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An Introduction to Computational Learning Theory. MIT Press, 1994. A textbook. M. Mohri, A. Rostamizadeh, and A. Talwalkar. Foundations of Machine Learning. MIT Press, 2018. Chapter 2 contains a detailed treatment of PAC-learnability. Readable through open access from the publisher. D. Haussler.
Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of statistical algorithms that can learn from data and generalize to unseen data, and thus perform tasks without explicit instructions. [1]
Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could be whether or not the mushrooms are edible.
A learning algorithm over is a computable map from to . In other words, it is an algorithm that takes as input a finite sequence of training samples and outputs a function from X {\displaystyle X} to Y {\displaystyle Y} .
Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. [ 1 ] [ 2 ] [ 3 ] Statistical learning theory deals with the statistical inference problem of finding a predictive function based on data.
Ensemble learning, including both regression and classification tasks, can be explained using a geometric framework. [15] Within this framework, the output of each individual classifier or regressor for the entire dataset can be viewed as a point in a multi-dimensional space.
Empirically, for machine learning heuristics, choices of a function that do not satisfy Mercer's condition may still perform reasonably if at least approximates the intuitive idea of similarity. [6] Regardless of whether k {\displaystyle k} is a Mercer kernel, k {\displaystyle k} may still be referred to as a "kernel".
In general, the risk () cannot be computed because the distribution (,) is unknown to the learning algorithm. However, given a sample of iid training data points, we can compute an estimate, called the empirical risk, by computing the average of the loss function over the training set; more formally, computing the expectation with respect to the empirical measure: