Probably approximately correct learning

Probably approximately correct learning

In computational learning theory, probably approximately correct learning (PAC learning) is a framework for mathematical analysis of machine learning. It was proposed in 1984 by Leslie Valiant.[1]

In this framework, the learner receives samples and must select a generalization function (called the hypothesis) from a certain class of possible functions. The goal is that, with high probability (the "probably" part), the selected function will have low generalization error (the "approximately correct" part). The learner must be able to learn the concept given any arbitrary approximation ratio, probability of success, or distribution of the samples.

The model was later extended to treat noise (misclassified samples).

An important innovation of the PAC framework is the introduction of computational complexity theory concepts to machine learning. In particular, the learner is expected to find efficient functions (time and space requirements bounded to a polynomial of the example size), and the learner itself must implement an efficient procedure (requiring an example count bounded to a polynomial of the concept size, modified by the approximation and likelihood bounds).

Contents

Definitions and terminology

In order to give the definition for something that is PAC-learnable, we first have to introduce some terminology.[2] [3]

For the following definitions, two examples will be used. The first is the problem of character recognition given an array of n bits. The other example is the problem of finding an interval that will correctly classify points within the interval as positive and the points outside of the range as negative.

Let X be a set called the instance space or the encoding of all the samples, and each instance have length assigned. In the character recognition problem, the instance space is X = {0,1}n. In the interval problem the instance space is X=\mathbb{R}, where \mathbb{R} denotes the set of all real numbers.

A concept is a subset c \subset X. One concept is the set of all of the bits that encode for the letter "P" in X = {0,1}n. An example concept from the second example is the set of all of the numbers between π / 2 and \sqrt{10}. A concept class C is a set of concepts over X. This could be the set of all of the array of bits that are skeletonized 4-connected (width of the font is 1).

Let EX(c,D) be a procedure that draws an example, x, using a probability distribution D and gives the correct label c(x), that is 1 if x \in c and 0 otherwise.

Say that there is an algorithm A that given access to EX(c,D) and inputs \epsilon and δ that, with probability of at least 1 − δ, A outputs a hypothesis h \in C that has error less than or equal to \epsilon with examples drawn from X with the distribution D. If there is such an algorithm for every concept c \in C, for every distribution D over X, and for all 0<\epsilon<1/2 and 0 < δ < 1 / 2 then C is PAC learnable(or distribution-free PAC learnable). We can also say that A is a PAC learning algorithm for C.

An algorithm runs in time t if it draws at most t examples and requires at most t time steps. A concept class is efficiently PAC learnable if it is PAC learnable by an algorithm that runs in time polynomial in 1 / ε, 1 / δ and instance length.

Equivalence

Under some regularity conditions these three conditions are equivalent:

  1. The concept class C is PAC learnable.
  2. The VC dimension of C is finite.
  3. C is a uniform Glivenko-Cantelli class.

References

  1. ^ L. Valiant. A theory of the learnable. Communications of the ACM, 27, 1984.
  2. ^ Kearns and Vazirani, pg. 1-12,
  3. ^ Balas Kausik Natarajan, Machine Learning , A Theoretical Approach, Morgan Kaufmann Publishers, 1991

Further reading


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Probably Approximately Correct Learning — Wahrscheinlich Annähernd Richtiges Lernen (WARL) oder englisch Probably approximately correct learning (PAC learning) ist ein Framework für das maschinelle Lernen, das von Leslie Valiant in seinem Paper A theory of the learnable[1] eingeführt… …   Deutsch Wikipedia

  • Computational learning theory — In theoretical computer science, computational learning theory is a mathematical field related to the analysis of machine learning algorithms. Contents 1 Overview 2 See also 3 References 3.1 Surveys …   Wikipedia

  • Algorithmic learning theory — (or algorithmic inductive inference) is a framework for machine learning.The framework was introduced in E. Mark Gold s seminal paper Language identification in the limit . The objective of language identification is for a machine running one… …   Wikipedia

  • Supervised learning — is a machine learning technique for learning a function from training data. The training data consist of pairs of input objects (typically vectors), and desired outputs. The output of the functioncan be a continuous value (called regression), or… …   Wikipedia

  • Computational learning theory — Dieser Artikel wurde aufgrund von inhaltlichen Mängeln auf der Qualitätssicherungsseite der Redaktion Informatik eingetragen. Dies geschieht, um die Qualität der Artikel aus dem Themengebiet Informatik auf ein akzeptables Niveau zu bringen. Hilf… …   Deutsch Wikipedia

  • WARL — Wahrscheinlich Annähernd Richtiges Lernen (WARL) oder englisch Probably approximately correct learning (PAC learning) ist ein Framework für das maschinelle Lernen, das von Leslie Valiant in seinem Paper A theory of the learnable[1] eingeführt… …   Deutsch Wikipedia

  • Warl — Wahrscheinlich Annähernd Richtiges Lernen (WARL) oder englisch Probably approximately correct learning (PAC learning) ist ein Framework für das maschinelle Lernen, das von Leslie Valiant in seinem Paper A theory of the learnable[1] eingeführt… …   Deutsch Wikipedia

  • History of virtual learning environments — A virtual learning environment (VLE) is a system that creates an environment designed to facilitate teachers in the management of educational courses for their students, especially a system using computer hardware and software, which involves… …   Wikipedia

  • Вэлиант, Лесли — Лесли Вэлиант Leslie Valiant Дата рождения …   Википедия

  • Hypothesis Theory — is a psychological theory of learning developed during the 1960s and 1970 s. Experimental Framework In the basic experimental framework, the subject is presented with a series of multidimensional stimuli, and provided feedback about the class of… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”