SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Theorem 1. Let  $\delta >0$ ,  $f\in \mathcal{H}_k$ , and the corresponding  $\| \hat{f}\|_{\hat{k}}\leq b$ ,  $\max_{x\in \mathcal{X}}|f(x)| = M$ . Suppose the observation noise  $\zeta_{i} = y_{i} - f(x_{i})$  is  $\sigma_{\zeta}$ -sub-Gaussian, and thus with high probability  $|\zeta_i| < A$  for some  $A > 0$ . Assume that both  $k$  and  $P_{x}$  satisfy the conditions for  $\Delta f_{P_x}$  to be  $\sigma_{E}$ -sub-Gaussian, for a given  $\sigma_E > 0$ . Then, under Assumption 1 with  $\varepsilon >0$  and corresponding  $e_{\varepsilon}$ , setting  $\sigma^2 = 1 + \frac{2}{n}$ , running Gaussian Process with acquisition function
$$
\tilde {\alpha} (x \mid \mathcal {D} _ {n}) = \tilde {\mu} _ {n} (P _ {x}) + \beta_ {n} \tilde {\sigma} _ {n} (P _ {x}) \tag {15}
$$
$$
w h e r e \beta_ {n} = \left(b + \sqrt {\sigma_ {E} ^ {2} + \sigma_ {\zeta} ^ {2}} \sqrt {2 (\hat {\gamma} _ {n} + 1 - \ln \delta)}\right),
$$
we have that the uncertain-inputs cumulative regret satisfies:
$$
\tilde {R} _ {n} \in O \left(\sqrt {n \hat {\gamma} _ {n} (\hat {\gamma} _ {n} - \ln \delta)} + n ^ {2} \sqrt {(\hat {\gamma} _ {n} - \ln \delta) e _ {\varepsilon}} + n ^ {3} e _ {\varepsilon}\right) \tag {16}
$$
with probability at least  $1 - \delta - n\varepsilon$ . Here  $\tilde{R}_n = \sum_{t=1}^n \tilde{r}_t$ , and  $\tilde{r}_t = \max_{x \in \mathcal{X}} \mathbb{E}_{P_x}[f] - \mathbb{E}_{P_{x_t}}[f]$
The proof of our main theorem 1 can be found in appendix B.3.
The assumption that  $\zeta_{i}$  is  $\sigma_{\zeta}$ -sub-Gaussian is standard in  $\mathcal{GP}$  fields. The assumption that  $\Delta f_{P_x}$  is  $\sigma_{E}$ -sub-Gaussian can be met when  $P_{x}$  is uniformly bounded or Gaussian, as stated in Proposition 3, [26]. Readers may check the definition of sub-Gaussian in appendix, Definition 1.
To achieve an regret of order  $\tilde{R}_n\in O(\sqrt{n}\hat{\gamma}_n)$ , the same order as the exact Improved  $\mathcal{GP}$  regret (23), and ensure this with high probability, we need to take  $\varepsilon = O(\delta /n)$ ,  $e_{\varepsilon} = O(n^{-\frac{5}{2}}\hat{\gamma}_{n}(\hat{\gamma}_{n}^{-2}\wedge n^{-\frac{1}{2}}))$  and this requires a sample size  $m$  of order  $O(n^{5}\hat{\gamma}_{n}^{-2}(\hat{\gamma}_{n}^{4}\vee n)\log (n))$  for MCMC approximation, or with a same sample size  $m$  and a subsample size  $h$  of order  $O(n^{\frac{5}{2} +\nu}\hat{\gamma}_{n}^{-1 - \nu}(\hat{\gamma}_{n}^{2}\vee n^{\frac{1}{2}}))$  for Nyström approximation with some  $\nu >0$ . Note that (16) only offers an upper bound for cumulative regret, in real applications the calculated regret may be much smaller than this bound, as the approximation error  $e_{\epsilon}$  can be fairly small even with a few samples when the input noise is relatively weak.
To analysis the exact order of  $\hat{\gamma}_n$  could be difficult, as it is influenced by the specific choice of embedding kernel  $k$  and input uncertainty distributions  $P_{x_i}, x_i \in \mathcal{X}$ . Nevertheless, we can deduce the following result for a wide range of cases, showing that cumulative regret is sub-linear under mild conditions. One can check the proof in appendix B.4.
Theorem 2 (Bounding the Maximum information gain). Suppose  $k$  is  $r$ -th differentiable with bounded derivatives and translation invariant, i.e.,  $k(x,y) = k(x - y,0)$ . Suppose the input uncertainty is i.i.d., that is, the noised input density satisfies  $P_{x_i}(x) = P_0(x - x_i), \forall x_i \in \mathcal{X}$ . Then if the space  $\mathcal{X}$  is compact in  $\mathbb{R}^d$ , the maximum information gain  $\hat{\gamma}_n$  satisfies
$$
\hat {\gamma} _ {n} = O \left(n ^ {\frac {d (d + 1)}{r + d (d + 1)}} \log (n)\right).
$$
Thus, when  $r > d(d + 1)$ , the accumulate regret is sub-linear respect to  $n$ , with sufficiently small  $e_{\varepsilon}$ .
(a) GP posterior with a Gaussian input uncertainty  $P = \mathcal{N}(0,0.01)$
(b) MMDGP posterior with an input uncertainty  $P = \mathcal{N}(0,0.01)$ .
Figure 2: Modeling results under different types of input uncertainties.
(c) MMDGP posterior with a variance-changing beta distribution.
(d) MMDGP posterior with a Chi-squared distribution of changing DoF.
In this section, we first experimentally demonstrate AIRBO's ability to model uncertain inputs of arbitrary distributions, then validate the Nyström-based inference acceleration for GP posterior, followed by experiments on robust optimization of synthetic functions and real-world benchmark.
5.1 Robust Surrogate
Modeling arbitrary uncertain inputs: We demonstrate MMDGP's capabilities by employing an RKHS function as the black-box function and randomly selecting 10 samples from its input domain. Various types of input randomness are introduced into the observation and produce training datasets of  $\mathcal{D} = \{(x_i, f(x_i + \delta_i)) | \delta_i \sim P_{x_i}\}_{i=1}^{10}$  with different  $P_x$  configurations. Figure 2a and 2b compare the modeling results of a conventional GP and MMDGP under a Gaussian input uncertainty  $P_x = \mathcal{N}(0, 0.01^2)$ . We observe that the GP model appears to overfit the observed samples without recognizing the input uncertainty, whereas MMDGP properly incorporates the input randomness into its posterior.
To further examine our model's ability under complex input uncertainty, we design the input distribution to follow a beta distribution with input-dependent variance:  $P_{x} = \beta \alpha (\alpha = 0.5,\beta = 0.5,\sigma = 0.9(\sin 4\pi x + 1))$ . The MMDGP posterior is shown in Figure 2c. As the input variance  $\sigma$  changes along  $x$ , inputs from the left and right around a given location  $x_{i}$  yield different MMD distances, resulting in an asymmetric posterior (e.g., around  $x = 0.05$  and  $x = 0.42$ ). This suggests that MMDGP can precisely model the multimodality and asymmetry of the input uncertainty.
Moreover, we evaluated MMDGP using a step-changing Chi-squared distribution  $P_{x} = \chi^{2}(g(x),\sigma = 0.01)$ , where  $g(x) = 0.5$  if  $x\in [0,0.6]$ , and  $g(x) = 7.0$  otherwise. This abrupt change in  $g(x)$  significantly alters the input distribution from a sharply peaked distribution to a flat one with a long tail. Figure 2d illustrates that our model can accurately capture this distribution shape variation, as evidenced by the sudden posterior change around  $x = 0.6$ . This demonstrates our model can thoroughly quantify the characteristics of complex input uncertainties.
Comparing with the other surrogate models: We also compare our model with the other surrogate models under the step-changing Chi-squared input distribution. The results are reported in Figure 7 and they demonstrate our model outperforms obviously under such a complex input uncertainty (see Appendix D.1 for more details)
5.2 Accelerating the Posterior Inference
Estimation variance of MMD: We first examine the variance of MMD estimation by employing two beta distributions  $P = \text{beta}(\alpha = 0.4, \beta = 0.2, \sigma = 0.1)$  and  $Q = \text{beta}(\alpha = 0.4, \beta = 0.2, \sigma = 0.1) + c$ , where  $c$  is an offset value. Figure 3a shows the empirical MMDs computed via Eq. 8 with varying sampling sizes as  $Q$  moves away from  $P$ . We find that a sampling size of 20 is inadequate, leading to high estimation variance, and increasing the sampling size to 100 stabilizes the estimation.
We further utilize this beta distribution  $P$  as the input distribution and derive the MMDGP posterior via empirical estimator in Figure 3b. Note that the MMD instability caused by inadequate sampling subsequently engenders a fluctuating posterior and culminates in a noisy acquisition function, which prevents the acquisition optimizer (e.g., L-BFGS-B in this experiment) from identifying the optima. Although Figure 3c shows that this issue can be mitigated by using more samples during empirical
(a) The empirical MMD and covariance values between two beta distributions  $P$  and  $Q$  as  $Q$  moves away from  $P$ .
(b) The noisy posterior derived from a sampling size of 20 (upper) traps the acq. optimizer at  $x = 0$  (lower).
Figure 3: The posterior derived from the empirical and Nyström MMD approximators with varying sampling sizes.
(c) The posterior becomes smoother with a sampling size of 100 and acq. optimizer can easily identify the optima.
(d) The Nyström estimator with less memory consumption also produces a smooth posterior that is easy to optimize.
Table 1: Performance of Posterior inference for 512 samples.
Method	Sampling Size	Sub-sampling Size	Inference Time (seconds)	Batch Size (samples)
Empirical	20	-	1.143 ± 0.083	512
Empirical	100	-	8.117 ± 0.040	128
Empirical	1000	-	840.715 ± 2.182	1
Nystrom	100	10	0.780 ± 0.001	512
Nystrom	1000	100	21.473 ± 0.984	128
MMD estimation, it is crucial to note that a larger sampling size significantly increases GPU memory usage because of its quadratic space complexity of  $O(MNm^2)$  ( $M$  and  $N$  are the sample number of training and testing,  $m$  is the sampling size for MMD estimation). This limitation severely hinders parallel inference for multiple samples and slows the overall speed of posterior computation.
Table 1 summarizes the inference time of MMDGP posteriors at 512 samples with different sampling sizes. We find that, for beta distribution defined in this experiment, the Nyström MMD estimator with a sampling size of 100 and sub-sampling size of 10 already delivers a comparable result to the empirical estimator with 100 samples (as seen in the acquisition plot of Figure 3d). Also, the inference time is reduced from 8.117 to 0.78 seconds by enabling parallel computation for more samples. For the cases that require much more samples for MMD estimation (e.g., the input distribution is quite complex or high-dimensional), this Nyström-based acceleration can have a more pronounced impact.
Effect of Nyström estimator on optimization: To investigate the effect of Nyström estimator on optimization, we also perform an ablation study in Appendix D.2, the results in Figure 8 suggest that Nyström estimator slightly degrades the optimization performance but greatly improves the inference efficiency.
5.3 Robust Optimization
Experiment setup: To experimentally validate AIRBO's performance, we implement our algorithm based on BoTorch [2] and employ a linear combination of multiple rational quadratic kernels [6] to compute the MMD as Eq. 9. We compare our algorithm with several baselines: 1) uGP-UCB [26] is a closely related work that employs an integral kernel to model the various input distributions. It has a quadratic inference complexity of  $O(MNm^2)$ , where  $M$  and  $N$  are the sample numbers of the training and testing set, and  $m$  indicates the sampling size of the integral kernel. 3) GP-UCB is the standard GP with UCB acquisition, which represents a broad range of existing methods that
The code will be available on https://github.com/huawei-noah/HEBO, and more implementation details can be found in Appendix C.1.
(a) The RKHS function
(b) Robust regrets of RKHS function.
Figure 4: Robust optimization results on synthetic functions.
(c) The double-peak function
(d) Robust regrets of double-peak function
focus on non-robust optimization. 3) SKL-UCB employs symmetric Kullback-Leibler divergence to measure the distance between the uncertain inputs [20]. Its closed-form solution only exists if the input distributions are the Gaussians. 4) ERBF-UCB is the robust GP with the expected Radial Basis Function kernel proposed in [13]. It computes the expected kernel under input distribution using the Gaussian integrals. Assuming the input distributions are sub-Gaussians, this method can efficiently find the robust optimum. Since all the methods use UCB acquisition, we simply distinguish them by their surrogate names in the following tests.
At the end of the optimization, each algorithm needs to decide a final outcome  $x_{n}^{r}$ , perceived to be the robust optimum under input uncertainty at step  $n$ . For a fair comparison, we employ the same outcome policy across all the algorithms:  $x_{n}^{r} = \arg \max_{x\in \mathcal{D}_{n}}\hat{\mu}_{*}(x)$ , where  $\hat{\mu}_*(x)$  is the posterior mean of robust surrogate at  $x$  and  $\mathcal{D}_n = \{(x_i,f(x_i + \delta_i))|\delta_i\sim P_{x_i}\}$  are the observations so far. The optimization performance is measured in terms of robust regret as follows:
$$
r \left(x _ {n} ^ {r}\right) = \mathbb {E} _ {\delta \sim P _ {x ^ {*}}} \left[ f \left(x ^ {*} + \delta\right) \right] - \mathbb {E} _ {\delta \sim P _ {x _ {n} ^ {r}}} \left[ f \left(x _ {n} ^ {r} + \delta\right) \right], \tag {17}
$$
where  $x^{*}$  is the global robust optimum and  $x_{n}^{r}$  represents the outcome point at step  $n$ . For each algorithm, we repeat the optimization process 12 times and compare the average robust regret.
1D RKHS function: We begin the optimization evaluation with an RKHS function that is widely used in previous BO works [1, 24, 10]. Figure 4a shows its exact global optimum resides at  $x = 0.892$  while the robust optimum is around  $x = 0.08$  when the inputs follow a Gaussian distribution  $\mathcal{N}(0,0.01^2)$ . According to Figure 4b, all the robust BO methods work well with Gaussian uncertain inputs and efficiently identify the robust optimum, but the GP-UCB stagnates at a local optimum due to its neglect of input uncertainty. Also, we notice the regret of our method decreases slightly slower than uGP works in this low-dimensional and Gaussian-input case, but later cases with higher dimension and more complex distribution show our method is more stable and efficient.
1D double-peak function: To test with more complex input uncertainty, we design a blackbox function with double peaks and set the input distribution to be a multi-modal distribution  $P_{x} = \beta \alpha (\alpha = 0.4,\beta = 0.2,\sigma = 0.1)$ . Figure 4c shows the blackbox function (black solid line) and the corresponding function expectations estimated numerically via sampling from the input distribution (i.e., the colored lines). Note the true robust optimum is around  $x = 0.251$  under the beta distribution, but an erroneous location at  $x = 0.352$  may be determined if the input uncertainty is incorrectly presumed to be Gaussian. This explains the results in Figure 4d: the performance of SKL-UCB and ERBF-UCB are sub-optimal due to their misidentification of inputs as Gaussian variables, while our method accurately quantifies the input uncertainty and outperforms the others.
10D bumped-bowl function: we also extend our evaluation to a 10D bumped-bowl function [27] under a concatenated circular distribution. Figure 9 demonstrates AIRBO scales efficiently to high dimension and outperforms the others under complex input uncertainty(see Appendix D.3).
Robust robot pushing: To evaluate AIRBO in a real-world problem, we employ a robust robot pushing benchmark from [31], in which a ball is placed at the origin point of a 2D space and a robot learns to push it to a predefined target location  $(g_x, g_y)$ . This benchmark takes a 3-dimensional input  $(r_x, r_y, r_t)$ , where  $r_x, r_y \in [-5, +5]$  are the 2D coordinates of the initial robot location and  $r_t \in [0, 30]$  controls the push duration. We set four targets in separate quadrants, i.e.,  $g_1 = (-3, -3)$ ,  $g_2 = (-3, 3)$ ,  $g_3 = (4.3, 4.3)$ , and a "twin" target at  $g_3' = (5.1, 3.0)$ , and describe the input uncertainty via a two-component Gaussian Mixture Model (defined in Appendix D.4). Following [7, 10], this blackbox benchmark outputs the minimum distance to these 4 targets under squared and linear distances:  $loss = \min(d^2(g_1, l), d(g_2, l), d(g_3, l), d(g_3', l))$ , where  $d(g_i, l)$  is the Euclidean distance between the ball's ending location  $l$  and the  $i$ -th target. This produces a loss landscape as
(a) Contour of the robot push world
(b) Robust regrets of different algorithms
Figure 5: Robust optimization of the robot push problem.
Figure 6: The robot's initial locations and push times found by different algorithms
shown in Figure 5a. Note that  $g_{2}$  is a more robust target than  $g_{1}$  because of its linear-form distance while pushing the ball to quadrant I is the best choice as the targets,  $g_{3}$  and  $g_{3}^{\prime}$ , match the dual-mode pattern of the input uncertainty. According to Figure 5b, our method obviously outperforms the others because it efficiently quantifies the multimodal input uncertainty. This can be further evidenced by the push configurations found by different algorithms in Figure 6, in which each dot represents the robot's initial location and its color represents the push duration. We find that AIRBO successfully recognizes the targets in quadrant I as an optimal choice and frequently pushes from quadrant III to quadrant I. Moreover, the pushes started close to the origin can easily go far away under input variation, so our method learns to push the ball from a corner with a long push duration, which is more robust in this case.
In this work, we generalize robust Bayesian Optimization to an uncertain input setting. The weight-space interpretation of GP inspires us to empower the GP kernel with MMD and build a robust surrogate for uncertain inputs of arbitrary distributions. We also employ the Nyström approximation to boost the posterior inference and provide theoretical regret bound under approximation error. The experiments on synthetic blackbox function and benchmarks demonstrate our method can handle various input uncertainty and achieve state-of-the-art optimization performance.
There are several interesting directions that worth to explore: though we come to current MMD-based kernel from the weight-space interpretation of GP and the RKHS realization of MMD, our kernel design exhibits a deep connection with existing works on kernel over probability measures [22, 11]. Along this direction, as our theoretic regret analysis in Section 4 does not assume any particular form of kernel and the Nystrom acceleration can also be extended to the other kernel computation, it is possible that AIRBO can be further generalized to a more rich family of kernels. Moreover, the MMD used in our kernel is by no means limited to its RKHS realization. In fact, any function class  $\mathcal{F}$  that comes with uniform convergence guarantees and is sufficiently rich can be used, which renders different realizations of MMD. With proper choice of function class  $\mathcal{F}$ , MMD can be expressed as the Kolmogorov metric or other Earth-mover distances [15]. It is also interesting to extend AIRBO with the other IPMs.
We sincerely thank Yanbin Zhu and Ke Ma for their help on formulating the problem. Also, a heartfelt appreciation goes to Lu Kang for her constant encouragement and support throughout this work.
[23] Alfred Müller. "Integral probability metrics and their generating classes of functions". In: Advances in applied probability 29.2 (1997), pp. 429-443.
[24] Jose Nogueira et al. "Unscented Bayesian Optimization for Safe Robot Grasping". In: 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). Daejeon, South Korea: IEEE, 2016, pp. 1967-1972. ISBN: 978-1-5090-3762-9. DOI: 10.1109/IROS.2016.7759310.
[25] Rafael Oliveira, Lionel Ott, and Fabio Ramos. "Bayesian Optimisation under Uncertain Inputs". In: Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics. PMLR, 2019, pp. 1177–1184.
[26] Rafael Oliveira, Lionel Ott, and Fabio Ramos. Bayesian optimisation under uncertain inputs. 2019. arXiv: 1902.07908.
[27] Nicholas D. Sanders et al. Bayesian Search for Robust Optima. 2021. arXiv: 1904.11416.
[28] Bobak Shahriari et al. "Taking the Human Out of the Loop: A Review of Bayesian Optimization". In: Proceedings of the IEEE 104.1 (Jan. 2016), pp. 148-175. DOI: 10.1109/JPROC.2015.2494218.
[29] Jasper Snoek, Hugo Larochelle, and Ryan P Adams. "Practical bayesian optimization of machine learning algorithms". In: Advances in neural information processing systems 25 (2012).
[30] Niranjan Srinivas et al. "Gaussian Process Optimization in the Bandit Setting: No Regret and Experimental Design". In: International Conference on Machine Learning. 2009. URL: https://api-semanticscholar.org/CorpusID:59031327.
[31] Zi Wang and Stefanie Jegelka. "Max-value entropy search for efficient Bayesian optimization". In: International Conference on Machine Learning. PMLR. 2017, pp. 3627-3635.
[32] Colin White, Willie Neiswanger, and Yash Savani. "Bananas: Bayesian optimization with neural architectures for neural architecture search". In: Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 35. 12. 2021, pp. 10293-10301.
[33] Christopher Williams and Matthias Seeger. "Using the Nyström Method to Speed Up Kernel Machines". In: Advances in Neural Information Processing Systems. Vol. 13. MIT Press, 2000.
[34] Christopher KI Williams and Carl Edward Rasmussen. Gaussian processes for machine learning. Vol. 2. 3. MIT press Cambridge, MA, 2006.
A Nyström Estimator Error Bound
Nyström estimator can easily approximate the kernel mean embedding  $\psi_{p_1},\psi_{p_2}$  as well as the MMD distance between two distribution density  $p_1$  and  $p_2$ . We need first assume the boundedness of the feature map to the kernel  $k$ :
Assumption 2. There exists a positive constant  $K \leq \infty$  such that  $\sup_{x \in \mathcal{X}} \| \phi(x) \| \leq K$
The true MMD distance between  $p_1$  and  $p_2$  is denoted as  $\mathrm{MMD}(p_1, p_2)$ . The estimated MMD distance when using a Nyström sample size  $m_i$ , sub-sample size  $h_i$  for  $p_i$  respectively, is denoted as  $\mathrm{MMD}_{(p_i, m_i, h_i)}$ . Then the error
$$
\operatorname {E r r} _ {\left(p _ {i}, m _ {i}, h _ {i}\right)} := \left| \operatorname {M M D} \left(p _ {1}, p _ {2}\right) - \operatorname {M M D} _ {\left(p _ {i}, m _ {i}, h _ {i}\right)} \right|
$$
and now we have the lemma from Theorem 5.1 in [8]
Lemma 1. Let Assumption 2 hold. Furthermore, assume that for  $i \in 1,2$ , the data points  $X_{1}^{i}, \dots, X_{m_{i}}^{i}$  are drawn i.i.d. from the distribution  $\rho_{i}$  and that  $h_{i} \leq m_{i}$  sub-samples  $\tilde{X}_{1}^{i}, \dots, \tilde{X}_{h_{i}}^{i}$  are drawn uniformly with replacement from the dataset  $\{X_{1}^{i}, \dots, X_{m_{i}}^{i}\}$ . Then, for any  $\delta \in (0,1)$ , it holds with probability at least  $1 - 2\delta$
$$
E r r _ {(p _ {i}, m _ {i}, h _ {i})} \leq \sum_ {i = 1, 2} \left(\frac {c _ {1}}{\sqrt {m _ {i}}} + \frac {c _ {2}}{h _ {i}} + \frac {\sqrt {\log (h _ {i} / \delta)}}{h _ {i}} \sqrt {\mathcal {N} ^ {p _ {i}} \left(\frac {1 2 K ^ {2} \log (h _ {i} / \delta)}{h _ {i}}\right)}\right),
$$
provided that, for  $i\in \{1,2\}$
$$
h _ {i} \geq \max  (6 7, 1 2 K ^ {2} \| C _ {i} \| _ {\mathcal {L} (\mathcal {H})} ^ {- 1}) \log (h _ {i} / \delta)
$$
where  $c_{1} = 2K\sqrt{2\log(6 / \delta)}, c_{2} = 4\sqrt{3} K\log (12 / \delta)$  and  $c_{4} = 6K\sqrt{\log(12 / \delta)}$ . The notation  $\mathcal{N}^{p_i}$  denotes the effective dimension associated to the distribution  $p_k$ .
Specifically, when the effective dimension  $\mathcal{N}$  satisfies, for some  $c\geq 0$
- either  $\mathcal{N}^{\rho_i}(\sigma^2) \leq c\sigma^{2 - \gamma}$  for some  $\gamma \in (0,1)$ ,
- or  $\mathcal{N}^{\rho_i}(\sigma^2) \leq \log (1 + c / \sigma^2) / \beta$ , for some  $\beta > 0$ .
Then, choosing the subsample size  $m$  to be
-  $h_i = m_i^{1/(2 - \gamma)} \log(m_i / \delta)$  in the first case
- or  $h_i = \sqrt{m_i} \log \left( \sqrt{m_i} \max \left( \frac{1}{\delta}, c / (6K^2) \right) \right)$  in the second case,
we get  $Err_{(\rho_i, m_i, h_i)} = O(1 / \sqrt{m_i})$
B.1 Exact Kernel Uncertainty  $\mathcal{GP}$  Formulating
Following the same notation in Section 4, now we can construct a Gaussian process  $\mathcal{GP}(0,\hat{k})$  modelling functions over  $\mathcal{P}$ . This  $\mathcal{GP}$  model can then be applied to learn  $\hat{f}$  from a given set of observations  $\mathcal{D}_n = \{(P_i,y_i)\}_{i = 1}^n$ . Under zero mean condition, the value of  $\hat{f} (P_{*})$  for a given  $P_{*}\in \mathcal{P}$  follows a Gaussian posterior distribution with
$$
\hat {\mu} _ {n} \left(P _ {*}\right) = \hat {\mathbf {k}} _ {n} \left(P _ {*}\right) ^ {T} \left(\hat {\mathbf {K}} _ {n} + \sigma^ {2} \mathbf {I}\right) ^ {- 1} \mathbf {y} _ {n} \tag {18}
$$