SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Sebastian Shenghong Tay $^{1,2}$ , Chuan Sheng Foo $^{2,3}$ , Daisuke Urano $^{4}$ , Richalynn Chiu Xian Leong $^{4}$ , Bryan Kian Hsiang Low $^{1}$
$^{2}$ Institute for Infocomm Research (I2R), A*STAR, Singapore
<sup>3</sup>Centre for Frontier AI Research (CFAR), A*STAR, Singapore
4Temasek Life Sciences Laboratory, Singapore
sebastian.tay@u.nus.edu, foo_chuan_sheng@i2r.a-star.edu.sg, daisuke@tll.org.sg, richalynn@tll.org.sg, lowkh@comp.nus.edu.sg
We introduce the problem of Bayesian optimization with cost-varying variable subsets (BOCVS) where in each iteration, the learner chooses a subset of query variables and specifies their values while the rest are randomly sampled. Each chosen subset has an associated cost. This presents the learner with the novel challenge of balancing between choosing more informative subsets for more directed learning versus leaving some variables to be randomly sampled to reduce incurred costs. This paper presents a novel Gaussian process upper confidence bound-based algorithm for solving the BOCVS problem that is provably no-regret. We analyze how the availability of cheaper control sets helps in exploration and reduces overall regret. We empirically show that our proposed algorithm can find significantly better solutions than comparable baselines with the same budget.
Bayesian optimization (BO) is a powerful framework for the sample-efficient optimization of costly-to-evaluate black-box objective functions [11] and has been successfully applied to many experimental design problems of significance such as hyperparameter optimization [6, 39], chemical synthesis [30], and particle accelerator control [29], among others. Conventional BO assumes that the learner has full control over all query variables (i.e., all variables in the input to the objective function). However, in many real-world optimization problems, some of the query variables may be subject to randomness affecting their values. In some cases, the randomness affecting a specific variable can be eliminated (by allowing the learner to select its value), but at a cost. We illustrate with a few concrete scenarios: In precision agriculture, consider a farm aiming to find the optimal conditions for largest crop yield where the query variables are a set of soil nutrient concentrations (e.g., Ca, B,  $\mathrm{NH}_3$ , K) and pH. The farm may rely on the naturally-occurring quantities of these nutrients in the available soil, but these quantities will be randomly sampled. Alternatively, they may control some subset of these quantities (via manufactured soil and fertilizers) at a higher cost. In advanced manufacturing where random variation occurs in every operation [34], certain specifications of a product may be left unspecified by the manufacturer and randomly determined, or specified but at a higher cost. In ad revenue maximization or crowdsourcing where information is gathered from a large number of individuals via ad platforms or crowdsourcing platforms such as Amazon Mechanical Turk, suppose that the query variables describe the demographics of the individual, such as country of origin or income level. The learner may allow the platform to randomly assign the task to any individuals, or the learner may demand a specific subgroup of individuals at a higher cost. In all these practical scenarios, the goal is to find the maximizer with as little incurred cost as possible. At each query iteration, the learner is
faced with the non-trivial problem of deciding which variables to specify (for more directed learning) vs. which variables to allow to be randomly sampled (to reduce incurred costs), in addition to the usual BO problem of deciding the specified variables' values.
To the best of our knowledge, there are no existing works that tackle this problem precisely. The work of Hayashi et al. [13] introduced the problem of BO with partially specified queries (BOPSQ) in which the subset of deterministically selected variables (control set) and randomly sampled variables (random set) can also be chosen by the learner, but it does not consider the costs incurred by such choices. This is a non-trivial limitation as the presence of costs can significantly alter the learner's decisions. Under such a formulation, if a control set is a strict subset of another, then the former will never be chosen as there is no benefit to having variable values be randomly sampled instead of chosen by the learner. Consequently, if there exists a control set that includes all the variables in a query, then all other control sets will not be used and the problem reduces to conventional BO. In practice, however, the availability of other control sets confers an advantage if these other control sets are cheaper. Having access to cheaper but more random control sets allows the learner to explore the query space cheaply and then use costlier but more deterministic control sets to exploit high-value regions. BOPSQ in its current formulation excludes the analysis of such strategies and is akin to multi-fidelity BO [15] but without modeling the costs of the different information sources: In this case, the learner would simply choose the highest-fidelity information source all the time, thus making the problem setting trivial.
This paper introduces the problem of BO with cost-varying variable subsets (BOCVS) that explicitly models the cost of each control set and is more useful in practical scenarios. Our work generalizes BOPSQ and argues that BOCVS problems are much richer when analyzed from a similar perspective as multi-fidelity BO, and the various control sets are treated as information sources with different levels of usefulness and costs. By using cheap control sets for exploration and expensive control sets for exploitation, we show that with an appropriately designed algorithm, a learner can find significantly better solutions with a lower cost expenditure. To achieve this, we leverage the Gaussian process upper confidence bound (GP-UCB) acquisition function [7, 32] to design a novel no-regret algorithm, i.e., its incurred simple regret tends to 0 as the number of iterations tends to infinity, and the algorithm's best chosen query converges to the optimal solution. We additionally analyze the impact of the availability of cheaper control sets on the regret incurred by the most expensive control set. We observe that our algorithm generally outperforms the non-cost-aware baselines, while simple extensions based on Thompson sampling, maximizing UCB or expected improvement-based acquisition scores per unit cost [31, Sec. 3.2] either fail to converge or fail to utilize cheap control sets effectively. Concretely, the contributions of our work in this paper include the following:
- We introduce the BOCVS problem (Sec. 4) and solve it by designing a novel UCB-based algorithm (Sec. 4.1) with a theoretical analysis of its properties, including the conditions under which it is provably no-regret and the impact of the availability of cheaper control sets on the regret incurred by the most expensive control set, and discuss the practical considerations (Sec. 4.2);
- We empirically evaluate the performance of our proposed algorithm against the baselines under several experimental settings with synthetic and real-world datasets (Sec. 5), including a plant growth dataset and an airfoil self-noise dataset corresponding, respectively, to the precision agriculture and advanced manufacturing use cases motivated earlier in this section.
The work of Hayashi et al. [13] introduced BO with partially specified queries (BOPSQ) and tackled the problem with Thompson sampling. However, it fails to consider the relative costs of control sets, which hinders the learner's ability to take advantage of all control sets even in the presence of more deterministic control sets. The work of Oliveira et al. [25] proposed BO with uncertain inputs in which the executed query is sampled from a probability distribution depending on the proposed query. Though related, its problem setting is motivated more by uncertainty in the input query even post-observation and does not involve variable subset selection. These two works are part of a line of research investigating BO in situations where the learner may not have full control over all variables in a query, which includes BO for expected values [38], risk-averse BO [5, 21, 22], and distributionally robust BO [17, 24, 37]. These works also do not consider variable subset selection. Our treatment of the BOCVS problem is inspired by multi-fidelity BO in which the learner has access to cheap, low-fidelity surrogates of the true objective function [15, 27, 35, 36]. In such works (and in
ours), modeling costs is crucial as the learner would simply choose the highest-fidelity information source (in ours, the maximally deterministic control set) otherwise. While the general idea of paying less for potentially less informative queries is similar, our problem setting is fundamentally different: The lack of informativeness comes from the uncertainty of the executed query as opposed to a bias in the observed function values.
The BOCVS setting may be viewed as a special case of causal BO as formulated by Aglietti et al. [1] and continued in several works [2, 4]. Specifically, our setting is a case in which there are no 'non-manipulative' variables and the causal DAG is such that all input variables have no parents and are parents of the output variable. Nevertheless, we believe our focus on this special case has value as it allows us to derive useful theoretical results such as algorithm regret bounds that, to the best of our knowledge, do not exist for the completely general causal BO setting at the time of writing. The work of Sussex et al. [33] includes a regret bound, but is also a special case of [1], and has little overlap with our work as it does not consider costs of control sets or explicit probability distributions over input variables. We believe that our work is sufficiently general to be useful for practical scenarios (where the full causal BO apparatus may be unnecessary), and is also a stepping stone towards theory for the general case.
3 BO and Gaussian Processes
We will first give a brief review of conventional BO [11]. Given a query set  $\mathcal{X}$  and an objective function  $f: \mathcal{X} \to \mathbb{R}$ , a learner wishes to find the maximizing query  $\mathbf{x}^* := \operatorname{argmax}_{\mathbf{x} \in \mathcal{X}} f(\mathbf{x})$ . However,  $f$  is black-box (i.e., not available in closed form) and can only be learned by submitting a query  $\mathbf{x}_t \in \mathcal{X}$  in each iteration  $t$  for function evaluation and receiving a noisy observation  $y_t := f(\mathbf{x}_t) + \xi_t$  where each  $\xi_t$  is i.i.d.  $\sigma$ -sub-Gaussian noise with zero mean. Each function evaluation is assumed to be expensive in some way, such as in terms of money or time spent. So, the learner must be sample-efficient and find  $\mathbf{x}^*$  in as few iterations as possible. BO achieves sample efficiency by leveraging a Bayesian model to represent a probabilistic belief of the function values at unobserved regions of  $\mathcal{X}$  in a principled manner. While any Bayesian model may be used for BO, Gaussian processes (GPs) [42] are a common choice as they enable exact posterior inference: The GP posterior belief of  $f$  at any query  $\mathbf{x} \in \mathcal{X}$  after  $t$  iterations is a Gaussian with posterior mean and variance given by
$$
\mu_ {t} (\mathbf {x}) := \mathbf {k} _ {t} (\mathbf {x}) ^ {\top} \left(\mathbf {K} _ {t} + \lambda \mathbf {I}\right) ^ {- 1} \mathbf {y} _ {t}, \quad \sigma_ {t} ^ {2} (\mathbf {x}) := k (\mathbf {x}, \mathbf {x}) - \mathbf {k} _ {t} (\mathbf {x}) ^ {\top} \left(\mathbf {K} _ {t} + \lambda \mathbf {I}\right) ^ {- 1} \mathbf {k} _ {t} (\mathbf {x}) \tag {1}
$$
where  $\mathbf{y}_t \coloneqq (y_j)_{j=1}^t \in \mathbb{R}^t$ ,  $k$  is a positive semidefinite kernel (covariance function),  $\mathbf{k}_t(\mathbf{x}) \coloneqq (k(\mathbf{x}, \mathbf{x}_j))_{j=1}^t \in \mathbb{R}^t$ ,  $\mathbf{K}_t \coloneqq (k(\mathbf{x}_j, \mathbf{x}_{j'})))_{j,j' = 1}^t \in \mathbb{R}^{t \times t}$ , and  $\lambda$  is an algorithm parameter; if the noise is a Gaussian with variance  $\sigma^2$ , then the true posterior is recovered with  $\lambda = \sigma^2$ . The kernel  $k$  is an important modeling choice as the GP posterior mean will reside in the reproducing kernel Hilbert space (RKHS) associated with  $k$ . For simplicity, we assume w.l.o.g. that  $k(\mathbf{x}, \mathbf{x}') \leq 1$  for any pair of queries  $\mathbf{x}, \mathbf{x}' \in \mathcal{X}$ . Kernel  $k$  affects the maximum information gain (MIG) defined as
$$
\gamma_{T}(\mathcal{X}):= \max_{\{\mathbf{x}_{t}\}_{t = 1}^{T}\subseteq \mathcal{X}}0.5\log \left|\mathbf{I} + \lambda^{-1}\mathbf{K}_{T}\right|.
$$
The MIG characterizes the statistical complexity of a problem and plays an integral role in the theoretical analysis. For the commonly used squared exponential kernel,  $\gamma_T(\mathcal{X}) = \mathcal{O}((\log T)^{d + 1})$  while for the Matérn kernel with  $\nu > 1$ ,  $\gamma_T(\mathcal{X}) = \mathcal{O}(T^{d(d + 1) / (2v + d(d + 1))}(\log T))$  [32]. Importantly,  $\gamma_T(\mathcal{X})$  is increasing in the volume of  $\mathcal{X}$  [32, Theorem 8].
4 BO with Cost-varying Variable Subsets (BOCVS)
The BOCVS problem consists of a compact query set  $\mathcal{X} \subset \mathbb{R}^d$  and an objective function  $f: \mathcal{X} \to \mathbb{R}$  in the RKHS of  $k$  with the RKHS norm upper bounded by  $B$ . For simplicity, assume w.l.o.g. that  $\mathcal{X} = [0,1]^d$ . Let  $[d] := \{1,2,\dots,d\}$ . The learner is given a collection  $\mathcal{I} \subseteq 2^{[d]}$  of control sets indexed by  $1,2,\ldots,m := |\mathcal{I}|$ . Each control set  $i \in [m]$ , denoted by  $\mathcal{I}_i \subseteq [d]$ , indicates the variables in a query with values that can be chosen by the learner. The complement  $\overline{\mathcal{I}}_i := [d] \setminus \mathcal{I}_i$  of  $\mathcal{I}_i$  is the corresponding random set indicating the variables in a query with values that will be randomly sampled from some distribution. A query  $\mathbf{x} \in \mathcal{X}$  can be represented by a combination of partial queries  $[\mathbf{x}^i, \mathbf{x}^{-i}]$  comprising the control partial query  $\mathbf{x}^i := (x_\ell)_{\ell \in \mathcal{I}_i}$  (i.e.,  $\mathbf{x}^i$  collects the variables
Iteration  $t$ :
$$
i _ {t} = 1
$$
$$
\mathbf {x} ^ {i _ {t}} = (x _ {t, 1})
$$
$$
\mathbf {X} ^ {- i _ {t}} = (X _ {t, 2}, X _ {t, 3})
$$
Figure 1: Two iterations in a BOCVS problem setting. The grey boxes are isometric views of a query set  $\mathcal{X} \subset \mathbb{R}^3$ . The blue regions depict the probability densities of random vectors  $[\mathbf{x}^{i_t}, \mathbf{X}^{-i_t}]$  and  $[\mathbf{x}^{i_{t+1}}, \mathbf{X}^{-i_{t+1}}]$ . In iteration  $t$ , the learner chooses the control set  $i_t = 1$  and specifies the value (of the first variable  $x_{t,1}$ ) in control partial query  $\mathbf{x}^{i_t}$ , while the last two variables  $X_{t,2}, X_{t,3}$  in random partial query  $\mathbf{X}^{-i_t}$  will be randomly sampled. In iteration  $t + 1$ , the learner chooses the control set  $i_{t+1} = 2$  and specifies the values (of the first two variables  $x_{t,1}, x_{t,2}$ ) in control partial query  $\mathbf{x}^{i_{t+1}}$ , while the last variable  $X_{t,3}$  in random partial query  $\mathbf{X}^{-i_{t+1}}$  will be randomly sampled.
Iteration  $t + 1$  ..
$$
i _ {t + 1} = 2
$$
$$
\mathbf {x} ^ {i _ {t + 1}} = (x _ {t + 1, 1}, x _ {t + 1, 2})
$$
$$
\mathbf {X} ^ {- i _ {t + 1}} = \left(X _ {t + 1, 3}\right)
$$
indexed by  $\mathcal{I}_i$ ) and the random partial query  $\mathbf{x}^{-i} \coloneqq (x_{\ell})_{\ell \in \overline{\mathcal{I}}_i}$  where  $x_{\ell}$  denotes the  $\ell$ -th variable in the query vector  $\mathbf{x}$ . Note that  $[\mathbf{x}^i, \mathbf{x}^{-i}]$  is not a simple vector concatenation as the variables may need to be reordered according to their indices. Furthermore, let  $\mathcal{X}^i \coloneqq \{\mathbf{x}^i \mid \mathbf{x} \in \mathcal{X}\}$ .
In iteration  $t$ , the learner chooses control set  $i_t \in \mathcal{I}$  and specifies the values in control partial query  $\mathbf{x}^{i_t}$ . The random partial query  $\mathbf{x}^{-i_t}$  will then be randomly sampled from the environment. For example, if  $d = 4$  and  $\mathcal{I}_{i_t} = \{1,3\}$ , then  $\overline{\mathcal{I}}_{i_t} = \{2,4\}$  and the learner will be able to choose the values in  $\mathbf{x}^{i_t}$  (i.e., the  $1^{\mathrm{st}}$  and  $3^{\mathrm{rd}}$  variables) but not those in  $\mathbf{x}^{-i_t}$  (i.e., the  $2^{\mathrm{nd}}$  and  $4^{\mathrm{th}}$  variables). The full query in iteration  $t$  is then  $\mathbf{x}_t = [\mathbf{x}^{i_t}, \mathbf{x}^{-i_t}] = (x_{t,\ell})_{\ell \in [d]}$ . Each observed variable  $x_{t,\ell}$  for  $\ell \in \overline{\mathcal{I}}_{i_t}$  is a realization of a random variable  $X_{t,\ell} \sim \mathcal{P}_{\ell}$ . The observed  $\mathbf{x}^{-i_t}$  is then a realization of the random vector  $\mathbf{X}^{-i_t} := (X_{t,\ell})_{\ell \in \overline{\mathcal{I}}_{i_t}} \sim \mathbb{P}^{-i_t}$  where  $\mathbb{P}^{-i_t}$  is the product measure  $\times_{\ell \in \overline{\mathcal{I}}_{i_t}} \mathcal{P}_{\ell}$ . In other words, each variable in a random partial query is independently sampled from a probability distribution that governs that variable. All distributions are assumed to be known. The learner then observes  $y_t := f(\mathbf{x}_t) + \xi_t$  where each  $\xi_t$  is i.i.d.  $\sigma$ -sub-Gaussian noise with a zero mean. Fig. 1 illustrates two iterations in a BOCVS problem setting.
The learner wishes to find the optimal control set  $i^*$  and specified values in control partial query  $\mathbf{x}^{i^*}$  that maximize the expected value of  $f([\mathbf{x}^i, \mathbf{X}^{-i}])$  where the expectation is w.r.t.  $\mathbf{X}^{-i} \sim \mathbb{P}^{-i}$ :
$$
\left(i ^ {*}, \mathbf {x} ^ {i ^ {*}}\right) := \operatorname * {a r g m a x} _ {\left(i, \mathbf {x} ^ {i}\right) \in [ m ] \times \mathcal {X} ^ {i}} \mathbb {E} \left[ f \left(\left[ \mathbf {x} ^ {i}, \mathbf {X} ^ {- i} \right]\right) \right].
$$
The learner has an initial budget  $C \in \mathbb{R}^{+}$  and every control set  $\mathcal{I}_i$  has an associated cost  $c_{i} > 0$  for all  $i \in [m]$ . Let the control set indices be defined such that  $c_{1} \leq c_{2} \leq \ldots \leq c_{m}$ .<sup>1</sup>
In every iteration  $t$ , the learner pays  $c_{i_t}$ . The learning procedure ends after  $T$  iterations when  $C - \sum_{t=1}^{T} c_{i_t} < c_{i_{T+1}}$ , i.e., the learner has not enough budget left to pay for the chosen control set.  $T$  will now be a random variable depending on the algorithm and the random outcomes of the learning procedure. The cost-varying cumulative regret is defined as
$$
R _ {T} := \sum_ {t = 1} ^ {T} c _ {i _ {t}} \left(\mathbb {E} \left[ f \left(\left[ \mathbf {x} ^ {i ^ {*}}, \mathbf {X} ^ {- i ^ {*}} \right]\right) \right] - \mathbb {E} \left[ f \left(\left[ \mathbf {x} ^ {i _ {t}}, \mathbf {X} ^ {- i _ {t}} \right]\right) \right]\right).
$$
The regret incurred by choosing a sub-optimal control set and specifying sub-optimal values in the control partial query is weighted by the cost of that control set. This naturally incorporates the notion that the penalty for sub-optimal plays is lower if the play was cheap, while also penalizing using the entire budget on sub-optimal plays, regardless of whether those plays are cheap or expensive. Intuitively, to minimize the cost-varying regret, a learner would attempt to use the cheap control sets (i.e., low  $c_{i}$ , low  $\mathbb{E}\big[f([\mathbf{x}^i,\mathbf{X}^{-i}])\big]$ ) to explore the query space, and use the expensive control sets (i.e., high  $c_{i}$ , high  $\mathbb{E}\big[f([\mathbf{x}^i,\mathbf{X}^{-i}])\big]$ ) to exploit control partial queries with high expected function values. When all  $c_{i} = 1$ , we recover the BOPSQ problem [13], and  $C$  is simply the number of iterations
<sup>1</sup>While our problem definition does not require that  $c_{i} \leq c_{j} \Leftrightarrow \max_{\mathbf{x}^{i} \in \mathcal{X}^{i}} \mathbb{E}\left[f(\left[\mathbf{x}^{i}, \mathbf{X}^{-i}\right])\right] \leq \max_{\mathbf{x}^{j} \in \mathcal{X}^{j}} \mathbb{E}\left[f(\left[\mathbf{x}^{j}, \mathbf{X}^{-j}\right])\right]$ , one might reasonably expect this to be the case in real-world problems, i.e., "better" control sets cost more to specify. This also implies that  $\mathcal{I}_{i} \subseteq \mathcal{I}_{j} \Rightarrow c_{i} \leq c_{j}$ .
Algorithm 1 UCB-CVS
1: Input: GP with kernel  $k$ , budget  $C$ , control sets  $\mathcal{I}$ , costs  $(c_{i})_{i=1}^{m}$ ,  $\epsilon$ -schedule  $(\epsilon_{t})_{t=1}^{\infty}$   
2: for iteration  $t = 1$  to  $\infty$  do  
3:  $g_{t} := \max_{(i,\mathbf{x}^{i}) \in [m] \times \mathcal{X}^{i}} \mathbb{E}[u_{t-1}([x^{i}, X^{-i}])]$   
4:  $\mathcal{S}_{1} := \{i \in [m] \mid \max_{x^{i} \in \mathcal{X}^{i}} \mathbb{E}[u_{t-1}([x^{i}, X^{-i}])] + \epsilon_{t} \geq g_{t}\}$   
5:  $\mathcal{S}_{2} := \{i \in \mathcal{S}_{1} \mid c_{i} = \min_{j \in \mathcal{S}_{1}} c_{j}\}$   
6:  $(i_{t}, x^{it}) := \operatorname{argmax}_{(i, x^{i}) \in S_{2} \times \mathcal{X}^{i}} \mathbb{E}[u_{t-1}([x^{i}, X^{-i}])]$   
7: break if  $C - \sum_{\tau=1}^{t-1} c_{i\tau} < c_{it}$   
8: Observe  $x^{-i_{t}}$  drawn from  $\mathbb{P}^{-i_{t}}$   
9: Observe  $y_{t} := f(x_{t}) + \xi_{t}$   
10:  $\mathcal{D}_{t} := \{(x_{\tau}, y_{\tau})\}_{\tau=1}^{t}$   
11: end for  
12: return  $\mathcal{D}_{t}$
in the learning trajectory. In fact, BOPSQ reduces to a simpler problem if there exists a full query control set that allows the learner to choose the values of all  $d$  variables. If  $[d] \in \mathcal{I}$ , then  $\mathcal{I}_{i^*} = [d]$  and  $\mathbb{E}\big[f([\mathbf{x}^{i^*},\mathbf{X}^{-i^*}])\big] = \max_{\mathbf{x} \in \mathcal{X}} f(\mathbf{x})$  since expectations of a function are never greater than the maximum value of the function. In other words, the full query control set is guaranteed to be the optimal control set and the BOPSQ problem reduces to one of conventional BO. In general, under BOPSQ, any control set that is a strict subset of another will never be chosen.
4.1 UCB-CVS
Alg. 1 describes our UCB-CVS algorithm for solving the BOCVS problem. In iteration  $t$ , it uses the GP posterior belief of  $f$  to construct an upper confidence bound (UCB)  $u_{t-1}$  of  $f$ :
$$
u _ {t - 1} (\mathbf {x}) = \mu_ {t - 1} (\mathbf {x}) + \beta_ {t} \sigma_ {t - 1} (\mathbf {x})
$$
where the sequence  $(\beta_{t})_{t\geq 1}$  is an algorithm parameter that controls the tradeoff between exploration and exploitation. UCB-based algorithm design is a classic strategy in the stochastic bandits [19, Ch. 7] and BO literature [7, 32] and makes use of the "optimism in the face of uncertainty" (OFU) principle [18]: Queries with a large posterior standard deviation (i.e., high uncertainty) are given high acquisition scores as the function values at those queries may be potentially high. UCB-CVS adapts this strategy by taking the expectation of the UCB as part of the acquisition process. Due to the monotonicity of expectation, if  $u_{t - 1}$  is an upper bound of  $f$  (i.e.,  $u_{t - 1}(\mathbf{x})\geq f(\mathbf{x})$  for any  $\mathbf{x}\in \mathcal{X}$ ), then  $\mathbb{E}\big[u_{t - 1}([\mathbf{x}^i,\mathbf{X}^{-i}])\big]$  is also an upper bound of  $\mathbb{E}\big[f([\mathbf{x}^i,\mathbf{X}^{-i}])\big]$  for any  $i\in [m],\mathbf{x}^i\in \mathcal{X}^i$ .