Paper: Bayesian Optimization through Gaussian Cox Process Models for Spatio-temporal Data (ICLR-2024)

Reviewer comment (Writing):
My first question is general. In a standard BO setting, we aim at minimizing a function which is costly to evaluate. It seems that this is not the objective of the presented BO problem. It would have been good to make a clear distinction between the two problems.

Author response:
Thank you for the question. We would like to clarify that this is indeed our objective. However, expensive-to-evaluate functions relating to discrete point observations over time and space, such as arrivals, events, and occurrences, must be modeled as a Gaussian Cox process, which is a Poisson process modulated by a latent Gaussian process. Standard BO relies on a Gaussian prior and cannot model such expensive-to-evaluate functions with respect to point observation and spatio-temporal data. It requires using doubly stochastic process models, i.e., Gaussian Cox process models considered in this paper. 

We clarify that the paper makes two related contributions: (1) a Gaussian Cox process model for expensive-to-evaluate functions relating to discrete point observations over time and space, and (2) a BO framework that builds on such models.

We will also add two illustrative examples to demonstrate the unique advantages of using Gaussian Cox process models for BO. (1) Consider a problem where we need to find the time and location of maximum truck arrivals from video feeds collected by tens of thousands of geo-distributed cameras in a city. This is an expensive-to-evaluate function since it requires performing analytics on a huge volume of video feeds. It is natural to model the truck arrivals (over time and location) as a Gaussian Cox process. We can sample a small set of video clips and use them to estimate the latent intensity, thus supporting efficient BO. (2) Consider the problem of finding the time and location of maximum bird-related incidents in a city. Again, this is an expensive-to-evaluate function, and we have to estimate a latent intensity function from limited samples. We can use the proposed method to model the incidents through a Gaussian Cox process model and then support a BO to solve this problem. For problems requiring estimating a latent intensity function from point observations, we need to use BO with Gaussian Cox process models as proposed in this paper.

Reviewer comment (Novelty):
In the literature review, the authors write that "existing works mostly concentrate on the mean estimation". I am not an expert in Cox models, but it seems that this has been investigated in [1]. This means that this approach could have also been tested in the numerical experiments with Bayesian Optimization. It also seems that this cited paper uses some tools (Laplace approximation, eigenfunctions decomposition) similar to the ones in the paper. Would it be possible to highlight the main modeling differences with this paper? What is the theoretical/computational benefit of the paper's approach compared to the cited paper?

Author response:
Thanks for the question. In the paper, we have compared our results with those reported in [1]. Specifically, this is included in Table 2 with respect to three commonly used synthetic functions and the IQL metric. Regarding the approaches and modeling differences, we would like to clarify that the previous paper [1] uses path integral to obtain an approximation of the solution. In contrast, our approach is to recast the problem into a new RKHS through a change of kernel technique so it becomes easier to handle. It also allows applying the representer theorem to ensure the uniqueness of the estimated mean and covariance. Additionally, our approach enables the use of Nystrom approximation for efficient numerical computation (in Section 3.3), given the change of kernel and the new RKHS. These fundamental differences allow our proposed solution to achieve more favorable results than [1], as demonstrated in Table 2, and make our framework align more with the BO design.

Reviewer comment (Reproducibility):
The authors claim that they can use the Nystrom approximation to compute the next approximation from new samples in an incremental fashion. Is this step used in the experiment? How does it work in practice?

Author response:
In the experiments, we utilized the Nystrom method, and the results are provided in the evaluation section, as shown by the mean and variance estimates in Fig.2. We note that this reduces the computational complexity to $O(n^2m+m^3)$ where $m \ll n$ [3]. To do this, first, we discretize the time span into a uniform grid. The pseudocode for this approximation is given by Algorithm 2 in the appendix. Once we have the approximated kernel matrix, we can compute the posterior mean and covariance for BO.

Reviewer comment (Theory):
After reading the proof of Lemma 2, there is a point I did not understand. Why does the function $h$ belong to $\mathcal{H}$? This fact does not look straightforward to me. Perhaps I missed something in the development.

Author response:
RKHS is a functional space. Given a non-empty domain and a positive definite kernel $k$, we have a unique RKHS $\mathcal{H}_k$ of functions $h$. In $\mathcal{H}_k$, we can leverage the classical representer theorem to optimize $h$, as described in Section 3.2. By constructing a unique RKHS of $h$, we can utilize the property in $\mathcal{H}_k$ and derive a closed-form solution for posterior mean and covariance.

Reviewer comment (Experiments):
In the numerical experiments, the authors do not report the results of PIF in Table 2. Is there a reason for that? I've not found it in the paper.

Author response:
Since the PIF code in [1] has not been made publicly available due to some regulation reasons, we can only compare with the results based on numbers reported in Tabel S3 in [1], which only contains the IQL metric. Nevertheless, our method comfortably outperforms this baseline for the IQL comparison.

Reviewer comment (Presentation):
In figure 2c, it looks like the maximum of the acquisition function is around t=40. This means that we have to sample around $t=40$. However I don't see any sample around $t=40$ in figure 2d.

Author response:
Sorry about the confusion. The black vertical bars in Fig. 2 at the bottom of each subplot are the synthetic ground-truth events of the given ground-truth function. The red vertical bars beneath the black ones illustrate the cumulative observations till the current steps, representing the union of all observations from the initial step onward. Our BO algorithm is oblivious to ground-truth events (black vertical bars), which are shown for comparison with observations (red vertical bars). We state their definitions in Appendix G and will also point them out at the beginning of the experiment section.

The green curve depicting the UCB function anticipates the region of interest for the next step based on our posterior estimates of the current step. Since there are no ground-truth events (black vertical bars) in this area, even though the acquisition function indicates $t=40$ as the next sampling region of step 15, no actual observations will be collected. Moreover, we provided the completed consecutive step-wise figure (Fig. 8 in Appendix L), demonstrating the BO procedure comprehensively.

Reviewer comment (Reproducibility):
Is the code publicly available?

Author response:
The code is made available via the [link](https://anonymous.4open.science/r/gaussian_cox_bo-6926).

Reviewer comment (Experiments):
Although the strengths lie in both the theoretical and numerical aspects, the motivation for performing BO in point processes lacks practical utility. For example, in the spatio-temporal application describing tornadoes in the USA, I fail to see how new events (tornadoes involving damage) can be sampled sequentially to promote active learning of the intensity function $\lambda$. I assume that for illustrative purposes, the authors considered adding the "closest event" available in the database that matches the BO's suggestion. Is this correct? If this is the case, and if the size of the database allows tractable implementations, we can consider all events for inference of $\lambda$. If the model cannot handle the whole database, the BO schema is an interesting idea that promotes a threshold between inference quality and the number of observed point events. However, what can be done if no similar events are recorded in the database? Can the authors give further details on the practical utility of their framework?

Author response:
***Response***:  Thank you for the question. We will certainly clarify our motivation in the final version. BO based on Gaussian Cox process models is useful when the target function (or dataset, as the reviewer mentioned) is expensive to evaluate. 

We will add two illustrative examples where the target function is hard to evaluate (either because it requires performing expensive analytics on a known dataset or due to the fact that obtaining the complete dataset is expensive). (1) Consider a problem where we need to find the time and location of maximum truck arrivals from video feeds collected by tens of thousands of geo-distributed cameras in a city. This is an expensive-to-evaluate function since it requires performing analytics on a huge volume of video feeds. It is natural to model the truck arrivals (over time and location) as a Gaussian Cox process. We can sample a small set of video clips and use them to estimate the latent intensity, thus supporting efficient BO. (2) Consider the problem of finding the time and location of maximum bird-related incidents in a city. Again, this is an expensive-to-evaluate function, and we have to estimate a latent intensity function from limited samples. We can use the proposed method to model the incidents through a Gaussian Cox process model and then support a BO to solve this problem.

We also need to clarify that our proposed BO leverages Gaussian Cox process models to estimate the target function (or dataset as mentioned by the reviewer) from limited samples available. It is a Bayesian method and does not require similar data. The results support the development of acquisition functions that balance exploration (by collecting new samples) and exploitation (to optimize performance objectives) in BO.

Reviewer comment (Presentation):
In Figure 2, at the initial step, the UCB acquisition function suggests adding new events at $t > 90$ (since we seek to maximize such criterion) but they are added somewhere else. Similarly, in step 14, the UCB targets the instants around $t = 40$ but events are again added somewhere else. Besides the authors argue that "the algorithm keeps sampling by maximizing UCB acquisition function and then improving the estimation based on new samples observed", the plots do not validate their point. Can the authors further explain the results while clarifying my concern? Is it possible to add extra plots at consecutive steps (e.g. steps 1 and 2) for a better understanding of the BO's choice?

Author response:
***Response***: Sorry about the confusion. In Fig. 2, the black vertical bars at the bottom of each subplot denote the ground-truth events that are not visible to BO. Only the red vertical bars are regions sampled already and thus visible in BO. For instance, at the initial step, our proposed algorithm tries to build a Gaussian Cox process model based on the observed events shown by red vertical bars ($t=25$ and $t=60$).

The green curve depicting the UCB acquisition function anticipates the region of interest for the next step based on our posterior estimates of the current step. In Fig. 2(c), the maximum UCB acquisition value indicates that the algorithm plans to explore around $t=40$ in the upcoming step 15. However, since no events are observed near $t=40$ when we sample, no new observations (i.e., vertical red bars) are added at $t=40$. The Gaussian Cox process model is adjusted based on this new information, and the UCB acquisition function is updated accordingly to guide future samples/steps. We have added the complete consecutive step-wise figure (Fig. 8 in Appendix L, providing a comprehensive demonstration of the BO procedure.

Reviewer comment (Experiments):
In the experiments, the choice of the hyperparameters $w_1, w_2, w_3$ is not discussed. Can the authors precise their values in each experiment and explain how they were tuned?

Author response:
***Response***: $\omega_1, \omega_2, \omega_3$ are weights for mean and covariance terms to balance their contributions in the acquisition function. When these values are small, BO will place higher emphasis on the mean, favoring exploitation in the process. In contrast, when they are large, BO favors exploration by placing higher emphasis on large variance. In our experiments, we maintain the fixed values for all these parameters, specifically as 0.8.

Furthermore, additional experiments were conducted to assess the impact. In the experiments, we add two extra levels (0.6 and 1.0) for $\omega_3$ to compare with the original $\omega_3=0.8$ in Fig. 6 as an example. The results are shown in Fig. 9 in Appendix M, where if we increase the $\omega_3$ value, the acquisition function will encourage exploring areas with high uncertainty earlier.

Reviewer comment (Experiments):
The authors approximate the integral $\int_{\mathcal{S}} \kappa(g(t)) dt$ using an $m$-partition Riemann sum to obtain a closed-form of the posterior covariance. Since such approximation depends on $m$, can the authors discuss the quality of the approximation in terms of $m$ and precise how they tune that value in the experimental setup? Can they also discuss the scalability of the approximation when $d$ increases?

Author response:
***Response***: Since the exact closed-form solutions are not tractable, we discretize the integral term for obtaining the computable closed-form covariance. In the experiment, we set $m$ as the grid size. Therefore, a higher $m$ leads to a finer granularity but higher computation complexity; otherwise a lower $m$ will increase the computational efficiency but with slightly lower quality. To explore the impact of this value, we conduct an additional experiment and show the results in the following table, where we used the synthetic intensity and ran BO for 10 steps. In the table, the runtime will increase with a higher $m$, while the performance does not deviate much.

| $m$ | 200 | 400 | 600 | 800 |
| :---  | :----: | :---: | :---: | :---: | 
| step 5 $l_2$-norm | 8.35 (3.1) | 10.54 (3.43) | 12.99 (2.78) | 11.91 (2.98) |
| step 10 $l_2$-norm | 4.68 (1.84) | 5.38 (2.21) | 4.42 (1.02) | 4.08 (1.12) |
| runtime | 6.03 (0.12) | 15.98 (0.23) | 34.81 (0.09) | 71.61 (0.20) |

Regarding the scalability of the approximation, our solution computes the kernel matrix of each dimension separately and then applies the Kronecker product to construct the final kernel matrix as $K = K_1 \otimes \dots \otimes K_d$. For example, given $d=2$ where we have two kernel matrices with size $m_1$ and $m_2$, the time complexity will be $O(m_1^2 m_2^2)$.

Reviewer comment (Evaluation):
The limitations of the proposed framework are not discussed in the paper. Can the authors add a remark on this subject?

Author response:
***Response***: Thanks for the question. For future work, we will analyze more complex scenarios with the high-dimensional Gaussian Cox process model, and we will try to add some discussions about this in the final version.

Reviewer comment (Experiments):
Because many other people have not used Gaussian Cox Process models for BO before, I wonder how much modelling the latent intensity actually helps. I did not see any results or discussion on this, but it feels like a useful comparison to make to show that using GCP is actually more performant than standard BO.

Author response:
Thanks for your question. The Gaussian Cox process is the suitable model for addressing problems involving discrete point observations over time and space, such as arrivals, events, and occurrences. This process, characterized by a Poisson process modulated by a latent Gaussian process, proves essential for scenarios where the standard BO relying on a Gaussian prior falls short. The latter cannot effectively model expensive-to-evaluate functions in the context of point observation and spatio-temporal data. Thus, the estimation of latent intensity from discrete point observations becomes necessary, requiring doubly stochastic process models. Gaussian Cox process models are considered as the golden standard for spatio-temporal data [1,2]. This is what motivates our work.

We will provide two illustrative examples demonstrating the unique advantages of using Gaussian Cox process models for BO. (1) Consider a problem where we need to find the time and location of maximum truck arrivals from video feeds collected by tens of thousands of geo-distributed cameras in a city. This is an expensive-to-evaluate function since it requires analyzing a huge volume of video feeds. It is natural to model the truck arrivals (over time and location) as a Gaussian Cox process, as proposed in this paper. We can sample a small set of video clips and use them to estimate the latent intensity, thus supporting efficient BO. (2) Consider the problem of finding the time and location of maximum bird-related incidents in a city. Again, this is an expensive-to-evaluate function, and we need to estimate a latent intensity function from limited samples. We can use the proposed method to model the incidents through a Gaussian Cox process model and then support a BO to solve this problem. For problems requiring estimating a latent intensity function from point observations, we need to use the Gaussian Cox process models proposed in this paper.

Reviewer comment (Theory):
I'm slightly confused about Section 3.4 in that it seems like one can choose any acquisition function that would solve their problem. What about using Gaussian Cox enables us to do this in contrast to standard BO?

Author response:
As mentioned in the previous answer, for problems relating to discrete point observations over time and space, such as arrivals, events, and occurrences, they must be modeled as a (doubly stochastic) Gaussian Cox process. Gaussian process models in Standard BO are insufficient. In this paper, we first develop a Gaussian Cox process model to estimate the latent intensity from point observations and then, in Section 3.4, show that various acquisition functions can be developed in our proposed BO framework, supporting standard acquisition functions like UCB as well as more specialized acquisition functions defined on the latent intensity function, such as maximum idle time detection and change point detection, fully exploiting the temporal properties of the underlying data.

Reviewer comment (Experiments):
Were there any experiments done which could find the posterior mean and covariance in closed form (without Nystrom approximation)? I don't have good intuition for how much expressivity is lost in doing this approximation.

Author response:
This is a great question. We have conducted additional experiments to assess the approximation gap. It uses a synthetic set of point events and an RBF kernel, comparing the mean intensity estimations with and without Nystrom approximation. As shown in the following table, the approximation is quite accurate.

| Trials    | 1 | 2 | 3 | average |
| ----------- | :----------: | :-----------: | :-----------: | :-----------: |
| $l_2$-norm | 0.79| 1.33 | 0.89 | 1.00 (0.23) |

We would also like to note that Nystrom approximation can significantly reduce the computational complexity from $O(n^3)$ to $O(n^2m+m^3)$, where $m$ is the number of grid size and smaller than the number of events $n$ [3]. The theoretical analysis of the approximation gap has been discussed in [4]. Additionally, the Nystrom approximation proves beneficial when obtaining a closed-form solution for a specific kernel is not feasible through explicit Mercer’s expansion. We will cite these results and explain them in the final version.

Reviewer comment (Experiments):
The paper analyzes this method on spatial-temporal data, but couldn't I use this method with any temporal data?

Author response:
Yes. Considering temporal data is a special case of spatial-temporal data, our proposed method can be directly applied. In fact, Fig. 2 on page 8 of the paper shows an example of applying the proposed method to synthetic and real-world temporal data.