Probably approximately correct learning

In computational learning theory, probably approximately correct (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).

Definitions and terminologyEdit

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

For the following definitions, two examples will be used. The first is the problem of character recognition given an array of   bits encoding a binary-valued image. 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   be a set called the instance space or the encoding of all the samples. In the character recognition problem, the instance space is  . In the interval problem the instance space,  , is the set of all bounded intervals in  , where   denotes the set of all real numbers.

A concept is a subset  . One concept is the set of all patterns of bits in   that encode a picture of the letter "P". An example concept from the second example is the set of open intervals,  , each of which contains only the positive points. A concept class   is a collection of concepts over  . This could be the set of all subsets of the array of bits that are skeletonized 4-connected (width of the font is 1).

Let   be a procedure that draws an example,  , using a probability distribution   and gives the correct label  , that is 1 if   and 0 otherwise.

Now, given  , assume there is an algorithm   and a polynomial   in   (and other relevant parameters of the class  ) such that, given a sample of size   drawn according to  , then, with probability of at least  ,   outputs a hypothesis   that has an average error less than or equal to   on   with the same distribution  . Further if the above statement for algorithm   is true for every concept   and for every distribution   over  , and for all   then   is (efficiently) PAC learnable (or distribution-free PAC learnable). We can also say that   is a PAC learning algorithm for  .


Under some regularity conditions these conditions are equivalent: [3]

  1. The concept class C is PAC learnable.
  2. The VC dimension of C is finite.
  3. C is a uniformly Glivenko-Cantelli class.[clarification needed]
  4. C is compressible in the sense of Littlestone and Warmuth

See alsoEdit


  1. ^ L. Valiant. A theory of the learnable. Communications of the ACM, 27, 1984.
  2. ^ Kearns and Vazirani, pg. 1-12,
  3. ^ Blumer, Anselm; Ehrenfeucht, Andrzej; David, Haussler; Manfred, Warmuth (October 1989). "Learnability and the Vapnik-Chervonenkis Dimension". Journal of the Association for Computing Machinery. 36 (4): 929–965. doi:10.1145/76359.76371. S2CID 1138467.

Further readingEdit