In computational learning theory, probably approximately correct learning (PAC learning) is a framework for mathematical analysis of machine learning. It was proposed in 1984 byLeslie 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
[hide] - 1 Definitions and terminology
- 2 Equivalence
- 3 References
- 4 Further reading
Definitions and terminology[edit]
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
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
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
. In the interval problem the instance space is
, 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 all of the numbers between
and
. A concept class
is a set 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.
Say that there is an algorithm
that given access to
and inputs
and
that, with probability of at least
,
outputs a hypothesis
that has error less than or equal to
with examples drawn from
with the distribution
. If there is such an algorithm for every concept
, for every distribution
over
, and for all
and
then
is PAC learnable (or distribution-free PAC learnable). We can also say that
is a PAC learning algorithm for
.
An algorithm runs in time
if it draws at most
examples and requires at most
time steps. A concept class is efficiently PAC learnable if it is PAC learnable by an algorithm that runs in time polynomial in
,
and instance length.
Equivalence[edit]
Under some regularity conditions these three conditions are equivalent:
- The concept class C is PAC learnable.
- The VC dimension of C is finite.
- C is a uniform Glivenko-Cantelli class.
References[edit]
- Jump up^ L. Valiant. A theory of the learnable. Communications of the ACM, 27, 1984.
- Jump up^ Kearns and Vazirani, pg. 1-12,
- Jump up^ Balas Kausik Natarajan, Machine Learning , A Theoretical Approach, Morgan Kaufmann Publishers, 1991
Further reading[edit]