2 Algorithms and minimal errors

In general, a deterministic algorithm $A_n\colon \, F\to Y$ is an arbitrary mapping of the form $A_n=\varphi _n\circ N_n$ with $N_n\colon \, F\to \mathbb {R}^n$ being the information mapping, and $\varphi _n\colon \, \mathbb {R}^n\to Y$ the reconstruction mapping. We mostly pose no restriction at all on the mappings $\varphi _n$ and focus on the form of $N_n$ ; see also Section 6. The most general form we consider is that an information mapping is given recursively by

$$\begin{align*}N_n(f) = \left( N_{n-1}(f),\, L_n(f) \right), \end{align*}$$

where the choice of the n-th linear functional $L_n=L_n(\cdot ,N_{n-1}(f))$ may depend on the first $n-1$ measurements. This is called an adaptive choice of information, and we denote the collection of all such algorithms by $\mathcal {A}_n^{\mathrm {det}} (F,Y)$ , or just $\mathcal {A}_n^{\mathrm {det}}$ .

An algorithm is called nonadaptive if $N_n=(L_1,\dots ,L_n)$ , that is, the same functionals are used for every input, and we denote by $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}}$ the corresponding class of algorithms.

Let us add that the assumption that measurements are given by linear functionals is very common in numerical analysis and approximation theory. However, also other concepts are possible. We shortly discuss this in Section 6.2.

For an algorithm $A_n\in \mathcal {A}_n^*$ with $*\in \{\mathrm {det},\mathrm {det}\text{-}\mathrm{non}\}$ , a mapping $S\colon \, X\to Y$ and a set $F\subset X$ , we define the worst-case error of $A_n$ for approximating S over F by

$$\begin{align*}e(A_n,S,F) \;:=\; \sup_{f\in F} \|S(f)-A_n(f)\|_Y. \end{align*}$$

(Note that we omit the Y in $\|\cdot \|_Y$ when no confusion is possible.)

Randomized algorithms are random variables whose realizations are deterministic algorithms as described above.

A randomized algorithm is a family of deterministic algorithms $A_n=(A_n^\omega )_{\omega \in \Omega }\subset \mathcal {A}_n^{\mathrm {det}}(F,Y)$ which is indexed by a probability space $(\Omega ,\mathcal {A},\mathbb {P})$ . For technical reasons, we assume that the mapping $(f,\omega ) \mapsto \|S(f)-A_n^\omega (f)\|_Y$ is $(\mathcal {B}_F\otimes \mathcal {A},\,\mathcal {B}_Y)$ -measurable, where $\mathcal {B}_Y$ denotes the Borel $\sigma $ -algebra on Y, the set F is assumed to be convex, and $\mathcal {B}_F$ denotes the Borel $\sigma $ -algebra of the topology associated with F, that is, with respect to the semi-norm whose unit ball is the convex and symmetric set $F-F$ . Then, formally, the desirable statement $\mathcal {A}_n^{\mathrm {det}}\subset \mathcal {A}_n^{\mathrm {ran}}$ is not correct since we do not assume that a deterministic algorithm has to be measurable. See [Reference Novak and Woźniakowski51, Section 4.3.3] and Section 6 for a discussion of this technicality.

We denote the class of all such (possibly adaptive) algorithms by $\mathcal {A}_n^{\mathrm {ran}}(F,Y)$ and let $\mathcal {A}_n^{\mathrm {ran}\text{-}\mathrm{non}}(F,Y)$ be the class of randomized algorithms whose realizations are nonadaptive. Again, we may omit the dependence on F and Y. We define the worst-case error of a randomized algorithm $A_n\in \mathcal {A}_n^{\mathrm {ran}}(F,Y)$ for approximating S over F by

$$\begin{align*}e(A_n,S,F) \;:=\; \sup_{f\in F}\, \mathbb{E}\, \|S(f)-A_n(f)\|_Y. \end{align*}$$

In order to compare the power of the just introduced types of algorithms, we now define the n-th minimal worst-case error for approximating S over F by

$$\begin{align*}e_n^{*}(S,F) \;:=\; \inf_{A_n\in \mathcal{A}_n^{*}}\, e(A_n,S,F), \end{align*}$$

where $*\in \{\mathrm {det}, \mathrm {det}\text{-}\mathrm{non}, \mathrm {ran}, \mathrm {ran}\text{-}\mathrm{non}\}$ .

The respective concepts can indeed lead to very different minimal errors. Several examples, remarks, and open problems will be presented in Section 6.

3 Widths and s-numbers

Widths have a long tradition in approximation theory and there is a whole range of widths of sets within normed spaces. See, for example, Tikhomirov [Reference Tikhomirov66] and Ismagilov [Reference Ismagilov26] for early treatments, and Pinkus [Reference Pinkus58] or Lorentz et al. [Reference Lorentz, Golitschek and Makovoz40] for books on the subject. A somehow competing concept are s-numbers of operators which play an important role in operator theory and geometry of Banach spaces, see Pietsch [Reference Pietsch55, Reference Pietsch56]. A short account of their history and potential differences can be found in [Reference Pietsch56, 6.2.6], see also [Reference Edmunds and Lang13, Reference Heinrich20]. Some of these widths and numbers have an obvious relation to algorithms, and hence to information-based complexity, while others are seemingly unrelated. We will discuss some known relations in Section 4. However, we first study the relation of the relevant numbers among each other.

We start by providing a common generalization of the above concepts. That is, we introduce various s-numbers of a mapping $S\in \mathcal {L}(X,Y)$ on a subset $F\subset X$ . Alternatively, one may call them widths of a set $F\subset X$ with respect to a mapping $S\in \mathcal {L}(X,Y)$ .

The original definitions of the corresponding widths of sets $F\subset X$ are obtained by considering the s-numbers of the identity $\mathrm {id}_X$ on $F\subset X$ (or the width of F w.r.t. $\mathrm {id}_X$ ), while s-numbers of the operator S are recovered by considering $F=B_X$ (or the width of $B_X$ w.r.t. S).

Here and in the following, the (closed) unit ball of X is denoted by $B_X$ and the continuous dual space of X by $X'$ .

We define the Gelfand numbers of $S\in \mathcal {L}(X,Y)$ on $F\subset X$ by

$$ \begin{align*} c_n(S,F) :=&\ \inf_{L_1,\dots,L_{n}\in X'}\, \sup_{\substack{f,g\in F: \\ L_k(f)=L_k(g)}} \frac12\, \big\|S(f)-S(g)\big\| \\= &\ \inf_{\substack{M \subset X \text{ closed} \\ \operatorname{\mathrm{codim}}(M)\le n}}\ \sup_{\substack{f,g\in F: \\ f-g\in M}}\, \frac12\, \big\|S(f)-S(g)\big\|. \end{align*} $$

In particular, $c_0(S,F)=\frac {1}{2}\mathrm {diam}\big (S(F)\big )$ . Note that in the theory of s-numbers, there is usually an index shift of one and “ $s_n$ ” is only considered for $n\ge 1$ (such that $s_1(S)=\|S\|=\frac {1}{2}\mathrm {diam}\big (S(B_X)\big )$ ). We use a different convention here because n is used for the amount of information. It is well-known, and we will present the details in Section 4, that the Gelfand numbers $c_n$ are closely related to $e_n^{\mathrm {det}\text{-}\mathrm{non}}$ .

Other quantities that will serve as lower bounds for all minimal errors are the Bernstein numbers of S on F, which are defined by

$$ \begin{align*} b_n(S,F) \,:= \sup_{\substack{\dim(V)=n+1 \\ S \text{ injective on } V}} \sup\big\{ &r>0 \,:\, g+ B \subset F \text{ for some } g\in F \\[-17pt] &\text{and a ball } B \text{ of radius } r \text{ in } (V,\Vert\cdot\Vert_S) \big\}. \end{align*} $$

Here, we consider the norm on the linear space V that is induced by S, that is, $\Vert x \Vert _S := \Vert Sx \Vert _Y$ . If F is convex and symmetric, it suffices to consider balls centered at the origin in the above definition. We note again that these numbers coincide with the classical Bernstein widths if S is the identity on X. In the special case that F is a bounded subset of X, it is not hard to verify that we have the handy formula

$$ \begin{align*} b_n(S,F) \,&=\, \sup_{\substack{V\subset X \text{ affine}\\\dim(V)=n+1}}\, \sup_{g\in F\cap V} \,\inf_{f\in V\cap (X\setminus F)} \, \|S(f)-S(g)\|. \end{align*} $$

We will see in Section 4, that one obtains bounds on $e_n^{\mathrm {\mathrm {det}\text{-}\mathrm{non}}}/e_n^{\mathrm {ran}}$ from corresponding bounds involving $c_n$ and $b_n$ , which is the approach of this paper. For this, we want to employ proof ideas that have already been used for bounding the maximal difference between s-numbers, see, for example, [Reference Pietsch55, Reference Pietsch56]. Inspired by Hilbert numbers, see [Reference Bauhardt4], which are the smallest s-numbers, we introduce the Hilbert numbers of $S\in \mathcal {L}(X,Y)$ on $F\subset X$ by

$$ \begin{align*} h_n(S,F) \,:=\, \sup\Big\{&c_n\big(BSA, B_{\ell_2}\big)\colon\, B\in\mathcal{L}(Y,\ell_2) \text{ with } \Vert B\Vert \le 1,\\ &A\in \mathcal{L}(\ell_2,X) \text{ and } x\in F \text{ with } A(B_{\ell_2}) + x \subset F \Big\}. \end{align*} $$

In this definition, we can replace $c_n$ with $b_n$ since both numbers coincide for operators $T\in \mathcal {L}(\ell _2,\ell _2)$ ; they are both equal to the $(n+1)$ -st singular value of T, see, for example, [Reference Pietsch53].

One of the key ingredients to our results will be bounds between these numbers. First, note that they are related in the same way as the corresponding s-numbers, see [Reference Bauhardt4, Reference Pietsch53].

Proposition 3.2. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every convex $F\subset X$ and $n\in \mathbb {N}_0$ , we have

$$\begin{align*}h_n(S,F) \;\le\; b_n(S,F) \;\le\; c_n(S,F). \end{align*}$$

Equalities hold if X and Y are Hilbert spaces and $F=B_X$ .

Proof of Proposition 3.2

In order to prove $b_n \ge h_n$ , let $B\in \mathcal {L}(Y,\ell _2)$ with $\Vert B \Vert \le 1$ as well as $A\in \mathcal {L}(\ell _2,X)$ and $g\in X$ with $A(B_{\ell _2}) + g \subset F$ . For any $\beta < b_n\big (BSA, B_{\ell _2}\big )$ , there exists an $(n+1)$ -dimensional linear space $V \subset \ell _2$ such that $BSA$ is injective on V and for any $v\in V$ it holds that $\Vert BSAv \Vert _2 \le \beta $ implies $v\in B_{\ell _2}$ . First, we observe that the injectivity of $BSA$ implies that S is injective on $W=A(V)$ and that W is $(n+1)$ -dimensional. Moreover, let $f\in W$ with $\Vert Sf\Vert _Y \le \beta $ . Choose $v\in V$ with $f=Av$ . We have

$$\begin{align*}\Vert BSAv\Vert_2 \le \Vert SAv \Vert_Y = \Vert Sf \Vert_Y \le \beta \end{align*}$$

and hence $v\in B_{\ell _2}$ . By assumption, this implies $f+g\in F$ for all such f. Hence, F contains an $\Vert \cdot \Vert _S$ -ball of radius $\beta $ in W and we have $b_n(S,F) \ge \beta $ . Taking the supremum over all $\beta $ gives

$$\begin{align*}b_n(S,F) \,\ge\, b_n(BSA, B_{\ell_2}) \end{align*}$$

and taking the supremum over all B, A, and g as above gives

$$\begin{align*}b_n(S,F) \,\ge\, h_n(S,F). \end{align*}$$

In order to show $c_n \ge b_n$ , let $\beta <b_n(S,F)$ be arbitrary and let $V\subset X$ be an $(n+1)$ -dimensional subspace such that S is injective on V as well as $m\in F$ such that $h\in V$ and $\Vert Sh \Vert _Y \le \beta $ imply $m+h \in F$ . Now, for all $L_1,\dots ,L_n\in X'$ , there must be some $h\in V\setminus \{0\}$ with $L_i(h)=0$ for all $i\le n$ . We choose h such that $\Vert Sh \Vert _Y=\beta $ , which implies $f=m+h \in F$ and $g=m-h \in F$ . Note that $L_i(f)=L_i(g)$ for all $i\le n$ . Moreover, $\frac 12 \Vert Sf-Sg \Vert = \beta $ and hence $c_n(S,F)\ge \beta $ .

Here, we prove the following reverse inequalities, which are reminiscent to the corresponding bounds for s-numbers, see Remark 3.4.

Theorem 3.3. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every convex $F\subset X$ and $n\in \mathbb {N}$ , we have

$$\begin{align*}c_{n-1}(S,F) \;\le\; \left(\prod_{k=0}^{n-1} c_k(S,F)\right)^{1/n} \;\le\; n^{3/2}\,\left(\prod_{k=0}^{n-1} h_k(S,F)\right)^{1/n}. \end{align*}$$

In special cases, the following improvements hold:

1. a) if F is symmetric, we can replace the exponent $3/2$ with $1$ ,

2. b) if Y is a Hilbert space, we can replace the exponent $3/2$ with $1$ ,

3. c) if F is symmetric and Y a Hilbert space, replace $3/2$ with $1/2$ ,

4. d) if X is a Hilbert space and F its unit ball, we can replace the exponent $3/2$ with $1/2$ if we also replace all $c_{k}$ with $c_{2k}$ .

Proof of Theorem 3.3

Let $S\in \mathcal {L}(X,Y)$ and $F\subset X$ be convex.

General case: We first show that, for fixed $\varepsilon>0$ , we can find $f_0,g_0$ , $f_1,g_1,\ldots \in F$ and $L_0,L_1,\ldots \in {X'}$ such that, with $p_k:=\frac {f_k-g_k}{2}$ , we have $L_j(p_k)=0$ for $j<k$ and $(1+\varepsilon ) L_k(p_k)>c_k(S,F)$ for $k=0,1,\ldots $ . See [Reference Novak48, p. 132] for a similar proof.

The proof is by induction. Let $k\in \mathbb {N}_0$ and assume that $f_j,g_j$ and $L_j$ for $j<k$ are already found. Define

$$\begin{align*}M_k \,:=\, \Big\{p\in X\colon\, L_j(p)=0 \;\;\text{ for }\; j< k\Big\}. \end{align*}$$

Since $\operatorname {\mathrm {codim}}{M_k}\le k$ , we can choose $f_k,g_k\in F$ with $p_k\in M_k$ and

(3.1) $$ \begin{align} (1+\varepsilon)\,\|Sp_k\| \,\ge\, c_k(S,F). \end{align} $$

By the Hahn-Banach theorem, there is $\lambda _k\in B_{Y'}$ with $\lambda _k(Sp_k)=\|Sp_k\|$ and hence

(3.2) $$ \begin{align} \lambda_k(Sp_k) \;\ge\; (1+\varepsilon)^{-1} \, c_k(S,F). \end{align} $$

We finish the induction step by setting $L_k=\lambda _k\circ S\in X'$ .

For $n\in \mathbb {N}$ , we now define $g=\frac {1}{n}\sum _{i<n} \frac {f_i+g_i}{2} \in F$ and the operators

(3.3) $$ \begin{align} A(\xi) \,:=\, \frac{1}{n}\sum_{i<n} \xi_i p_i \in X, \quad \xi=(\xi_i)_{i<n} \in\ell_2^n, \end{align} $$

and

$$\begin{align*}B(y) \,:=\, \frac{1}{\sqrt{n}} \big({\lambda_i}(y)\big)_{i<n}\in\ell_2^n, \qquad y\in Y, \end{align*}$$

and consider the mapping $S_n:=BSA$ . We observe that $\|B\|\le 1$ and, for all $\xi \in [-1,1]^n$ , due to convexity, it holds

$$\begin{align*}A(\xi) + g \,=\, \frac{1}{n} \sum_{i<n} \left( \frac{1+\xi_i}{2} f_i + \frac{1-\xi_i}{2} g_i \right) \in F.\end{align*}$$

In particular, $A(B_{\ell _2^n})+g \subset F$ . This gives, for any $k< n$ ,

$$\begin{align*}c_k(S_n,B_{\ell_2^n})\le h_k(S,F). \end{align*}$$

Our bounds are obtained by considering the determinant of $S_n\colon \, \ell _2^n\to \ell _2^n$ . Since $S_n$ is generated by the triangular matrix $n^{-3/2} (L_j(p_i))_{i,j<n}$ , we have

$$\begin{align*}\det(S_n) \,\ge\, \prod_{k<n} \frac{c_k(S,F)}{n^{3/2}(1+\varepsilon)}. \end{align*}$$

On the other hand, the determinant is multiplicative and equals the product of the singular values, which in turn equal the Gelfand widths $c_k(S_n,B_{\ell _2^n})$ . Therefore, we obtain

$$\begin{align*}\prod_{k<n} \frac{c_k(S,F)}{n^{3/2}(1+\varepsilon)} \;\le\;\det(S_n) \;=\; \prod_{k<n} c_k(S_n,B_{\ell_2}) \;\le\; \prod_{k<n} h_k(S,F). \end{align*}$$

Taking the infimum over all $\varepsilon>0$ and using $c_k(S,F) \ge c_{n-1}(S,F)$ for $k<n$ , we obtain

$$\begin{align*}c_{n-1}(S,F) \;\le\; n^{3/2}\,\bigg(\prod_{k<n} h_k(S,F) \bigg)^{1/n}. \end{align*}$$

$\underline {F\text{ symmetric}}$ : If F is additionally symmetric, then one has $p_i=\frac {f_i-g_i}{2} \in F$ such that we can redefine A by $A(\xi ):= \frac {1}{\sqrt {n}}\sum _{i<n} \xi _i p_i$ to have $A(B_{\ell _2}) \subset F$ . Hence, we can continue with the triangular matrix $S_n=n^{-1}(L_j(p_i))_{i,j<n}$ to obtain the improved bound.

$\underline {Y\text{ Hilbert space}}$ : If Y is a Hilbert space, we can choose the functionals $L_k$ from the Hahn-Banach theorem explicitly as

$$ \begin{align*} L_k\,:=\, \bigg\langle\,\cdot\ ,\, \frac{Sp_k}{\Vert Sp_k\Vert_Y} \bigg\rangle, \end{align*} $$

where $\left \langle \cdot ,\cdot \right \rangle $ is the inner product in Y. Hence, by the definition of the sets $M_k$ , we see that the $Sp_k$ for $k\le n$ are pairwise orthogonal. We can hence skip the factor $n^{-1/2}$ in the definition of B and put $B(y):=(\lambda _i(y))_{i<n}$ while preserving the property $\Vert B \Vert \le 1$ . Thus, also in this case, we can continue with the triangular matrix $S_n=n^{-1}(L_j(p_i))_{i,j<n}$ .

$\underline {\text{F symmetric}, Y\text{ Hilbert space}}$ : We combine the modifications from the previous two cases and work with the matrix $S_n=n^{-1/2}(L_j(p_i))_{i,j<n}$ .

$\underline {F\text{ unit ball of a Hilbert space}}$ : The proof is similar to the general case. We show by induction that for fixed $\varepsilon>0$ , there are orthogonal vectors $p_0,p_1,\ldots \in B_X$ and $L_0,L_1,\ldots \in X'$ such that $L_j(p_k)=0$ for $j<k$ and $(1+\varepsilon ) L_k(p_k)\ge c_{2k}(S,B_X)$ for $k=1,2,\ldots $

Assume that $p_j,L_j$ for $j<k$ are already found, and define

$$\begin{align*}M_n \,:=\, \Big\{p\in X\colon\, L_j(p)=0 \text{ and } \langle p_k,p \rangle = 0 \text{ for } j< k\Big\}. \end{align*}$$

Since $\operatorname {\mathrm {codim}}{M_k}\le 2k$ , we can choose $p_k\in B_X$ with $p_k\in M_k$ and

$$\begin{align*}(1+\varepsilon) \|Sp_k\| \,\ge\, c_{2k}(S,B_X). \end{align*}$$

By the Hahn-Banach theorem, there is $\lambda _k\in B_{Y'}$ with $\lambda _k(Sp_k)=\|Sp_k\|$ .

For $n\in \mathbb {N}$ , we define the operators $A \in \mathcal {L}(\ell _2^n,X)$ and $B\in \mathcal {L}(Y,\ell _2^n)$ by

$$\begin{align*}A(\xi) \,:=\, \sum_{i<n} \xi_i p_i, \qquad B(y) \,:=\, \frac{1}{\sqrt{n}}\big(\lambda_i(y)\big)_{i<n} \end{align*}$$

such that $\Vert A \Vert \le 1$ and $\|B\|\le 1$ . The mapping $S_n:=BSA$ is generated by the triangular matrix $n^{-1/2}(L_j(p_i))_{i,j<n}$ , where $L_j:=\lambda _j\circ S$ . This gives

$$\begin{align*}\prod_{k<n} \frac{c_{2k}(S,B_X)}{n^{1/2}(1+\varepsilon)} \;\le\;\det(S_n) \;=\; \prod_{k<n} c_k(S_n,B_{\ell_2^n}) \;\le\; \prod_{k<n} h_k(S,B_X). \end{align*}$$

Taking the infimum over $\varepsilon>0$ and using $c_{2k} \le c_{2n-2}$ for $k<n$ , we get

$$\begin{align*}c_{2n-2}(S,F) \;\le\; n^{1/2}\,\bigg(\prod_{k<n} h_k(S,F) \bigg)^{1/n}.\\[-47pt] \end{align*}$$

It may be desirable to compare $c_n$ directly with $h_n$ instead of the geometric mean of $h_0,\ldots ,h_n$ . Under the regularity condition that $h_k/h_{2k}$ is bounded, such a comparison is obtained from the following lemma.

Lemma 3.5. Let $n\in \mathbb {N}$ be even and $z_1 \ge \ldots \ge z_n>0$ . Moreover, let $c>0$ with $z_k \le c\, z_{2k}$ for all $k\le n/2$ . Then

$$\begin{align*}\bigg(\prod_{k=1}^n z_k \bigg)^{1/n} \;\le\; c^4\,z_n. \end{align*}$$

Proof of Lemma 3.5

Choose $v\in \mathbb {N}_0$ such that $n/2 < 2^v \le n$ . For $2^j \le k \le n$ , we get

$$\begin{align*}z_k \le z_{2^j} \le c^{v-j} z_{2^v} \le c^{v-j} z_{n/2} \le c^{v-j+1} z_n\end{align*}$$

so that

$$\begin{align*}\bigg(\prod_{k=1}^n z_k \bigg)^{1/n} \le \bigg( \prod_{j=0}^{v} \prod_{2^j \le k < 2^{j +1}} c^{v-j+1} \bigg)^{1/n} \cdot z_n = c^\kappa \, z_n \end{align*}$$

with

$$\begin{align*}\kappa = \frac1n \sum_{j=0}^{v} (v-j+1) 2^j = \frac{2^{v+1}}{n} \sum_{i=1}^{v+1} i 2^{-i} \le 4.\\[-49pt] \end{align*}$$

Lemma 3.5 is a fitting estimate for sequences of polynomial decay: If $z_n=n^{-\alpha }$ , then $c=2^\alpha $ is independent of n. For sequences of super-polynomial decay, it might be better to use the simple estimate

(3.4) $$ \begin{align} \bigg(\prod_{k=1}^n z_k \bigg)^{1/n} \;\le\; \sqrt{\phantom{l} z_1 \cdot z_{n/2}}. \end{align} $$

In the case that Y is a Hilbert space, there also is the following alternative bound which works for individual n without any regularity condition as in Lemma 3.5. On the downside, the upper bound is in terms of the (possibly larger) Bernstein numbers instead of the Hilbert numbers.

Theorem 3.6. Let X be a Banach space, H be a Hilbert space and $S\in \mathcal {L}(X,H)$ . For every convex $F\subset X$ and $n\in \mathbb {N}_0$ , we have

$$\begin{align*}c_n(S,F) \;\le\; (n+1)\cdot b_n(S,F). \end{align*}$$

We can replace $(n+1)$ by $\sqrt {n+1}$ if F is additionally symmetric.

The paper [Reference Pukhov59] contains bounds similar to Theorem 3.6 with the Gelfand widths replaced by the Kolmogorov widths. Since the target space is a Hilbert space, the Kolmogorov widths are larger than the Gelfand widths, see, for example, [Reference Pinkus58, Prop. 5.2] and Section 6. This means that the bounds of Theorem 3.6 are known up to constants. We presented the proof anyway since the result follows with little effort from our other observations.

Due to their relations with the previously defined minimal worst-case errors (as discussed in the next section), the Gelfand, Bernstein, and Hilbert numbers as considered above are of particular interest to us. Nonetheless, in Section 6, we will mention some other types of widths that may be of independent interest and discuss how Theorem 3.3 applies to these widths.

4 Widths versus minimal errors

In this section, we discuss how the Gelfand, Bernstein, and Hilbert numbers are related to minimal errors and hence obtain bounds between the different types of minimal errors.

First, note that the Gelfand numbers characterize the minimal worst-case error of deterministic algorithms up to a factor of two. This is a special case of a classical result in information-based complexity, see, e.g, [Reference Novak and Woźniakowski51, Sec. 4.1]. In our setting, the result reads as follows.

Proposition 4.1. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every $F\subset X$ and $n\in \mathbb {N}_0$ , we have

$$\begin{align*}c_n(S,F) \,\le\, e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F) \,\le\, 2\, c_n(S,F). \end{align*}$$

If F is convex and symmetric then

$$\begin{align*}c_n(S,F) \,\le\, e_n^{\det} (S,F) \, \le \, e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F) \,\le\, 2\, c_n(S,F). \end{align*}$$

We turn to the relation of Bernstein numbers and minimal errors. It is known, see [Reference Novak48], that $b_n(S,F)$ may serve as a lower bound for the error of adaptive deterministic algorithms.

Proposition 4.2. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every $F\subset X$ and $n\in \mathbb {N}_0$ , we have

$$\begin{align*}e_n^{\det}(S,F) \, \ge \, b_n(S,F). \end{align*}$$

In the case that X and Y are Hilbert spaces and F is the unit ball of X, it is shown in [Reference Novak46] that the Bernstein numbers also yield lower bounds for randomized adaptive algorithms. Here we use a slightly different error criterion and hence formulate a lemma.

Lemma 4.3. Let H and G be Hilbert spaces and $S\in \mathcal {L}(H,G)$ . For every $n\in \mathbb {N}_0$ , we have

(4.1) $$ \begin{align} e_n^{\mathrm{ran}}(S,B_H) \,\ge\, \frac{1}{2}\, b_{2n-1}(S,B_H) \,=\, \frac{1}{2}\, e_{2n-1}^{\mathrm{det}\text{-}\mathrm{non}}(S,F). \end{align} $$

With a different constant, Lemma 4.3 is implied by [Reference Heinrich21, Cor. 2].

Proof of Lemma 4.3

Let $0<b<b_{2n-1}(S,F)$ . There exists a subspace $V \subset H$ with dimension $2n$ such that $\Vert Sf \Vert _G \ge b \Vert f \Vert _H$ for all $f\in V$ . Let $W:=S(V)$ . We choose $R\in \mathcal {L}(\mathbb {R}^{2n},V)$ and $Q\in \mathcal {L}(W,\mathbb {R}^{2n})$ , where $\mathbb {R}^{2n}$ is considered with the Euclidean norm, such that $\Vert R \Vert \le 1$ and $\Vert Q \Vert \le b^{-1}$ and such that $QSR$ equals the identity $\mathrm {id}_{2n}$ on $\mathbb {R}^{2n}$ . If $A_n$ is a randomized algorithm for S with error less than $b/2$ , then $QA_nR$ is a randomized algorithm for $\mathrm {id}_{2n}$ with error less than $1/2$ . It hence suffices to prove the lower bound $1/2$ in the case $S=\mathrm {id}_{2n}$ .

We let $P_{2n}$ be the uniform distribution on the sphere of $\mathbb {R}^{2n}$ . An application of Fubini’s theorem (known as Bakhvalov’s proof technique, see [Reference Bakhvalov3] or [Reference Novak and Woźniakowski51, Section 4.3.3]) gives

$$\begin{align*}e_n^{\mathrm{ran}}(S,F) \,\ge\, \inf_{A_n}\, \int \Vert f - A_n(f) \Vert \, \mathrm{d} P_{2n}(f), \end{align*}$$

where the infimum runs over all deterministic and measurable algorithms $A_n\in \mathcal {A}_n^{\mathrm {det}}$ . Hence let $A_n = \varphi \circ N_n$ be a measurable deterministic algorithm with adaptively chosen information $N_n=(L_1, \dots , L_n)$ and let f be distributed according to $P_{2n}$ . Assume that the functionals $L_i$ are chosen orthonormal; this is no restriction. For each y in the unit ball of $\mathbb {R}^n$ , the information $N_n(f)=y$ defines a sphere $\mathbb {S}_y$ of radius $r_y=\sqrt {1-\Vert y\Vert ^2}$ . We have

$$\begin{align*}\int \Vert f - A_n(f) \Vert \, \mathrm{d} P_{2n}(f) \,=\, \int \int \Vert f - \varphi(y) \Vert \, \mathrm{d} \mu_y(f) \, \mathrm{d} \nu(y) \end{align*}$$

where $\mu _y$ is the uniform distribution on $\mathbb {S}_y$ and $\nu $ is the distribution of $N_n(f)$ . The inner integral is minimized if $\varphi (y)$ equals the center of $\mathbb {S}_y$ , so that we have

$$\begin{align*}\int \Vert f - A_n(f) \Vert \, \mathrm{d} P_{2n}(f) \,\ge\, \int r_y \, \mathrm{d} \nu(y) \,\ge\, \int r_y^2 \, \mathrm{d} \nu(y). \end{align*}$$

From the symmetry of $P_{2n}$ it follows that $\nu $ does not depend on $N_n$ . We choose and get

$$\begin{align*}\int \Vert f - A_n(f) \Vert \, \mathrm{d} P_{2n}(f) \,\ge\, \int \sum_{i=n+1}^{2n} f_i^2 \, \mathrm{d} P_{2n}(f) \,=\, \frac{1}{2}. \end{align*}$$

The last identity holds since the $f_i^2$ are identically distributed so that their expected value equals $1/(2n)$ . See Proposition 3.2 for the equality $b_n=e_n^{\mathrm {det}\text{-}\mathrm{non}}$ .

More recently, it has been shown in [Reference Kunsch, Cools and Nuyens36, Reference Kunsch37] that also for Banach spaces X and Y, it holds that

(4.2) $$ \begin{align} e_n^{\mathrm{ran}}(S,F) \,\ge\, \frac{1}{30}\, b_{2n-1}(S,F). \end{align} $$

The result of [Reference Kunsch37] is proven only in the symmetric case but it remains valid if F is only convex. On the other hand, the result (4.1) for the Hilbert case easily implies the following.

Proposition 4.4. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every convex $F\subset X$ and $n\in \mathbb {N}$ , we have

$$\begin{align*}e_n^{\mathrm{ran}}(S,F) \,\ge\, \frac{1}{2}\, h_{2n-1}(S,F). \end{align*}$$

Proof of Proposition 4.4

Let $A_n \in \mathcal {A}_n^{\mathrm {ran}}(X,Y)$ and let $B\in \mathcal {L}(Y,\ell _2)$ with $\Vert B \Vert \le 1$ as well as $A\in \mathcal {L}(\ell _2,X)$ and $g\in F$ with $A(B_{\ell _2}) + g \subset F$ . Then we have $BA_nA \in \mathcal {A}_n^{\mathrm {ran}}(\ell _2,\ell _2)$ . This algorithm uses information of the form $L_k'=L_k A\in \ell _2'$ , if $L_k\in X'$ is the information used by $A_n$ . Note that A is continuous in the norm induced by F due to $A(B_{\ell _2}) \subset F-F$ and hence the algorithm is measurable. By (4.1), we have

$$\begin{align*}e(BA_nA,BSA,B_{\ell_2}) \,\ge\, \frac12\, b_{2n-1}(BSA,B_{\ell_2}). \end{align*}$$

On the other hand,

$$ \begin{align*} e(BA_nA,BSA,B_{\ell_2}) \,&=\, e(BA_n(A+g),BS(A+g),B_{\ell_2})\\ \,&\le\, e(BA_n,BS,F) \,\le\, e(A_n,S,F), \end{align*} $$

where we used $(A+g)(B_{\ell _2}) \subset F$ in the first and $\Vert B \Vert \le 1$ in the second inequality. So,

$$\begin{align*}e(A_n,S,F) \,\ge\, \frac12\, b_{2n-1}(BSA,B_{\ell_2}). \end{align*}$$

Taking the supremum over all B, A and g as above gives the result.

We point out that the Hilbert numbers can be much smaller than the Bernstein numbers. For example, if S is the identity on $\ell _1$ and F the unit ball of $\ell _1$ , then the Bernstein numbers are equal to one, see [Reference Pietsch53], while the Hilbert numbers are of order $n^{-1/2}$ , see [Reference Pietsch55, 2.9.19]. So in general, one should prefer the bound (4.2) over Proposition 4.4. However, since our upper bounds are in terms of the Hilbert numbers anyway, we will obtain a better constant in the overall comparison if we work with Proposition 4.4 instead of (4.2).

5 The main result

We now arrive at our main result, Theorem 1.1, which we present here in a slightly stronger form.

Theorem 5.1. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every convex $F\subset X$ and $n\in \mathbb {N}$ , we have

$$\begin{align*}\left(\prod_{k<2n} e_k^{\mathrm{det}\text{-}\mathrm{non}}(S,F)\right)^{1/(2n)} \,\le\, 2^{7/2}\, n^{3/2} \, \left(\prod_{k<n} e_k^{\mathrm{ran}}(S,F)\right)^{1/n}. \end{align*}$$

In special cases, the following improvements hold:

1. a) if F is symmetric, we can replace $n^{3/2}$ with n,

2. b) if Y is a Hilbert space, we can replace $n^{3/2}$ with n,

3. c) if F is symmetric and Y a Hilbert space, replace $n^{3/2}$ with $n^{1/2}$ ,

4. d) if X is a Hilbert space and F its unit ball, we can replace $n^{3/2}$ with $n^{1/2}$ if we also replace the range $k<2n$ with $k<4n$ .

Proof of Theorems 1.1 and 5.1

A successive application of Proposition 4.1, Theorem 3.3, the monotonicity of the Hilbert widths, and Proposition 4.4 (in the weaker form $h_{2k} \le 2 e_k^{\mathrm {ran}}$ for all $k\in \mathbb {N}_0$ ) gives

$$ \begin{align*} \prod_{k<2n} &e_k^{\mathrm{det}\text{-}\mathrm{non}}(S,F) \,\le\, 2^{2n} \cdot \prod_{k<2n} c_k(S,F) \,\le\, 2^{5n} n^{3n} \cdot \prod_{k<2n} h_k(S,F)\\ &\le\, 2^{5n} n^{3n} \cdot \prod_{k<n} h_{2k}(S,F)^2 \,\le\, 2^{7n} n^{3n} \cdot \prod_{k<n} e_k^{\mathrm{ran}}(S,F)^2. \end{align*} $$

The modifications in the special cases are obvious.

Since estimates in terms of geometric means might be unfamiliar to the reader, we present a corollary of Theorem 1.1 which is reminiscent of Carl’s inequality.

Corollary 5.2. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every convex $F\subset X$ , $n\in \mathbb {N}$ and $\alpha>0$ , we have

$$\begin{align*}e_{2n-1}^{\mathrm{det}\text{-}\mathrm{non}}(S,F) \;\le\; C_\alpha \, n^{-\alpha+3/2}\cdot \sup_{k< n}\left( (k+1)^{\alpha}\, e_k^{\mathrm{ran}}(S,F)\right), \end{align*}$$

where $C_\alpha \le 12^{\alpha +1}$ . In accordance with the special cases given in Theorem 1.1, the exponent $3/2$ can be replaced with $1$ or $1/2$ .

We also write explicitly the implication for the polynomial rate of convergence, which has been presented in the introduction as Table 1. The polynomial rate of convergence of a sequence $(z_n) \subset [0,\infty )$ is defined by

(5.1) $$ \begin{align} \mathrm{rate}(z_n) \,:=\, \sup\Big\{ \alpha>0 \ \Big|\ \exists C\ge 0: \forall n\in\mathbb{N}: z_n \le C n^{-\alpha} \Big\}. \end{align} $$

We only give the result for the symmetric case, where the bounds are sharp up to logarithmic factors. It should be obvious enough what the corresponding results in the nonsymmetric case look like.

Corollary 5.3. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every convex and symmetric $F\subset X$ , we have

$$\begin{align*}\mathrm{rate}\Big(e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F)\Big) \;\ge\; \mathrm{rate}\Big(e_n^{\mathrm{ran}}(S,F)\Big) \,-\, 1. \end{align*}$$

If either Y is a Hilbert space or F is the unit ball of a Hilbert space X, we even have

$$\begin{align*}\mathrm{rate}\Big(e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F)\Big) \;\ge\; \mathrm{rate}\Big(e_n^{\mathrm{ran}}(S,F)\Big) \,-\, 1/2. \end{align*}$$

Moreover, in each of these cases, equality can occur.

6 Examples and related problems

In this section we give further details and extensions of our result and discuss related problems. In particular, we analyze the individual influence of adaption and randomization on the minimal worst-case error, and present those examples which exhibit the largest gain known to us. In addition, we show how our results apply to other nonlinear widths and to approximation based on standard information, that is, function evaluations, as shown in Theorem 1.2. We also present a list of open problems.

6.1 The individual power of adaption and randomization

By the results of the previous sections, we know how much adaption and randomization can help if they are allowed together. That is, we have a good understanding of the maximal gain from $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}}$ to $\mathcal {A}_n^{\mathrm { ran}}$ . In the symmetric case, we even know that our bounds are optimal up to logarithmic factors. However, our knowledge about the individual power of randomization or adaption still has several gaps.

Let us first talk about upper bounds. Intuitively, it is clear that the gain of randomization or adaption alone cannot be larger than the gain of adaption and randomization together. Let us make this a corollary.

Corollary 6.1. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every convex and bounded $F\subset X$ and $n\in \mathbb {N}$ , we have

$$\begin{align*}e_{2n-1}^*(S,F) \;\le\; C n^{3/2}\,\bigg(\prod_{k<n} e_k^\square(S,F) \bigg)^{1/n}, \end{align*}$$

where $*,\square \in \{\mathrm {det}, \mathrm {det}\text{-}\mathrm{non}, \mathrm {ran}, \mathrm {ran}\text{-}\mathrm{non}\}$ and C is a universal constant. If X or Y is a Hilbert space or if F is symmetric, the improvements of Theorem 1.1 apply.

This corollary is not as obvious as it seems at first glance. The problem is that, due to the assumed measurability of randomized algorithms, we do not have the relation $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}} \subset \mathcal {A}_n^{\mathrm {ran}\text{-}\mathrm{non}}$ . What we do have is the relation $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}\text{-}\mathrm{mb}} \subset \mathcal {A}_n^{\mathrm {ran}\text{-}\mathrm{non}}$ , where $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}\text{-}\mathrm{mb}}$ denotes the class of all $(\mathcal {B}_F,\mathcal {B}_Y)$ -measurable deterministic and nonadaptive algorithms with the corresponding minimal worst-case error denoted by $e_n^{\mathrm {det}\text{-}\mathrm{non}\text{-}\mathrm{mb}}$ . The issue is fixed by the following lemma, which shows that measurable deterministic and nonadaptive algorithms are (roughly) as good as arbitrary deterministic nonadaptive algorithms, that is, there is no real difference between $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}\text{-}\mathrm{mb}}$ and $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}}$ .

Lemma 6.2. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . For every bounded and convex $F\subset X$ and $n\in \mathbb {N}_0$ , we have

$$\begin{align*}e_n^{\mathrm{det}\text{-}\mathrm{non}\text{-}\mathrm{mb}}(S,F) \,\le\, 8\, e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F). \end{align*}$$

Lemma 6.2 is proven in [Reference Mathé41, Thm. 11(v)] for the case that F is the unit ball of the space X. (In fact, it is shown that continuous algorithms are almost optimal.) We show below how it can be transferred to general convex classes. It is open whether the factor 8 can be removed and to what extent measureable algorithms are as good as nonmeasurable algorithms also in other settings, see [51, Section 4.3.3] and [Reference Novak and Ritter50].

Proof of Lemma 6.2

Without loss of generality we assume that $0\in F$ ; the error numbers do not change if we shift F. Then we have $F\subset F-F$ . The class $F-F$ is convex, bounded and symmetric, and hence the unit ball of a norm on X. By [Reference Mathé41, Thm. 11(v)], it holds that

$$\begin{align*}e_n^{\mathrm{det}\text{-}\mathrm{non}\text{-}\mathrm{mb}}(S,F) \,\le\, e_n^{\mathrm{det}\text{-}\mathrm{non}\text{-}\mathrm{mb}}(S,F-F) \,\le\, 2\,e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F-F). \end{align*}$$

On the other hand, [Reference Novak and Woźniakowski51, Lemma 4.3] and Proposition 4.1 give

$$ \begin{align*}e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F-F) \,\le\, 2\,c_n(S,F-F) \,=\, 4\,c_n(S,F) \,\le\, 4\,e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F).\\[-37pt] \end{align*} $$

We can now prove Corollary 6.1.

Proof of Corollary 6.1

All the minimal errors are bounded from below by $\frac 12 \cdot h_{2n-1}(S,F)$ . For $\mathcal A_n^{\mathrm {ran}}$ and $\mathcal A_n^{\mathrm {ran}\text{-}\mathrm{non}}$ , this follows from Proposition 4.4. For $\mathcal A_n^{\mathrm {det}}$ and $\mathcal A_n^{\mathrm {det}\text{-}\mathrm{non}}$ , it follows from [Reference Novak48, Prop. 1] and Proposition 3.2. On the other hand, all the minimal errors are bounded from above by $16\cdot c_n(S,F)$ . For $\mathcal A_n^{\mathrm {det}}$ and $\mathcal A_n^{\mathrm {det}\text{-}\mathrm{non}}$ , this follows from Proposition 4.1. Lemma 6.2 implies that the upper bound also holds for $A_n^{\mathrm {det}\text{-}\mathrm{non}\text{-}\mathrm{mb}}$ and thus for $\mathcal A_n^{\mathrm {ran}}$ and $\mathcal A_n^{\mathrm {ran}\text{-}\mathrm{non}}$ . Hence, the statement follows from Theorem 3.3.

We summarize the state of the art for the maximal gain between the different classes of algorithms, see Table 2. For this, let us define the “maximal gain function”

$$\begin{align*}\mathrm{gain}(*,{\square}, \triangle) \,:=\, \sup_{\substack{S\in\mathcal{L}(X,Y)\\ F\subset X \text{ is } \triangle}} \left(\mathrm{rate}\Big(e_n^{\square}(S,F)\Big) - \mathrm{rate}\Big(e_n^{*}(S,F)\Big) \right), \end{align*}$$

where the rate function is defined in (5.1), $*,\square \in \{\mathrm {det}$ , $\mathrm {det}\text{-}\mathrm{non}$ , $\mathrm {ran}$ , $\mathrm {ran}\text{-}\mathrm{non}\}$ and $\triangle \in \{\text {convex}$ , $\text { convex+symmetric}\}$ .

Table 2 Maximal gain in the rate of convergence between different classes of algorithms using linear information.

The upper bounds in Table 2 are given by Corollary 6.1 and Proposition 4.1. We now turn to the lower bounds on the (individual) gain of adaption and randomization. For this, we collect specific examples:

• • F convex+symmetric:

  1. 1. Power of adaption, deterministic ( $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}}\to \mathcal {A}_n^{\mathrm {det}}$ ):

     There is no gain in the rate of convergence. As stated in Proposition 4.1, we have $e_n^{\mathrm {det}\text{-}\mathrm{non}}(S,F)\le 2\cdot e_n^{\mathrm {det}}(S,F)$ for any $S\in \mathcal {L}$ , see for example [Reference Creutzig and Wojtaszczyk9, Reference Novak49, Reference Novak and Woźniakowski51, Reference Traub, Wasilkowski and Woźniakowski68]. An example where adaptive algorithms are slightly better can be found in [Reference Kon and Novak28].

  2. 2. Power of randomization, nonadaptive ( $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}}\to \mathcal {A}_n^{\mathrm {ran}\text{-}\mathrm{non}}$ ):

     Randomization of nonadaptive algorithms can yield a gain of 1/2 for certain Sobolev embeddings. The simplest case is from $W^k_2([0,1])$ into $L_\infty ([0,1])$ , where the optimal rate with deterministic algorithms is $n^{-k+1/2}$ for $k> 1/2$ . This is a classical result of [Reference Solomjak and Tichomirov62]. Using nonadaptive randomized algorithms, one can get the upper bound $n^{-k} \log n$ , see [Reference Mathé42] and [Reference Byrenheid, Kunsch and Nguyen6, Reference Fang and Duan14, Reference Heinrich21] for further results and extensions.

  3. 3. Power of adaption, randomized ( $\mathcal {A}_n^{\mathrm {ran}\text{-}\mathrm{non}}\to \mathcal {A}_n^{\mathrm {ran}}$ ):

     The paper [Reference Kunsch, Novak and Wnuk38] shows that one may gain a factor $(n/\log n)^{1/2}$ for the embedding $S\colon \ell _1^m \to \ell _2^m$ and suitable (large) m and in [Reference Kunsch and Wnuk39] it is proved that one may gain, up to logarithmic terms, a factor of polynomial order n for the embedding $S\colon \ell _1^m \to \ell _\infty ^m$ with appropriate m. Note that this example (or more precisely, an infinite-dimensional version of it) shows the optimality of Corollary 5.3 and the factor n in Theorem 1.2. We do not know if a gain of 1 can also occur in the transitions $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}}\to \mathcal {A}_n^{\mathrm {ran}\text{-}\mathrm{non}}$ .

  4. 4. Power of randomization, adaptive ( $\mathcal {A}_n^{\mathrm {det}}\to \mathcal {A}_n^{\mathrm {ran}}$ ):

     If we employ Example (1), as well as $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}\text{-}\mathrm{mb}}\subset \mathcal {A}_n^{\mathrm {ran}\text{-}\mathrm{non}}$ and Lemma 6.2, we can take the examples from (3), to obtain the same gain.

• • F convex:

  1. 5. Power of adaption, deterministic ( $\mathcal {A}_n^{\mathrm {det}\text{-}\mathrm{non}}\to \mathcal {A}_n^{\mathrm {det}}$ ):

     Consider $S=\mathrm {id}\in \mathcal {L}(\ell _\infty ,\ell _\infty )$ , that is, approximation in $\ell _\infty $ , for inputs from

     $$ \begin{align*}\hspace{18mm} F= \{ x \in \ell_\infty \mid x_i \ge 0, \sum x_i \le 1, x_k \ge x_{2k}, x_k \ge x_{2k+1} \}. \end{align*} $$
     Then one can prove a lower bound $c (\sqrt {n} \log n)^{-1}$ for nonadaptive algorithms while a simple adaptive algorithm (using “function values,” that is, values of coordinates of x) gives the upper bound $(n+3)^{-1}$ ; see [Reference Novak47] for details. This shows a gain of 1/2.In the case of standard information, that is, function values, see also Section 6.3, there are more extreme examples, where adaption yields a gain up to the order n, see again [Reference Novak47].

  2. 6. Remaining Cases:

     Clearly, the examples given in the symmetric case also apply here. This gives the remaining lower bounds of Table 2. We do not know any example of nonsymmetric F, where the gain of adaptive over nonadaptive randomized algorithms, or of randomized over deterministic algorithms (adaptive or not), is larger than the corresponding gain in the symmetric case.

Let us highlight a few open problems indicated by Table 2 that we assume to be of particular interest. In particular, note that it is still possible that the maximal gain of adaption and randomization is $1$ also for nonsymmetric convex sets.

Open problems.

1. 1. Bounds for individual n: Verify whether $e_{2n}^{\mathrm {det}\text{-}\mathrm{non}}(S,F)\le C\, n\, e_n^{\mathrm {ran}}(S,F)$ for some $C>0$ and all convex and symmetric F. See also [Reference Pietsch53, Thm. 8.6]. Does a similar bound hold even for all convex classes F?

2. 2. Power of adaption: Is there some $S\in \mathcal {L}$ and convex F such that

   $$ \begin{align*}e_{2n}^{\mathrm{det}}(S,F)\le C \, n^{-1}\, e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F)\end{align*} $$
   for all $n\in \mathbb {N}$ ?

3. 3. Power of randomization: Is there some $S\in \mathcal {L}$ and convex F such that

   $$ \begin{align*}e_{2n}^{\mathrm{ran}\text{-}\mathrm{non}}(S,F)\le C \, n^{-1}\, e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F) \end{align*} $$
   for all $n\in \mathbb {N}$ ?

4. 4. Power of randomization for symmetric sets: Is there some $S\in \mathcal {L}$ and convex and symmetric F such that

   $$ \begin{align*}e_{2n}^{\mathrm{ran}\text{-}\mathrm{non}}(S,F)\le C \, n^{-\alpha}\, e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F) \end{align*} $$
   for all $n\in \mathbb {N}$ and some $\alpha>\frac 12$ ?

6.2 Other widths

Our main emphasis was to compare different classes of algorithms. However, the study of widths is clearly not only of interest in IBC. These notions are used in many areas, including approximation theory, geometry, and the theory of Banach spaces.

We indicate how our results apply here, and give further references.

6.2.1 Kolmogorov widths

Possibly most prominent among the widths are the Kolmogorov widths

$$\begin{align*}d_n(S,F) \,:=\, \inf_{\substack{M \subset Y\\ \dim(M)\le n}}\, \sup_{f\in F}\, \inf_{g\in M}\, \left\Vert S(f)-g\right\Vert, \end{align*}$$

which describe how well $S(f)$ for $f\in F$ can be approximated by elements from an affine linear subspace. Such a best approximation can generally not be found by an algorithm, and so, $d_n$ is usually not a suitable benchmark for algorithms. Still, it is a natural “geometric” quantity.

If $F=B_X$ , then it is known that the Kolmogorov numbers $d_n$ are dual to the Gelfand numbers, and that the Hilbert numbers are self-dual, see [Reference Bauhardt4, Reference Pietsch53, Reference Pietsch55]. Similar statements might hold for general F, and this could be used to extend our results to $d_n$ . However, one may also repeat the proofs of our results almost verbatim for $d_n$ . By this, we obtain for $S\in \mathcal {L}(X,Y)$ and convex $F\subset X$ that

(6.1) $$ \begin{align} d_n(S,F) \;\le\; (n+1)^{\alpha}\,\bigg(\prod_{k=0}^n h_k(S,F) \bigg)^{1/(n+1)} \end{align} $$

with $\alpha =1$ if F is symmetric and $\alpha =3/2$ otherwise. The symmetric case is proven in [Reference Ullrich70]; the modifications for the nonsymmetric case are analogous. We omit the details.

Of course, the same bound holds with the larger $b_n$ in place of the smaller $h_n$ . The ratio of Kolmogorov and Bernstein widths was studied at least since the paper [Reference Mityagin and Henkin43] from Mityagin and Henkin (1963). They proved that $d_n(S,F)\le (n+1)^2 \, b_n(S,F)$ for convex and symmetric F, and conjectured that one has indeed $d_n(S,F)\le (n+1)\, b_n(S,F)$ . See also [Reference Novak48] for the nonsymmetric case. The above considerations show that this old conjecture is true, at least for regular sequences and up to constants. Note again that this was known, see [Reference Pietsch56].

Similar problems appear in geometry, where often different notions are used, such as successive radii, or inner and outer radii. In this context, their ratio was mainly considered for sets in Hilbert spaces, see [Reference Merino18, Reference Merino19, Reference Perel’man52, Reference Pukhov59], where one can find further references. Results for general norms can be found, for example, in [Reference Merino19, Theorem 5.1]. These bounds are improved by the inequalities above.

Remark 6.3. For S being the identity on a Hilbert space, it is conjectured that the regular simplex provides the largest gap in the nonsymmetric case and the regular cube or the regular cross-polytope provides the largest gap in the symmetric case. The Kolmogorov and Bernstein widths of these sets are completely known [Reference Brandenberg5, Reference Pukhov60], but the proven general bounds are slightly weaker.

6.2.2 Linear widths

If we replace the requirement that the approximation space is linear by the requirement that the approximation procedure is linear, we end up with the approximation numbers of S on F, or linear widths of F with respect to S, that is,

$$ \begin{align*} a_n(S,F) \;:=\; \inf_{\substack{L_1,\dots, L_n\in X'\\ \varphi_0, \dots,\varphi_n\in Y}} \, \sup_{f\in F} \, \left\Vert S(f) - \varphi_0 - \sum_{i=1}^n L_i(f)\, \varphi_i\right\Vert. \end{align*} $$

This corresponds to the minimal worst-case error of affine algorithms that use at most n pieces of linear information.

If F is the unit ball of X, then it is known from [Reference Pietsch53, Thm. 8.4] (based on [Reference Kadets and Snobar27]) that

(6.2) $$ \begin{align} a_n(S,B_X) \,\le\, (1+\sqrt{n})\, c_n(S,B_X). \end{align} $$

Note that $a_n(S,B_X)$ and $c_n(S,B_X)$ are equal if X is a Hilbert space or if Y has the metric extension property, see [Reference Pietsch53] or [Reference Creutzig and Wojtaszczyk9, Reference Mathé41] for extensions. Together with Theorem 3.3, inequality (6.2) implies

$$ \begin{align*} a_n(S,B_X) \;\le\; 2\,(n+1)^{3/2} \,\bigg(\prod_{k=0}^n h_k(S,B_X) \bigg)^{1/(n+1)}. \end{align*} $$

The exponent $3/2$ can be replaced by $1$ in the aforementioned cases or in the case that Y is a Hilbert space since then we have the identity $a_n(S,B_X)=d_n(S,B_X)$ . Again, for this symmetric case, the result is essentially known, see [Reference Pietsch56, 6.2.3.14], and we only remove an oversampling constant compared to the known estimate. It is a major open problem, whether the exponent $3/2$ can be reduced to $1$ , see [Reference Pietsch57, Open Problem 5].

This problem is of particular interest in the theory of s-numbers, where F is assumed to be the unit ball of X. Note that the approximation numbers form the largest scale of s-numbers while the Hilbert numbers are the smallest scale of s-numbers, see [Reference Pietsch55, Ch. 2].

6.2.3 Nonlinear widths

In connection with nonlinear approximation, also several types of nonlinear widths appear in the literature, see, for example, [Reference Cohen, DeVore, Petrova and Wojtaszczyk7, Reference DeVore, Howard and Micchelli10, Reference DeVore, Kyriazis, Leviatan and Tikhomirov11, Reference Siegel61, Reference Stesin63, Reference Stesin64, Reference Yarotsky73]. Let us introduce two of them to illustrate the relation to our setting. First, the manifold widths of $F\subset X$ with respect to $S\in \mathcal {L}(X,Y)$ are defined by

$$\begin{align*}\delta_n(S,F) \,:=\, \inf_{\substack{N\in C(X,{\mathbb{R}^n}) \\ \varphi\in C(\mathbb{R}^n,Y)}}\, \sup_{f\in F}\, \big\|S(f)-\varphi\left(N(f)\right)\big\|, \end{align*}$$

where $C(X,Y)$ denotes the class of continuous mappings from X to Y. Moreover, the continuous co-widths of $F\subset X$ w.r.t. $S\in \mathcal {L}(X,Y)$ are

$$\begin{align*}\widetilde{c}_n(S,F) \,:=\, \inf_{N\in C(X,{\mathbb{R}^n})}\, \sup_{\substack{f,g\in F: \\ N(f)=N(g)}} \frac12\, \big\|S(f)-S(g)\big\|. \end{align*}$$

These numbers correspond (up to a factor 2) to minimal errors for approximating S over F based on n nonadaptive continuous measurements. Comparing these definitions to minimal errors and Gelfand numbers, we see that this approach is more general in the sense that the information mapping (or parameter selection map) N is not built from linear functionals, but less general in the sense that a discontinuous adaptive choice of the one-dimensional measurements (like, e.g., a bisection method) is not allowed. These quantities seem to appear in the literature only in the special case of $S\in \mathcal {L}(X,X)$ being the identity. We naturally extend the definitions, but only comment on this special case in the following.

First, note that another important (and very early) concept are the Aleksandrov widths, see [Reference Aleksandrov1, Reference DeVore, Kyriazis, Leviatan and Tikhomirov11], which replace $\mathbb {R}^n$ by more general n-dimensional complexes. However, it is shown in [Reference DeVore, Kyriazis, Leviatan and Tikhomirov11] that all these quantities are equivalent up to constants and oversampling.

Second, it is clear from the definitions that these widths are smaller than $e_n^{\mathrm {det}\text{-}\mathrm{non}}$ and $c_n$ , respectively. Moreover, it is shown in [Reference DeVore, Howard and Micchelli10] that they are lower-bounded by the Bernstein widths $b_n$ . So, we have everything we need to apply our technique: The upper bounds in Theorem 3.3 and in Table 1 are also applicable to the maximal gain in the rate of convergence when passing from linear to arbitrary continuous measurement maps $N\colon F \to \mathbb {R}^n$ , at least for S being the identity.

In fact, combining Theorem 3.3, in the form of Corollary 5.2, with [Reference DeVore, Howard and Micchelli10, Thm. 3.1] (and noting that they use the notation “ $d_n$ ” for “ $\delta _n$ ”), we obtain, for example, for all convex $F\subset X$ that

(6.3) $$ \begin{align} c_{2n}(\mathrm{id}_X,F) \;\le\; C_\alpha \, n^{-\alpha+3/2}\cdot \sup_{k< n} \, (k+1)^{\alpha}\, \delta_k(\mathrm{id}_X,F), \end{align} $$

where $C_\alpha \le 16^{\alpha +1}$ . Again, the exponent $3/2$ can be replaced by 1 for symmetric F and, taking Section 6.2.1 into account, this bound also holds with $d_{2n}$ in place of $c_{2n}$ .

For a very different (and surprising for us) result on adaptive continuous measurements that shows an exponential speed-up, see [Reference Krieg, Novak and Ullrich30].

6.3 Sampling numbers and other classes of information

As already indicated in the introduction, our proof technique can also be employed for sampling recovery in the uniform norm. In fact, our upper bound from Theorem 1.1 also holds if the class of deterministic, nonadaptive algorithms is further restricted to those using only certain restricted information.

For this, we consider a set of linear functionals $ \Lambda \subset X'$ , which we call the admissible information. The n-th minimal worst-case error for approximating S over F with information from $\Lambda $ is defined by

$$\begin{align*}e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F,\Lambda) \;:=\; \inf_{\substack{L_1,\dots, L_n\in\Lambda \\ \varphi\colon\, \mathbb{R}^n\to Y}} \, \sup_{f\in F} \, \left\Vert S(f) - \varphi(L_1(f),\dots, L_n(f))\right\Vert. \end{align*}$$

Moreover, we call $\mathcal N\subset X'$ a norming set of X if

$$\begin{align*}\left\Vert f\right\Vert=\sup\{\,|\lambda(f)|\colon\, \lambda\in \mathcal N\} \qquad \text{ for all } \quad f\in X. \end{align*}$$

With this, we obtain the following generalization of Theorem 1.1.

Theorem 6.4. Let X and Y be Banach spaces and $S\in \mathcal {L}(X,Y)$ . Moreover, let $\Lambda _S\subset \Lambda \subset X'$ , where $\Lambda _S$ is of the form $\Lambda _S=\{\lambda \circ S\colon \, \lambda \in \mathcal N\}$ and $\mathcal N\subset Y'$ is a norming set of Y. Then, for every convex $F\subset X$ and $n\in \mathbb {N}$ , we have

$$ \begin{align*} e_{2n-1}^{\mathrm{det}\text{-}\mathrm{non}}(S,F,\Lambda) \;&\le\; \left(\prod_{k<2n} e_k^{\mathrm{det}\text{-}\mathrm{non}}(S,F,\Lambda)\right)^{1/(2n)}\\ &\le\; 12\, n^{3/2}\, \bigg(\prod_{k<n} e_k^{\mathrm{ran}}(S,F) \bigg)^{1/n}. \end{align*} $$

The corresponding improvements from Theorem 1.1 apply if F is additionally symmetric or the unit ball of a Hilbert space X.

Proof of Theorem 6.4

First note that the bound

(6.4) $$ \begin{align} e_n^{\mathrm{det}\text{-}\mathrm{non}}(S,F,\Lambda) \,\le\, 2\, c_n(S,F,\Lambda) \end{align} $$

from Proposition 4.1 also holds for the generalized Gelfand numbers

$$\begin{align*}c_n(S,F,\Lambda) \,:=\, \inf_{L_1,\dots,L_{n}\in \Lambda}\, \sup_{\substack{f,g\in F: \\ L_k(f)=L_k(g)}} \frac12\, \big\|S(f)-S(g)\big\| \end{align*}$$

see [Reference Novak and Woźniakowski51, Sec. 4.1]. Hence it suffices to bound the modified Gelfand numbers in terms of the Hilbert numbers as in Theorem 3.3.

We proceed as in the proof of Theorem 3.3. By the definition of a norming set $\mathcal N\subset Y'$ , we can choose the functionals $\lambda _k$ in (3.2) from $\mathcal N$ , arguing with a slightly smaller $\varepsilon $ in the previous inequality. Hence, the functionals $L_k$ defining $M_k$ are from $\Lambda _S$ and we can choose $f_k, g_k$ such that (3.1) holds with the modified Gelfand widths. This shows that the assertion in the very beginning of the proof of Theorem 3.3 holds with $c_k(S,F)$ replaced by $c_k(S,F,\Lambda )$ .

The same replacement is possible for the corresponding assertion in the case that F is the unit ball of a Hilbert space. We can therefore copy the rest of the proof of Theorem 3.3 with $c_k(S,F)$ replaced by $c_k(S,F,\Lambda )$ .

Recall that the minimal errors on the right-hand side in Theorem 6.4 are for the (much bigger) class of randomized, adaptive algorithms that have access to arbitrary linear functionals, see Section 2. We do not consider the case that Y is a Hilbert space since we believe that norming sets in Hilbert spaces are too large to yield an interesting generalization of Theorem 1.1. Let us also note that our proof of Theorem 1.1 only employs information of the form $\Lambda _S$ . As such, Theorem 1.1 cannot catch the optimal behavior of $e_n^{\mathrm {det}\text{-}\mathrm{non}}(S,F)$ if the latter decay faster than $e_{n}^{\mathrm {det}\text{-}\mathrm{non}}(S,F,\Lambda _S)$ .

We discuss a few examples:

1. 1. Linear information:

   The case studied in Theorem 1.1 corresponds to $\mathcal N=B_{Y'}$ and $\Lambda =X'$ which satisfy the assumptions of the theorem. The class $B_{Y'}$ is a norming set by the Hahn-Banach theorem.

2. 2. Uniform approximation (and proof of Theorem 1.2):

   We consider $X=Y=B(D)$ and $S = \mathrm {APP}_{\infty }$ , that is, the identity on $B(D)$ , and observe that $\Lambda ^{\mathrm {std}}:=\{\delta _x\colon \, x\in D\}$ with the Dirac functionals $\delta _x(f)=f(x)=Sf(x)$ is a norming set of $B(D)$ . Since $g_n(S,F)= e_{n}^{\mathrm {det}\text{-}\mathrm{non}}(S,F,\Lambda ^{\mathrm {std}})$ , see [Reference Creutzig and Wojtaszczyk9] or [Reference Novak and Woźniakowski51, Thm. 4.8], we obtain Theorem 1.2 from Theorem 6.4. A special case is the space of bounded sequences $\ell _\infty =B(\mathbb {N})$ , where $\Lambda ^{\mathrm {std}}$ consists of evaluations of coordinates. Note that the factor 12 from Theorem 6.4 can be replaced with 6 in Theorem 1.2 since the factor 2 in (6.4) can be removed in this case, see again [Reference Novak and Woźniakowski51, Sec. 4.1]. A related bound on the sampling numbers in $B(D)$ , without the geometric mean and in terms of entropy numbers, was recently obtained in [Reference Ullrich71].

3. 3. $C^k$ -approximation:

   We can also apply Theorem 6.4 to function recovery in $C^k(D)$ , the space of k-times continuously differentiable functions on a compact domain $D\subset \mathbb {R}^d$ . That is, we consider the identity S on the space $X=Y=C^k(D)$ , which we equip with the norm

   $$\begin{align*}\Vert f \Vert_{C^k(D)} \,:=\, \max_{|\alpha|\le k}\ \max_{x\in D}\, \left\vert \frac{\partial^{|\alpha|} f(x)}{\partial_{x_1}^{\alpha_1} \cdots \partial_{x_d}^{\alpha_d}} \right\vert. \end{align*}$$
   Theorem 6.4 applies for the class $\Lambda $ of point evaluations of derivatives up to order k, that is, for
   $$\begin{align*}\Lambda \,=\, \left\{ \delta_x \circ \frac{\partial^{|\alpha|} }{\partial_{x_1}^{\alpha_1} \cdots \partial_{x_d}^{\alpha_d}} \ \Big\vert\ x\in D,\ \vert \alpha\vert \le k \right\}, \end{align*}$$
   which is a norming set on $C^k(D)$ .

We discuss two implications of Theorem 1.2.

Firstly, Theorem 1.2 contributes to the question on the power of adaption and randomization if only function values are available. Namely, considering algorithms for uniform approximation that only use standard information, Theorem 1.2 gives that adaption and randomization cannot lead to a speed-up larger than one, that is,

$$ \begin{align*} \mathrm{rate}\Big(e_n^{\mathrm{det}\text{-}\mathrm{non}}(\mathrm{APP}_{\infty},F, \Lambda^{\mathrm{std}} )\Big) \;\ge\; \mathrm{rate}\Big(e_n^{\mathrm{ran}}(\mathrm{APP}_{\infty} ,F, \Lambda^{\mathrm{std}} )\Big) \,-\, 1 \end{align*} $$

for any convex and symmetric $F\subset B(D)$ . This is in analogy to our result for linear information, see Corollary 5.3.

Secondly, Theorem 1.2 also contributes to the question on the power of standard information compared to arbitrary linear information. If we consider randomized algorithms for uniform approximation, Theorem 1.2 gives that a restriction to standard information causes a loss in the rate of convergence of no more than one, that is,

$$\begin{align*}{\mathrm{rate}}\Big(e_n{}^{\mathrm{ran}}(\mathrm{APP}_{\infty} ,F, \Lambda^{\mathrm{std}} )\Big) \;\ge\; \mathrm{rate}\Big(e_n{}^{\mathrm{ran}}(\mathrm{APP}_{\infty},F )\Big) \,-\, 1 \end{align*}$$

for any convex and symmetric $F\subset B(D)$ . The analogous result for deterministic algorithms has been proven in [Reference Novak45]. This has recently been improved to

(6.5) $$ \begin{align} {\mathrm{rate}}\Big(e_n{}^{\mathrm{det}}(\mathrm{APP}_{\infty} ,F, \Lambda^{\mathrm{std}} )\Big) \;\ge\; \mathrm{rate}\Big(e_n{}^{\mathrm{det}}(\mathrm{APP}_{\infty},F )\Big) \,-\, 1/2, \end{align} $$

see [Reference Krieg, Pozharska, Ullrich and Ullrich32]. Note that we even have equality of the rates if F is the unit ball of a certain kind of reproducing kernel Hilbert space, see [Reference Geng and Wang16, Reference Krieg, Pozharska, Ullrich and Ullrich31].

It is an interesting open problem whether (6.5) also holds in the randomized setting, and to what extent the results above hold for more general problems $S\in \mathcal {L}(X,Y)$ .

Remark 6.5 (Sampling numbers in $L_2$ )

Another problem where several new bounds have been obtained recently is the case that $S=\mathrm {APP}_2$ , that is, the embedding of X into the space $Y=L_2$ . In this case, there are various upper bounds for the error of nonadaptive algorithms based on function values in terms of the Kolmogorov numbers $d_n(S,F)$ , see [Reference Dolbeault, Krieg and Ullrich12, Reference Krieg and Ullrich34, Reference Krieg and Ullrich35, Reference Nagel, Schäfer and Ullrich44, Reference Temlyakov65, Reference Ullrich69] for deterministic and [Reference Cohen and Dolbeault8, Reference Krieg29, Reference Wasilkowski and Woźniakowski72] for randomized algorithms. On the other hand, the bound (6.1) and Lemma 4.3 give an upper bound on $d_n(S,F)$ in terms of the error of adaptive randomized algorithms. Hence, we may derive several bounds on the maximal gain of adaption and/or randomization for the problem of sampling recovery in $L_2$ .

We only mention the special case that F is the unit ball of a reproducing kernel Hilbert space $X=H$ with finite trace. Using that $\mathrm {rate}(e_n^{\mathrm {det}\text{-}\mathrm{non}}(\mathrm {APP}_2 ,B_H, \Lambda ^{\mathrm {std}} ))={\mathrm {rate}}(c_n(\mathrm {APP}_2 ,B_H))$ from [Reference Krieg and Ullrich34, Corollary 1] together with Lemma 4.3 and Proposition 3.2, we obtain that

$$\begin{align*}{\mathrm{rate}}\Big(e_n^{\mathrm{det}\text{-}\mathrm{non}}({\mathrm{APP}}_2,B_H, \Lambda^{\mathrm{std}} )\Big) \;=\;{\mathrm{rate}}\Big(e_n^{\mathrm{ran}}({\mathrm{APP}}_2 ,B_H )\Big). \end{align*}$$

That is, linear sampling algorithms are optimal (in the sense of order) among arbitrary adaptive, randomized algorithms that may use general linear information.

Remark 6.6 (Exponential decay)

For many classes F of smooth functions, the n-th minimal error has a super-polynomial decay and Theorem 1.2 together with (3.4) implies a bound of the form

$$\begin{align*}g_{cn}^{\mathrm{lin}}(\mathrm{APP}_{\infty},F) \;\le\; e_n{}^{\mathrm{ran}}(\mathrm{APP}_{\infty},F) \end{align*}$$

for all $n\ge n_0$ , where $n_0\in \mathbb {N}$ and $c\ge 1$ are (relatively small) constants. One such example is given by reproducing kernel Hilbert spaces with a Gaussian kernel, see, for example, [Reference Karvonen and Suzuki17, Thm 1.1]. This means that, for all such examples, there is no need for sophisticated algorithms that use randomization, adaption or general linear information, at least from the viewpoint of information complexity. In comparison to deterministic and nonadaptive algorithms that only use function evaluations, at most a factor c can be gained. A similar result for $L_2$ -approximation can be obtained along the lines of Remark 6.5, see also [Reference Krieg, Siedlecki, Ullrich and Woźniakowski33].