant in the basic vocabulary, concepts, and workflows that underlie ML. We'll introduce the standard ML problem types (classification and regression) and discuss prediction functions, feature extraction, learning algorithms, performance evaluation, cross-validation, sample bias, nonstationarity, overfitting, and hyperparameter tuning.

If you're already familiar with standard machine learning practice, you can skip this lecture.

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We have an interactive discussion about how to reformulate a real and subtly complicated business problem as a formal machine learning problem. The real goal isn't so much to solve the problem, as to convey the point that properly mapping your business problem to a machine learning problem is both extremely important and often quite challenging. This course doesn't dwell on how to do this mapping, though see Provost and Fawcett's book in the references.

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This is where our "deep study" of machine learning begins. We introduce some of the core building blocks and concepts that we will use throughout the remainder of this course: input space, action space, outcome space, prediction functions, loss functions, and hypothesis spaces. We present our first machine learning method: empirical risk minimization. We also highlight the issue of overfitting, which may occur when we find the empirical risk minimizer over too large a hypothesis space.
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A recurring theme in machine learning is that we formulate learning problems as optimization problems. Empirical risk minimization was our first example of this. To do learning, we need to do optimization. In this lecture we cover stochastic gradient descent, which is today's standard optimization method for large-scale machine learning problems.
We introduce the notions of approximation error, estimation error, and optimization error. While these concepts usually show up in more advanced courses, they will help us frame our understanding of the tradeoffs between hypothesis space choice, data set size, and optimization run times. In particular, these concepts will help us understand why "better" optimization methods (such as quasi-Newton methods) may not find prediction functions that generalize better, despite finding better optima.
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We introduce "regularization", our main defense against overfitting. We discuss the equivalence of the penalization and constraint forms of regularization (see ), and we introduce L1 and L2 regularization, the two most important forms of regularization for linear models. When L1 and L2 regularization are applied to linear least squares, we get "lasso" and "ridge" regression, respectively. We compare the "regularization paths" for lasso and ridge regression, and give a geometric argument for why lasso often gives "sparse" solutions. Finally, we present "coordinate descent", our second major approach to optimization. When applied to the lasso objective function, coordinate descent takes a particularly clean form and is known as the "shooting algorithm".

We continue our discussion of ridge and lasso regression by focusing on the case of correlated features, which is a common occurrence in machine learning practice. We will see that ridge solutions tend to spread weight equally among highly correlated features, while lasso solutions may be unstable in the case of highly correlated features. Finally, we introduce the "elastic net", a combination of L1 and L2 regularization, which ameliorates the instability of L1 while still allowing for sparsity in the solution. (Credit to Brett Bernstein for the excellent graphics.)

We start by discussing absolute loss and Huber loss. We consider them as alternatives to the square loss that are more robust to outliers. Next, we introduce our approach to the classification setting, introducing the notions of score, margin, and margin-based loss functions. We discuss basic properties of the hinge loss (i.e SVM loss), logistic loss, and the square loss, considered as margin-based losses. The interplay between the loss function we use for training and the properties of the prediction function we end up with is a theme we will return to several times during the course.
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We introduce the basics of convex optimization and Lagrangian duality. We discuss weak and strong duality, Slater's constraint qualifications, and we derive the complementary slackness conditions. As far as this course is concerned, there are really only two reasons for discussing Lagrangian duality: 1) The complementary slackness conditions will imply that SVM solutions are "sparse in the data" ( ), which has important practical implications for the kernelized SVMs (see the ). 2) Strong duality is a sufficient condition for the equivalence between the penalty and constraint forms of regularization (see ).

This mathematically intense lecture may be safely skipped.

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We define the soft-margin support vector machine (SVM) directly in terms of its objective function (L2-regularized, hinge loss minimization over a linear hypothesis space). Using our knowledge of Lagrangian duality, we find a dual form of the SVM problem, apply the complementary slackness conditions, and derive some interesting insights into the connection between "support vectors" and margin. Read the "SVM Insights from Duality" in the Notes below for a high-level view of this mathematically dense lecture.

Notably absent from the lecture is the hard-margin SVM and its standard geometric derivation. Although the derivation is fun, since we start from the simple and visually appealing idea of maximizing the "geometric margin", the hard-margin SVM is rarely useful in practice, as it requires separable data, which precludes any datasets with repeated inputs and label noise. One fixes this by introducing "slack" variables, which leads to a formulation equivalent to the soft-margin SVM we present. Once we introduce slack variables, I've personally found the interpretation in terms of maximizing the margin to be much hazier, and I find understanding the SVM in terms of "just" a particular loss function and a particular regularization to be much more useful for understanding its properties. That said, Brett Bernstein gives a very nice development of the geometric approach to the SVM, which is linked in the References below. At the very least, it's a great exercise in basic linear algebra.

Neither the lasso nor the SVM objective function is differentiable, and we had to do some work for each to optimize with gradient-based methods. It turns out, however, that gradient descent will essentially work in these situations, so long as you're careful about handling the non-differentiable points. To this end, we introduce "subgradient descent", and we show the surprising result that, even though the objective value may not decrease with each step, every step brings us closer to the minimizer.

This mathematically intense lecture may be safely skipped.

When using linear hypothesis spaces, one needs to encode explicitly any nonlinear dependencies on the input as features. In this lecture we discuss various strategies for creating features. Much of this material is taken, with permission, from Percy Liang's CS221 course at Stanford.

With linear methods, we may need a whole lot of features to get a hypothesis space that's expressive enough to fit our data -- there can be orders of magnitude more features than training examples. While regularization can control overfitting, having a huge number of features can make things computationally very difficult, if handled naively. For objective functions of a particular general form, which includes ridge regression and SVMs but not lasso regression, we can "kernelize", which can allow significant speedups in certain situations. In fact, with the "kernel trick", we can even use an infinite-dimensional feature space at a computational cost that depends primarily on the training set size.

In more detail, it turns out that even when the optimal parameter vector we're searching for lives in a very high-dimensional vector space (dimension being the number of features), a basic linear algebra argument shows that for certain objective functions, the optimal parameter vector lives in a subspace spanned by the training input vectors. Thus, when we have more features than training points, we may be better off restricting our search to the lower-dimensional subspace spanned by training inputs. We can do this by an easy reparameterization of the objective function. This result is referred to as the "representer theorem", and its proof can be given on one slide.

After reparameterization, we'll find that the objective function depends on the data only through the Gram matrix, or "kernel matrix", which contains the dot products between all pairs of training feature vectors. This is where things get interesting a second time: Suppose f is our featurization function. Sometimes the dot product between two feature vectors f(x) and f(x') can be computed much more efficiently than multiplying together corresponding features and summing. In such a situation, we write the dot products in terms of the "kernel function": k(x,x')=〈f(x),f(x')〉, which we hope to compute much more quickly than O(d), where d is the dimension of the feature space. The essence of a "kernel method" is to use this "kernel trick" together with the reparameterization described above. This allows one to use huge (even infinite-dimensional) feature spaces with a computational burden that depends primarily on the size of your training set. In practice, it's useful for small and medium-sized datasets for which computing the kernel matrix is tractable. Scaling kernel methods to large data sets is still an active area of research.

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This is our second "black-box" machine learning lecture. We start by discussing various models that you should almost always build for your data, to use as baselines and performance sanity checks. From there we focus primarily on evaluating classifier performance. We define a whole slew of performance statistics used in practice (precision, recall, F1, etc.). We also discuss the fact that most classifiers provide a numeric score, and if you need to make a hard classification, you should tune your threshold to optimize the performance metric of importance to you, rather than just using the default (typically 0 or 0.5). We also discuss the various performance curves you'll see in practice: precision/recall, ROC, and (my personal favorite) lift curves.

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So far we have studied the regression setting, for which our predictions (i.e. "actions") are real-valued, as well as the classification setting, for which our score functions also produce real values. With this lecture, we begin our consideration of "conditional probability models", in which the predictions are probability distributions over possible outcomes. We motivate these models by discussion of the "CitySense" problem, in which we want to predict the probability distribution for the number of taxicab dropoffs at each street corner, at different times of the week. Given this model, we can then determine, in real-time, how "unusual" the amount of behavior is at various parts of the city, and thereby help you find the secret parties, which is of course the ultimate goal of machine learning.

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In empirical risk minimization, we minimize the average loss on a training set. If our prediction functions are producing probability distributions, what loss functions will give reasonable performance measures? In this lecture, we discuss "likelihood", one of the most popular performance measures for distributions. We temporarily leave aside the conditional probability modeling problem, and focus on the simpler problem of fitting an unconditional probability model to data. We can use "maximum likelihood" to fit both parametric and nonparametric models. Once we have developed a collection of candidate probability distributions on training data, we select the best one by choosing the model that has highest "hold-out likelihood", i.e. likelihood on validation data.
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In this lecture we consider prediction functions that produce distributions from a parametric family of distributions. We restrict to the case of linear models, though later in the course we will show how to make nonlinear versions using gradient boosting and neural networks. We develop the technique through four examples: Bernoulli regression (logistic regression being a special case), Poisson regression, Gaussian regression, and multinomial logistic regression (our first multiclass method). We conclude by connecting this maximum likelihood framework back to our empirical risk minimization framework.
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We review some basics of classical and Bayesian statistics. For classical "frequentist" statistics, we define statistics and point estimators, and discuss various desirable properties of point estimators. For Bayesian statistics, we introduce the "prior distribution", which is a distribution on the parameter space that you declare before seeing any data. We compare the two approaches for the simple problem of learning about a coin's probability of heads. Along the way, we discuss conjugate priors, posterior distributions, and credible sets. Finally, we give the basic setup for Bayesian decision theory, which is how a Bayesian would go from a posterior distribution to choosing an action.

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In our earlier discussion of conditional probability modeling, we started with a hypothesis space of conditional probability models, and we selected a single conditional probability model using maximum likelihood or regularized maximum likelihood. In the Bayesian approach, we start with a prior distribution on this hypothesis space, and after observing some training data, we end up with a posterior distribution on the hypothesis space. For making conditional probability predictions, we can derive a predictive distribution from the posterior distribution. We explore these concepts by working through the case of Bayesian Gaussian linear regression. We also make a precise connection between MAP estimation in this model and ridge regression.

We begin our discussion of nonlinear models with tree models. We first describe the hypothesis space of decision trees, and we discuss some complexity measures we can use for regularization, including tree depth and the number of leaf nodes. The challenge starts when we try to find the regularized empirical risk minimizer (ERM) over this space for some loss function. It turns out finding this ERM is computationally intractable. We discuss a standard greedy approach to tree building, both for classification and regression, in the case that features take values in any ordered set. We also describe an approach for handling categorical variables (in the binary classification case) and missing values.

In this lecture, we define bootstrap sampling and show how it is typically applied in statistics to do things such as estimating variances of statistics and making confidence intervals. It can be used in a machine learning context for assessing model performance.

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We motivate bagging as follows: Consider the regression case, and suppose we could create a bunch of prediction functions, say B of them, based on B independent training samples of size n. If we average together these prediction functions, the expected value of the average is the same as any one of the functions, but the variance would have decreased by a factor of 1/B -- a clear win! Of course, this would require an overall sample of size nB. The idea of bagging is to replace independent samples with bootstrap samples from a single data set of size n. Of course, the bootstrap samples are not independent, so much of our discussion is about when bagging does and does not lead to improved performance. Random forests were invented as a way to create conditions in which bagging works better.

Although it's hard to find crisp theoretical results describing when bagging helps, conventional wisdom says that it helps most for models that are "high variance", which in this context means the prediction function may change a lot when you train with a new random sample from the same distribution, and "low bias", which basically means fitting the training data well. Large decision trees have these characteristics and are usually the model of choice for bagging. Random forests are just bagged trees with one additional twist: only a random subset of features are considered when splitting a node of a tree. The hope, very roughly speaking, is that by injecting this randomness, the resulting prediction functions are less dependent, and thus we'll get a larger reduction in variance. In practice, random forests are one of the most effective machine learning models in many domains.

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Gradient boosting is an approach to "adaptive basis function modeling", in which we learn a linear combination of M basis functions, which are themselves learned from a base hypothesis space H. Gradient boosting may be used with any subdifferentiable loss function and over any base hypothesis space on which we can do regression. Regression trees are the most commonly used base hypothesis space. It is important to note that the "regression" in "gradient boosted regression trees" (GBRTs) refers to how we fit the basis functions, not the overall loss function. GBRTs are routinely used for classification and conditional probability modeling. They are among the most dominant methods in competitive machine learning (e.g. Kaggle competitions).

If the base hypothesis space H has a nice parameterization (say differentiable, in a certain sense), then we may be able to use standard gradient-based optimization methods directly. In fact, neural networks may be considered in this category. However, if the base hypothesis space H consists of trees, then no such parameterization exists. This is where gradient boosting is really needed.

For practical applications, it would be worth checking out the GBRT implementations in and .

See the Notes below for fully worked examples of doing gradient boosting for classification, using the hinge loss, and for conditional probability modeling using both exponential and Poisson distributions. The code gbm.py illustrates L2-boosting and L1-boosting with decision stumps, for a one-dimensional regression dataset.

Here we consider how to generalize the score-producing binary classification methods we've discussed (e.g. SVM and logistic regression) to multiclass settings. We start by discussing "One-vs-All", a simple reduction of multiclass to binary classification. This usually works just fine in practice, despite the interesting failure case we illustrate. However, One-vs-All doesn't scale to a very large number of classes, since we have to train a separate model for each class. This is the real motivation for presenting the "compatibility function" approach described in this lecture. The approach presented here extends to structured prediction problems, where the output space may be exponentially large. We didn't have time to define structured prediction in the lecture, but please see the slides and the SSBD book in the references.
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Here we start our short unit on unsupervised learning. k-means clustering is presented first as an algorithm and then as an approach to minimizing a particular objective function. One challenge with clustering algorithms is that it's not obvious how to measure success. (See Section 22.5 of the SSBD book for a nice discussion.) When possible, I prefer to take a probabilistic modeling approach, as discussed in the next two lectures.
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A Gaussian mixture model (GMM) is a family of multimodal probability distributions, which is a plausible generative model for clustered data. We can fit this model using maximum likelihood, and we can assess the quality of fit by evaluating the model likelihood on holdout data. While the "learning" phase of Gaussian mixture modeling is fitting the model to data, in the "inference" phase, we determine for any point drawn from the GMM the probability that it came from each of the k components. To use a GMM for clustering, we simply assign each point to the component that it is most likely to have come from. k-means clustering can be seen as a limiting case of a restricted form of Gaussian mixture modeling.
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It turns out, fitting a Gaussian mixture model (GMM) by maximum likelihood is easier said than done: there is no closed form solution, and our usual gradient methods do not work well. The standard approach to maximum likelihood estimation in a Gaussian mixture model is the expectation maximization (EM) algorithm. In this lecture, we present the EM algorithm for a general latent variable model, of which GMM is a special case. We present the EM algorithm as a very basic "variational method" and indicate a few generalizations.
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In the context of this course, we view neural networks as "just" another nonlinear hypothesis space. On the practical side, unlike trees and tree-based ensembles (our other major nonlinear hypothesis spaces), neural networks can be fit using gradient-based optimization methods. On the theoretical side, a large enough neural network can approximate any continuous function. We discuss the specific case of the multilayer perceptron for multiclass classification, which we view as a generalization of multinomial logistic regression from linear to nonlinear score functions.
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Neural network optimization is amenable to gradient-based methods, but if the actual computation of the gradient is done naively, the computational cost can be prohibitive. Backpropagation is the standard algorithm for computing the gradient efficiently. We present the backpropagation algorithm for a general computation graph. The algorithm we present applies, without change, to models with "parameter tying", which include convolutional networks and recurrent neural networks (RNN's), the workhorses of modern computer vision and natural language processing. We illustrate backpropagation with one of the simplest models with parameter tying: regularized linear regression. Backpropagation for the multilayer perceptron, the standard introductory example, is presented in detail in .

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We point the direction to many other topics in machine learning that should be accessible to students of this course, but that we did not have time to cover.
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Assignments

GD, SGD, and Ridge Regression

Lasso Regression

SVM and Sentiment Analysis

Kernel Methods

Probabilistic Modeling

Multiclass, Trees, and Gradient Boosting

Computation Graphs, Backpropagation, and Neural Networks

The cover of Hands-On Machine Learning with Scikit-Learn and TensorFlow

Other tutorials and references

  • Carlos Fernandez-Granda's lecture notes provide a comprehensive review of the prerequisite material in linear algebra, probability, statistics, and optimization.
  • Brian Dalessandro's iPython notebooks from DS-GA-1001: Intro to Data Science
  • The Matrix Cookbook has lots of facts and identities about matrices and certain probability distributions.
  • Stanford CS229: "Review of Probability Theory"
  • Stanford CS229: "Linear Algebra Review and Reference"
  • Math for Machine Learning by Hal Daumé III

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David S. Rosenberg

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This repository contains the exercises, lab works and home works assignment for the Introduction to Machine Learning online class taught by Professor Leslie Pack Kaelbling, Professor Tomás Lozano-Pérez, Professor Isaac L. Chuang and Professor Duane S. Boning from Massachusett Institute of Technology

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Introduction to Machine Learning

Online course from MIT Open Learning Library Go to course page »

About The Course

Learning objective, professor leslie pack kaelbling, professor tomas lozano-perez, professor isaac l. chuang, professor duane s. boning, format of the course, recommended prerequisites, license type.

Machine Learning

This course introduces principles, algorithms, and applications of machine learning from the point of view of modeling and prediction. It includes formulation of learning problems and concepts of representation, over-fitting, and generalization. These concepts are exercised in supervised learning and reinforcement learning, with applications to images and to temporal sequences.

This section should list any major frameworks that you built your project using. Leave any add-ons/plugins for the acknowledgements section. Here are a few examples.

  • Understand the formulation of well-specified machine learning problems.
  • Learn how to perform supervised and reinforcement learning, with images and temporal sequences.

Course Instructors

Leslie Kaelbling

Leslie Pack Kaelbling is Professor of Computer Science and Engineering at MIT. She has previously held positions at Brown University, the Artificial Intelligence Center of SRI International, and at Teleos Research. In 2000, she founded the Journal of Machine Learning Research, a high-quality journal that is both freely available electronically as well as published in archival form; she currently serves as editor-in-chief.

Tomas Lozano-Perez

Tomas Lozano-Perez is currently the School of Engineering Professor in Teaching Excellence at the Massachusetts Institute of Technology (MIT), USA, where he is a member of the Computer Science and Artificial Intelligence Laboratory. He has been Associate Director of the Artificial Intelligence Laboratory and Associate Head for Computer Science of MIT's Department of Electrical Engineering and Computer Science.

Isaac L. Chuang

Isaac Chuang is Senior Associate Dean of Digital Learning, and Professor of Electrical Engineering & Computer Science, and Professor of Physics, at the Massachusetts Institute of Technology. He is associate director of the MIT Office of Digital Learning, and leads the quanta research group at the Center for Ultracold Atoms, in the MIT Research Laboratory of Electronics.

Duane S. Boning

Dr. Duane S. Boning is the Clarence J. LeBel Professor in Electrical Engineering, and Professor of Electrical Engineering and Computer Science in the EECS Department at MIT. He is affiliated with the MIT Microsystems Technology Laboratories, and serves as MTL Associate Director for Computation and CAD. He is also the Engineering Faculty Co-Director of the MIT Leaders for Global Operations (LGO) program.

This course includes lectures, lecture notes, exercises, labs, and homework problems.

  • Computer Programming with Python
  • Linear Algebra

Unless otherwise indicated, all content is © All Rights Reserved by the course instructor(s).

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Need Machine Learning Assignment Help?

Over the last ten years, machine learning has evolved as one of the most in-demand subjects in computer science. Increasingly, students are trying to learn and get hold of this new subject. As you dive deeper into machine learning, you might encounter certain challenges in understanding the subject or get stuck while working on university assignments. Machine learning assignments require a large amount of data collection and analysis. If you cannot take this pressure, you can take the expert machine learning help online.

At FavTutor, our experts provide machine learning assignment help, whether you are a beginner or trying to crack a complex problem. Our subject experts and help you complete your assignments at affordable rates. Not only do we offer exceptional quality of assignments, but also treat every student on equal priority. Any type of machine learning homework help you need, we are here for you. Moreover, we help you complete your machine learning assignments before the deadline.

What is Machine Learning?

Machine learning is an associate degree application of computer science that gives systems the flexibility to mechanically learn and improve from expertise while not being expressly programmed. The process of learning begins with observations or information, like examples, direct expertise, or instruction, to appear for patterns in the information and build higher selections within the future supported by the examples that we offer. The first aim is to permit the computers to learn mechanically while not human intervention or help and alter actions consequently.

Key Topics in ML

The most difficult concept to learn in Machine Learning is the classification of ML. ML is classified into 3 categories, let us understand them below:

  • Supervised Learning: Supervised learning is a variety of machine learning methodologies during which we offer sample tagged information to the machine learning system to coach it, and thereon basis, it predicts the output. Supervised learning relies on supervision, and it's similar to once a student learns things within the supervision of the teacher.
  • Unsupervised Learning: Unsupervised learning is a learning methodology during which a machine learns with none supervising. The coaching is provided to the machine with the set of knowledge that has not been tagged or classified, and also the formula must act thereon knowledge with none supervising.
  • Reinforcement Learning: Reinforcement learning is a feedback-based learning technique, during which a learning agent gets a gift for every right action and gets a penalty for every wrong action.

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Test prep and homework help from private online Machine Learning tutors

Our online Machine Learning tutors offer personalized, one-on-one learning to help you improve your grades, build your confidence, and achieve your academic goals.

Top 75 online Machine Learning tutors

Farhad's photo

14 years of tutoring

Vancouver , Canada

USD $ 59 /hr

University of British Columbia

Electrical Engineering, Machine Learning, Data Science, Python, Coding, Physics, Math, Statistics Tutor, a PhD holder from UBC in Electrical & Computer Engineering with 14+ years of experience

I have got my PhD in Electrical and Computer Engineering from UBC. I have 14+ years of tutoring and teaching experience in high schools, colleges, and universities. I was serving as a faculty member at Electrical Engineering Department of one of the Tehran's universities for 5 years. I have published more than 25 papers in different journals and conferences. For more details, please find my Google Scholar profile. Some of the subjects I teach/tutor for K-12, college, and university students are as follows: > Electric Circuits > Data Science > Machine Learning > Python > Coding > Algorithms > Physics > College/University Math courses (first year and second year) > Math 7, 8, 9, 10 > Pre-Calculus 11, 12 > IB (HL, SL) Math, Stat and Physics > AP Calculus AB & BC > AP Statistics > Advanced Calculus, Calculus I, II > Algebra > Probability and Statistics > Trigonometry and Geometry > Discrete Mathematics > SSAT (Math) Exam prep: lower level (for grades 4 and 5), middle level (for grades 6, 7, 8), and upper level (for grades 8, 9, 10, 11) > GRE (Quantitative) Exam Prep

Subjects : AP Calculus AB/BC, AP Statistics, Algebra, Calculus, Computer Science, Data Science, Data Structures & Algorithms, Electrical Circuit Analysis, Electrical Engineering, Elementary Math, Farsi, Machine Learning, Maths, Physics, Python

Etika's photo

8 years of tutoring

Bengaluru , India

USD $ 50 /hr

University of Notre Dame , University of Notre Dame , Indian Institute of Space Science and Technology

Experienced tutor with 10+ years of experience in MATLAB | Python | Aerospace | Control Systems | Probability | Machine Learning | Data Science

I am an expert tutor with a 5 star rating. My teaching journey started as a teaching assistant (TA) during my graduate school where I taught several under-graduate courses on dynamics, controls, signal processing, maths, optimization and MATLAB. I received the BEST TA AWARD in the year 2017 and never stopped teaching since. I believe that every student is different, and a little help with a personalized teaching method can go a long way. I will be more than happy to help you reinforce the concept understanding and handle those pestering coursework doubts. Happy learning!

Subjects : Aerospace Engineering, Data Science, Data Structures & Algorithms, Linear Control Systems, Linear Programming, MATLAB, Machine Learning, Mechanical Engineering, Physics (Newtonian Mechanics), Probability, Python

Nikhil's photo

5 years of tutoring

English , Hindi , Punjabi

Halifax Regional Municipality , Canada

CAD $ 35 /hr

Saint Mary's University

Expert in Physics, Python, SQL with a passion to teach

FREE 15 MIN FOR ALL STUDENTS. * I am Nikhil, I am a final year PhD student with a passion for teaching physics, maths and programming languages (SQL, Python) * I know what the teacher looks for in your answer and how to answer each question best. * I have been a Teaching Assistant for the past 3 years, I have designed and graded exams, and assignments for students. * I will teach you the art of thinking effectively, it is not the question that bothers you, it is the way you think about the problem that restricts you from solving it. Please send me a message to book a session today :) Thanks!

Subjects : AP Physics, C++, Data Analysis, Data Science, Data Visualization, Database, Machine Learning, Microsoft Power BI, MySQL, Nuclear Physics, Physics, Python, SQL

Buliao's photo

English (B2 level) , Chinese (Mandarin)

Cuenca , Ecuador

USD $ 40 /hr

National University of Defense Technology , National University of Defense Technology , University of Science and Technology of China

Experienced tutor for ALL University Math Courses, Professional Math

Hi there, With a Ph.D in Math, I know undergraduate math courses inside and out. I have 5+ years of experience focusing on tutoring math at University level. I can make ANY undergraduate math course crystal clear for you from an advanced standpoint. Professional math such as Math foundations for Machine Learning, PCA, etc are also my strength. I am proud that under my one year of tutoring (online) in 14 math courses, my student (math major) in *** University in England graduated with a First-Class Honours Degree in June 2021. Feel free to message me if you have any question in math (any level), or would like to book a session.

Subjects : AP Calculus AB/BC, Abstract Algebra, Algebraic topology, Calculus, Complex analysis, Differential Equations, Differential geometry, Integral Calculus, Linear Algebra, Multivariable Calculus, Number Theory, Ordinary and Partial Differential Equations, Real Analysis, Topology, Vector Calculus

Souvik's photo

2 years of tutoring

English , Hindi

Kitchener , Canada

CAD $ 20 /hr

University of Waterloo

Specialized in NLP Machine Learning

Taught Mathematics from grade 5 to 10 for 2 years. Studied mathematics till the engineering graduate program. Held Teaching assistant for 1 year at University of waterloo in programming. Expertise in NLP Machine Learning for more than 2 years.

Subjects : Calculus, Data Science, Data Visualization, Linear Algebra, Machine Learning, Pre-Calculus

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Frequently asked questions

Machine Learning

Homework help & tutoring.

Machine Learning

Our name 24HourAnswers means you can submit work 24 hours a day - it doesn't mean we can help you master what you need to know in 24 hours. If you make arrangements in advance, and if you are a very fast learner, then yes, we may be able to help you achieve your goals in 24 hours. Remember, high quality, customized help that's tailored around the needs of each individual student takes time to achieve. You deserve nothing less than the best, so give us the time we need to give you the best.

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Machine learning is a branch of artificial intelligence (AI). It uses statistical models and algorithms to get computers to imitate the way humans learn and improve their learning automatically. 

Students who study machine learning in college focus on advanced data science, math and computer science topics. You'll have to master many complicated subjects to succeed in your machine learning classes.

Our online tutors can help you with your machine learning coursework, whether you're trying to crack a complex concept or learn the basics. They are experts in the field and have a wealth of knowledge to share.

Online Machine Learning Tutors

Our experts are ready to help you with the machine learning concepts you're struggling with. You can choose from one-on-one tutoring or homework assistance to get the academic support you need on any machine learning topic.

Tutoring Sessions

Request a live, online tutoring session to get personalized academic support. Our tutors will develop a lesson tailored to your specific needs and learning style. They use any coursework you upload in advance as a starting point and provide helpful examples and other items to aid your learning. They'll also use our state-of-the-art whiteboard platform, which has desktop sharing, file upload, audio and video capabilities. 

Homework Help

Submit your machine learning assignments to get homework help from our experts. They'll review the problems and provide clearly written code or examples with accompanying explanations and documentation. You can use these materials to help you craft your own unique machine learning algorithm or statistical solution. 

You can also search our  Homework Library  to get answers fast. This database of solved problems includes many helpful machine learning resources.

Machine Learning Topics

You can get help on any aspect of machine learning, including:

  • Artificial intelligence :  Artificial intelligence deals with developing computers to perform tasks that typically require human intelligence, such as decision making, speech recognition and visual perception.
  • Computer science :  Computer science is the study of computers, including their software and hardware.
  • Algorithms :  An algorithm is a sequence of specific programming steps used to solve problems, particularly computer problems.
  • Python :  Python is a high-level computer programming language used for machine learning.
  • Supervised learning:  Supervised learning is a subcategory of machine learning that uses labeled datasets to teach algorithms to classify data or predict outcomes.
  • Unsupervised learning:  Unsupervised learning is a machine learning subcategory that uses algorithms to learn patterns from untagged data.

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Students choose us for our:

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We maintain a tutoring team of computer science veterans with machine learning expertise. They hold prestigious degrees, certifications and positions at top tech companies and universities such as the Massachusetts Institute of Technology (MIT). They'll help you understand machine learning with more developed, practical knowledge than you'll get from a peer tutor. 

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College Machine Learning Homework Help

Since we have tutors in all Machine Learning related topics, we can provide a range of different services. Our online Machine Learning tutors will:

  • Provide specific insight for homework assignments.
  • Review broad conceptual ideas and chapters.
  • Simplify complex topics into digestible pieces of information.
  • Answer any Machine Learning related questions.
  • Tailor instruction to fit your style of learning.

With these capabilities, our college Machine Learning tutors will give you the tools you need to gain a comprehensive knowledge of Machine Learning you can use in future courses.

24HourAnswers Online Machine Learning Tutors

Our tutors are just as dedicated to your success in class as you are, so they are available around the clock to assist you with questions, homework, exam preparation and any Machine Learning related assignments you need extra help completing.

In addition to gaining access to highly qualified tutors, you'll also strengthen your confidence level in the classroom when you work with us. This newfound confidence will allow you to apply your Machine Learning knowledge in future courses and keep your education progressing smoothly.

Because our college Machine Learning tutors are fully remote, seeking their help is easy. Rather than spend valuable time trying to find a local Machine Learning tutor you can trust, just call on our tutors whenever you need them without any conflicting schedules getting in the way.

Stanford University

Stanford engineering, s tanford e ngineering e verywhere, cs229 - machine learning, course details, course description.

This course provides a broad introduction to machine learning and statistical pattern recognition. Topics include: supervised learning (generative/discriminative learning, parametric/non-parametric learning, neural networks, support vector machines); unsupervised learning (clustering, dimensionality reduction, kernel methods); learning theory (bias/variance tradeoffs; VC theory; large margins); reinforcement learning and adaptive control. The course will also discuss recent applications of machine learning, such as to robotic control, data mining, autonomous navigation, bioinformatics, speech recognition, and text and web data processing. Students are expected to have the following background: Prerequisites: - Knowledge of basic computer science principles and skills, at a level sufficient to write a reasonably non-trivial computer program. - Familiarity with the basic probability theory. (Stat 116 is sufficient but not necessary.) - Familiarity with the basic linear algebra (any one of Math 51, Math 103, Math 113, or CS 205 would be much more than necessary.)

  • DOWNLOAD All Course Materials

FPO

Ng also works on machine learning algorithms for robotic control, in which rather than relying on months of human hand-engineering to design a controller, a robot instead learns automatically how best to control itself. Using this approach, Ng's group has developed by far the most advanced autonomous helicopter controller, that is capable of flying spectacular aerobatic maneuvers that even experienced human pilots often find extremely difficult to execute. As part of this work, Ng's group also developed algorithms that can take a single image,and turn the picture into a 3-D model that one can fly-through and see from different angles.

Course Handouts

Course Information
Course Schedule
Other AI Courses

Lecture Handouts

Linear Regression, Classification and logistic regression, Generalized Linear Models
Generative Learning algorithms
Support Vector Machines
Learning Theory
Regularization and model selection
The perceptron and large margin classifiers
The k-means clustering algorithm
Mixtures of Gaussians and the EM algorithm
The EM algorithm
Factor analysis
Principal components analysis
Independent Components Analysis
Reinforcement Learning and Control

Review Notes

Linear Algebra Review and Reference
Probability Theory Review
Matlab Review

Slides from Andrew's lecture on getting machine learning algorithms to work in practice can be found .

A list of last year's final projects can be found .

Here are a couple of Matlab tutorials that you might find helpful: and . For emacs users only: If you plan to run Matlab in emacs, here are , and a helpful .

For a free alternative to Matlab, check out . The official documentation is available . Some useful tutorials on Octave include and .

Depending on the computer you are using, you may be able to download a or for it if you don't already have one.

Assignments

AssignmentAssignment Data FilesSolutionSolution Data Files
 

Course Sessions (20):

Watch Online: Download: Duration:
Watch Now Download 1 hr 9 min
The Motivation & Applications of Machine Learning, The Logistics of the Class, The Definition of Machine Learning, The Overview of Supervised Learning, The Overview of Learning Theory, The Overview of Unsupervised Learning, The Overview of Reinforcement Learning

Transcripts

Watch Online: Download: Duration:
Watch Now Download 1 hr 16 min
An Application of Supervised Learning - Autonomous Deriving, ALVINN, Linear Regression, Gradient Descent, Batch Gradient Descent, Stochastic Gradient Descent (Incremental Descent), Matrix Derivative Notation for Deriving Normal Equations, Derivation of Normal Equations
Watch Online: Download: Duration:
Watch Now Download 1 hr 13 min
The Concept of Underfitting and Overfitting, The Concept of Parametric Algorithms and Non-parametric Algorithms, Locally Weighted Regression, The Probabilistic Interpretation of Linear Regression, The motivation of Logistic Regression, Logistic Regression, Perceptron
Watch Online: Download: Duration:
Watch Now Download 1 hr 13 min
Newton's Method, Exponential Family, Bernoulli Example, Gaussian Example, General Linear Models (GLMs), Multinomial Example, Softmax Regression
Watch Online: Download: Duration:
Watch Now Download 1 hr 16 min
Discriminative Algorithms, Generative Algorithms, Gaussian Discriminant Analysis (GDA), GDA and Logistic Regression, Naive Bayes, Laplace Smoothing
Watch Online: Download: Duration:
Watch Now Download 1 hr 14 min
Multinomial Event Model, Non-linear Classifiers, Neural Network, Applications of Neural Network, Intuitions about Support Vector Machine (SVM), Notation for SVM, Functional and Geometric Margins
Watch Online: Download: Duration:
Watch Now Download 1 hr 16 min
Optimal Margin Classifier, Lagrange Duality, Karush-Kuhn-Tucker (KKT) Conditions, SVM Dual, The Concept of Kernels
Watch Online: Download: Duration:
Watch Now Download 1 hr 17 min
Kernels, Mercer's Theorem, Non-linear Decision Boundaries and Soft Margin SVM, Coordinate Ascent Algorithm, The Sequential Minimization Optimization (SMO) Algorithm, Applications of SVM
Watch Online: Download: Duration:
Watch Now Download 1 hr 14 min
Bias/variance Tradeoff, Empirical Risk Minimization (ERM), The Union Bound, Hoeffding Inequality, Uniform Convergence - The Case of Finite H, Sample Complexity Bound, Error Bound, Uniform Convergence Theorem & Corollary
Watch Online: Download: Duration:
Watch Now Download 1 hr 13 min
Uniform Convergence - The Case of Infinite H, The Concept of 'Shatter' and VC Dimension, SVM Example, Model Selection, Cross Validation, Feature Selection
Watch Online: Download: Duration:
Watch Now Download 1 hr 22 min
Bayesian Statistics and Regularization, Online Learning, Advice for Applying Machine Learning Algorithms, Debugging/fixing Learning Algorithms, Diagnostics for Bias & Variance, Optimization Algorithm Diagnostics, Diagnostic Example - Autonomous Helicopter, Error Analysis, Getting Started on a Learning Problem
Watch Online: Download: Duration:
Watch Now Download 1 hr 14 min
The Concept of Unsupervised Learning, K-means Clustering Algorithm, K-means Algorithm, Mixtures of Gaussians and the EM Algorithm, Jensen's Inequality, The EM Algorithm, Summary
Watch Online: Download: Duration:
Watch Now Download 1 hr 15 min
Mixture of Gaussian, Mixture of Naive Bayes - Text clustering (EM Application), Factor Analysis, Restrictions on a Covariance Matrix, The Factor Analysis Model, EM for Factor Analysis
Watch Online: Download: Duration:
Watch Now Download 1 hr 21 min
The Factor Analysis Model,0 EM for Factor Analysis, Principal Component Analysis (PCA), PCA as a Dimensionality Reduction Algorithm, Applications of PCA, Face Recognition by Using PCA
Watch Online: Download: Duration:
Watch Now Download 1 hr 17 min
Latent Semantic Indexing (LSI), Singular Value Decomposition (SVD) Implementation, Independent Component Analysis (ICA), The Application of ICA, Cumulative Distribution Function (CDF), ICA Algorithm, The Applications of ICA
Watch Online: Download: Duration:
Watch Now Download 1 hr 13 min
Applications of Reinforcement Learning, Markov Decision Process (MDP), Defining Value & Policy Functions, Value Function, Optimal Value Function, Value Iteration, Policy Iteration
Watch Online: Download: Duration:
Watch Now Download 1 hr 17 min
Generalization to Continuous States, Discretization & Curse of Dimensionality, Models/Simulators, Fitted Value Iteration, Finding Optimal Policy
Watch Online: Download: Duration:
Watch Now Download 1 hr 17 min
State-action Rewards, Finite Horizon MDPs, The Concept of Dynamical Systems, Examples of Dynamical Models, Linear Quadratic Regulation (LQR), Linearizing a Non-Linear Model, Computing Rewards, Riccati Equation
Watch Online: Download: Duration:
Watch Now Download 1 hr 16 min
Advice for Applying Machine Learning, Debugging Reinforcement Learning (RL) Algorithm, Linear Quadratic Regularization (LQR), Differential Dynamic Programming (DDP), Kalman Filter & Linear Quadratic Gaussian (LQG), Predict/update Steps of Kalman Filter, Linear Quadratic Gaussian (LQG)
Watch Online: Download: Duration:
Watch Now Download 1 hr 17 min
Partially Observable MDPs (POMDPs), Policy Search, Reinforce Algorithm, Pegasus Algorithm, Pegasus Policy Search, Applications of Reinforcement Learning

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Assignments

Jump to: [Homeworks] [Projects] [Quizzes] [Exams]

There will be one homework (HW) for each topical unit of the course. Due about a week after we finish that unit.

These are intended to build your conceptual analysis skills plus your implementation skills in Python.

  • HW0 : Numerical Programming Fundamentals
  • HW1 : Regression, Cross-Validation, and Regularization
  • HW2 : Evaluating Binary Classifiers and Implementing Logistic Regression
  • HW3 : Neural Networks and Stochastic Gradient Descent
  • HW4 : Trees
  • HW5 : Kernel Methods and PCA

After completing each unit, there will be a 20 minute quiz (taken online via gradescope).

Each quiz will be designed to assess your conceptual understanding about each unit.

Probably 10 questions. Most questions will be true/false or multiple choice, with perhaps 1-3 short answer questions.

You can view the conceptual questions in each unit's in-class demos/labs and homework as good practice for the corresponding quiz.

There will be three larger "projects" throughout the semester:

  • Project A: Classifying Images with Feature Transformations
  • Project B: Classifying Sentiment from Text Reviews
  • Project C: Recommendation Systems for Movies

Projects are meant to be open-ended and encourage creativity. They are meant to be case studies of applications of the ML concepts from class to three "real world" use cases: image classification, text classification, and recommendations of movies to users.

Each project will due approximately 4 weeks after being handed out. Start early! Do not wait until the last few days.

Projects will generally be centered around a particular methodology for solving a specific task and involve significant programming (with some combination of developing core methods from scratch or using existing libraries). You will need to consider some conceptual issues, write a program to solve the task, and evaluate your program through experiments to compare the performance of different algorithms and methods.

Your main deliverable will be a short report (2-4 pages), describing your approach and providing several figures/tables to explain your results to the reader.

You’ll be assessed on effort, the sophistication of your technical approach, the clarity of your explanations, the evidence that you present to support your evaluative claims, and the performance of your implementation. A high-performing approach with little explanation will receive little credit, while a careful set of experiments that illuminate why a particular direction turned out to be a dead end may receive close to full credit.

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- Submitted homeworks may be either typed or handwritten. However, for ease of grading, please submit answers to the individual questions that make up each homework assignment on separate pieces of paper. When turning in code, please both print and attach a copy of your code to your homework and submit your code through the course blackboard website.
- We might reuse problem set questions from previous years, covered by papers and webpages, we expect the students not to copy, refer to, or look at the solutions in preparing their answers. Since this is a graduate class, we expect students to want to learn and not google for answers.

- Homeworks must be done individually, except where otherwise noted in the assignments. 'Individually' means each student must hand in their own answers, and each student must write their own code in the programming part of the assignment. It is acceptable, however, for students to collaborate in figuring out answers and helping each other solve the problems. We will be assuming that, as participants in a graduate course, you will be taking the responsibility to make sure you personally understand the solution to any work arising from such a collaboration. Students who collaborate in this way are expected to list the name of those they collaborate with in their homework submissions.

- of all late homework assignments to Sharon Cavlovich. Put down the date and time of submission on the HW sheets when submitting your assignments to Sharon. If she is not available, please slide your HW under her door.

Browse Course Material

Course info, instructors.

  • Rohit Singh
  • Prof. Tommi Jaakkola
  • Ali Mohammad

Departments

  • Electrical Engineering and Computer Science

As Taught In

  • Algorithms and Data Structures
  • Artificial Intelligence
  • Probability and Statistics
  • Cognitive Science

Learning Resource Types

Machine learning, assignments.

Ali Mohammad and Rohit Singh prepared the problem sets and solutions.

ASSIGNMENTS SOLUTIONS SUPPORTING FILES
Problem set 1 ( )

Section A ( )

Section B ( )

perceptron_test.m ( )

perceptron_train.m ( )

Errata ( )

p1.zip ( ) (The ZIP file contains: 3 .m files and 4 .dat files.)

p3.zip ( ) (The ZIP file contains: 3 .svm files.)

strimage.m ( )

Problem set 2 ( ) ( ) Errata ( )
Problem set 3 ( ) ( ) Errata ( )
Problem set 4 ( ) ( ) Errata ( )
Problem set 5 ( ) ( )

Errata ( )

Prob1 data ( ) (The ZIP file contains: 12 .m files, 1 .de file.)

Prob2 data ( ) (The ZIP file contains: 10 .m files and 2 .dat files.)

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Introduction to Machine Learning

Homework 1 -- numpy and ml.

Due: Wednesday, February 15, 2023 at 11:00 PM

Welcome to your first homework! Homeworks are designed to be our primary teaching and learning mechanism, with conceptual, math, and coding questions that are designed to highlight the critical ideas in this course. You may choose to tackle the questions in any order, but the homeworks are designed to be followed sequentially. Often, insights from the early problems will help with the later ones.

You have 'free checking'! That means you can check and submit your answer as many times as you want. Your best submission (the one that gives you the most points taking into account correctness and lateness) will be counted---you don't have to worry about it.

After submitting your answers, even if you have gotten a perfect score, we highly encourage you to hit 'View Answer' to look at the staff solution. You may find the staff solutions approached the problems in a different way than you did, which can yield additional insight. Be sure you have gotten your points before hitting 'View Answer', however. You will not be allowed to submit again after viewing the answer.

Each week, we'll provide a Colab notebook for you to use draft and debug your solutions to coding problems (you have better editing and debugging tools there); but you should submit your final solutions here to claim your points.

This week's Colab notebook can be found here: HW01 Colab Notebook (Click Me!)

The homework comes in two parts:

  • Learning to use numpy
  • Introduction to linear regression

Machine learning algorithms almost always boil down to matrix computations, so we'll need a way to efficiently work with matrices.

numpy is a package for doing a variety of numerical computations in Python that supports writing very compact and efficient code for handling arrays of data. It is used extensively in many fields requiring numerical analysis, so it is worth getting to know.

We will start every code file that uses numpy with import numpy as np , so that we can reference numpy functions with the np. precedent. The fundamental data type in numpy is the multidimensional array, and arrays are usually generated from a nested list of values using the np.array command. Every array has a shape attribute which is a tuple of dimension sizes.

In this class, we will use two-dimensional arrays almost exclusively. That is, we will use 2D arrays to represent both matrices and vectors! This is one of several times where we will seem to be unnecessarily fussy about how we construct and manipulate vectors and matrices, but this will make it easier to catch errors in our code. Even if [[1,2,3]] and [1,2,3] may look the same to us, numpy functions can behave differently depending on which format you use. The first has two dimensions (it's a list of lists), while the second has only one (it's a single list). Using only 2D arrays for both matrices and vectors gives us predictable results from numpy operations.

Using 2D arrays for matrices is clear enough, but what about column and row vectors? We will represent a column vector as a d\times 1 array and a row vector as a 1\times d array. So for example, we will represent the three-element column vector, x = \left[ \begin{array}{c} 1 \\ 5 \\ 3 \\ \end{array} \right], as a 3 \times 1 numpy array. This array can be generated with

~~~ x = np.array([[1],[5],[3]]),

or by using the transpose of a 1 \times 3 array (a row vector) as in,

~~~ x = np.transpose(np.array([[1,5,3]]),

where you should take note of the "double" brackets.

It is often more convenient to use the array attribute .T , as in

~~~ x = np.array([[1,5,3]]).T

to compute the transpose.

Before you begin, we would like to note that in this assignment we will not accept answers that use for or while loops. One reason for avoiding loops is efficiency. For many operations, numpy calls a compiled library written in C, and the library is far faster than interpreted Python (in part due to the low-level nature of C, optimizations like vectorization, and in some cases, parallelization). But the more important reason for avoiding loops is that using numpy library calls leads to simpler code that is easier to debug. So, we expect that you should be able to transform loop operations into equivalent operations on numpy arrays, and we will practice this in this assignment.

Of course, there will be more complex algorithms that require loops, but when manipulating matrices you should always look for a solution without loops.

You can find general documentation on numpy here .

Numpy functions and features you should be familiar with for this assignment:

  • np.transpose (and the equivalent method a.T )
  • np.ndarray.shape
  • np.dot (and the equivalent method a.dot(b) )
  • np.linalg.inv
  • Elementwise operators +, -, *, /

Note that in Python, np.dot(a, b) is the matrix product a @ b , not the dot product a^T b .

If you're unfamiliar with numpy and want to see some examples of how to use it, please see this link: Numpy Overview .

Array Basics

Creating Arrays

Provide an expression that sets A to be a 2 \times 3 numpy array ( 2 rows by 3 columns), containing any values you wish.

Write a procedure that takes an array and returns the transpose of the array. You can use np.transpose or the property T , but you may not use loops.

Note: as with other coding problems in 6.390 you do not need to call the procedure; it will be called/tested when submitted.

Shapes Hint: If you get stuck, code and run these expressions (with array values of your choosing), then print out the shape using A.shape

Let A be a 4\times 2 numpy array, B be a 4\times 3 array, and C be a 4\times 1 array. For each of the following expressions, indicate the shape of the result as a tuple of integers ( recall python tuples use parentheses, not square brackets, which are for lists, and a tuple with just one item x in it is written as (x,) with a comma ). Write "none" (as a Python string with quotes) if the expression is illegal.

For example,

  • If the result array was [45, 36, 75] , the shape is (3,)
  • If the result array was [[1,2,3],[4,5,6]] , the shape is (2,3)

Hint: for more compact and legible code, use @ for matrix multiplication, instead of np.dot . If A and B , are matrices (2D numpy arrays), then A @ B = np.dot(A, B) .

Indexing vs. Slicing The shape of the resulting array is different depending on if you use indexing or slicing. Indexing refers to selecting particular elements of an array by using a single number (the index) to specify a particular row or column. Slicing refers to selecting a subset of the array by specifying a range of indices.

If you're unfamiliar with these terms, and the indexing and slicing rules of arrays, please see the indexing and slicing sections of this link: Numpy Overview (Same as the Numpy Overview link from the introduction). You can also look at the official numpy documentation here .

In the following questions, let A = np.array([[5,7,10,14],[2,4,8,9]]) . Tell us what the output would be for each of the following expressions. Use brackets [] as necessary. If the operation is invalid, write the python string "none" .

Note: Remember that Python uses zero-indexing and thus starts counting from 0, not 1. This is different from R and MATLAB.

Indexing, revisited

Slicing, revisited

Lone Colon Slicing

Combining Indexing and Slicing

Combining Indexing and Slicing, revisited

Combining Indexing and Slicing, revisited again

Coding Practice

Now that we're familiar with numpy arrays, let's practice actually using numpy in our code!

In the following questions, you must get the shapes of the output correct for your answer to be accepted. If your answer contains the right numbers but the grader is still saying your answers are incorrect, check the shapes of your output. The number and placement of brackets need to match!

Write a procedure that takes a list of numbers and returns a 2D numpy array representing a row vector containing those numbers. Recall that a row vector in our usage will have shape (1, d) where d is the number of elements in the row.

Column Vector

Write a procedure that takes a list of numbers and returns a 2D numpy array representing a column vector containing those numbers. You can use the rv procedure.

Write a procedure that takes a column vector and returns the vector's Euclidean length (or equivalently, its magnitude) as a scalar . You may not use np.linalg.norm , and you may not use loops.

Remember that the formula for the Euclidean length for a vector \mathbf{x} is:

{\rm length}(\mathbf{x}) = \sqrt{x_1^2 + x_2^2 + ... + x_n^2} \\ = \sqrt{\sum_{i=1}^n{x^2_i}}

Write a procedure that takes a column vector and returns a unit vector (a vector of length 1 ) in the same direction. You may not use loops. Use your length procedure from above (you do not need to define it again).

Last Column

Write a procedure that takes a 2D array and returns the final column vector as a two dimensional array. You may not use loops. Hint: negative indices are interpreted as counting from the end of the array.

Matrix inverse

A scalar number x has an inverse x^{-1} , such that x^{-1} x = 1 , that is, their product is 1 . Similarly, a matrix A may have a well-defined inverse A^{-1} , such that A^{-1} A = I , where matrix multiplication is used, and I is the identity matrix. Such inverses generally only exist when A is a square matrix, and just as 0 has no well defined multiplicative inverse, there are also cases when matrices are "singular" and have no well defined inverses.

Write a procedure that takes a matrix A and returns its inverse, A^{-1} . Assume that A is well-formed, such that its inverse exists. Feel free to use routines from np.linalg .

Working with Data in Numpy

Representing data

Mat T. Ricks has collected weight and height data of 3 people and has written it down below:

Weight, Height 150, 5.8 130, 5.5 120, 5.3

He wants to put this into a numpy array such that each column represents one individual's weight and height (in that order), in the order of individuals as listed. Write code to set data equal to the appropriate numpy array:

We are beginning our study of machine learning with linear regression which is a fundamental problem in supervised learning. Please study Sections 2.1 through 2.4 of the Chapter 2 - Regression lecture notes before starting in on these problems.

A hypothesis in linear regression has the form y = \theta^T x + \theta_0 where x is a d \times 1 input vector, y is a scalar output prediction, \theta is a d \times 1 parameter vector and \theta_0 is a scalar offset parameter.

This week, just to get warmed up, we will consider a simple algorithm for trying to find a hypothesis that fits the data well: we will generate a lot of random hypotheses and see which one has the smallest error on this data, and return that one as our answer. (We don't recommend this method in actual practice, but it gets us started and makes some useful points.)

Here is a data-set for a regression problem, with d = 1 and n = 5 : \mathcal{D} = {([1], 2), ([2], 1), ([3], 4), ([4], 3), ([5], 5)} Recall from the notes that \mathcal{D} is a set of (x, y) (input, output) pairs.

Linear prediction

Assume we are given an input x as a column vector and the parameters specifying a linear hypothesis. Let's compute a predicted value.

Write a Python function which is given:

  • x : input vector d \times 1
  • th : parameter vector d \times 1
  • th0 : offset parameter 1 \times 1 or scalar

and returns:

  • y value predicted for input x by hypothesis th , th0

Lots of data!

Now assume we are given n points in an array, let's compute predictions for all the points.

  • X : input array d \times n
  • a 1\times n vector y of predicted values, one for each column of X for hypothesis th , th0

Try to make it so that your answer to this question can be used verbatim as an answer to the previous question.

Mean squared error

Given two 1 \times n vectors of output values, Y and Y_hat , compute a 1 \times 1 (or scalar) mean squared error.

  • Read about np.mean
  • Y : vector of output values 1 \times n
  • Y_hat : vector of output values 1 \times n
  • a 1\times 1 array with the mean square error

More mean squared error

Assume now that you have two k \times n arrays of output values, Y and Y_hat . Each row (0 \dots k-1) in a matrix represents the results of using a different hypothesis. Compute a k \times 1 vector of the mean-squared errors associate with each of the hypotheses (but averaged over all n data points, in each case.)

  • Read about the axis and keepdims arguments to np.mean

(Try to make it so that your answer to this question can be used verbatim as an answer to the previous question.)

  • Y : vector of output values k \times n
  • Y_hat : vector of output values k \times n
  • a k\times 1 vector of mean squared error values

Linear prediction error

Use the mse and lin_reg_predict procedures to implement a procedure that takes

  • X : d \times n input array representing n points in d dimensions
  • Y : 1 \times n output vector representing output values for n points
  • th0 : offset 1 \times 1 (or scalar)

and returns

1 \times 1 (or scalar) value representing the MSE of hypothesis th , th0 on the data set X , Y .

Read about the axis argument to np.mean

Our first machine learning algorithm!

The code is below. It takes in

  • X : d\times n input array representing n points in d dimensions
  • Y : 1\times n output vector representing output values for n points
  • k : a number of hypotheses to try

And generates as output

  • the tuple ((th, th0), error) where th , th0 is a hypothesis and error is the MSE of that hypothesis on the input data.

def random_regress(X, Y, k): 1 d, n = X.shape 2 thetas = 2 *np.random.rand(d, k) - 1 3 th0s = 2* np.random.rand(1, k) - 1 4 errors = lin_reg_err(X, Y, thetas, th0s.T) 5 i = np.argmin(errors) 6 theta, th0 = thetas[:,[i]], th0s[:,[i]] return (theta, th0), errors[i]

Note that in this code we use np.random.rand rather than np.random.randn as we will see in the lab. So some of the behavior will be different, and we'll ask some questions about that below.

  • Read about np.random.rand
  • Read about np.argmin

Rather than asking you to write the code, we are going to ask you some questions about it.

c. When we call lin_reg_err in line 4, we have objects with the following dimensions:

  • X : d \times n
  • ths : d\times k
  • th0s : 1\times k

If we want to get a matrix of predictions of all the hypotheses on all the data points, we can write np.dot(ths.T, X) + th0s.T But if we do the dimensional analysis here, there's something fishy.

(The form below is to help us improve/calibrate for future assignments; submission is encouraged but not required. Thanks!)

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  9. Machine Learning 10-701/15-781: Homework

    Late homework policy -. Late homeworks will be penalized according to the following policy: Homework is worth full credit at the beginning of class on the due date. It is worth half credit for the next 48 hours. It is worth zero credit after that. Turn in hardcopies of all late homework assignments to Sharon Cavlovich.

  10. Machine Learning

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  11. CS229

    The course will also discuss recent applications of machine learning, such as to robotic control, data mining, autonomous navigation, bioinformatics, speech recognition, and text and web data processing.

  12. Mathematics of Machine Learning Assignment 1

    This resource contains information regarding Mathematics of machine learning assignment 1.

  13. Assignments

    Homeworks There will be one homework (HW) for each topical unit of the course. Due about a week after we finish that unit. These are intended to build your conceptual analysis skills plus your implementation skills in Python. HW0: Numerical Programming Fundamentals HW1: Regression, Cross-Validation, and Regularization HW2: Evaluating Binary Classifiers and Implementing Logistic Regression HW3 ...

  14. Machine Learning 10-601: Homework

    Homework 2 Corrections and Clarifications: The original homework assignment stated there was a third optional question. This was incorrect. There are only two required (and no optional) questions. When using the MAP estimate for question 2.5, note that hallucinating each word appearing Beta times in the training set corresponds to having a Dirichlet prior with all parameters equal to (Beta + 1 ...

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  16. Homework 1 -- Numpy and ML

    Homework 1 -- Numpy and ML. Due: Wednesday, February 14, 2024 at 11:00 PM. Welcome to your first homework! Homeworks are designed to be our primary teaching and learning mechanism, with conceptual, math, and coding questions that are designed to highlight the critical ideas in this course. You may choose to tackle the questions in any order ...

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  18. Assignments

    The assignments section provides problem sets, solutions, and supporting files from the course.

  19. Introduction To Machine Learning 3rd Edition Textbook Solutions

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  22. Machine Learning Assignment

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  23. Homework 1 -- Numpy and ML

    Homework 1 -- Numpy and ML. Due: Wednesday, February 15, 2023 at 11:00 PM. Welcome to your first homework! Homeworks are designed to be our primary teaching and learning mechanism, with conceptual, math, and coding questions that are designed to highlight the critical ideas in this course. You may choose to tackle the questions in any order ...