Liberating the Dimension for Function Approximation and Integration

Transcription

1 Liberating the Dimension for Function Approximation and Integration G W Wasilkowski Abstract We discuss recent results on the complexity and tractability of problems dealing with -variate functions Such problems, especially path integrals, arise in many areas including mathematical finance, quantum physics and chemistry, and stochastic differential equations It is possible to replace the -variate problem by one that has only d variables since the difference between the two problems diminishes with d approaching infinity Therefore, one could use algorithms obtained in the Information-Based Complexity study, where problems with arbitrarily large but fixed d have been analyzed However, to get the optimal results, the choice of a specific value of d should be a part of an efficient algorithm This is why the approach discussed in the present paper is called liberating the dimension Such a choice should depend on the cost of sampling d-variate functions and on the error demand ε Actually, as recently observed for a specific class of problems, optimal algorithms are from a family of changing dimension algorithms which approximate -variate functions by a combination of special functions, each depending on a different set of variables Moreover, each such set contains no more than d(ε) = O(ln(1/ε)/ ln(ln(1/ε))) variables This is why the new algorithms have the total cost polynomial in 1/ε even if the cost of sampling a d-variate function is exponential in d 1 Introduction We discuss some recent results on computational problems dealing with functions of infinitely many variables, which are called -variate functions Such problems arise in many areas including mathematical finance, quantum physics and chemistry, and in solving deterministic and stochastic differential equations The main source of such problems is probably given by path integrals A partial list of references include [3, 4, 5, 6, 7, 12, 13, 14, 17, 36] Actually, all problems involving expectations of stochastic processes X(t) can be viewed as integration problems for -variate functions, ie, path integration Indeed, consider the expectation E(V(X(t))) for a function V and a Gaussian process X, eg, Brownian motion Due to the Karhulen-Loéve expansion, X(t) = x j g j (t) for iid N (0,1) random variables x j and some functions g j, the expectation is the integral of the -variate function ( ) f(x 1,x 2,) := V x j g j (t) with respect to the probability density functions ρ(x i )=e x2 i /2 / 2π One of the main tools used so far in practice is a variant of the Monte Carlo algorithm; however, it may be slow Since typical -variate functions could be approximated by functions with finite but sufficiently large number d of variables, the volume of results from Information-Based Complexity (IBC for short) could be applied, see, eg, [21] However, we believe that such an approach to -variate problems would not yield the most efficient algorithms G W Wasilkowski Department of Computer Science, University of Kentucky, Lexington, KY 40506, USA, greg@csukyedu 1

2 2 G W Wasilkowski Indeed, the majority of IBC papers on the complexity of multivariate problems consider spaces of functions with d variables for finite yet arbitrarily large d, see again [21] and papers cited there A typical question addressed in these papers is: How does the cost depend on the error demand ε and d? There are many positive results However, since d may be arbitrarily large independently of ε, there are also many negative results We are convinced that when dealing with -variate problems the selection of d should be a part of efficient algorithms and, in particular, should depend on the cost of sampling d-variate functions which is is denoted here by $(d) For instance, sampling a d-variate polynomial of degree 2 requires $(d)=o(d 2 ) arithmetic operations, whereas sampling polynomials of degree 10 is more expensive, $(d)=o(d 10 ) When simulating the Brownian path X(t), the Karhulen-Loéve expansion is usually truncated For instance, we may have X(t;x 1,x 2,) 2/π d sin(( j 1/2)π t/t) x j j 1/2 Hence, again, the cost depends on d and it would be reasonable to take $(d)=o(d) in this case Equally importantly, the value of d should depend on the error demand ε More precisely, d= d(ε) should be a function of ε so that the cost of computing an ε-approximation to the original -variate problem is minimized As we shall see later, for some problems, d(ε) increases surprisingly slowly with decreasing ε This point is important and shows the difference between the study of -variate problems and the study of tractability of multivariate problems For -variate problems, we are interested in algorithms that have good properties (eg, small cost) only for the pairs (ε,d(ε)) for ε (0,1), whereas for multivariate problems, good properties should hold for all the pairs (ε,d) for ε (0,1) and d = 1,2, This is why there are problems with negative tractability results if all pairs(ε, d) are considered and positive results if only pairs(ε,d(ε)) are of interest Such an approach for -variate functions was considered in [24, 31] for approximating Feynman-Kac type of integrals, and more recently in [2, 8, 9, 10, 16, 18, 19, 23] for approximating more general integrals, as well as in [33, 34] for function approximation The presentation of this paper is based on results from [16, 23, 33, 34] As in the four papers mentioned above, the functions to be integrated or approximated belong to a quasireproducing kernel Hilbert space (or Q-RKH space for short) This means that function evaluation may be a discontinuous functional for some sampling points We restrict the attention to the changing dimension algorithms (or CD algorithms for short) introduced in [16] since they provide optimal results, modulo logarithmic terms, with sharp bounds on the tractability exponents The CD-algorithms approximate only the most important terms from a special Fourier expansion of the function being approximated Each term depends on a different set of variables Quite surprisingly, each set contains at most O(ln(1/ε)/ ln(ln(1/ε))) variables This allows efficient algorithms even when the cost function $(d) is exponential in d The approach of using optimal algorithms for approximating the original -variate problem without prespecifying the value of d is what we call liberating the dimension

3 Liberating the Dimension for Function Approximation and Integration 3 2 Basic Concepts In this section, we recall basic definitions/concepts used in the paper For more detailed discussions, we refer to [16, 23, 33, 34] 21 Quasi-Reproducing Kernel Hilbert Spaces We follow here the model introduced in [16] and extended in [33] The spaces F γ of -variate functions are defined as weighted sums of tensor products of a space of univariate real functions More precisely, for a Borel measurable set D R, let F be a separable reproducing kernel Hilbert space (or RKH space for short) of real functions with the domain D whose kernel is denoted by K 1 To omit the trivial case, we always assume that K 0 To stress that F is generated by K, we will often write F = H(K) We will assume throughout the paper that 1 / F, (1) where 1 denotes the constant function f 1 When the information used by algorithms is restricted to function values, we will additionally assume that K(a,a)=0 (2) for a point a D called an anchor We are ready to define the class F γ Let D be the set of infinite sequences x=[x 1,x 2,] with x i D For a finite subset u ofn + ={1,2,}, define the reproducing kernel K u : D D R by The RKH space generated by K u is denoted by K u (x,y) := K(x j,y j ) for all x,y D, with K /0 1 j u H u = H(K u ), where H /0 is the space of constant functions Although, formally, the functions from H u have D as their domain, they depend only on the variables whose indices are listed in u Such variables are referred to as active variables In a number of important applications, consecutive variables of the functions have diminishing importance and/or the spaces of functions have small effective dimension, see eg, [1, 27, 28] Such function spaces can be modeled by using weights γ = {γ u } u, where γ u is a non-negative number The role of γ u is to quantify the importance of the group of variables with indices in u; the larger γ u the more important the group In particular, γ u = 0 means that the corresponding group of variables does not contribute to the functions For example, when γ u = 0 if u 3 then f is a sum of functions with each term depending on at most two variables Although results of [33, 34] hold for general weights, for simplicity of presentation we will restrict the attention to the product weights of the form γ u = γ j and γ /0 = 1, j u where γ j are positive numbers Without loss of generality, we assume that they are ordered, γ j γ j+1 for j 1 Consider next H γ as the pre-hilbert space spanned by the spaces H u and equipped with the inner-product f u, g u := γu 1 f u,g u Hu 1 The results of [33] hold for general Hilbert spaces F We restrict the attention to RKH spaces to simplify the presentation

4 4 G W Wasilkowski for u N+ γu 1 f u 2 H u < and u N+ γu 1 g u 2 H u < Finally, the space F γ is the completion of H γ with respect to the inner-product introduced above Since 1 H u for all u /0, the subspaces H u are mutually orthogonal and any function f F γ has the unique representation f = f u with f u H u (3) Clearly, F γ is also separable Moreover, it is a RKH space iff γ u K u (x,x)< for all x D (4) Since u N+ γ u K u (x,x)= (1+γ j K(x j,x j )), the condition (4) holds iff supk(x,x)< x D and γ j <, and then K γ (x,y) := γ u K u (x,y)= u (1+γ j K(x j,y j )) is well defined and it is the reproducing kernel of F γ If (4) does not hold then function sampling, L x ( f) := f(x), is a discontinuous or ill-defined functional for some x D However, even then, L x is continuous when x has only finitely many components different from the anchor a Indeed, for given x D and u, let[x; u] be a short hand notation for the point with active variables listed inu, ie, { x [x;u] := y=[y 1,y 2,] with y j := j if j u, (5) a if j / u Then f([x;u])= f v (x) and v u L [x;u] 2 = γ v K v (x,x)< v u Of course,[x; /0]=a=[a,a,] and f([x; /0])= f /0 for any x D and any f F γ If (4) does not hold then we refer to such spaces as quasi-reproducing kernel Hilbert spaces (Q-RKH spaces for short) Important examples of such spaces are provided by those generated by Wiener kernel discussed in the following example Example Consider K(x,y)=min(x,y) with D=[0,1] or D=[0, ) In this case, F consists of (locally) absolutely continuous functions with f(0)=0 and f L 2 (D), and the anchor equals a=0 Clearly, if γ j <, then F γ is a RKH space when D=[0,1], and it is only a Q-RKH space when D=[0, ) since sup x [0, ) K(x,x)= 22 Integration Problem Let ρ be a given probability density (p d) function on D We are interested in approximating integrals INT( f) := lim d D d f(x 1,,x d,a,a,) d ρ(x j )d(x 1,,x d ) for f F γ We assume that INT is a well defined and continuous functional on F γ Then INT 2 = γ u C u 0 = (1+γ j C 0 )<, (6)

5 Liberating the Dimension for Function Approximation and Integration 5 where Thus INT < iff C 0 := ρ(x) K(x,y) ρ(y)dydx D D C 0 < and γ j < (7) This is why we will assume that (7) holds whenever the integration problem is considered Moreover, we will also assume that C 0 > 0 since, otherwise, INT( f)= f(a) for all functions from F γ which makes the integration problem trivial 23 Function Approximation Problem As in the previous section, ρ is a given probability desnity on D Without loss of generality, we assume that it is positive almost everywhere on D Then L 2 (D,ρ) endowed with the norm f 2 L 2 (D,ρ) := D f(x) 2 ρ(x)d(x), is a well defined Hilbert space Following [33, 34], we assume that H(K) is continuously imbedded in L 2 (D,ρ), ie, H(K) L 2 (D,ρ) and f 2 L C 1 := sup 2 (D,ρ) f H(K) f 2 < with the convention 0 0 = 0 H(K) Next, consider the space G consisting of functions from F γ with the norm defined by f u Note that the last norm is always finite We are interested in approximating the imbedding operator 2 G := f u 2 L 2 (ρ u,d u ) (8) APP : F γ G given by APP( f)= f The problem is well defined if APP is continuous, and this holds iff APP 2 = sup γ u C u 1 < It is well known that C 1 is is the largest eigenvalue of the integral operator W : H(K) H(K) given by W( f)(x) := f(t) K(t,x) ρ(t)dt (9) D We want to stress that the space G is very special and, perhaps, not always interesting from a practical point of view In particular, it can happen that the approximation problem is easier than the integration problem, and that INT( f) sup < does not hold in general (10) f F f G Indeed, take the reproducing kernel K such that C 0 > 0 and C 1 < Note that D K(x,x) ρ(x) d(x) < implies that C 1 < For γ j = j β with β (0,1], we then have

6 6 G W Wasilkowski INT = while APP = max k N Ck 1 /(k!)β < We chose such a space G in [33, 34] as the first step in the study of approximation for -variate functions The results obtained there will be used in a forthcoming paper [35], see also Section 43, where function approximation is considered with G replaced by the Hilbert space L 2 (D,ρ ) whose norm is given by f 2 L 2 (D,ρ ) = lim d D d f(x 1,,x d,a,a,) 2 d ρ(x j )d(x 1,,x d ) (11) Of course, we will need stronger assumptions on F γ for f L2 (D,ρ ) to be well defined for all f F γ However, then INT( f) f L2 (D,ρ ) for all f F γ, and integration is no harder than approximation 24 Algorithms Let T be the solution operator whose values T ( f) we want to approximate; T = INT for the integration problem, and T = APP for the approximation problem Since F γ is a Hilbert space, we may restrict the attention to linear algorithms, see eg, [26], A n ( f)= n i=1 L i ( f) g i (12) Here the L i s are continuous linear functionals and their values {L i ( f)} n i=1 provide information about the specific function f The elements g i s are numbers for integration and functions from G for approximation If L i s may be arbitrary continuous linear functionals, then we deal with unrestricted linear information In many applications, including integration, only function samplings L i ( f)= f(t i ) are allowed Then A n ( f)= n i=1 f(t i ) g i with t i D and this corresponds to standard information Since in general, F γ is only a Q-RKH space, the sampling points t i used by the algorithms have to be restricted to those that have only finitely many active variables, see (5), ie, t i =[x i,u i ] for some x i D andu i That is, the algorithms using standard information are of the form A n ( f)= n i=1 f([x i,u i ]) g i (13) We believe that the cost of evaluating f at t i =[x i,u i ] should depend on the number u i of active variables in t i That is why we assume that the cost equals $( u i ) for a given cost function $ :N [1, ] At this moment, we only require that $ is monotonically non-decreasing Examples of $ include $(d)=(1+d) α, $(d)=e dα, and $(d)=e edα for α 0 The (information) cost of A n is defined as the total cost of sampling f at the points t i =[x i,u i ], ie, cost(a n ) := n i=1 $( u i )

7 Liberating the Dimension for Function Approximation and Integration 7 For algorithms that use linear functionals L i ( f), the definition of the cost is extended in a natural way with the cost of evaluating L i given as follows Let L i ( f)= f,h i Fγ, where h i F γ is the generator of L i For any h F γ, let Var(h) :={u : h u 0} for h= h u Then Var(h) is the number of active variables in h and the cost of L i ( f) is defined as $( Var(h i ) ) We say that an algorithm A n of the form (12) with L f ( f)= f,h i Fγ, is of a fixed dimension (FD), if there is a finite set V ofn + such that Var(h i )= V for all i=1,2,,n For example, we may have Var(h i ) = {1,,d}, for all i Otherwise, the algorithm is of a changing dimension (CD) As observed in [16], CD algorithms may be significantly superior to FD algorithms In the worst case setting, the error of A n is defined by error wor (A n )=error wor (A n ;F γ,t ) := sup T ( f) A n ( f) G f Fγ 1 In the randomized setting, the choice of the functionals L i or function sample points [x i ;u i ] may be random Then the error of a randomized algorithm is defined by ( 1/2, error ran (A n )=error ran (A n ;F γ,t ) := sup E T ( f) A n ( f) G) 2 f Fγ 1 whereedenotes the expectation with respect to all random parameters in the randomized algorithm A n 25 Complexity and Tractability For a given error demand ε > 0, let comp sett (ε)=comp sett (ε;f γ,t ) := inf { cost(a n ) : error sett (A n ) ε } be the minimal cost among algorithms with errors not exceeding ε Here and elsewhere, sett {wor,ran} denotes the setting When only standard information is allowed, we consider of course only algorithms that use function values To distinguish the complexities with standard and unrestricted linear information, we will sometimes write comp sett (ε;λ) or comp sett (ε;λ,f γ,t ) with Λ = Λ std for standard information and Λ = Λ all for unrestricted linear information We say that the problem is weakly tractable if the complexity is not exponential in 1/ε, ie, lim ε ln( comp sett (ε) ) = 0 ε 0 A stronger notion is polynomial tractability which means that there are some non-negative C and p such that comp sett (ε) C ε p for all ε > 0 The smallest (or more precisely, infimum of) such p is called the exponent of polynomial tractability, ie, p sett := limsup ε 0 ln(comp sett (ε)) ln(1/ε) We sometimes write p sett = p sett (Λ) or p sett (Λ,F γ,t ) with Λ {Λ all,λ std } to stress what type of information is used

8 8 G W Wasilkowski 3 Results for Integration We present in this section selected results from [23] for CD algorithms Recall that these algorithms were defined for the first time in [16] and have the following form A n ( f)= n i=1 f([x i ;u i ]) g i for some points x i, sets of active variables u i, and the numbers g i which may depend on n Moreover, in the randomized setting, all parameters x i,u i, and g i may be chosen randomly In what follows, the operator I is the integration operator for functions from H(K), Theorem 1 Let sett {wor, ran} Suppose that the product weights satisfy I( f)= f(x) ρ(x)dx D γ j = O( j β ) for β > 1, (14) there exists a sequence of algorithms{a n } n for the univariate problem and positive constants α,c such that A n uses at most n function evaluations and the error of A n for the univariate integration problem over the space H(K) satisfies error sett (A n ;H(K),I) c n α for all n N (15) Then there are algorithms{a ε } ε for the -variate integration problem such that error sett (A ε ;F γ,int) ε for all ε > 0 with the following bounds on their cost If $(d) = O ( e k d) ) for some k 0, then for all p > max( 1α 2, β 1 there exists a number C depending, in particular, on p such that cost(a ε ) C ε p for all ε > 0 This means that the -variate integration problem is polynomially tractable with the exponent at most ( ) 1 max α, 2 β 1 Furthermore, in the worst case setting, the exponent is equal to the maximum above if α and β are sharp, and $(d)=ω(d) ( If $(d)=o e ek d) for some k 0, then lim ε 0 ε ln(cost(a ε))=0 This means that the -variate integration problem is weakly tractable We now comment on this theorem As shown in [16] for the integration problem in the worst case setting for the Wiener kernel and D=[0,1], the assumption (14) is necessary for polynomial tractability If the algorithms A n are deterministic then so are the algorithms A ε The proof is constructive Algorithms A ε are based on Smolyak s construction from [25] and results from [30] Moreover, the algorithms A ε use function values at points that have at most d(ε) = o(ln(1/ε)) active variables This is why the problem is polynomially tractable even when the cost function $ is exponential, and is weakly tractable even when the cost function $ is doubly exponential

9 Liberating the Dimension for Function Approximation and Integration 9 Assume now that the complexity of the univariate integration problem over H(K) is Θ(ε p ) Then we can find algorithms A n for which (15) holds with α = 1/p and this value of α is the largest one Then the exponent of polynomial tractability equals p sett = α 1 = p whenever β 1+2/p (16) In this case, the -variate problem is roughly of the same complexity as the univariate problem If β (1,1+2/p) then the -variate problem is harder than the univariate problem but still we have polynomial tractability In this case, however, the exponent can be arbitrarily large The proof that the exponent is sharp also in this case is based on a lower bound from [16] for the -integration problem in the worst case setting We illustrate the theorem for the Wiener kernel Example (continued) For K(x,y)=min(x,y), D=[0,1], and ρ 1, the condition (15) holds with α = 1 in the worst case setting and with α = 3/2 in the randomized setting, and both values are sharp Hence ( ) ( ) p wor = max 1, and β 1 3 pran max 3, 2 β 1 Note that the exponent in the randomized setting is smaller that the exponent in the worst case setting if β > 3 It is open what is the actual value of p ran for β (1,4) 4 Results for Approximation We present in this section selected results from [33] for unrestricted linear information and from [34] for standard information All of them are for the worst case setting and for the range space G We next discuss extensions of the results to the randomized setting and to the range space L 2 (D,ρ ) Recall that for the approximation problem, APP is the imbedding operator from F γ to G We will also use S to denote the imbedding from H(K) to L 2 (D,ρ) 41 Unrestricted Linear Information Consider the operator W defined by (9) It is well known, see eg, [26], that a necessary condition for polynomial tractability of the approximation problem is a polynomial dependence of the eigenvalues λ j of W, ie, λ j = O ( j 2 α) for some α > 0 (17) This is because the errors of optimal algorithms A n for the univariate approximation over H(K) are equal to or equivalently, error wor (A n ;H(K),S)= λ n+1 = O ( n α), comp wor (ε;λ all,h(k),s)=$(1) inf { n : λ n+1 ε 2} One of the results in [33] is the construction of optimal algorithms for the -variate problem which allows to get a necessary and sufficient condition on the polynomial tractability for general weights γ u Here we state one special result for the product weights Theorem 2 Consider the worst case setting Suppose that the product weights satisfy ( γ j = O j β) for β > 0 (18)

10 10 G W Wasilkowski and the eigenvalues satisfy (17) Then there are algorithms{a ε } ε for the -variate approximation problem such that error wor (A ε ;F γ,app) ε with the following bounds on their cost If $(d)=o ( e k d) ) for some k 0, then for all p>max( 1α, 2 β there exists a number C depending, in particular, on p such that cost(a ε ) C ε p for all ε > 0 This means that the -variate problem is polynomially tractable with the exponent at most ( 1 max α, 2 ) β Furthermore, ( the exponent is equal to the maximum above if α and β are sharp, and $(d) = Ω(d) If $(d)=o e ek d) for some k 0, then This means that the problem is weakly tractable lim ε ln(cost(a ε))=0 ε 0 As before, the proof is constructive Moreover, A ε uses inner products with generators having at most d(ε) active variables, where now ( ) ln(1/ε) d(ε)=o ln(ln(/ε)) Example (continued) For the Wiener kernel, D=[0,1], and ρ 1, we have α = 1 and, hence, ( p wor (Λ all )=max 1, 2 ) β 42 Standard Information We have a similar result for algorithms using standard information, see [34, Thm7] Theorem 3 Consider the worst case setting Suppose that the product weights satisfy (18) and there exists a sequence of algorithms{a n } n, each using at most n function evaluations, such that their errors for the univariate approximation problem over the space H(K) satisfy error wor (A n ;H(K),S) c n α for α > 0 (19) Then there are algorithms{a ε } ε for the -variate approximation problem using standard information such that error wor (A ε ;F γ,app) ε with the following bounds on their cost If $(d)=o ( e k d) ) for some k 0, then for all p>max( 1α, 2 β there exists a number C depending, in particular, on p such that cost(a ε ) C ε p for all ε (0,1) This means that the problem is polynomially tractable with the exponent at most

11 Liberating the Dimension for Function Approximation and Integration 11 ( 1 max α, 2 ) β Furthermore, ( the exponent is equal to the maximum above if α and β are sharp, and $(d) = Ω(d) If $(d)=o e ek d) for some k 0, then lim ε ln(cost(a ε))=0 ε 0 This means that the -variate problem is weakly tractable Again, the proof is constructive and the sampling points used by A ε have at most d(ε) active variables with ( ) ln(1/ε) d(ε)=o ln(ln(1/ε)) We stress that the parameters α in (17) and (19) are not necessarily the same The parameter α in (17) describes the power of unrestricted linear information given by the decay of the eigenvalues λ j The parameter α in (19) describes the power of standard information given by the best speed of convergence of algorithms using n function evaluations There are examples of spaces H(K) for which the values of α for unrestricted linear and standard information are different, see [11] There is still an open problem whether they are the same if we assume that α > 1/2 for unrestricted linear information, see [22] for more details Example (continued) For the Wiener kernel, D=[0,1], and ρ 1, the conditions (17) and (19) hold with the same α = 1 Therefore, the exponent of polynomial tractability for standard information is the same as for unrestricted linear information, ( p wor (Λ std )= p wor (Λ all )=max 1, 2 ) β For this particular space, standard and unrestricted linear information are equally powerful Note that for β (0, 3), the exponent for the approximation problem is smaller than the corresponding exponent for the integration problem This is due to the special form of the space G, see Section L 2 -Approximation As already mentioned, the space G was chosen for the approximation problem since it has a relatively simple structure of the eigenpairs of the operator W = APP APP In the forthcoming paper [35], we will present results for the L 2 -approximation problem with the space G replaced by the L 2 = L 2 (D,ρ ) space whose norm is given by (11) Here are some results for product weights It is easy to see that L 2 = G if C 0 = 0 This is why we assume that C 0 > 0 also for the L 2 -approximation problem The first result of [35] is for $(d)= ( e k d) Then the L 2 -approximation problem is polynomially tractable iff the exponent β satisfies β > 1 Recall that for the G -approximation problem, we only need β > 0 Next, if β > 1 then the exponent of polynomial tractability of the L 2 -approximation problem is bounded by ( ) 1 p wor (Λ) max α, 2, β 1 where α is from Theorem 2 for Λ = Λ all and from Theorem 3 for Λ = Λ std Moreover, if α and β are sharp and Ω(d)=$(d)=O ( e k d) then

12 12 G W Wasilkowski ( ) 1 p wor (Λ std )=max α, 2 β 1 ( Furthermore, if $(d) = O e ek d) then the L 2 -approximation problem is weakly tractable In another words, we have similar results for L 2 -approximation as for G -approximation with β replaced by β 1 44 Randomized Setting It has been known for quite some time, see [20, 29], that randomization does not help for multivariate approximation defined over Hilbert spaces when unrestricted linear information is allowed More precisely, for a Hilbert space F d of d-variate functions and the the space G d with norm f 2 G d = f(x) 2 ρ d (x)dx, D d consider the problem of approximating the corresponding imbedding operator S d : F d G d Let sett {wor,ran} Denote by κ sett (Λ all,s d ) the order of convergence of optimal algorithms in the worst case and randomized settings, respectively That is, κ sett (Λ all,s d ) is the supremum of α for which the worst case (or randomized) error of an optimal algorithm using n linear functionals is of order n α Then the results of [20, 29] imply that κ ran (Λ all,s d )=κ wor (Λ all,s d ) More recently, it has been constructively proved in [32] that the standard information is as powerful as Λ all in the randomized setting That is, if κ sett (Λ std,s d ) denotes the order of convergence of optimal algorithms using standard information then κ ran (Λ std,s d )=κ ran (Λ all,s d )=κ wor (Λ all,s d ) As already mentioned, the power of standard information in the worst case setting is not yet completely known It terms of the orders of convergence, it was recently shown, see [11], that there exist reproducing kernel Hilbert spaces F d for which However, it is still open whether κ wor (Λ std,s d )=0 and κ wor (Λ all,s d )= 1 2 κ wor (Λ std,s d )=κ wor (Λ all,s d ) if κ wor (Λ all,s d )> 1 2, see [22] for more details Since in the study of multivariate problems the cost is measured by the number of linear functional or function values used by an algorithm, these and further results translate into complexity and tractability results Namely, the complexity and tractability of{s d } in the worst case setting with Λ all are equivalent to complexity and tractability in the randomized setting with Λ all and/or Λ std It turns out that similar results hold for -variate approximation problem with the cost depending on the number of active variables More precisely, we have the following theorem Theorem 4 Assume that the cost function $(d)=o ( e k d) for some k 0, the eigenvalues λ j = O ( j 2 α) for some α > 0, the product weights are γ j = O( j β ) with the exponent β > 0 for the G -approximation problem, and β > 1 for the L 2 -approximation problem Then p ran (Λ std )= p ran (Λ all )= p wor (Λ all )

13 Liberating the Dimension for Function Approximation and Integration 13 We now outline the proof of this theorem We will do it only for the G -approximation problem since similar arguments and arguments similar to those in [34] can be used for the L 2 -approximation For this purpose, we need to recall some facts about the optimal algorithms for G -approximation in the worst case setting with Λ all The optimal algorithm A ε whose error is at most ε has the form A ε ( f)= A u,n(u,ε) ( f u ) for u U(ε) f = u U γ f u, where A u,n(u,ε) are special projections into H u and they use n(u,ε) linear functional evaluations The set U(ε) is a special finite subset of U γ In particular, A ε ( f v )=0 for f v with v / U(ε) Furthermore, for all u U(ε) we have u d(ε), where d(ε) is the maximal number of active variables As already mentioned, we have d(ε)= max u U(ε) u =O ( ln(1/ε) ln(ln(1/ε)) ) The cost of A ε is given by cost(a ε )= $( u ) n(u,ε) $(d(ε)) n(u,ε) u U(ε) u U(ε) Each algorithm A u,ε can be replaced by the corresponding randomized algorithm that uses standard information due to the already cited result from [32] However, these randomized algorithms need to evaluate functions f u for u U(ε) instead of the whole function f As shown in [15], a value of f u can be obtained by computing at most 2 u values of f at points with at most u active variables Note that ) ln (2 d(ε) 2 u 2 d(ε) and (ln(1/ε)) c/ln(ln(1/ε)) 1 ln(1/ε) for a positive constant c This implies that To show the opposite inequality, ie, note that since ln(cost(a ε )) lim sup ε 0 ln(1/ε) limsup ε 0 p ran (Λ std ;APP) p wor (Λ all ;APP) p wor (Λ all ;APP) p ran (Λ all ;APP), ln ( u U(ε) n(u,ε) ) ) + ln ($(d(ε)) 2 d(ε) ln(1/ε) ) ln ($(d(ε)) 2 d(ε) lim sup lim(ln(1/ε)) c /ln(ln(1/ε)) 1 = 0 ε 0 ln(1/ε) ε 0 ln ( u U(ε) n(u,ε) ) = limsup ε 0 ln(1/ε) This means that the cost function $ does not contribute to the tractability exponents p sett (Λ all ;APP) and we can replace it by $(d) 1 For such a constant cost function, the worst case ε-complexity is the same as the complexity with respect the space F d given by the span of H u for u U(ε/2) as follows from the proof in [33] Moreover, the complexity in the randomized setting is bounded from below if F γ is replaced by F d Hence the results from [20, 29] complete the proof for the G -approximation Acknowledgements I would like to thank Henryk Woźniakowski for valuable comments and suggestions to this paper

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