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[section] [theorem] [theorem]Lemma [subsection]Machine Learning Chapter

[section] [theorem] [theorem]Lemma [subsection]

Preamble

\label{section-preamble} Most of the discussions in this chapter are based on information extracted from two distinguished textbooks: [bishop - patternRecognition] and \ref{hastie-elemstatslearn}. Some fundamental concepts and material were also used from \label{scikit-learn}. While other texts were also used, it is not necessary that we point them out here as they have been correctly cited. Where not specified, the reader can assume that the proof is based from said texts.

[section] [theorem] [theorem]Lemma [subsection]

Brief ovierview to machine learning and supervised classification problems

\label{section-introduction}

Machine learning is a subfield of computer science with broad theoeretical intersections with statistics and mathematical optimization. At present time it has a wide range of applications. A non-exhausitve list of its use includes self-driving cars, spam detection systems, face and/or voice recognition, temperature prediction in weather, AI opponents in games, disease prediction in patients, text and audio language translation, stock pricing, movie recommendation systems, etc. Examples of these machine learning programs are now widespread to the point where their use today has direct impact on the lives of millions of people. Due to this, machine learning has practical intersections with data and software engineering.

The most widely used definition of machine learning is attributed to Tom Mitchell: “A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P if its performance at tasks in T, as measured by P, improves with experience E” (Mitchell 1997). To our purpose it is clear though that this definition 1 is not formally well-defined. However it serves to convey the idea of algorithms that automatically learn to do a a specific task better over time and with more data. Note that the “goodness” of their performance is inherently subjected to the evaluation criteria chosen for the task. Because of this,learning is less associated with a cognitive definition in this context and more to a performance based approach.

Machine Learning is divided into two main categories: Supervised and Unsupervised Learning. Let $$Y \in \mathbb{R}^n$$ be the outputs of the model and $$X \in \mathbb{R}^{n \times p}$$ be the inputs: for supervised learning algorithms are set to produce outputs from input data i.e. for each instance $$n$$, the computer has access to examples of outputs and tries to reproduce them based on information contained in $$X$$. In this context, the algorithm is generally referred to as a learner.

The second class of problems is where the output $$Y$$ is missing altogether from the data. In this scenario the most common objectives are clustering samples, density estimation and data compression. Linear regression and K-means clustering are examples of algorithms in each of these categories respectively.

In supervised learning tasks can be sub-categorized by the nature of the problem. When the type of the output variable $$Y$$ takes a (generally small )discrete set of values then it is said that it is a supervised classification problem. On the other hand, when the output takes continuous (or dense in an open set of $$\mathbb{R}$$) range of quantitative values, the problem is said to be of supervised regression. Note that regression problems can be encoded into classification problems by grouping the output values into categories by taking ranges of output values.

Suppose our objective is to predict $$y$$ given a new sample $$x$$. In supervised regression problems, $$y$$ will comprise a continuous variable. for classification problems on the other hand, $$y$$ will represent a label for a certain class. For the case of $$K$$ classes, $$y$$ most commonly takes values in the ranges $$0$$ through $$K-1$$ or $$1$$ through $$K$$. In both cases, the joint probability distribution $$p(X, Y)$$, called the true distribution, gives all of the information we need on these two variables. However the values of this probability is most often unknown. The idea is to then user estimations and inference on the most likely values for new samples and take decisions with the information at hand. These decisions will be based on the most probabilistic characterization of the data we have. In this work we will focus on the classification aspect of machine learning.

In these type of problems the theoretical and the computational aspects are both of interest. The algorithms used need to consider the technical requirements of the software and hardware at hand, as well as the time or resource constrains imposed by the problem setting. As such, they are expected to be executed in a reasonable amount of time, imposed by the task specification, and limited by the computing power available. 2 There can be problems which require that the algorithm output predictions in real-time, to the resolution of milliseconds. Picture a system where a credit-card transaction needs to be approved or labeled as fraud. Here the system needs to respond in short time if the transaction is fraudulent.

Other use cases might require the system to process a big volume of data at once, not a single event, but a batch of these and output this answer. The system in use needs to be prepared to run lean with a big inflow of data, without overloading the hardware capacity. These examples show that for a given problem, multiple algorithms are available to use but while all of them are theoretically doing the same task, we must also consider the practical advantages of these. Computational efficiency and scalability are relevant when working with these problems. Even though we won’t delve into these aspects in this work, it has an important consideration in the application of Machine Learning solutions.

In its essence, a machine learning method is a probabilistic model built from data so it is very similar to a statistical model. However, it differs specially in that its focus is generally on the models’ predictive abilities more than in the model’s parameters estimates.(Breiman 2001) The algorithms will be built and used to try and replicate as best as possible a given phenomena, without really identifying the true nature of the mechanisms behind this phenomena. As such, most applications will try to imitate the task’s behavior rather than try to identify the real system behind it. These subtle differences in the way machine learning approaches a problem is also reflected in the terminology used by the field. Here most terms have equivalent or similar notations to those used in classic statistics. As a start, the dependent $$Y$$ is called the target or label and the independent variables, covariates or input variables are named features in this case.

1. Other authors might reference machine learning as statistical learning. See (Hastie 2009) as an example.

2. Here the word reasonable is used in a broad sense. It will depend entirely on time constraints, computational capacity, usage and other aspects of each learning application.