SlowGuess/NeurIPS_2023_OCR huggingface dataset:
For the synthetic experiment, some important experimental details have been discussed in Sec. 6. The reward function  $f$  used in the synthetic experiment is sampled from a GP with the SE kernel with a length scale of 0.1. In the AutoML experiment, an SVM is used for diabetes diagnosis. That is, we adopt the diabetes diagnosis dataset which can be found at https://www.kaggle.com/uciml/pima-indians-diabetes-database, which is under the CC0 license. The task involves using 8 features to predict whether the patient had diabetes or not, which constitutes a two-class classification problem. We use  $70\%$  of the dataset as the training set and the remaining dataset as the validation set. We aim to tune two hyperparameters of the SVM: the penalty parameter and the RBF kernel parameter, both within the range of  $[10^{-4}, 1]$ . For simplicity, for each of the two hyperparameters, we discretize its domain into 5 points with equal spacing. This results in a domain with  $|\mathcal{X}| = 25$ . Here we adopt the SE kernel:  $k(x, x') \triangleq \sigma_0^2 \exp(-\left\|x - x'\right\|_2^2 / (2l^2))$  with  $\sigma_0^2 = 0.5$ . Since we have implemented our algorithms (both classical GP-UCB and our Q-GP-UCB) using RFF approximation as discussed above, we performed a grid search for the length scale  $l$  within  $\{0.1, 0.2, \dots, 1\}$  for both GP-UCB and Q-GP-UCB when generating the random features. We found that  $l = 1.0$  leads to the best performance for GP-UCB and  $l = 0.2$  works the best for Q-GP-UCB. This is likely because for our Q-GP-UCB, the effective noise is much smaller (thanks to the accurate estimation by the QMC algorithm), which allows us to use a smaller length scale (which can model more complicated functions) to accurately model the reward function.
(a) synthetic, Bernoulli noise
(b) synthetic, Gaussian noise
(a) synthetic, Gaussian noise
Figure 3: Cumulative regret for (a) the synthetic experiment with Bernoulli noise, (b) the synthetic experiment with Gaussian noise, and (c) the AutoML experiment, with the additional comparisons with our Q-GP-UCB algorithm after considering quantum noise.
Figure 2: Cumulative regret for the synthetic experiment with (a) Bernoulli noise and (b) Gaussian noise, with the additional comparisons with LinUCB and Q-LinUCB.
(c) AutoML
All our experiments are run on a computer with Intel(R) Xeon(R) Gold 6326 CPU @ 2.90GHz, with 64 CPUs. No GPUs are needed.
More Experimental Results. Fig. 2 additionally shows the cumulative regret for the LinUCB and Q-LinUCB algorithms in the synthetic experiment. Consistent with the results in Fig. 1 (d) in the main paper (Sec. 6) which has included the results for LinUCB and Q-LinUCB in the AutoML experiment, here LinUCB and QLinUCB again underperform significantly in Fig. 2 for both types of noises. These results further verify that in realistic experiments with non-linear reward functions, algorithms based on linear bandits are significantly outperformed by those based on BO/kernelized bandits. Fig. 3 additionally shows the results of our Q-LinUCB after considering quantum noise. As we have discussed in the main paper (Sec. 6), the results show that the quantum noise leads to slightly worse performances for our Q-GP-UCB, yet they are still able to significantly outperform classical GP-UCB.
Results Using A Real Quantum Computer. Here we additionally perform an experiment using a real quantum computer. Specifically, we ran our QMC subroutine on a real IBM quantum computer based on superconducting qubit technology (7 qubit, Falcon r5.11H processor) through the IBM Quantum cloud service. We ran our synthetic experiment with Gaussian noise on the system for only 1 trial (instead of 10 as in other simulations) due to time and resource constraints. The results in Fig. 4 show that although the performance of our Q-GP-UCB on real quantum computers is worse than those obtained without quantum noise and with simulated quantum noise, our Q-GP-UCB run on real quantum computers is still able to significantly outperform classical GP-UCB. This experiment provides a further demonstration for the potential of our Q-GP-UCB algorithm to lead to quantum speedup real BO applications.
Figure 4: Results using a real IBM quantum computer. Cumulative regret for the synthetic experiment with Gaussian noise with  $\sigma = 0.4$ .