|
|
| Line 1: |
Line 1: |
| {{Machine learning bar}}
| | Anyone invest loads of cash things like controls or alternatively memory cards, appear on the internet for a secondhand variation. Occasionally a store will probably be out of used-game hardware, which could be inexpensive. Make sure you look recorded at a web-based seller's feedback to be able to the purchase so this whether you are turning into what you covered.<br><br> |
|
| |
|
| == Introduction ==
| | [https://www.vocabulary.com/dictionary/Game+titles Game titles] are fun to have fun with your kids. This helps you learn much more info on your kid's interests. Sharing interests with your kids like this can likewise create great conversations. It also gives an opportunity to monitor continuing development of their skills.<br><br>Video games are very well-liked in many homes. The associated with people perform online online games to pass through time, however, some blessed people are paid to experience clash of clans sur pc. Games is going to grow to be preferred for some opportunity into the future. Here's more on clash of clans hack no survey ([http://prometeu.net find out here]) have a look at the website. These tips will help you if you are intending to try out online.<br><br>Seen the evaluations and see currently the trailers before buying another video game. Help it become one thing you have decided you're before you get in which. These video games aren't low-cost, and also you likely to get nearly as abundant cash whenever you companies inside a employed cd which you have solitary utilized several times.<br><br>A blog is offering Clash for Clans hack tool personal trainer to users who demand it. The website offering this tool remains safe and secure and it guarantees most useful software. There will also other sites which secure the tool. But would be the either incomplete or of bad quality. when users download these incomplete hack tools, instead of accomplishing well they end up in trouble. So, players are advised pick the tool from a website that offers complete shows.Users who are finding it tough to mongrel the hurdles can find a site that allows people download the cheats. Most of the websites allow free download a few websites charge fees. Users can locate an online site from where they can obtain good quality software.<br><br>Using this information, we're accessible to actually alpha dog substituting values. Application Clash of Clans Cheats' data, let's say during archetype you appetite 1hr (3, 600 seconds) to bulk 20 gems, and 1 day (90, 900 seconds) to help group 260 gems. May appropriately stipulate a task for this kind relating to band segment.<br><br>In order to conclude, clash of clans hack tool no surveys must not be encouraged to get in approach of the bigger question: what makes we perfect? Putting this aside the truck bed cover's of great importance. It replenishes the self, provides financial security and also always chips in. |
| | |
| '''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]], which attempts to explain the learning process from a statistical point of view.
| |
| | |
| VC theory is related to '''statistical learning theory''' and to [[empirical processes]]. [[Richard M. Dudley]] and [[Vladimir Vapnik]] himself, among others, apply VC-theory to [[empirical processes]].
| |
| | |
| VC theory covers at least four parts (as explained in ''The Nature of Statistical Learning Theory''{{ref|nslt}}):
| |
| *Theory of consistency of learning processes
| |
| **What are (necessary and sufficient) conditions for consistency of a learning process based on the [[empirical risk minimization]] 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 [[Machine learning#Generalization|generalization]] 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]]. One of its main applications in statistical learning theory, is to provide [[Machine learning#Generalization|generalization]] conditions for learning algorithms. From this point of view, VC theory is related to '''[[Stability (learning theory)|stability]]''', which is an alternative approach for characterizing generalization.
| |
| | |
| In addition, VC theory and [[VC dimension]] 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"{{ref|wcep}}.
| |
| | |
| == Overview of VC theory in Empirical Processes ==
| |
| | |
| === Background on Empirical Processes ===
| |
| | |
| Let <math>X_1,X_2,\ldots,X_n</math> are random elements defined on a measurable space <math>(\mathcal{X}, \mathcal{A})</math>. Define the empirical measure <math>\mathbb{P}_n = n^{-1} \sum_{i = 1}^n \delta_{X_i}</math>, where <math>\delta</math> here stands for the [[dirac measure]]. Denote with <math>Qf = \int f dQ</math> for a measure <math>Q</math>. Measurability issues, will be ignored here. For more technical detail consult{{ref|nslt}}.
| |
| | |
| Let <math>\mathcal{F}</math> be a class of measurable functions <math>f:\mathcal{X} \rightarrow \mathbb{R}</math>. The empirical measure induces a map from <math>\mathcal{F}</math>to <math>\mathbb{R}</math> given by:
| |
| | |
| <center><math>f \mapsto \mathbb{P}_n f</math></center>
| |
| | |
| Let <math>\vert\vert Q\vert\vert_{\mathcal{F}} = \sup \{\vert Qf \vert: f \in \mathcal{F} \}</math>. Empirical Processes theory aims at identifying classes <math>\mathcal{F}</math> for which statements like:
| |
| | |
| * <math> \vert \vert \mathbb{P}_n - P\vert \vert_{\mathcal{F}} \rightarrow 0 </math>, aka uniform law of large numbers
| |
| * <math> \mathbb{G}_n = \sqrt{n} (\mathbb{P}_n - P) \rightsquigarrow \mathbb{G}, \quad </math> in <math>\ell^{\infty}(\mathcal{F})</math>, aka uniform central limit theorem
| |
| | |
| hold. Here <math>P</math> is the underlying true distribution of the data, which is unknown in practice. In the former case the class <math>\mathcal{F}</math> is called ''Glivenko-Cantelli'', and in the latter case (under the assumption <math>\sup_{f \in \mathcal{F}}\vert f(x) - Pf \vert < \infty. \forall x</math>) the class <math>\mathcal{F}</math> is called ''Donsker'' or <math>P</math>-Donsker. Obviously, a Donsker class is Glivenko-Cantelli in probability by an application of [[Slutsky's theorem]] .
| |
| | |
| These statements are true for a single <math>f</math>, by standard [[Law of large numbers|LLN]], [[Central limit theorem|CLT]] arguments under regularity conditions, and the difficulty in the Empirical Processes comes in because joint statements are being made for all <math>f \in \mathcal{F}</math>. Intuitively then, the set <math>\mathcal{F}</math> cannot be too large, and as it turns out that the geometry of <math>\mathcal{F}</math> plays a very important role.
| |
| | |
| One way of measuring how big the function set <math>\mathcal{F}</math> is to use the so-called [[covering number]]s. The covering number <math>N(\varepsilon, \mathcal{F}, ||\cdot||)</math> is the minimal number of balls <math>\{g: ||g - f|| < \varepsilon \}</math> needed to cover the set <math>\mathcal{F}</math> (here it is obviously assumed that there is an underlying norm on <math>\mathcal{F}</math>). The entropy is the logarithm of the covering number.
| |
| | |
| Two sufficient conditions are provided below, under which it can be proved that the set <math>\mathcal{F}</math> is Glivenko-Cantelli or Donsker.
| |
| | |
| A class <math>\mathcal{F}</math> is <math>P</math>-Glivenko-Cantelli if it is <math>P</math>-measurable with envelope <math>F</math> such that <math>P^{\ast} F < \infty</math> and satisfies:
| |
| | |
| <center><math>\sup_{Q} N(\varepsilon ||F||_Q, \mathcal{F}, L_1(Q)) < \infty</math> for every <math>\varepsilon > 0</math>.</center>
| |
| | |
| The next condition is a version of the celebrated [[Dudley's theorem]]. If <math>\mathcal{F}</math> is a class of functions such that
| |
| | |
| <center><math>\int_0^{\infty} \sup_{Q} \sqrt{\log N(\varepsilon ||F||_{Q,2}, \mathcal{F}, L_2(Q))}d \varepsilon < \infty</math></center>
| |
| | |
| then <math>\mathcal{F}</math> is <math>P</math>-Donsker for every probability measure <math>P </math> such that <math>P^{\ast} F^2 < \infty</math>. In the last integral, the notation means <math>||f||_{Q,2} = \left(\int |f|^2 d Q\right)^{1/2}</math>.
| |
| | |
| === 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:
| |
| | |
| <center><math>f \mapsto (\mathbb{P}_n - P)f = \dfrac{1}{n} \sum_{i = 1}^n (f(X_i) - Pf) </math></center>
| |
| | |
| Turns out that there is a connection between the empirical and the following symmetrized process:
| |
| | |
| <center><math>f \mapsto \mathbb{P}^0_n = \dfrac{1}{n} \sum_{i = 1}^n \varepsilon_i f(X_i) </math></center>
| |
| | |
| The symmetrized process is a Rademacher process, conditionally on the data <math>X_i </math>. Therefore it is a sub-Gaussian process by [[Hoeffding's inequality]].
| |
| | |
| '''Lemma (Symmetrization).''' For every nondecreasing, convex <math>\Phi : \mathbb{R} \rightarrow \mathbb{R}</math> and class of measurable functions <math>\mathcal{F}</math>,
| |
| | |
| <center><math> \mathbb{E} \Phi (||\mathbb{P}_n - P||_{\mathcal{F}}) \leq \mathbb{E} \Phi (2 ||\mathbb{P}^0_n||_{\mathcal{F}}) </math></center>
| |
| | |
| The proof of the Symmetrization lemma relies on introducing independent copies of the original variables <math>X_i</math> (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 different signs could be introduced (hence the name symmetrization) without changing the expectation. The proof can be found below because of its instructive nature.
| |
| | |
| <div class="NavFrame collapsed">
| |
| <div class="NavHead">[Proof]</div>
| |
| <div class="NavContent">
| |
| Introduce the "ghost sample" <math>Y_1,\ldots, Y_n</math> to be independent copies of <math>X_1,\ldots, X_n</math>. For fixed values of <math>X_1,\ldots, X_n</math> one has:
| |
| <center><math>||\mathbb{P}_n - P||_{\mathcal{F}} = \sup_{f \in \mathcal{F}} \dfrac{1}{n} \left|\sum_{i = 1}^n [f(X_i) - \mathbb{E} f(Y_i)] \right| \leq \mathbb{E}_{Y} \sup_{f \in \mathcal{F}} \dfrac{1}{n} \left|\sum_{i = 1}^n [f(X_i) - f(Y_i)] \right|</math></center>
| |
| Therefore by Jensen's inequality:
| |
| <center><math>\Phi(||\mathbb{P}_n - P||_{\mathcal{F}}) \leq \mathbb{E}_{Y} \Phi \left(\left|\left| \dfrac{1}{n}\sum_{i = 1}^n [f(X_i) - f(Y_i)] \right|\right|_{\mathcal{F}} \right)</math></center>
| |
| Taking expectation with respect to <math>X</math> gives:
| |
| <center><math>\mathbb{E}\Phi(||\mathbb{P}_n - P||_{\mathcal{F}}) \leq \mathbb{E}_{X} \mathbb{E}_{Y} \Phi \left(\left|\left| \dfrac{1}{n}\sum_{i = 1}^n [f(X_i) - f(Y_i)] \right|\right|_{\mathcal{F}}\right)</math></center>
| |
| Note that adding a minus sign in front of a term <math>[f(X_i) - f(Y_i)]</math> doesn't change the RHS, because it's a symmetric function of
| |
| <math>X</math> and <math>Y</math>. Therefore the RHS remains the same under "sign perturbation":
| |
| <center><math>\mathbb{E} \Phi \left( \left|\left| \dfrac{1}{n}\sum_{i = 1}^n e_i[f(X_i) - f(Y_i)] \right|\right|_{\mathcal{F}} \right) </math></center>
| |
| for any <math>(e_1,e_2,\ldots,e_n) \in \{-1,1\}^n</math>. Therefore:
| |
| <center><math>\mathbb{E}\Phi(||\mathbb{P}_n - P||_{\mathcal{F}}) \leq \mathbb{E}_{\varepsilon} \mathbb{E} \Phi \left( \left|\left| \dfrac{1}{n}\sum_{i = 1}^n \varepsilon_i [f(X_i) - f(Y_i)] \right|\right|_{\mathcal{F}} \right)</math>
| |
| </center>
| |
| Finally using first triangle inequality and then convexity of <math>\Phi</math> gives:
| |
| <center><math>\mathbb{E}\Phi(||\mathbb{P}_n - P||_{\mathcal{F}}) \leq \dfrac{1}{2}\mathbb{E}_{\varepsilon} \mathbb{E} \Phi \left( 2 \left|\left| \dfrac{1}{n}\sum_{i = 1}^n \varepsilon_i f(X_i)\right|\right|_{\mathcal{F}} \right) + \dfrac{1}{2}\mathbb{E}_{\varepsilon} \mathbb{E} \Phi \left( 2 \left|\left| \dfrac{1}{n}\sum_{i = 1}^n \varepsilon_i f(Y_i)\right|\right|_{\mathcal{F}} \right)</math>
| |
| Where the last two expressions on the RHS are the same, which concludes the proof.
| |
| </center>
| |
| </div>
| |
| </div>
| |
| | |
| A typical way of proving empirical CLTs, first uses symmetrization to pass the empirical process to <math>\mathbb{P}_n^0</math> 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 <math>\mathcal{F}</math> and the entropy numbers. Uniform covering numbers can be controlled by the notion of ''Vapnik-Cervonenkis classes of sets'' - or shortly ''VC sets''.
| |
| | |
| Take a collection of subsets of the sample space <math>\mathcal{X}</math> -<math>\mathcal{C}</math>. A collection of sets <math>\mathcal{C}</math> is said to ''pick out'' a certain subset of the finite set <math>S = \{x_1,\ldots, x_n\} \subset \mathcal{X}</math> if <math>S = S \cap C</math> for some <math>C \in \mathcal{C}</math>. <math>\mathcal{C}</math> is said to ''shatter'' <math>S</math> if it picks out each of its <math>2^n</math> subsets. The ''VC-index'' (similar to [[VC dimension]] + 1 for an appropriately chosen classifier set) <math>V(\mathcal{C})</math> of <math>\mathcal{C}</math> is the smallest <math>n</math> for which no set of size <math>n</math> is shattered by <math>\mathcal{C}</math>.
| |
| | |
| [[Sauer–Shelah lemma|Sauer's lemma]] then states that the number <math>\Delta_n(\mathcal{C}, x_1, \ldots, x_n)</math> of subsets picked out by a VC-class <math>\mathcal{C}</math> satisfies:
| |
| | |
| <center><math>\max_{x_1,\ldots, x_n} \Delta_n(\mathcal{C}, x_1, \ldots, x_n) \leq \sum_{j = 0}^{V(\mathcal{C}) - 1} {n \choose j} \leq \left( \frac{n e}{V(\mathcal{C}) - 1}\right)^{V(\mathcal{C}) - 1}</math></center>
| |
| | |
| Which is a polynomial number <math>O(n^{V(\mathcal{C}) - 1})</math> of subsets rather than an exponential number. Intuitively this means that a finite VC-index implies that <math>\mathcal{C}</math> 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 <math>f : \mathcal{X} \mapsto \mathbb{R}</math> the [[Hypograph (mathematics)|''subgraph'']] is a subset of <math>\mathcal{X} \times \mathbb{R}</math> such that: <math>\{(x,t): t < f(x)\}</math>. A collection of <math>\mathcal{F}</math> is called a VC subgraph class if all subgraphs form a VC-class.
| |
| | |
| Consider a set of indicator functions <math> \mathcal{I}_{\mathcal{C}} = \{1_C: C \in \mathcal{C} \}</math> in <math>L_1(Q)</math> for discrete empirical type of measure <math>Q</math> (or equivalently for any probability measure <math>Q</math>). It can then be shown that quite remarkably, for <math>r \geq 1</math>:
| |
| | |
| <center><math>N(\varepsilon, \mathcal{I}_{\mathcal{C}}, L_r(Q)) \leq KV(\mathcal{C}) (4e)^{V(\mathcal{C})} \left(\dfrac{1}{\varepsilon}\right)^{r (V(\mathcal{C}) - 1)}</math></center>
| |
| | |
| Further consider the ''symmetric convex hull'' of a set <math>\mathcal{F}</math>: <math>\operatorname{sconv}\mathcal{F}</math> being the collection of functions of the form <math>\sum_{i =1}^m \alpha_i f_i</math> with <math>\sum_{i =1}^m |\alpha_i| \leq 1</math>. Then if
| |
| | |
| <center><math>N(\varepsilon||F||_{Q,2}, \mathcal{F}, L_2(Q)) \leq C \left(\dfrac{1}{\varepsilon}\right)^V</math></center>
| |
| | |
| the following is valid for the convex hull of <math>\mathcal{F}</math>:
| |
| | |
| <center><math>\log N(\varepsilon||F||_{Q,2}, \operatorname{sconv}\mathcal{F}, L_2(Q)) \leq K \left(\dfrac{1}{\varepsilon}\right)^{\frac{2V}{V + 2}}</math></center>
| |
| | |
| The important consequence of this fact is that the power of <math>1/\varepsilon</math> -- <math>2V/(V + 2)</math> is strictly less than 2, which is just enough so that the entropy integral is going to converge, and therefore the class <math>\operatorname{sconv}\mathcal{F}</math> is going to be <math>P</math>-Donsker.
| |
| | |
| Finally an example of a VC-subgraph class is considered. Any finite-dimensional vector space <math>\mathcal{F}</math> of measurable functions <math>f:\mathcal{X} \mapsto \mathbb{R}</math> is VC-subgraph of index smaller than or equal to <math>\dim(\mathcal{F}) + 2</math>.
| |
| | |
| <div class="NavFrame collapsed">
| |
| <div class="NavHead">[Proof]</div>
| |
| <div class="NavContent">
| |
| Take <math>n = \dim(\mathcal{F}) + 2 </math> points <math>(x_1, t_1), \ldots, (x_n, t_n)</math>. The vectors:
| |
| <center><math>(f(x_1), \ldots, f(x_n)) - (t_1, \ldots, t_n)</math></center>
| |
| are in a <math>n - 1</math> dimensional subspace of <math>\mathbb{R}^n</math>. Take <math>a \neq 0</math>, a non-zero vector that is orthogonal to this subspace. Therefore:
| |
| <center><math>\sum_{a_i > 0} a_i (f(x_i) - t_i) = \sum_{a_i < 0} (-a_i) (f(x_i) - t_i), \quad \forall f \in \mathcal{F}</math></center>
| |
| Consider the set <math>S = \{(x_i, t_i): a_i > 0\}</math>. This set cannot be picked out since if there is some <math>f</math> such that <math>S = \{(x_i,t_i): f(x_i) > t_i\}</math> that would imply
| |
| that the LHS is strictly positive but the RHS is non-negative.
| |
| </div>
| |
| </div>
| |
| | |
| There are generalizations of the notion VC subgraph class, e.g. there is the notion of pseudo-dimension. The interested reader can look into{{ref|pollard}}.
| |
| | |
| == VC Inequality ==
| |
| | |
| A similar setting is considered, which is more common to [[machine learning]]. Let <math>\mathcal{X}</math> is a feature space and <math>\mathcal{Y} = \{0,1\}</math>. A function <math>f : \mathcal{X} \mapsto \mathcal{Y}</math> is called a classifier. Let <math>\mathcal{F}</math> be a set of classifiers. Similarly to the previous section, define the ''[[Shattered set|shattering coefficient]]'' <math>S(\mathcal{F},n) = \max_{x_1,\ldots, x_n} |\{(f(x_1), \ldots, f(x_n)), f \in \mathcal{F}\}|</math>. The shattering coefficient is also known as growth function. Note here that there is a 1-1 mapping between each of the functions in <math>\mathcal{F}</math> and the set on which the function is 1. Therefore in terms of the previous section the shattering coefficient is precisely <math>\max_{x_1,\ldots, x_n} \Delta_n(\mathcal{C}, x_1, \ldots, x_n)</math> for <math>\mathcal{C}</math> being the collection of all sets described above. Now for the same reasoning as before, namely using Sauer's Lemma it can be shown that <math>S(\mathcal{F},n)</math> is going to be polynomial in <math>n</math> provided that the class <math>\mathcal{F}</math> has a finite VC-dimension or equivalently the collection <math>\mathcal{C}</math> has finite VC-index.
| |
| | |
| Let <math>D_n = \{(X_1, Y_1), \ldots, (X_n,Y_m)\}</math> is an observed dataset. Assume that the data is generated by an unknown probability distribution <math>P_{XY}</math>. Define <math>R(f) = P(f(X) \neq Y)</math> to be the expected 0/1 loss. Of course since <math>P_{XY}</math> is unknown in general, one has no access to <math>R(f) </math>. However the ''empirical risk'', given by:
| |
| | |
| <center><math>\hat{R}_n(f) = \dfrac{1}{n}\sum_{i = 1}^n \mathbb{I}(f(X_n) \neq Y_n)</math></center>
| |
| | |
| 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:
| |
| | |
| <center><math>P\left(\sup_{f \in \mathcal{F}} |\hat{R}_n(f) - R(f)| >\varepsilon \right) \leq 8 S(\mathcal{F},n) e^{-n\varepsilon^2/32} </math></center>
| |
| | |
| and
| |
| | |
| <center><math>\mathbb{E}\left[\sup_{f \in \mathcal{F}} |\hat{R}_n(f) - R(f)| \right] \leq 2 \sqrt{\dfrac{\log S(\mathcal{F},n) + \log 2}{n}}</math></center>
| |
| | |
| In words the VC inequality is saying that as the sample increases, provided that <math>\mathcal{F}</math> has a finite VC dimension, the empirical 0/1 risk becomes a good proxy for the expecred 0/1 risk. Note that both RHS of the two inequalities will converge to 0, provided that <math>S(\mathcal{F},n)</math> grows polynomially in <math>n</math>.
| |
| | |
| The connection between this framework and the Empirical Process framework is evident. Here one is dealing with a modified empirical process <math>|\hat{R}_n - R|_{\mathcal{F}}</math> 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]]). The interested reader can check the book {{ref|aptpp}} Theorems 12.4 and 12.5.
| |
| | |
| == References ==
| |
| *{{note|nslt}}{{cite book
| |
| | last=Vapnik
| |
| | first=Vladimir N
| |
| | authorlink = Vladimir Vapnik
| |
| | title=The Nature of Statistical Learning Theory
| |
| | publisher = [[Springer-Verlag]]
| |
| | series=Information Science and Statistics
| |
| | year = 2000
| |
| | isbn=978-0-387-98780-4}}
| |
| *{{cite book
| |
| | last=Vapnik
| |
| | first=Vladimir N
| |
| | authorlink = Vladimir Vapnik
| |
| | title=''Statistical Learning Theory''
| |
| | publisher = [[John Wiley & Sons|Wiley-Interscience]]
| |
| | year = 1989
| |
| | isbn=0-471-03003-1}}
| |
| *{{note|wcep}} {{cite book |first=Aad W. |last=van der Vaart |first2=Jon A. |last2=Wellner |title=Weak Convergence and Empirical Processes: With Applications to Statistics |edition=2nd |publisher=Springer |year=2000 |isbn=978-0-387-94640-5 }}
| |
| *{{note|aptpp}}{{cite book |first=L.|last=Gyorfi |first2=L. |last2=Devroye| first3=G.|last3 = Lugosi |title=A probabilistic theory of pattern recognition.|edition=1st |publisher=Springer |year=1996 |isbn= 978-0387946184}}
| |
| * See references in articles: [[Richard M. Dudley]], [[empirical processes]], [[Shattered set]].
| |
| *{{note|pollar}}{{cite book
| |
| | last=Pollard
| |
| | first=David
| |
| | title=''Empirical Processes: Theory and Applications''
| |
| | publisher = NSF-CBMS Regional Conference Series in Probability and Statistics Volume 2
| |
| | year = 1990
| |
| | isbn= 0-940600-16-1
| |
| }}
| |
| *{{cite paper
| |
| | last1=Bousquet
| |
| | first1=O.
| |
| | last2 = Boucheron
| |
| | first2 = S.
| |
| | last3 = Lugosi
| |
| | first3 = G.
| |
| | title=Introduction to Statistical Learning Theory
| |
| | journal = Advanced Lectures on Machine Learning Lecture Notes in Artificial Intelligence 3176, 169-207. (Eds.) Bousquet, O., U. von Luxburg and G. Ratsch, Springer
| |
| | year = 2004
| |
| }}
| |
| *{{cite paper
| |
| | last1=Vapnik
| |
| | first1=V.
| |
| | last2 = Chervonenkis
| |
| | first2 = A.
| |
| | title=On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
| |
| | journal = Theory Probab. Appl., 16(2), 264–280.
| |
| | year = 2004
| |
| }}
| |
| | |
| {{DEFAULTSORT:Vapnik-Chervonenkis theory}}
| |
| [[Category:Computational learning theory]]
| |
| [[Category:Empirical process]]
| |