Vapnik–Chervonenkis theory (also known as VC theory) was developed during 1960–1990 by
Vladimir Vapnik and
Alexey Chervonenkis. The theory is a form of
computational learning theory
In computer science, computational learning theory (or just learning theory) is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms.
Overview
Theoretical results in machine learning m ...
, which attempts to explain the learning process from a statistical point of view.
Introduction
VC theory covers at least four parts (as explained in ''The Nature of Statistical Learning Theory''
[):
*Theory of ]consistency
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
of learning processes
**What are (necessary and sufficient) conditions for consistency of a learning process based on the empirical risk minimization
In statistical learning theory, the principle of empirical risk minimization defines a family of learning algorithms based on evaluating performance over a known and fixed dataset. The core idea is based on an application of the law of large num ...
principle?
*Nonasymptotic theory of the rate of convergence of learning processes
**How fast is the rate of convergence of the learning process?
*Theory of controlling the generalization ability of learning processes
**How can one control the rate of convergence (the generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
ability) of the learning process?
*Theory of constructing learning machines
**How can one construct algorithms that can control the generalization ability?
VC Theory is a major subbranch of statistical learning theory
Statistical learning theory is a framework for machine learning drawing from the fields of statistics and functional analysis. Statistical learning theory deals with the statistical inference problem of finding a predictive function based on da ...
. One of its main applications in statistical learning theory is to provide generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteri ...
conditions for learning algorithms. From this point of view, VC theory is related to stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Exponential stability
** Linear stability
**Lyapunov stability
** Marginal s ...
, which is an alternative approach for characterizing generalization.
In addition, VC theory and VC dimension
VC may refer to:
Military decorations
* Victoria Cross, a military decoration awarded by the United Kingdom and other Commonwealth nations
** Victoria Cross for Australia
** Victoria Cross (Canada)
** Victoria Cross for New Zealand
* Victorious ...
are instrumental in the theory of empirical processes, in the case of processes indexed by VC classes. Arguably these are the most important applications of the VC theory, and are employed in proving generalization. Several techniques will be introduced that are widely used in the empirical process and VC theory. The discussion is mainly based on the book ''Weak Convergence and Empirical Processes: With Applications to Statistics''.[
]
Overview of VC theory in empirical processes
Background on empirical processes
Let be a measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
It captures and generalises intuitive notions such as length, area, an ...
. For any measure on , and any measurable functions , define
:
Measurability issues will be ignored here, for more technical detail see.[ Let be a class of measurable functions and define:
:
Let be independent, identically distributed random elements of . Then define the empirical measure
:
where here stands for the ]Dirac measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.
...
. The empirical measure induces a map given by:
:
Now suppose is the underlying true distribution of the data, which is unknown. Empirical Processes theory aims at identifying classes for which statements such as the following hold:
*uniform law of large numbers
In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law o ...
:
:That is, as ,
:
:uniformly for all .
*uniform central limit theorem
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
:
::
In the former case is called ''Glivenko–Cantelli'' class, and in the latter case (under the assumption ) the class is called ''Donsker'' or -Donsker. A Donsker class is Glivenko–Cantelli in probability by an application of Slutsky's theorem
In probability theory, Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables.
The theorem was named after Eugen Slutsky. Slutsky's theorem is also attributed to ...
.
These statements are true for a single , by standard LLN, CLT arguments under regularity conditions, and the difficulty in the Empirical Processes comes in because joint statements are being made for all . Intuitively then, the set cannot be too large, and as it turns out that the geometry of plays a very important role.
One way of measuring how big the function set is to use the so-called covering number
In mathematics, a covering number is the number of balls of a given size needed to completely cover a given space, with possible overlaps between the balls. The covering number quantifies the size of a set and can be applied to general metric spac ...
s. The covering number
:
is the minimal number of balls needed to cover the set (here it is obviously assumed that there is an underlying norm on ). The entropy is the logarithm of the covering number.
Two sufficient conditions are provided below, under which it can be proved that the set is Glivenko–Cantelli or Donsker.
A class is -Glivenko–Cantelli if it is -measurable with envelope such that and satisfies:
:
The next condition is a version of the celebrated Dudley's theorem. If is a class of functions such that
:
then is -Donsker for every probability measure such that . In the last integral, the notation means
:.
Symmetrization
The majority of the arguments of how to bound the empirical process rely on symmetrization, maximal and concentration inequalities, and chaining. Symmetrization is usually the first step of the proofs, and since it is used in many machine learning proofs on bounding empirical loss functions (including the proof of the VC inequality which is discussed in the next section) it is presented here.
Consider the empirical process:
:
Turns out that there is a connection between the empirical and the following symmetrized process:
:
The symmetrized process is a Rademacher process, conditionally on the data . Therefore, it is a sub-Gaussian process by Hoeffding's inequality
In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wass ...
.
Lemma (Symmetrization). For every nondecreasing, convex and class of measurable functions ,
:
The proof of the Symmetrization lemma relies on introducing independent copies of the original variables (sometimes referred to as a ''ghost sample'') and replacing the inner expectation of the LHS by these copies. After an application of Jensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier p ...
different signs could be introduced (hence the name symmetrization) without changing the expectation. The proof can be found below because of its instructive nature. The same proof method can be used to prove the Glivenko–Cantelli theorem
In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the fundamental theorem of statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, describes the asymptotic behaviour of the empirica ...
.[Devroye, L., Gyorfi, L. & Lugosi, G. A Probabilistic Theory of Pattern Recognition. ''Discrete Appl Math'' 73, 192–194 (1997).]
A typical way of proving empirical CLTs, first uses symmetrization to pass the empirical process to and then argue conditionally on the data, using the fact that Rademacher processes are simple processes with nice properties.
VC Connection
It turns out that there is a fascinating connection between certain combinatorial properties of the set and the entropy numbers. Uniform covering numbers can be controlled by the notion of ''Vapnik–Chervonenkis classes of sets'' – or shortly ''VC sets''.
Consider a collection of subsets of the sample space . is said to ''pick out'' a certain subset of the finite set if for some . is said to ''shatter'' if it picks out each of its subsets. The ''VC-index'' (similar to VC dimension
VC may refer to:
Military decorations
* Victoria Cross, a military decoration awarded by the United Kingdom and other Commonwealth nations
** Victoria Cross for Australia
** Victoria Cross (Canada)
** Victoria Cross for New Zealand
* Victorious ...
+ 1 for an appropriately chosen classifier set) of is the smallest for which no set of size is shattered by .
Sauer's lemma then states that the number of subsets picked out by a VC-class satisfies:
:
Which is a polynomial number of subsets rather than an exponential number. Intuitively this means that a finite VC-index implies that has an apparent simplistic structure.
A similar bound can be shown (with a different constant, same rate) for the so-called ''VC subgraph classes''. For a function the ''subgraph'' is a subset of such that: . A collection of is called a VC subgraph class if all subgraphs form a VC-class.
Consider a set of indicator functions in for discrete empirical type of measure (or equivalently for any probability measure ). It can then be shown that quite remarkably, for :
:
Further consider the ''symmetric convex hull'' of a set : being the collection of functions of the form with . Then if
:
the following is valid for the convex hull of :
:
The important consequence of this fact is that
:
which is just enough so that the entropy integral is going to converge, and therefore the class is going to be -Donsker.
Finally an example of a VC-subgraph class is considered. Any finite-dimensional vector space of measurable functions is VC-subgraph of index smaller than or equal to .
Proof: Take points . The vectors:
:
are in a dimensional subspace of . Take , a vector that is orthogonal to this subspace. Therefore:
:
Consider the set . This set cannot be picked out since if there is some such that that would imply that the LHS is strictly positive but the RHS is non-positive.
There are generalizations of the notion VC subgraph class, e.g. there is the notion of pseudo-dimension.[
]
VC inequality
A similar setting is considered, which is more common to machine learning
Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of Computational statistics, statistical algorithms that can learn from data and generalise to unseen data, and thus perform Task ( ...
. Let is a feature space and . A function is called a classifier. Let be a set of classifiers. Similarly to the previous section, define the '' shattering coefficient'' (also known as growth function):
:
Note here that there is a 1:1 go between each of the functions in and the set on which the function is 1. We can thus define to be the collection of subsets obtained from the above mapping for every . Therefore, in terms of the previous section the shattering coefficient is precisely
:.
This equivalence together with Sauer's Lemma implies that is going to be polynomial in , for sufficiently large provided that the collection has a finite VC-index.
Let is an observed dataset. Assume that the data is generated by an unknown probability distribution . Define to be the expected 0/1 loss. Of course since is unknown in general, one has no access to . However the ''empirical risk'', given by:
:
can certainly be evaluated. Then one has the following Theorem:
Theorem (VC Inequality)
For binary classification and the 0/1 loss function we have the following generalization bounds:
:
In words the VC inequality is saying that as the sample increases, provided that has a finite VC dimension, the empirical 0/1 risk becomes a good proxy for the expected 0/1 risk. Note that both RHS of the two inequalities will converge to 0, provided that grows polynomially in .
The connection between this framework and the Empirical Process framework is evident. Here one is dealing with a modified empirical process
:
but not surprisingly the ideas are the same. The proof of the (first part of) VC inequality, relies on symmetrization, and then argue conditionally on the data using concentration inequalities (in particular Hoeffding's inequality
In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wass ...
). The interested reader can check the book [ Theorems 12.4 and 12.5.
]
References
* See references in articles: Richard M. Dudley, empirical processes, Shattered set.
* This is a translation by B. Seckler, of the 1968 note.
** Reprinted in
** They obtained results in a draft form in July 1966 and announced in 1968 in their note
** The paper was first published properly in Russian as
*
*
{{DEFAULTSORT:Vapnik-Chervonenkis theory
Computational learning theory
Empirical process