SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Definition 6 ( $\gamma$ -Graph Dimension). Let  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}, \ell : \mathbb{R}^2 \to [0,1]$ . We say that a sample  $S \in \mathcal{X}^n$  is  $\gamma$ -graph shattered by  $\mathcal{H}$  if there exists  $f : [n] \colon \to [0,1]$  such that for all  $b \in \{0,1\}^n$  there exists  $h_b \in \mathcal{H}$  such that:
-  $h_b(x_i) = f(i), \forall i \in [n]$  such that  $b_i = 0$ .
-  $\ell(h_b(x_i), f(i)) > \gamma, \forall i \in [n]$  such that  $b_i = 1$ .
The  $\gamma$ -graph dimension  $\mathbb{D}_{\gamma}^{\mathrm{G}}$  is defined to be the maximum size of a  $\gamma$ -graph shattered set.
We mention that the asymmetry in the above definition is crucial. In particular, replacing the equality  $h_b(x_i) = f(i)$  with  $\ell(h_b(x_i), f(i)) \leq \gamma$  fails to capture the properties of the graph dimension. Intuitively, the equality in the definition reflects the assumption of realizability, i.e., the guarantee that there exists a hypothesis  $h^*$  that exactly fits the labels. Before stating our main result, we can collect some useful observations about this new combinatorial measure. In particular, we provide examples inspired by [DSS14, DSBDSS15] which show (i) that there exist gaps between different ERM learners (see Example 3) and (ii) that any learning algorithm with a close to optimal sample complexity must be improper (see Example 4).
Our first main result relates the scaled graph dimension with the learnability of any class  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}$  using a (worst case) ERM learner. This result is the scaled analogue of known multiclass results [DSS14, DSBDSS15] but its proof for the upper bound follows a different path. For the formal statement of our result and its full proof, we refer the reader to Appendix B.
Informal Theorem 2 (Informal, see Theorem 1). Any  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}$  is PAC learnable in the realizable regression setting by a worst-case ERM learner if and only if  $\mathbb{D}_{\gamma}^{\mathrm{G}}(\mathcal{H}) < \infty$  for all  $\gamma \in (0,1)$ .
Proof Sketch. The proof of the lower bound follows in a similar way as the lower bound regarding learnability of binary hypothesis classes that have infinite VC dimension [VC71, BEHW89]. If  $\mathcal{H}$  has infinite  $\gamma$ -graph dimension for some  $\gamma \in (0,1)$ , then for any  $n \in \mathbb{N}$  we can find a sequence of  $n$  points  $x_1, \ldots, x_n$  that are  $\gamma$ -graph shattered by  $\mathcal{H}$ . Then, we can define a distribution  $\mathcal{D}$  that puts most of its mass on  $x_1$ , so if the learner observes  $n$  samples then with high probability it will not observe at least half of the shattered points. By the definition of the  $\gamma$ -graph dimension this shows
that, with high probability, there exists at least one ERM learner that is  $\gamma$ -far on at least half of these points. The result follows by taking  $n \to \infty$ .
The most technically challenging part of our result is the upper bound. There are three main steps in our approach. First, we argue that by introducing a "ghost" sample of size  $n$  on top of the training sample, which is also of size  $n$ , we can bound the true error of any ERM learner on the distribution by twice its error on the ghost sample. This requires a slightly more subtle treatment compared to the argument of [BEHW89] for binary classification. The second step is to use a "random swaps" type of argument in order to bound the performance of the ERM learner on the ghost sample as follows: we "merge" these two samples by considering any realizable sequence  $S$  of  $2n$  points, we randomly swap elements whose indices differ by  $n$ , and then we consider an ERM learner who gets trained on the first  $n$  points. We can show that if such a learner does not make many mistakes on the unseen part of the sequence, in expectation over the random swaps, then it has a small error on the true distribution. The main advantage of this argument is that it allows us to bound the number of mistakes of such a learner on the unseen part of the sequence without using any information about  $\mathcal{D}$ . The last step of the proof is where we diverge from the argument of [BEHW89]. In order to show that this expectation is small, we first map  $\mathcal{H}$  to a partial concept class  $\overline{\mathcal{H}}$  [AHHM22], then we project  $\overline{\mathcal{H}}$  to the sequence  $S$ , and finally we map  $\overline{\mathcal{H}}$  back to a total concept class by considering a disambiguation of it. Through these steps, we can show that there will be at most  $(2n)^{O(\mathbb{D}_{\gamma}^{\mathrm{G}}(\mathcal{H})\log(2n))}$  many different functions in this "projected" class. Then, we can argue that, with high probability, any ERM learner who sees the first half of the sequence makes, in expectation over the random swaps, a small number of mistakes on the second half of the sequence. Finally, we can take a union bound over all the  $(2n)^{O(\mathbb{D}_{\gamma}^{\mathrm{G}}(\mathcal{H})\log(2n))}$  possible such learners to conclude the proof.
2.4  $\gamma$ -One-Inclusion Graph Dimension
In this section, we provide a minimax optimal learner for realizable PAC regression. We first review a fundamental construction which is a crucial ingredient in the design of our learner, namely the one-inclusion (hyper)graph (OIG) algorithm  $\mathbb{A}^{\mathrm{OIG}}$  for a class  $\mathcal{H} \subseteq \mathcal{V}^{\mathcal{X}}$  [HLW94, RBR09, DSS14,  $\mathrm{BCD}^{+}22$ ]. This algorithm gets as input a training set  $(x_{1},y_{1}),\ldots,(x_{n},y_{n})$  realizable by  $\mathcal{H}$  and an additional example  $x$ . The goal is to predict the label of  $x$ . Let  $S = \{x_{1},\ldots,x_{n},x\}$ . The idea is to construct the one-inclusion graph  $G_{\mathcal{H}|S}^{\mathrm{OIG}}$  induced by the pair  $(S,\mathcal{H})$ . The node set  $V$  of this graph corresponds to the set  $\mathcal{H}|_S$  (projection of  $\mathcal{H}$  to  $S$ ) and, so,  $V \subseteq \mathcal{V}^{[n+1]}$ . For the binary classification case, two vertices are connected with an edge if they differ in exactly one element  $x$  of the  $n+1$  points in  $S$ . For the case where  $\mathcal{V}$  is discrete and  $|\mathcal{V}| > 2$ , the hyperedge is generalized accordingly.
Definition 7 (One-Inclusion Hypergraph [HLW94, RBR09, BCD+22]). Consider the set  $[n]$  and a hypothesis class  $\mathcal{H} \subseteq \mathcal{V}^{[n]}$ . We define a graph  $G_{\mathcal{H}}^{\mathrm{OIG}} = (V, E)$  such that  $V = \mathcal{H}$ . Consider a direction  $i \in [n]$  and a mapping  $f: [n] \setminus \{i\} \to \mathcal{Y}$ . We introduce the hyperedge  $e_{i,f} = \{h \in V : h(j) = f(j), \forall j \in [n] \setminus \{i\}\}$ . We define the edge set of  $G_{\mathcal{H}}^{\mathrm{OIG}}$  to be the collection
$$
E = \left\{e _ {i, f}: i \in [ n ], f: [ n ] \setminus \{i \} \rightarrow \mathcal {Y}, e _ {i, f} \neq \emptyset \right\}.
$$
In the regression setting, having created the one-inclusion graph with  $\mathcal{V} = [0,1]$ , the goal is to orient the edges; the crucial property is that "good" orientations of this graph yield learning algorithms with low error. An orientation is good if the maximum out-degree of the graph is small (cf. Definition 8). Informally, if the maximum out-degree of any node is  $M$ , then we can create a predictor whose expected error rate is at most  $M / (n + 1)$ . Note that in the above discussion we have not addressed the issue that we deal with regression tasks and not classification and, hence, some notion of scale is required in the definition of the out-degree.
Intuitively, the set  $S$  that induces the vertices  $\mathcal{H}|_S$  of the OIG consists of the features of the training examples  $\{x_1,\dots,x_n\}$  and the test point  $x$ . Hence, each edge of the OIG should be thought of as the set of all potential labels of the test point that are realizable by  $\mathcal{H}$ , so it corresponds to the set of all possible meaningful predictions for  $x$ . An orientation  $\sigma$  maps every edge  $e$  to a vertex  $v\in e$  and, hence, it is equivalent to the prediction of a learning algorithm. We can now formally define the notion of an orientation and the scaled out-degree.
For an introduction to partial concept classes and their disambiguations we refer the reader to Appendix B.1.
Definition 8 (Orientation and Scaled Out-Degree). Let  $\gamma \in [0,1], n \in \mathbb{N}, \mathcal{H} \subseteq [0,1]^{[n]}$ . An orientation of the one-inclusion graph  $G_{\mathcal{H}}^{\mathrm{OIG}} = (V, E)$  is a mapping  $\sigma: E \to V$  so that  $\sigma(e) \in e$  for any  $e \in E$ . Let  $\sigma_i(e) \in [0,1]$  denote the  $i$ -th entry of the orientation.
For a vertex  $v \in V$ , corresponding to some hypothesis  $h \in \mathcal{H}$  (see Definition 7), let  $v_i$  be the  $i$ -th entry of  $v$ , which corresponds to  $h(i)$ . The (scaled) out-degree of a vertex  $v$  under  $\sigma$  is  $\mathrm{outdeg}(v; \sigma, \gamma) = |\{i \in [n] : \ell(\sigma_i(e_{i,v}), v_i) > \gamma\}|$ . The maximum (scaled) out-degree of  $\sigma$  is  $\mathrm{outdeg}(\sigma, \gamma) = \max_{v \in V} \mathrm{outdeg}(v; \sigma, \gamma)$ .
Finally, we introduce the following novel dimension in the context of real-valued regression. An analogous dimension was proposed in the context of learning under adversarial robustness [MHS22].
Definition 9 ( $\gamma$ -OIG Dimension). Consider a class  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}$  and let  $\gamma \in [0,1]$ . We define the  $\gamma$ -one-inclusion graph dimension  $\mathbb{D}_{\gamma}^{\mathrm{OIG}}$  of  $\mathcal{H}$  as follows:
$\mathbb{D}_{\gamma}^{\mathrm{OIG}}(\mathcal{H}) = \sup \{n\in \mathbb{N}:\exists S\in \mathcal{X}^n$  such that  $\exists$  finite subgraph  $G = (V,E)$  of  $G_{\mathcal{H}|S}^{\mathrm{OIG}} = (V_n,E_n)$  such that  $\forall$  orientations  $\sigma ,\exists v\in V$  where  $\mathrm{outdeg}(v;\sigma ,\gamma) > n / 3\}$ .
We define the dimension to be infinite if the supremum is not attained by a finite  $n$ .
In words, it is the largest  $n \in \mathbb{N}$  (potentially  $\infty$ ) such that there exists an (unlabeled) sequence  $S$  of length  $n$  with the property that no matter how one orients some finite subgraph of the one-inclusion graph, there is always some vertex for which at least  $1/3$  of its coordinates are  $\gamma$ -different from the labels of the edges that are attached to this vertex. We remark that the hypothesis class in Example 1 has  $\gamma$ -OIG dimension equal to  $O(1)$ . Moreover, a finite fat-shattering dimension of hypothesis class implies a finite OIG dimension of roughly the same size (see Appendix C). We also mention that the above dimension satisfies the "finite character" property and the remaining criteria that dimensions should satisfy according to  $[\mathrm{BDHM}^{+}19]$  (see Appendix F). As our main result in this section, we show that any class  $\mathcal{H}$  is learnable if and only if this dimension is finite and we design an (almost) optimal learner for it.
Informal Theorem 3 (Informal, see Theorem 2). Any  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}$  is PAC learnable in the realizable regression setting if and only if  $\mathbb{D}_{\gamma}^{\mathrm{OIG}}(\mathcal{H}) < \infty$  for all  $\gamma \in (0,1)$ .
The formal statement and its full proof are postponed to Appendix C.
Proof Sketch. We start with the lower bound. As we explained before, orientations of the one-inclusion graph are, in some sense, equivalent to learning algorithms. Therefore, if this dimension is infinite for some  $\gamma > 0$ , then for any  $n \in \mathbb{N}$ , there are no orientations with small maximum out-degree. Thus, for any learner we can construct some distribution  $\mathcal{D}$  under which it makes a prediction that is  $\gamma$ -far from the correct one with constant probability, which means that  $\mathcal{H}$  is not PAC learnable.
Let us now describe the proof of the converse direction, which consists of several steps. First, notice that the finiteness of this dimension provides good orientations for finite subgraphs of the one-inclusion graph. Using the compactness theorem of first-order logic, we can extend them to a good orientation of the whole, potentially infinite, one-inclusion graph. This step gives us a weak learner with the following property: for any given  $\gamma$  there is some  $n_0\in \mathbb{N}$  so that, with high probability over the training set, when it is given  $n_0$  examples as its training set it makes mistakes that are of order at least  $\gamma$  on a randomly drawn point from  $\mathcal{D}$  with probability at most  $1 / 3$ . The next step is to boost the performance of this weak learner. This is done using the "median-boosting" technique [Keg03] (cf. Algorithm 2) which guarantees that after a small number of iterations we can create an ensemble of weak learners such that, a prediction rule according to their (weighted) median will not make any  $\gamma$ -mistakes on the training set. However, this is not sufficient to prove that the ensemble of these learners has small loss on the distribution  $\mathcal{D}$ . This is done by establishing sample compression schemes that have small length. Essentially, such schemes consist of a compression function  $\kappa$  which takes as input a training set and outputs a subset of it, and a reconstruction function  $\rho$  which takes as input the output of  $\kappa$  and returns a predictor whose error on every point of the training set  $S$  is at most  $\gamma$ . Extending the arguments of [LW86] from the binary setting to the real-valued setting we show that the existence of such a scheme whose compression function returns a set of "small" cardinality implies generalization properties of the underlying learning rule. Finally, we show that our weak learner combined with the boosting procedure admits such a sample compression scheme.
Before proceeding to the next section, one could naturally ask whether there is a natural property of the concept class that implies finiteness of the scaled OIG dimension. The work of [Men02] provides
a sufficient and natural condition that implies finiteness of our complexity measure. In particular, Mendelson shows that classes that contain functions with bounded oscillation (as defined in [Men02]) have finite fat-shattering dimension. This implies that the class is learnable in the agnostic setting and hence is also learnable in the realizable setting. As a result, the OIG-based dimension is also finite. So, bounded oscillations are a general property that guarantees that the finiteness of OIG-based dimension and fat-shattering dimension coincide. We also mention that deriving bounds for the OIG-dimension for interesting families of functions is an important yet non-trivial question.
2.5  $\gamma$ -DS Dimension
So far we have identified a dimension (cf. Definition 9) that characterizes the PAC learnability of realizable regression. However, it might not be easy to calculate it in some settings. Our goal in this section is to introduce a relaxation of this definition which we conjecture that also characterizes learnability in this context. This new dimension is inspired by the DS dimension, a combinatorial dimension defined by Daniely and Shalev-Shwartz in [DSS14]. In a recent breakthrough result,  $\left[\mathrm{BCD}^{+}22\right]$  showed that the DS dimension characterizes multiclass learnability (with a possibly unbounded number of labels and the 0-1 loss). We introduce a scaled version of the DS dimension. To this end, we first define the notion of a scaled pseudo-cube.
Definition 10 (Scaled Pseudo-Cube). Let  $\gamma \in [0,1]$ . A class  $\mathcal{H} \subseteq [0,1]^d$  is called a  $\gamma$ -pseudo-cube of dimension  $d$  if it is non-empty, finite and, for any  $f \in \mathcal{H}$  and direction  $i \in [d]$ , the hyper-edge  $e_{i,f} = \{g \in \mathcal{H} : g(j) = f(j) \forall j \in [d], i \neq j\}$  satisfies  $|e_{i,f}| > 1$  and  $\ell(g_1(i), g_2(i)) > \gamma$  for any  $g_1, g_2 \in e_{i,f}, g_1 \neq g_2$ .
Pseudo-cubes can be seen as a relaxation of the notion of a Boolean cube (which should be intuitively related with the Natarajan dimension) and were a crucial tool in the proof of  $\left[\mathrm{BCD}^{+}22\right]$ . In our setting, scaled pseudo-cubes will give us the following combinatorial dimension.
Definition 11 ( $\gamma$ -DS Dimension). Let  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}$ . A set  $S \in \mathcal{X}^n$  is  $\gamma$ -DS shattered if  $\mathcal{H}|_S$  contains an  $n$ -dimensional  $\gamma$ -pseudo-cube. The  $\gamma$ -DS dimension  $\mathbb{D}_{\gamma}^{\mathrm{DS}}$  is the maximum size of a  $\gamma$ -DS-shattered set.
Extending the ideas from the multiclass classification setting, we show that the scaled-DS dimension is necessary for realizable PAC regression. Simon (Section 6, [Sim97]) left as an open direction to "obtain supplementary lower bounds [for realizable regression] (perhaps completely unrelated to the combinatorial or Natarajan dimension)". Our next result is a novel contribution to this direction.
Informal Theorem 4 (Informal, see Theorem 3). Any  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}$  is PAC learnable in the realizable regression setting only if  $\mathbb{D}_{\gamma}^{\mathrm{DS}}(\mathcal{H}) < \infty$  for all  $\gamma \in (0,1)$ .
The proof is postponed to Appendix D. We believe that finiteness of  $\mathbb{D}_{\gamma}^{\mathrm{DS}}(\mathcal{H})$  is also a sufficient condition for realizable regression. However, the approach of  $[\mathrm{BCD}^{+}22]$  that establishes a similar result in the setting of multiclass classification does not extend trivially to the regression setting.
Conjecture 1 (Finite  $\gamma$ -DS is Sufficient). Let  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}$ . If  $\mathbb{D}_{\gamma}^{\mathrm{DS}}(\mathcal{H}) < \infty$  for all  $\gamma \in (0,1)$ , then  $\mathcal{H}$  is PAC learnable in the realizable regression setting.
3 Online Learnability for Realizable Regression
In this section we will provide our main result regarding realizable online regression. Littlestone trees have been the workhorse of online classification problems [Lit88]. First, we provide a definition for scaled Littlestone trees.
Definition 12 (Scaled Littlestone Tree). A scaled Littlestone tree of depth  $d \leq \infty$  is a complete binary tree of depth  $d$  defined as a collection of nodes
$$
\bigcup_ {0 \leq \ell <   d} \left\{x _ {u} \in \mathcal {X}: u \in \{0, 1 \} ^ {\ell} \right\} = \left\{x _ {\emptyset} \right\} \cup \left\{x _ {0}, x _ {1} \right\} \cup \left\{x _ {0 0}, x _ {0 1}, x _ {1 0}, x _ {1 1} \right\} \cup \dots
$$
and real-valued gaps
$$
\bigcup_ {0 \leq \ell <   d} \left\{\gamma_ {u} \in [ 0, 1 ]: u \in \{0, 1 \} ^ {\ell} \right\} = \left\{\gamma_ {\emptyset} \right\} \cup \left\{\gamma_ {0}, \gamma_ {1} \right\} \cup \left\{\gamma_ {0 0}, \gamma_ {0 1}, \gamma_ {1 0}, \gamma_ {1 1} \right\} \cup \dots
$$
such that for every path  $\mathbf{y} \in \{0,1\}^d$  and finite  $n < d$ , there exists  $h \in \mathcal{H}$  so that  $h(x_{\mathbf{y}_{\leq \ell}}) = s_{\mathbf{y}_{\leq \ell + 1}}$  for  $0 \leq \ell \leq n$ , where  $s_{\mathbf{y}_{\leq \ell + 1}}$  is the label of the edge connecting the nodes  $x_{\mathbf{y}_{\leq \ell}}$  and  $x_{\mathbf{y}_{\leq \ell + 1}}$  and  $\ell(s_{\mathbf{y}_{\leq \ell, 0}}, s_{\mathbf{y}_{\leq \ell, 1}}) = \gamma_{\mathbf{y}_{\leq \ell}}$ .
In words, scaled Littlestone trees are complete binary trees whose nodes are points of  $\mathcal{X}$  and the two edges attached to every node are its potential classifications. An important quantity is the gap between the two values of the edges. We define the online dimension  $\mathbb{D}^{\mathrm{onl}}(\mathcal{H})$  as follows.
Definition 13 (Online Dimension). Let  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}$ . Let  $\mathcal{T}_d$  be the space of all scaled Littlestone trees of depth  $d$  (cf. Definition 12) and  $\mathcal{T} = \bigcup_{d=0}^{\infty} \mathcal{T}_d$ . For any scaled tree  $T = \bigcup_{0 \leq \ell \leq \mathrm{dep}(T)} \left\{(x_u, \gamma_u) \in (\mathcal{X}, [0,1]): u \in \{0,1\}^\ell \right\}$ , let  $\mathcal{P}(T) = \{y = (y_0, \dots, y_{\mathrm{dep}(T)}): y_i \in \{0,1\}^i)\}$  be the set of all paths in  $T$ . The dimension  $\mathbb{D}^{\mathrm{onl}}(\mathcal{H})$  is
$$
\mathbb {D} ^ {\text {o n l}} (\mathcal {H}) = \sup  _ {T \in \mathcal {T}} \inf  _ {y \in \mathcal {P} (T)} \sum_ {i = 0} ^ {\operatorname {d e p} (T)} \gamma_ {y _ {i}}. \tag {1}
$$
In words, this dimension considers the tree that has the maximum sum of label gaps over its path with the smallest such sum, among all the trees of arbitrary depth. Note that we are taking the supremum (infimum) in case there is no tree (path) that achieves the optimal value. Providing a characterization and a learner with optimal cumulative loss  $\mathfrak{C}_T$  for realizable online regression was left as an open problem by [DG22]. We resolve this question (up to a factor of 2) by showing the following result.
Informal Theorem 5 (Informal, see Theorem 4). For any  $\mathcal{H} \subseteq [0,1]^{\mathcal{X}}$  and  $\varepsilon > 0$ , there exists a deterministic learner with  $\mathfrak{C}_{\infty} \leq \mathbb{D}^{\mathrm{onl}}(\mathcal{H}) + \varepsilon$  and any, potentially randomized, learner satisfies  $\mathfrak{C}_{\infty} \geq \mathbb{D}^{\mathrm{onl}}(\mathcal{H}) / 2 - \varepsilon$ .
The formal statement and the full proof of our results can be found in Appendix E. First, we underline that this result holds when there is no bound on the number of rounds that the learner and the adversary interact. This follows the same spirit as the results in the realizable binary and multiclass classification settings [Lit88, DSBDSS15]. The dimension that characterizes the minimax optimal cumulative loss for any given  $T$  follows by taking the supremum in Definition 13 over trees whose depth is at most  $T$  and the proof follows in an identical way (note that even with finite fixed depth  $T$  the supremum is over infinitely many trees). Let us now give a sketch of our proofs.
Proof Sketch. The lower bound follows using similar arguments as in the classification setting: for any  $\varepsilon > 0$ , the adversary can create a scaled Littlestone tree  $T$  that achieves the sup inf bound, up to an additive  $\varepsilon$ . In the first round, the adversary presents the root of the tree  $x_{\emptyset}$ . Then, no matter what the learner picks the adversary can force error at least  $\gamma_{\emptyset} / 2$ . The game is repeated on the new subtree. The proof of the upper bound presents the main technical challenge to establish Theorem 4. The strategy of the learner can be found in Algorithm 3. The key insight in the proof is that, due to realizability, we can show that in every round  $t$  there is some  $\widehat{y}_t \in [0,1]$  the learner can predict so that, no matter what the adversary picks as the true label  $y_t^\star$ , the online dimension of the class under the extra restriction that  $h(x_t) = y_t^\star$ , i.e., the updated version space  $V = \{h \in \mathcal{H} : h(x_\tau) = y_\tau^\star, 1 \leq \tau \leq t\}$ , will decrease by  $\ell(y_t^\star, \widehat{y}_t)$ . Thus, under this strategy of the learner, the adversary can only distribute up to  $\mathbb{D}^{\mathrm{onl}}(\mathcal{H})$  across all the rounds of the interaction. We explain how the learner can find such a  $\widehat{y}_t$  and we handle technical issues that arise due to the fact that we are dealing with sup inf instead of max min in the formal proof (cf. Appendix E).