SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
p _ {i} = \frac {k _ {i}}{\sum_ {j} k _ {j}}
$$
where  $k_{j}$  is the degree of node  $j$
Watts-Strogatz model (WS). The WS model was introduced to explain the "small-world" phenomena in a variety of networks. It achieves this by interpolating between a randomized structure close to ER graphs and a regular ring lattice. Given a mean degree  $K$  and a parameter  $\beta \in [0,1]$ . An undirected graph is constructed with  $N$  nodes and  $NK/2$  edges as follows:
1. Constructs a regular one-dimensional network with only local connections of range  $K$ , meaning each node is connected to its  $K/2$  nearest neighbours on each side.
2. For every node  $i = 0, \dots, N - 1$  take every edge connecting  $i$  to its  $K/2$  rightmost neighbours. Rewire each of these edges with probability  $\beta$  to random nodes while avoiding self-loop and link duplicates.
B.2 Baseline Algorithms
Breadth first search (BFS) and depth-first search (DFS). These algorithms aim to explore the whole graph data structure. It starts with a root node and explores according to the depth or breadth of the graph. In the former, the algorithm explores as far as possible along each branch before backtracking. In the latter, the algorithm explores all nodes at the present depth prior to moving on to the nodes at the next depth level.
Random search. In this algorithm, at each time step, a random node is selected for evaluation of our objective function.
Local search. In this algorithm, at each time step, we sample and query a random node from a neighbour of the node of the maximum value encountered so far, and we move to a neighbour node if the queried value is better than the incumbent best. When the algorithm reaches a local optimum (i.e., all neighbours have worse values than the current optimum), we allow our algorithm to restart at a random, unvisited node in the graph.
Figure 12: Betweeness/Eigenvector Centrality on BA/WS graphs.
B.3 Synthetic Optimisation Tasks
B.3.1 Maximising network centrality
Centrality measures were introduced in network analysis to study the importance of certain vertices with respect to desired characteristics. In this paper, we used two centrality measures: betweenness centrality and eigenvalue centrality. We show examples of these functions on sample BA/WS graphs in Fig. 12.
Betweenness centrality. This centrality focuses not just on overall connectedness but on the occupying positions that are pivotal to the network's connectivity. The following formula gives the betweenness of node  $v$ :
$$
g(v) = \sum_{\substack{s,t\in \mathcal{V}\setminus \{v\} \\ s\neq t}}\frac{\sigma_{st}(v)}{\sigma_{st}},
$$
where  $\sigma_{st}$  is the total number of shortest paths from node  $s$  to node  $t$  and  $\sigma_{st}(v)$  is the number of those paths that pass through  $v$ .  $\mathcal{V} \backslash \{v\}$  denotes the set of neighbouring nodes of  $v$  except the node  $v$  itself.
Eigenvector centrality. This centrality measure accounts for the influence of a particular node within the network. The centrality score for the whole set of vertices, represented as a vector  $\mathbf{x}$ , is a solution to the equation:
$$
\mathbf {A} \mathbf {x} = \lambda \mathbf {x},
$$
where the matrix  $\mathbf{A}$  represents the adjacency matrix and  $\lambda$  represents the largest eigenvalue of the adjacency matrix.
B.3.2 Synthetic test functions
In this subsection, we describe the Rosenbrock and Ackley test functions used for our task, both of which are discretised versions of their original, continuous function forms. The mathematical definitions of the test functions are listed below and are visualized in Fig. 13.
Rosenbrock function.
$$
f (x, y) = 1 0 0 (y - x ^ {2}) ^ {2} + (x - 1) ^ {2}
$$
Ackley function.
$$
f (x, y) = - 2 0 \exp \left(- 0. 2 \sqrt {0 . 5 (x ^ {2} + y ^ {2})}\right) - \exp \left(- 0. 5 (\cos 2 \pi x + \cos 2 \pi y)\right) + 2 0 + \exp (1)
$$
Additive noise. In order to alter the smoothness property of our graph signal defined over the grid, we add random noise governed by noise standard deviation  $\sigma_{n}$  to be added to the loss function  $\hat{f} (x,y) = f(x,y) + \epsilon$  with  $\epsilon \sim \mathcal{N}(0,\sigma_n^2)$ . In our experiment, we vary the noise standard deviation  $\sigma_{n}\in \{0,0.1,1,5\}$  for the Ackley function and  $\sigma_{n}\in \{0,0.1,0.5,1\}$  for the Rosenbrock function.
Figure 13: Test function values taken on a regular graph corresponding to the input space: Rosenbrock (left); Ackley (right).
Figure 14: Simulation of the SIR process on BA/WS graphs. It is worth noting that some local optima are visible due to the  $\epsilon$  parameter.
B.4 Real-World Optimisation Tasks
B.4.1 Finding patient zero in a contact network
In this task, we simulate the diffusion processes over a graph via epidemics SIR models Kermack & McKendrick [22]. We slightly modify this model to allow for a parameter  $\epsilon$  representing the probability of spontaneous infection from unknown factors.
More formally, given a graph  $\mathcal{G} = \{\mathcal{V},\mathcal{E}\}$ , our model has three parameters. Parameter  $\beta$  encodes the probability of infection,  $\beta$  the probability of recovery  $\epsilon$  the probability of spontaneous infection, and  $T$  the time spent since the beginning of the outbreak. Let time  $t = 1,\dots T$  be the current time,  $\mathbf{x}_{v,t}\in \{I,S,R\}$  the node status (Infected, Susceptible, Recovered) and  $S_{I,t},S_{S,t},S_{R,t}$  the set of nodes in each category at time t. We have:
$$
\forall v \in \mathcal {S} _ {I, t}, \left\{ \begin{array}{l} \mathbb {P} [ \mathbf {x} _ {v, t + 1} = R ] = \gamma \\ \mathbb {P} [ \mathbf {x} _ {v, t + 1} = I ] = 1 - \gamma . \end{array} \right. \tag {3}
$$
$$
\forall v \in \mathcal {S} _ {S, t}, \left\{ \begin{array}{l} \mathbb {P} [ \mathbf {x} _ {v, t + 1} = I ] = 1 - (1 - \epsilon) \times (1 - \beta) ^ {| N (v) \cap \mathcal {S} _ {I, t} |} \\ \mathbb {P} [ \mathbf {x} _ {v, t + 1} = S ] = (1 - \epsilon) \times (1 - \beta) ^ {| N (v) \cap \mathcal {S} _ {I, t} |}. \end{array} \right. \tag {4}
$$
$$
\forall v \in \mathcal {S} _ {R, t}, \mathbb {P} \left[ \mathbf {x} _ {v, t + 1} = R \right] = 1. \tag {5}
$$
Given such a process, we construct an objective function indicating how close a certain node is to the source of the infection as follows. At time  $T$ , where we consider the diffusion to have ended (or corresponding to the present moment when looking for patient zero), for every node in  $S_{R,T} \cup S_{I,T}$  we denote by  $\tau_v$  the first time of infection. The objective function is then defined as:
$$
\forall v \in \mathcal {V}, f (v) = \left\{ \begin{array}{l l} 0 & \text {i f} v \in \mathcal {S} _ {S, T} \\ \left(1 - \frac {\tau_ {v}}{T}\right) ^ {2} & \text {i f} v \in \mathcal {S} _ {I, T} \cup \mathcal {S} _ {R, T} \end{array} \right. \tag {6}
$$
This function takes value in  $[0,1]$  and is maximised when the node corresponds to the patient zero. We expect local methods to perform well in this setting as local behaviour can trace the source of the infection through diffusion, and the variation of the functions on the graph is relatively smooth. The introduction of parameter  $\epsilon$  nevertheless adds some sources of "local" minima in the graph objective function - we show some examples of such phenomenon in Fig. 14, where we give exemplary graph signals induced by the generative process we described in this section.
B.4.2 Finding influential users in a social network
In this task, we consider a common problem in identifying the most influential person within a social network. Influence, in this context, is often quantified approximately using degree centrality, which may account for, for example, the number of followers or connections an individual possesses. However, the enormity of social network graphs often restricts our access to complete graph information, necessitating alternative search approaches. To validate our methodology, we conduct tests on various real-world graphs derived from diverse social networks. These include the Enron email network, which represents email communication between members of a corporation; the Facebook page network, which is a network of interconnected Facebook pages; and the Twitch social network, which provides insight into the relationships among users on the Twitch platform.
B.4.3 Team optimisation
In this task, we aim to tackle the problem of optimising the performance of a team of individuals with different skills. We will assume that a team would perform most effectively when 1) all skills are covered by combining individual skills and 2) some individuals master every skill. More formally, we will consider a pool of  $N$  individuals. Each individual can be represented by a vector of skills  $\mathbf{x}_i \in [0,1]^K$  where  $K$  is the number of skills. This setup fits our framework well in the scenario where the skills of individuals are unknown, and the pool of potential candidates is also unknown in advance. A sample graph generated from this problem is shown in Fig. 15.
Skill generative process. In each experiment, we will assume individual skills to be generated according to a Dirichlet distribution with parameter  $\alpha$ :
Figure 15: An exemplary graph induced from the team optimisation problem.
$$
\mathbf {x} _ {i} \sim \operatorname {D i r} (\boldsymbol {\alpha}), \boldsymbol {\alpha} = \left[ \alpha_ {1},..., \alpha_ {D} \right] ^ {\top}
$$
The parameter  $\alpha$  encodes the sparsity of skill expertise in the general population. Small  $\alpha$  generates individuals with specialised skills with more probability than large  $\alpha$  where all skill levels concentrate to a score of 0.5.
Graph construction. To solve this task and allow flexible exploration of teams with a varying number of individuals, we construct a graph where nodes represent teams and edges are based on the Jaccard index between each pair of team member sets. More specifically, given two teams  $s_1 \subset \mathbb{N}$  and  $s_2 \subset \mathbb{N}$  the similarity between them is computed as  $w(s_1, s_2) = \frac{s_1 \cap s_2}{s_1 \cup s_2}$ . Given  $N$  teams, we can then construct an undirected graph with edges:
$$
\forall s _ {1}, s _ {2} \subset [ N ], (s _ {1}, s _ {2}) \in \mathcal {E} \iff w (s _ {1}, s _ {2}) > \text {M e d i a n} (\{w (s _ {1}, s _ {2}): s _ {1}, s _ {2} \subset [ N ] \})
$$
Objective function. To model the two desirable properties in terms of team composition, we choose the following objective function:
$$
\forall s \subset [ N ]: f (s) = H _ {k} [ \mathbb {E} _ {n} [ \mathbf {x} ] ] - \mathbb {E} _ {n} [ H _ {k} [ \mathbf {x} ] ]
$$
Intuitively, the first term of the objective corresponds to the entropy of the skill distribution of the whole team, which is maximised when the skill distribution is close to the uniform distribution. The second term corresponds to the expected entropy of the distribution of skills of each individual, which is minimised (and the objective maximised) when each individual specialises in one skill. As a result, we can expect this objective to be well suited for modelling an ideal composition of a team.
C.1 Kernel Validation
Complementary to Fig. 4 in the main text, we conduct further regression analyses to confirm the expressive power of the investigated kernels. The results are shown in Fig. 16 and Fig. 17.
Figure 16: Expressiveness of kernels on a grid graph of size  $n = 200$  nodes. Refer to Fig. 4 for more explanations.
Figure 17: Expressiveness of kernels on a WS graph of size  $n = 200$  nodes. Refer to Fig. 4 for more explanations.
C.2 Centrality Maximisation on Large Graphs
In this section, we consider a similar problem of centrality maximisation as described in App. B.3.1, but on significantly larger graphs: we use BA and WS random graph generators similar to the experiments in Fig. 5 and Fig. 6 in the main text, but we generate graphs with  $10^{6}$  nodes instead and increase the query budget. We show the results in Fig. 18, and we find that the superiority of BayesOptG methods persists in this setting over the baseline methods.
C.3 Finding Patient Zero Task in Real-World Graphs
Setup. In this section, we consider several SIR diffusion problems as described in App. B.4.1 where we aim to find the patient zero on a real-world interaction network from the Copenhagen Networks Study [35]. This network represents physical proximity among participants (estimated via Bluetooth signal strength) in a population of more than 700 university students and thus is a good testbed to examine the diffusion process of a hypothetical epidemic outbreak. The visualization of the function on this graph is given by Fig. 19.
(a) Ackley;  $\sigma = 0$  1000 2000 3000 (e) BA
(f) BA
g) WS
(h) WS
Figure 18: Maximising centrality scores with the BA/WS random graph model and  $n = 10^6$  nodes.
Figure 19: Diffusion objective function on the real-world interaction network.
Results. The performance of each algorithm is presented in Fig. 20 - 23 where we use different values of initially infected population fraction and probability of recovery - it is clear that due to the increased complexity as revealed in Fig. 19, there is some performance degradation in all algorithms considered. However, we can see that in most cases, our method performs at least as well as the local search baseline.
C.4 Team Optimisation
We show additional results for the team optimisation tasks in Fig. 24 to 26. We can observe that the key findings from the main text on this problem (Fig. 10) largely hold true for these tasks induced by different parameters.
D Ablation and Sensitivity Studies
In this section, we perform a thorough ablation and sensitivity study on how much the additionally introduced hyperparameters affect the algorithm's performance. We report sensitivity analyses to the most important hyperparameters below, namely  $Q_{0}$  (initial trust region size), fail_tol,  $\eta$
Figure 20: Identifying the patient zero task with different SIR model hyperparameters  $\beta \in \{0.1, 0.2, 0.3\}$  and  $\gamma \in \{0.005, 0.015, 0.15\}$ . A fraction of  $\mathbf{0.0003}$  of the initial population was infected initially. The probability of recovery  $\epsilon$  is set to  $\mathbf{0}$ .
(order of the kernels) and  $\gamma$  (the trust region multiplier in case of successive successes or failures). We also additionally study the effect of introducing the trust region in this section. We perform ablation experiments in the setting with BA graphs and synthetic function optimization. We show the sensitivity analysis in Fig. 27 to 29 – it is evident that our algorithm is largely robust to the choice of hyperparameters as long as a value within a sensible range is chosen.
Use of trust regions. We compared, on some relatively small graphs (1000 nodes) in Fig. 31 – as observed, while there is a small drop in performance because of the use of local modelling compared to constructing a surrogate model on the whole graph, it is worth noting that, as shown in Fig. 3 in the main text, the full Bayesian optimisation procedure with kernels defined on the whole graph becomes too prohibitive, even for a relatively small graph of size 1000, and that when the graph is unknown, it is impossible in the first place to construct a whole-graph GP. This verifies that the trust region strikes a promising balance between efficiency and performance.
Figure 21: Identifying the patient zero task with different SIR model hyperparameters  $\beta \in \{0.1, 0.2, 0.3\}$  and  $\gamma \in \{0.005, 0.015, 0.15\}$ . A fraction of 0.0003 of the initial population was infected initially. The probability of recovery  $\epsilon$  is set to 0.005.
Figure 22: Identifying the patient zero task with different SIR model hyperparameters  $\beta \in \{0.1, 0.2, 0.3\}$  and  $\gamma \in \{0.005, 0.015, 0.15\}$ . A fraction of  $\mathbf{0.003}$  of the initial population was infected initially. The probability of recovery  $\epsilon$  is set to  $\mathbf{0}$ .
Figure 23: Identifying the patient zero task with different SIR model hyperparameters  $\beta \in \{0.1, 0.2, 0.3\}$  and  $\gamma \in \{0.005, 0.015, 0.15\}$ . A fraction of 0.003 of the initial population was infected initially. The probability of recovery  $\epsilon$  is set to 0.005.
Figure 24: Team optimisation task with  $s \in \{2, 4\}$  and  $\alpha \in \{1, 10\}$  with Jaccard index threshold of 0.1 (refer to App. B.4.3 for explanations)
Figure 25: Team optimisation task with  $s \in \{2,4\}$  and  $\alpha \in \{1,10\}$  with Jaccard index threshold of 0.2 (refer to App. B.4.3 for explanations)
Figure 26: Team optimisation task with  $s \in \{2,4\}$  and  $\alpha \in \{1,10\}$  with Jaccard index threshold of 0.3 (refer to App. B.4.3 for explanations)
Figure 27: Sensitivity of performance to  $Q_{0}$  on different tasks and kernels. From left to right: Centrality maximisation on Enron, Facebook and Twitch networks. Kernels from top to bottom: polynomial, sum-of-inverse polynomials, diffusion (with ARD), diffusion (without ARD), and graph Matérn.
Figure 28: Sensitivity of performance to fail_tol on different tasks and kernels. Refer to Fig. 27 for additional explanations.
Figure 29: Sensitivity of performance to  $\gamma$  on different tasks and kernels. Refer to Fig. 27 for additional explanations.
Figure 30: Sensitivity of performance to  $\eta$  on different tasks and kernels. Refer to Fig. 27 for additional explanations. Note that only Polynomial and Sum-of-inverse-polynomial kernels requiring non-trivial  $\eta$  selection are included.
Figure 31: Influence of trust region method performance compared to full graph knowledge in a WS graph of 400 (Left) and 1000 (right) nodes. It is worth noting that while the use of trust regions leads to some trade-off in terms of optimisation performance, without trust regions, the algorithms are significantly more computationally prohibitive. In the graph of 1000 nodes (right), the computational cost is more than  $20 \times$  the cost of a local algorithm (shown by Fig. 3 in the main text which focuses on wall-clock time comparison).