Tractability properties of the discrepancy in Orlicz norms

10/28/2019
by   Josef Dick, et al.
0

We show that the minimal discrepancy of a point set in the d-dimensional unit cube with respect to Orlicz norms can exhibit both polynomial and weak tractability. In particular, we show that the ψ_α-norms of exponential Orlicz spaces are polynomially tractable.

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1 Introduction and main results

The discrepancy of an -element point set in the unit cube measures the deviation of the empirical distribution of from the uniform measure. This concept has important applications in numerical analysis, where so-called Koksma-Hlawka inequalities establish a deep connection between norms of the discrepancy function and worst case errors of quasi-Monte Carlo integration rules determined by the point set . For a comprehensive introduction and exposition on this subject we refer the reader to [5, 10, 12] and the references cited therein.

To define the concept of discrepancy, we first introduce the local discrepancy function defined as

where for and stands for the -dimensional Lebesgue measure. We now apply a norm to the local discrepancy function to obtain the discrepancy of the point set with respect to the norm . Of particular interest are the norms on the usual Lebesgue spaces () of -integrable functions on the unit cube . Those lead to the central notions of -discrepancy for , and the -discrepancy, which is usually called the star-discrepancy, when .

The minimal discrepancy with respect to the norm in dimension is the best possible discrepancy over all point sets of size in the -dimensional unit cube , i.e.,

We compare this value with the initial discrepancy given by the discrepancy of the empty point set . Since the initial discrepancy may depend on the dimension, we use it to normalize the minimal discrepancy when we study the dependence of on the dimension . We therefore define the inverse of the minimal discrepancy in dimension as the number which is the smallest number such that a point set with points exists that reduces the initial discrepancy at least by a factor of ,

In this paper we are interested in how depends simultaneously on and the dimension . In general, the dependence of the inverse of the minimal discrepancy can take different forms. For instance, if the dependence on the dimension or on is exponential, then we call the discrepancy intractable. If the inverse of the minimal discrepancy grows exponentially fast in

, then the discrepancy is said to suffer from the curse of dimensionality. On the other hand, if

increases at most polynomially in and , as increases and tends to zero, then the discrepancy is said to be polynomially tractable. This leads us to the following definition.

Definition 1.

The discrepancy with respect to the norm is polynomially tractable if there are numbers , , and such that

(1)

The infimum over all exponents such that a bound of the form (1) holds is called the -exponent of polynomial tractability.

To cover cases between polynomial tractability and intractability, we now introduce the concept of weak tractability, where is not exponential in and . This encodes the absence of intractability.

Definition 2.

The discrepancy with respect to the norm is weakly tractable, if

The subject of tractability of multivariate problems is a very popular and active area of research and we refer the reader to the books [15, 16] by Novak and Woźniakowski for an introduction into tractability studies of discrepancy and an exhaustive exposition.

A famous result by Heinrich, Novak, Wasilkowski, and Woźniakowski [8] based on the theory of empirical processes and Talagrand’s majorizing measure theorem shows that the star-discrepancy is polynomially tractable. In fact, they show that in Definition 1 can be set to one and hence in this case the inverse of the star-discrepancy depends at most linearly on the dimension . It was shown in [8] and [9] that is the minimal possible in Definition 1 for the star-discrepancy. Determining the optimal exponent for is an open problem. On the other hand, the -discrepancy is known to be intractable, as shown by Woźniakowski [17] (see also [16]). The behavior of the inverse of the -discrepancy in between, where , seems to be unknown.

Note that due to the normalization with the initial discrepancy, we cannot infer a continuous change in the behavior of as goes from to . A natural assumption seems to be that the -discrepancy is intractable for any . If correct, this would mean that there is a sharp change from intractability to polynomial tractability as one goes from to . A natural question which hence arises is what happens between those two cases and .

To study this question, we introduce for the exponential Orlicz norms , which for a measurable function defined on are given by

where . The assumption guarantees the convexity of . These norms play an important role in the study of the concentration of mass in high-dimensional convex bodies and we refer the reader to [3, 4] and [13] for more information. An introduction to the theory of Orlicz spaces can be found in [11]. As we shall see later, the discrepancy with respect to -norms turns out to be polynomially tractable as well.

In our context it is interesting to also study variations of these norms exhibiting different types of behavior of as a function of the dimension . In fact, we may write as the series

and consider the more general case where is replaced by a function

(2)

for a non-decreasing function with . Note that the growth condition on guarantees, according to the ratio test, the absolute convergence of the series (2) for all . Choosing takes us back to the -norm, which is therefore a special case of the more general setting.

Below we will characterize functions for which the discrepancy with respect to , given by

is polynomially tractable and weakly tractable. In general, if is zero in zero, increasing, convex, and satisfies , then is called an -function and is a norm. The limit assumptions simply guarantee that the convex-dual is again an -function. Such types of norms are known as Luxemburg norms, named after W. A. J. Luxemburg [14]. One typically just speaks of Orlicz functions and Orlicz norms.

The aim of this paper is to show the following result.

Theorem 1.

Let . Then the following hold:

  1. The discrepancy with respect to the -norm is polynomially tractable.

  2. For any for which there exists an and a constant such that for all

    (3)

    the discrepancy with respect to is polynomially tractable. The -exponent of polynomial tractability is at most .

  3. For any which satisfies

    (4)

    the discrepancy with respect to is weakly tractable.

Remark 1.

Note that by choosing we obtain the classical -norm. In this case and for all and otherwise. This choice of does not satisfy any of the conditions in Theorem 1.

An example of a function that satisfies condition (4) for weak tractability is with some . This function does not satisfy condition (3).

We can in fact provide a more accurate estimate for the exponential Orlicz norms and the

-exponent of polynomial tractability.

Theorem 2.

For any , we have

where

In particular, the -exponent of polynomial tractability is at most .

This upper bound on shows that for the inverse of the star-discrepancy depends linearly on the dimension, thereby matching the result of Heinrich, Novak, Wasilkowski, and Woźniakowski [8].

In the following Section 2 we present the proofs of our main results, where we start by establishing an equivalence between the norms and an expression involving a supremum of classical -norms. Subsection 2.1 is then devoted to the proof of Theorem 1. The proof of Theorem 2 will be presented in Subsection 2.2.

2 The proofs

For the proofs of Theorems 1 and 2 we define another norm which we show to be equivalent to the Orlicz norm , namely

(5)

with . In the special case of exponential Orlicz norms such an equivalence is a classical result in asymptotic geometric analysis and may be found, without explicit constants, in the monographs [3, Lemma 3.5.5] and [4, Lemma 2.4.2]. In the context of this paper it is important that these constants do not depend on the dimension .

Lemma 1.

Let and . For any measurable function , we have the estimates

(6)

In particular, for any , we have

(7)

where .

Proof.

Using the series expansion of , we obtain

By choosing

we obtain

Therefore, we have

This implies the upper bound in (6) for all .

In the opposite direction, we need to choose such that

which implies that

If , and , then

Hence,

In any case, for all , we have that

which implies the result since .

The bound (7) for the -norms can be shown using similar arguments together with Stirling’s formula

(8)

We are now prepared to present the proofs of our main results.

2.1 The proof of Theorem 1

An important consequence of Lemma 1 is that the constants do not depend on the dimension, and hence the Orlicz norm discrepancy satisfies the same tractability properties as the discrepancy with respect to the norm . Therefore in the following proof we will only use the latter norm.

It is well known and easily checked (see, e.g., [16, p. 54]) that for every , the initial -discrepancy in dimension satisfies

If , then the initial discrepancy is for every dimension . This implies that

where we used the choice to obtain the last inequality.

From [8] we know that

(9)

for some absolute constant . Aistleitner [1] showed that one can choose , but according to [7] the constant may be reduced to .

Hence, we have

where stands for the discrepancy with respect to the norm introduced in (5). This implies that

(10)

where for , . This concludes the proof of the second statement in Theorem 1. As mentioned above, if we choose , then we obtain the -norm. Using Stirling’s formula (8) together with the previous result, we can deduce the first part of Theorem 1.

In order to prove the third part of Theorem 1, we apply the logarithm to . From (2.1) we obtain that

for some only depending on . Hence,

This implies weak tractability of the discrepancy with respect to .

2.2 The proof of Theorem 2

First we show the corresponding result for which is based on the norm . Recall that for a measurable function , we defined . Let us start with a lower bound for the initial discrepancy. We have

(11)

where we have chosen . The final estimate follows from the fact that

attains its minimum in with minimal value .

Now let . Then from Gnewuch [6, Theorem 3] we obtain that

and from Aistleitner and Hofer [2, Corollary 1] that for any

where the expectation and probability are with respect to the point set

consisting of independent and uniformly distributed points. Now Markov’s inequality implies that there exists an

-element point set in such that

provided that

For this point set , we obtain

(12)

Combining (11) and (12), we obtain the upper bound

for all . Note that we may choose leading to .

Using the second part of Lemma 1, we obtain

with

From this we finally obtain the upper bound for .

References

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