SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Figs. 11a and b plot the constructed groundtruth mean and noise variance functions, including the locations of some selected inputs (the selected inputs after every 4 iterations are included) as well as their corresponding number of replications  $n_t$ 's. Similarly, Figs. 11c and d show the queried inputs and the  $n_t$ 's of Mean-Var-BTS-RED ( $\omega = 0.2$ ), shown on the heat maps of the mean-variance objective (c) and noise variance functions (d). The figures show that similar to Fig. 3 for the precision agriculture experiment, the majority of our input queries fall into regions with large (either mean or
Figure 11: Groundtruth (a) mean and (b) noise variance functions for hyperparameter tuning of SVM, including some selected queries (white stars) and the corresponding  $n_t$ 's. (c, d): The corresponding plots for mean-variance optimization ( $\omega = 0.2$ ) for hyperparameter tuning of SVM.
Figure 12: AutoML experiment using different budgets  $\mathbb{B} = 100$  and  $\mathbb{B} = 30$ .
mean-variance) objective function values (Figs. 11a and c) and that the selected number of replications  $n_t$  is generally larger at those input locations with larger noise variance (Figs. 11b and d).
H.4 Additional Experiments with Higher-Dimensional Inputs
The Lunar-Lander experiment requires tuning  $d = 12$  parameters of a heuristic controller used to control the Lunar-Lander environment from OpenAI Gym [4]. The controller can be found at https://github.com/openai/gym/blob/8a96440084a6b9be66b2216b984a1c170e4a061c/gym/envs/box2d/lunar_lander.py#L447. The robot pushing task was introduced by [45], and we defer a more detailed introduction to the experimental settings to the work of [45]. Both experiments are widely used benchmarks for high-dimensional BO experiments [16, 20]. In both experiments, for
(a) Lunar landing.
(b) Robot pushing.
Figure 13: Experiments with higher-dimensional input spaces.
every evaluated controller parameters (i.e., every evaluated input  $\mathbf{x}$ ), the observation (i.e., cumulative rewards) is noisy due to random environmental factors. In addition, the noise may be heteroscedastic. For example, an effective set of parameters which can reliably and consistently control the robot is likely to induce small noise variance, whereas some ineffective sets of parameters may cause radically varying behaviors and hence large noise variances. Therefore, these experiments are also suitable for the application of our algorithms. Since the domain is continuous in these two experiments, here we maximize the acquisition function in every iteration via a combination of random search and L-BFGS-B: in every iteration when we need to maximize the acquisition function, we firstly randomly sample 10,000 inputs/points in the entire domain and find the point with the largest acquisition function value, and then further refine the search via L-BFGS-B with 100 random re-starts.