SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $\{e_{ij}\}_{i = 1}^{k - 1}$  is the previously designed experiments, and is fixed when learning  $e_k$ .
What's wrong with the greedy method. The theoretical foundations of the greedy method is demonstrated by the previous work [19] as follows:
Theorem 2. If  $I(\pmb{x}; \phi_M | e, D)$  is (1) submodular and (2) non-decreasing, then
$$
I \left(\left\{\boldsymbol {x} _ {i} ^ {g} \right\} _ {i = 1} ^ {n}; \phi_ {M} \mid \left\{\boldsymbol {e} _ {i} ^ {g} \right\} _ {i = 1} ^ {n}, D\right) \geq \left(1 - \frac {1}{e}\right) I \left(\left\{\boldsymbol {x} _ {i} ^ {*} \right\} _ {i = 1} ^ {n}; \phi_ {M} \mid \left\{\boldsymbol {e} _ {i} ^ {*} \right\} _ {i = 1} ^ {n}, D\right), \tag {9}
$$
where  $\{e_i^g\}_{i = 1}^n$  is the solution obtained from the greedy method, and  $\{e_i^*\}_{i = 1}^n$  is the optimal solution.
However, due to the introduction of multi-fidelity, in our model,  $I(\pmb{x}; \phi_M | e, D)$  is actually not submodular. More specifically, in the proof of submodular, two samples  $\pmb{x}_s$  and  $\pmb{x}_t$  has to be independent given  $\phi_M$  (e.g., see B.4 in [8]). In the single-fidelity setting, this requirement naturally holds, since  $\phi_M$  is exactly the parameter used to sample  $\pmb{x}_s$  and  $\pmb{x}_t$ . However, when we introduce multi-fidelity,  $\pmb{x}_s$  and  $\pmb{x}_t$  may not be independent given  $\phi_M$ , since they are only directly influenced by their own oracles (see Figure 1).
An improved greedy method tailored to MFACD. To alleviate the above problem, in this section we design an improved greedy method tailored to our task. In specific, we first define several new concepts, and then build theories based on these concepts to inspire our model designs.
Definition 2 ( $\epsilon$ -independent). For random variables  $A, B$  and  $C$ , if their mutual information satisfy  $I(A;B|C) \leq \epsilon$ , then we say  $A$  and  $B$  are  $\epsilon$ -independent given  $C$ .
Definition 3 ( $\epsilon$ -submodular). Suppose  $f(\cdot)$  is a set function on  $\Omega$ . If for any  $A, B \subseteq \Omega$ ,  $A \subseteq B$  and  $x \in \Omega \setminus B$ ,  $f(A \cup \{x\}) - f(A) \geq f(B \cup \{x\}) - f(B) - \epsilon$ , then we say  $f$  is  $\epsilon$ -submodular on  $\Omega$ .
Based on the above two definitions, we have:
Theorem 3. For any two experiments  $\mathbf{e}_s$  and  $\mathbf{e}_t$ , if the corresponding samples  $\mathbf{x}_s$  and  $\mathbf{x}_t$  are  $\epsilon$ -independent given  $\phi_M, \{\mathbf{e}_s, \mathbf{e}_t\}$  and  $D$ , then  $I(\cdot; \phi_M|, D)$  is  $\epsilon$ -submodular.
Theorem 4. If  $I(\cdot ;\phi_M|\cdot ,D)$  is  $\epsilon$ -submodular on  $X$  and non-decrease, for any  $i,j,\frac{\lambda_i}{\lambda_j}\leq B_\lambda$ , then
$$
I (\{\pmb {x} _ {i} ^ {g} \} _ {i = 1} ^ {n}; \phi_ {M} | \{\pmb {e} _ {i} ^ {g} \} _ {i = 1} ^ {n}, D) \geq (1 - e ^ {- \frac {1}{B _ {\lambda}}}) I (\{\pmb {x} _ {i} ^ {*} \} _ {i = 1} ^ {n}; \phi_ {M} | \{\pmb {e} _ {i} ^ {*} \} _ {i = 1} ^ {n}, D) - B,
$$
where  $B = \frac{\epsilon}{B_{\lambda}}\cdot \sum_{i = 1}^{n}(1 - \frac{1}{B_{\lambda}n})^{i - 1}$
Following this theory, we improve objective (6) to a constraint-based ELBO as follows to capture the degree of independence between different experiment results:
$$
\begin{array}{l} \max  \mathbb {E} _ {\Phi \sim q (\Phi)} \left[ \log p (D | \Phi) - \log q (\Phi) + \log p (\Phi) \right], \\ \text {s . t .} \sum_ {\left\{\boldsymbol {e} _ {s}, \boldsymbol {e} _ {t} \right\}} I \left(\boldsymbol {x} _ {s}; \boldsymbol {x} _ {t} \mid \phi_ {M}, \left\{\boldsymbol {e} _ {s}, \boldsymbol {e} _ {t} \right\}, D\right) \leq \epsilon . \tag {10} \\ \end{array}
$$
The proofs of the above theories are presented in the appendix, and similar to objective (2), we use Monte Carlo method to approximate the mutual information in objective (10).
Bayesian Active Causal Discovery. Causal discovery [20-22] refers to recovering causality in a set of variables, especially trying to find a directed acyclic graph (DAG) that can represent the relationship between variables in a system. Active causal discovery was first proposed in [23, 24] with the assumption that the data is discrete-valued. In active causal discovery, the experimenter attempts to intervene on the variables in the system at each step, utilizes the interventional data to recover causal relation, and finally identifies the entire causal structure with minimal cost [25]. Many methods have been discussed in different settings over the past decades, including continuous linear Bayesian networks [26, 27], non-linear causal models [28], and large-scale causal models [8].
Multi-fidelity Settings. The fidelity commonly refers to how accurate the model or environment can be when providing information. Higher-fidelity models are more accuracy but cost much, while lower ones are less accurate but cheaper. In order to combine the strength of each model, multi-fidelity models are proposed to achieve an accurate model with lower costs. Multi-fidelity models can be divided into two main categories [29]: Multi-fidelity Surrogate Models (MFSM) [30-32] and Multi-fidelity Hierarchical Models (MFHM) [33-35]. In addition to optimization, multi-fidelity models can also be used for uncertainty propagation [36] and statistical inference [37]. Recent works also start to study the multi-fidelity settings when conducting Bayesian experiments [38-40], and adopt deep learning frameworks to solve corresponding problems.
Our paper makes a first step towards multi-fidelity active causal discovery, and solve the non-trivial challenges when combining the above two fields, which, to the best of our knowledge, is the first time in the causal inference domain.
In this section, we conduct experiments to demonstrate the effectiveness of our model, where we focus on the following problems: (1) whether our model can achieve better performance than the previous ACD methods? (2) Whether the constraint in objective (10) is necessary? (3) How the DAG regularization coefficient influences the model performance? In the following, we first introduce the experiment setup, and then present and analyze the results.
Figure 2: Results of the overall performance on different datasets and budgets. Lower SHD, RMSE or larger AUPRC indicate better performances. We conduct each experiment for ten times, and report the average performances and error bars.
We experiment with three commonly used causal discovery datasets, including Erdős-Rényi graph (ER) [41], Scale-Free graph (SF) [42] and DREAM [43]. To demonstrate the effectiveness of our model, we compare it with AIT [44] and CBED [8], which are the recent state-of-the-art models in this field. Since they cannot select different fidelities, we design two variants for each of the baseline, that is, X-REAL and X-RANDOM, which means that the model always interacts with the ground truth oracle  $\phi_M$  or randomly select the oracles. Here "X" is AIT or CBED. For the evaluation metrics, we use SHD [45], AUPRC [46] and RMSE to evaluate different models. The first two metrics aim to evaluate the accuracy of the learned topological structure, and the last one measures the performance of functional relations. For single intervention, we first generate several observational samples to initialize the model. Then, we indicate a total intervention budget, and let the model interact with the causal graph with different fidelities until the budget runs out. For each interaction, the model will provide an intervention, and correspondingly obtain a sample from the oracles, which is used to update the model. Finally, the model outputs the estimated causal graph, which is leveraged to evaluate the performance. For batch intervention, we indicate the total budget for each intervention step, and the model determines  $n$  interventions simultaneously, which are delivered to the oracles to obtain the samples. We present more detailed settings in the appendix.
5.2 Overall Performance
In this experiment, we evaluate the models under different total budgets, and we present the results on ER and SF datasets with 10 graph nodes. The experiments on DREAM and more nodes are presented in the appendix. From the results shown in Figure 2, we can see: as the total budget becomes larger, the performances of all the models tend to increase on both datasets. This is not surprising, since more experiment budgets enable us to conduct more interventions or query more accurate oracles, which can reveal more information about the ground truth and facilitate more accurate causal discovery. In most cases, our model can perform better than the baselines across different datasets, evaluation metrics and intervention budgets. These results demonstrate the effectiveness of our model. In specific, on the metrics of SHD, AUPRC and RMSE, our model can on average improve performance of the best baseline by about  $27.74\%$ ,  $82.35\%$  and  $22.74\%$  on ER, and  $17.69\%$ ,  $60.27\%$  and  $21.62\%$
(a) Batch Intervention with 30 Budget
Model	SHD ↓	AUPRC (%) ↑	RMSE (%) ↓
AIT-REAL	20.10 ± 2.26	17.74 ± 0.32	3.11 ± 0.00
AIT-RANDOM	19.92 ± 2.81	17.57 ± 0.43	3.53 ± 0.01
CBED-REAL	20.12 ± 2.17	20.25 ± 0.19	3.51 ± 0.00
CBED-RANDOM	19.34 ± 1.72	24.32 ± 0.47	3.74 ± 0.01
Licence (w/o reg)	17.10 ± 3.71	32.55 ± 1.81	3.12 ± 0.02
Licence	16.75 ± 5.68	37.94 ± 1.90	2.66 ± 0.01
(b) Influence of  $\beta$
Figure 3: (a) The results of experiments on ER graph with 10 graph nodes under the batch intervention scenario. The average performance and error bars are provided. (b) The results of Licence model with different DAG regulation coefficient  $\beta$ 's. The experiment is conducted based on ER graph with 10 graph nodes.
on SF, respectively. If we look more carefully, we find that, for both AIT and CBED, randomly querying the oracles can sometimes perform better than always interacting with the ground truth oracles. This result suggests that lower fidelity oracles can be helpful to trade-off the performance and cost. However, the random method is still suboptimal, and designing more principled and tailored strategies to select the fidelities is necessary, which is evidenced by the lowered performance of "X-RANDOM" than our model.
5.3 Necessity of the Mutual Information Constraint in Objective (10)
In our model, the mutual information constraint in objective (10) aims to make the greedy method valid. In this section, we study whether it is necessary by experiments. To achieve this goal, we introduce a variant of our model Licence (w/o reg), where we remove the mutual information constraint. We evaluate the models based on the dataset of ER with 10 graph nodes, and the total budget is set as 30. The results are presented in Figure 3(a). We can see, in some cases, "X-RANDOM" performs better than "X-REAL", which is consistent with the above experiments, and demonstrates that always querying the ground truth oracle may not lead to better performance under limited budget. By comparing our model with the variant Licence (w/o reg), we find that our model can consistently achieve better performances on all the evaluation metrics. In specific, Licence can improve the performance of Licence (w/o reg) by about  $2.05\%$ ,  $16.57\%$  and  $14.92\%$  on SHD, AUPRC and RMSE respectively. These results demonstrate that the mutual information constraint is necessary in our model, which empirically verifies the theories proposed in section 3.3.
5.4 Influence of the DAG regularization coefficient
In this section, we analysis the influence of the DAG regulation coefficient  $\beta$  in equation (4). The results are reported based on AUPRC. The coefficient  $\beta$  reflects the importance of DAG regulation when updating the model. As  $\beta$  increases, the DAGness is more emphasized for the causal graph. In this subsection, we conduct experiments for various  $\beta$ , ranging from 0.0 to 1.0, and the results are shown in Figure 3(b). We can see as  $\beta$  increases, the performance of AUPRC improves as well, and peaks at  $\beta = 0.8$ . That is probably because lower  $\beta$  will decrease the acyclic property of causal graphs, which is incompatible with the prior knowledge of true causal graphs. However, higher coefficient may also impact the optimization process, which leads to sub-optimal results. We think a trade-off between mild constraint for easy optimization and solid constraint for DAG property is supposed to take into consider for different tasks and settings.
This paper formally defines the task of active causal discovery with multi-fidelity oracles, which, to our knowledge, is the first time in the causal discovery domain. To solve this task, we propose a Bayesian framework, which is composed of a mutual information based acquisition function and a cascading fidelity model. We also extend our framework to the batch intervention scenario, and propose a constraint-based fidelity model to validate the efficient greedy method.
This paper actually makes an initial step toward considering different oracles in active causal discovery. There is much room left for improvement. To begin with, one can design more advanced batch intervention strategies, which can bypass the greedy method and does not need to introduce the mutual information constraint in the fidelity model. In addition, since the experiments in active causal discovery are conducted sequentially, and the former experiment results may influence the latter ones, it is interesting to consider the experiment designs as a Markov decision process, and leverage reinforcement learning to optimize the total information gains of all the experiments in a more principled manner.
This work is supported in part by National Key R&D Program of China (2022ZD0120103), National Natural Science Foundation of China (No. 62102420), Beijing Outstanding Young Scientist Program NO. BJWZYJH012019100020098, Intelligent Social Governance Platform, Major Innovation & Planning Interdisciplinary Platform for the "Double-First Class" Initiative, Renmin University of China, Public Computing Cloud, Renmin University of China, fund for building world-class universities (disciplines) of Renmin University of China, Intelligent Social Governance Platform.
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A Monte Carlo Approximation for  $f(j,v,m)$  15
A.1 Derivation Process for  $f(j,v,m)$  15
A.2 Sampling from  $p(\phi_m|D)$  16
A.3 Sampling from  $p(\phi_m|\phi_M,D)$  19
A.4 Calculation of  $p(\pmb {x}|\pmb {e},\pmb {\phi}_m)$  20
B Bayesian Optimization for Determining  $(j^{*},v^{*},m^{*})$  21
C Detailed Training Process of ELBO 22
C.1 Derivation Process of ELBO 22
C.2 Estimation of ELBO 22
C.3 Gaussian Reparameterization Trick 24
C.4 Gumbel-softmax Reparameterization Trick 24
C.5 Optimization of ELBO 25
D Training Process of Constraint based ELBO 25
E Proof of Theory 3 26
F Proof of Theory 4 26
G Algorithm 28
H More Experiments 28
H.1 Experimental Settings 28
H.1.1 Datasets 28
H.1.2Baselines 28
H.1.3 Metrics 29
H.2 Simulation of Oracles with Different Fidelities 30
H.3 Details of Configurations and Computation 30
H.4 Experiments on DREAM Dataset 30
H.5 Experiments on More Nodes 30
H.6 Supplementary Experiments on MAE Metric 31
H.7 Supplementary Experiments on Different Oracle Settings 31
H.8 Supplementary Experiments on Regularization Coefficient  $\lambda$  32
H.9 Supplementary Experiments on Ablation Studies 32
I Potentially Negative Social Impact 32
J Limitations 34
A Monte Carlo Approximation for  $f(j,v,m)$
A.1 Derivation Process for  $f(j,v,m)$
Considering that the mutual information is not directly tractable, we approximate  $f(j,v,m)$  by:
$$
\begin{array}{l} f (j, v, m) = - \frac {1}{\lambda_ {m} \cdot K _ {1} \cdot L _ {1}} \sum_ {k _ {1} = 1} ^ {K _ {1}} \sum_ {l _ {1} = 1} ^ {L _ {1}} \log \left[ \frac {1}{C} \sum_ {c _ {1} = 1} ^ {C _ {1}} p \left(\boldsymbol {x} _ {m} ^ {\left(k _ {1}, l _ {1}\right)} \mid \boldsymbol {\phi} _ {m} ^ {\left(c _ {1}\right)}, \boldsymbol {e}\right) \right] \\ + \frac {1}{\lambda_ {m} \cdot K _ {2} \cdot L _ {2} \cdot C _ {2}} \sum_ {k _ {2} = 1} ^ {K _ {2}} \sum_ {l _ {2} = 1} ^ {L _ {2}} \sum_ {c _ {2} = 1} ^ {C _ {2}} \log \left[ p (\pmb {x} _ {m} ^ {(k _ {2}, l _ {2}, c _ {2})} | \phi_ {M} ^ {(k _ {2})}, \pmb {e}) \right], \\ \end{array}
$$
where  $\pmb{e} = \{(j,v),m\}$  is the experiment to be designed,  $\phi_m^{(c_1)},\phi_m^{(k_1)}\sim p(\phi_m|D),\pmb{x}_m^{(k_1,l_1)}\sim p(\pmb {x}|\phi_m^{(k_1)},\pmb {e}),\phi_M^{(k_2)}\sim p(\phi_M|D),\phi_m^{(k_2,l_2)}\sim p(\phi_m|\phi_M^{(k_2)},D)$  and  $\pmb{x}_{m}^{(k_{2},l_{2},c_{2})}\sim p(\pmb {x}|\phi_{m}^{(k_{2},l_{2})},\pmb {e}).$
We present the detailed approximation process as follows:
$$
\begin{array}{l} f (j, v, m) = \frac {1}{\lambda_ {m}} I (\boldsymbol {x}; \phi_ {M} | \boldsymbol {e}, D) \\ = \frac {1}{\lambda_ {m}} [ H (\boldsymbol {x} | \boldsymbol {e}, D) - H (\boldsymbol {x} | \phi_ {M}, \boldsymbol {e}, D) ] \\ = \frac {1}{\lambda_ {m}} \left[ \right. - \mathbb {E} _ {p (\boldsymbol {x} | \boldsymbol {e}, D)} \left[ \log p (\boldsymbol {x} | \boldsymbol {e}, D) + \mathbb {E} _ {p (\phi_ {M} | D)} \left[ \mathbb {E} _ {p (\boldsymbol {x} | \phi_ {M}, \boldsymbol {e})} \left[ \log p (\boldsymbol {x} | \boldsymbol {e}, \phi_ {M}) ] \right]\right]\right] \\ = \underbrace {\frac {1}{\lambda_ {m}} \left[ - \mathbb {E} _ {p (\boldsymbol {x} | \boldsymbol {e} , D)} \left[ \log \mathbb {E} _ {p \left(\phi_ {m} | \boldsymbol {e} , D\right)} \left[ p (\boldsymbol {x} | \boldsymbol {e} , \phi_ {m}) \right] \right] \right]} _ {E} \\ + \underbrace {\frac {1}{\lambda_ {m}} \left[ \mathbb {E} _ {p (\phi_ {M} | D)} \left[ \mathbb {E} _ {p (\boldsymbol {x} | \boldsymbol {e} , \phi_ {M})} \left[ \log p (\boldsymbol {x} | \boldsymbol {e} , \phi_ {M}) \right] \right] \right]} _ {F} \\ \end{array}
$$
For part  $E$ , we can estimate it by
$$
E = - \frac {1}{\lambda_ {m} \cdot K _ {1} \cdot L _ {1}} \sum_ {k _ {1} = 1} ^ {K _ {1}} \sum_ {l _ {1} = 1} ^ {L _ {1}} \log \left[ \frac {1}{C} \sum_ {c _ {1} = 1} ^ {C _ {1}} p \left(\boldsymbol {x} _ {m} ^ {\left(k _ {1}, l _ {1}\right)} \mid \boldsymbol {\phi} _ {m} ^ {\left(c _ {1}\right)}, \boldsymbol {e}\right) \right],
$$
where for the first expectation on  $p(\phi_m|e,D)$ , we first sample  $\phi_{m}^{(k_{1})}$  from  $\phi_{m}^{(k_{1})}\sim p(\phi_{m}|e,D)$  for  $K_{1}$  times, and then for each  $\phi_{m}^{(k_{1})}$ , we sample  $\pmb{x}_{m}^{(k_{1},l_{1})}$  from  $\pmb{x}_{m}^{(k_{1},l_{1})}\sim p(\pmb {x}|\phi_{m}^{(k_{1})},\pmb {e})$  for  $L_{1}$  times. For the second expectation on  $p(\phi_m|e,D)$ , we sample  $\phi_m^{(c_1)}\sim p(\phi_m|e,D)$  for  $C_1$  times.
For part  $F$ , we have
$$
\begin{array}{l} F = \frac {1}{\lambda_ {m}} \cdot \left[ \mathbb {E} _ {p (\boldsymbol {\phi} _ {M} | D)} \left[ \mathbb {E} _ {p (\boldsymbol {x} | \boldsymbol {\phi} _ {M}, \boldsymbol {e})} \left[ \log p (\boldsymbol {x} | \boldsymbol {\phi} _ {M}, \boldsymbol {e}) \right] \right] \right] \\ = \frac {1}{\lambda_ {m}} \cdot \left[ \mathbb {E} _ {p (\phi_ {M} | D)} \left[ \int p (\boldsymbol {x} | \phi_ {M}, \boldsymbol {e}) \log p (\boldsymbol {x} | \phi_ {M}, \boldsymbol {e}) d \boldsymbol {x} \right] \right] \\ = \frac {1}{\lambda_ {m}} \cdot \left[ \mathbb {E} _ {p (\phi_ {M} | D)} \left[ \int \int p (\boldsymbol {x} | \phi_ {m}, \boldsymbol {e}) p (\phi_ {m} | \phi_ {M}) d \phi_ {m} \log p (\boldsymbol {x} | \phi_ {M}, \boldsymbol {e}) d \boldsymbol {x} \right] \right] \\ = \frac {1}{\lambda_ {m}} \cdot \left[ \mathbb {E} _ {p (\phi_ {M} | D, e)} \left[ \int \int p (\boldsymbol {x} | \phi_ {m}, e) p (\phi_ {m} | \phi_ {M}) \log p (\boldsymbol {x} | \phi_ {M}, e) d \boldsymbol {x} d \phi_ {m} \right] \right] \\ = \frac {1}{\lambda_ {m}} \cdot \left[ \mathbb {E} _ {p (\boldsymbol {\phi} _ {M} | D, \boldsymbol {e})} \left[ \int \mathbb {E} _ {p (\boldsymbol {x} | \boldsymbol {\phi} _ {m}, \boldsymbol {e})} \left[ p \left(\boldsymbol {\phi} _ {m} | \boldsymbol {\phi} _ {M}\right) \log p (\boldsymbol {x} | \boldsymbol {\phi} _ {M}, \boldsymbol {e}) \right] d \boldsymbol {\phi} _ {m} \right] \right] \\ = \frac {1}{\lambda_ {m}} \cdot \left[ \mathbb {E} _ {p (\boldsymbol {\phi} _ {M} | D, \boldsymbol {e})} \left[ \mathbb {E} _ {p (\boldsymbol {\phi} _ {m} | \boldsymbol {\phi} _ {M})} \left[ \mathbb {E} _ {p (\boldsymbol {x} | \boldsymbol {\phi} _ {m}, \boldsymbol {e})} \left[ \log p (\boldsymbol {x} | \boldsymbol {\phi} _ {M}, \boldsymbol {e}) \right] \right] \right] \right]. \\ \end{array}
$$
It can be estimated by
$$
\frac {1}{\lambda_ {m} \cdot K _ {2} \cdot L _ {2} \cdot C _ {2}} \sum_ {k _ {2} = 1} ^ {K _ {2}} \sum_ {l _ {2} = 1} ^ {L _ {2}} \sum_ {c _ {2} = 1} ^ {C _ {2}} \log \left[ p (\pmb {x} _ {m} ^ {(k _ {2}, l _ {2}, c _ {2})} | \phi_ {M} ^ {(k _ {2})}, \pmb {e}) \right],
$$