Uniform convergence in probability

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Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family converge to their theoretical probabilities. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory.

The law of large numbers says that, for each single event, its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. But in some applications, we are interested not in a single event but in a whole family of events. We would like to know whether the empirical frequency of every event in the family converges to its theoretical probability simultaneously. The Uniform Convergence Theorem gives a sufficient condition for this convergence to hold. Roughly, if the event-family is sufficiently simple (its VC dimension is sufficiently small) then uniform convergence holds.

Definitions[edit]

For a class of predicates defined on a set and a set of samples , where , the empirical frequency of on is

The theoretical probability of is defined as

The Uniform Convergence Theorem states, roughly, that if is "simple" and we draw samples independently (with replacement) from according to any distribution , then with high probability, the empirical frequency will be close to its expected value, which is the theoretical probability.

Here "simple" means that the Vapnik–Chervonenkis dimension of the class is small relative to the size of the sample. In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole.

The Uniform Convergence Theorem was first proved by Vapnik and Chervonenkis[1] using the concept of growth function.

Uniform convergence theorem[edit]

The statement of the uniform convergence theorem is as follows:[2]

If is a set of -valued functions defined on a set and is a probability distribution on then for and a positive integer, we have:

where, for any ,
and . indicates that the probability is taken over consisting of i.i.d. draws from the distribution .
is defined as: For any -valued functions over and ,

And for any natural number , the shattering number is defined as:

From the point of Learning Theory one can consider to be the Concept/Hypothesis class defined over the instance set . Before getting into the details of the proof of the theorem we will state Sauer's Lemma which we will need in our proof.

Sauer–Shelah lemma[edit]

The Sauer–Shelah lemma[3] relates the shattering number to the VC Dimension.

Lemma: , where is the VC Dimension of the concept class .

Corollary: .

Proof of uniform convergence theorem[edit]

[1] and [2] are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof.

  1. Symmetrization: We transform the problem of analyzing into the problem of analyzing , where and are i.i.d samples of size drawn according to the distribution . One can view as the original randomly drawn sample of length , while may be thought as the testing sample which is used to estimate .
  2. Permutation: Since and are picked identically and independently, so swapping elements between them will not change the probability distribution on and . So, we will try to bound the probability of for some by considering the effect of a specific collection of permutations of the joint sample . Specifically, we consider permutations which swap and in some subset of . The symbol means the concatenation of and .
  3. Reduction to a finite class: We can now restrict the function class to a fixed joint sample and hence, if has finite VC Dimension, it reduces to the problem to one involving a finite function class.

We present the technical details of the proof.

Symmetrization[edit]

Lemma: Let and

Then for , .

Proof: By the triangle inequality,
if and then .

Therefore,

since and are independent.

Now for fix an such that . For this , we shall show that

Thus for any , and hence . And hence we perform the first step of our high level idea.

Notice, is a binomial random variable with expectation and variance . By Chebyshev's inequality we get

for the mentioned bound on . Here we use the fact that for .

Permutations[edit]

Let be the set of all permutations of that swaps and in some subset of .

Lemma: Let be any subset of and any probability distribution on . Then,

where the expectation is over chosen according to , and the probability is over chosen uniformly from .

Proof: For any

(since coordinate permutations preserve the product distribution .)

The maximum is guaranteed to exist since there is only a finite set of values that probability under a random permutation can take.

Reduction to a finite class[edit]

Lemma: Basing on the previous lemma,

.

Proof: Let us define and which is at most . This means there are functions such that for any between and with for

We see that iff for some in satisfies, . Hence if we define if and otherwise.

For and , we have that iff for some in satisfies . By union bound we get

Since, the distribution over the permutations is uniform for each , so equals , with equal probability.

Thus,

where the probability on the right is over and both the possibilities are equally likely. By Hoeffding's inequality, this is at most .

Finally, combining all the three parts of the proof we get the Uniform Convergence Theorem.

References[edit]

  1. ^ a b Vapnik, V. N.; Chervonenkis, A. Ya. (1971). "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities". Theory of Probability & Its Applications. 16 (2): 264. doi:10.1137/1116025.  This is an English translation, by B. Seckler, of the Russian paper: "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities". Dokl. Akad. Nauk. 181 (4): 781. 1968.  The translation was reproduced as: Vapnik, V. N.; Chervonenkis, A. Ya. (2015). "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities". Measures of Complexity. p. 11. doi:10.1007/978-3-319-21852-6_3. ISBN 978-3-319-21851-9. 
  2. ^ a b Martin Anthony Peter, l. Bartlett. Neural Network Learning: Theoretical Foundations, pages 46–50. First Edition, 1999. Cambridge University Press ISBN 0-521-57353-X
  3. ^ Sham Kakade and Ambuj Tewari, CMSC 35900 (Spring 2008) Learning Theory, Lecture 11