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Causal Discovery via Bayesian Optimization

T0 review · 2 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read DrBO leverages Bayesian optimization over a low-rank, acyclicity-free DAG representation with dropout-network surrogates to recover high-scoring causal DAGs from observational data in fewer trials and less time than state-of-the-art…

desk verdict A solid BO-based DAG learner with genuinely strong empirical results; the unproven low-rank sufficiency and the overbroad 'less time' claim are the main caveats. read the letter →

arxiv 2501.14997 v1 pith:K76TUSKB submitted 2025-01-25 cs.LG stat.ML

Bao Duong , Sunil Gupta , Thin Nguyen This is my paper · ORCID
classification cs.LGstat.ML
keywords causaldiscoveryBayesianoptimizationscore-basedDAGlearninglow-rankrepresentationdropoutneuralnetworkscontinualobservationaldatastructuralHammingdistance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that score-based causal discovery from purely observational data can be made substantially more sample-efficient by treating DAG search as a Bayesian optimization problem. It proposes DrBO, which optimizes the BIC score over a low-dimensional unconstrained representation of DAGs: each node gets a potential and a rank-$k$ embedding vector, and an edge exists only when the potential increases and the embedding dot product is positive. Dropout neural networks learn the node-wise score components from past evaluations and combine them into an acquisition function, so the method spends its evaluations on the most promising DAG candidates. Across dense 30- and 100-node linear-Gaussian graphs, nonlinear additive-noise data, and several real structures, DrBO is claimed to reach near-zero structural error in fewer evaluations and less wall-clock time than gradient-based and reinforcement-learning baselines. If true, this would make accurate causal discovery practical in settings where score evaluations are expensive or trial budgets are tight.

What carries the argument

The central object is the low-rank DAG parametrization $\tau(p,R):=H(\operatorname{grad}(p))\odot H(RR^{\top})$, where $H$ is the entry-wise Heaviside step, $\operatorname{grad}(p)$ records differences $p_j-p_i$, and $R\in\mathbb{R}^{d\times k}$ with $k\ll d$. It converts the combinatorial acyclicity constraint into a potential ordering: edges only go from lower to higher potential, so cycles are impossible, and the embedding-dot-product factor decides which of those potential-consistent edges are present. This map is what makes Bayesian optimization feasible, because the search variable $z=(p,R)$ has only $d(1+k)$ dimensions and lies in a known bounded box. The other load-bearing component is the surrogate: $d$ dropout single-layer networks, each predicting one node's local score component from the binary parent-set indicator, trained continually with a reservoir replay buffer, so that Thompson-sampling-style acquisition is cheap and scales with the number of evaluations.

What would settle it

Take a ground-truth adjacency $A$ from the paper's own 30-node ER8 benchmark and solve the feasibility problem: does there exist $p\in\mathbb{R}^{d}$ and $R\in\mathbb{R}^{d\times 8}$ such that $\tau(p,R)=A$? If even one such graph is infeasible at the default rank $k=8$, then unlimited evaluations cannot recover that exact DAG, and the paper's sufficiency claim for 'very complex graphs' would need qualification.

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Extended reading notes

Core claim

The central claim is that the high-scoring DAG search can be solved effectively by Bayesian optimization once the DAG space is re-parametrized into a continuous, low-rank, acyclicity-free form and the score is modeled node-wise rather than globally. The paper proves that the map $\tau(p,R)=H(\operatorname{grad}(p))\odot H(RR^{\top})$ always outputs a DAG and is scale-invariant, so optimization can be restricted to a fixed hypercube of dimension $d(1+k)$. On top of this, DrBO trains $d$ independent dropout networks, one per node, to predict local score components such as the log-MSE terms of BIC, then forms acquisition values by combining sampled local scores according to the BIC rule; a continual-training replay buffer keeps retraining cost low. The paper's empirical conclusion is that this design finds near-optimal DAGs, often with structural Hamming distance close to zero where comparison methods remain well above 100, using fewer trials and less time.

Load-bearing premise

The load-bearing premise is that a fixed low rank, by default $k=8$, is rich enough to represent the true DAG, or at least some equally high-scoring DAG, in the $\tau$ family; the paper gives empirical evidence that it suffices for its test graphs, but concedes that this is not guaranteed when $k<d$.

Editorial extensions

If this is right

  • Score-based causal discovery can be driven by Bayesian optimization instead of gradient or reinforcement-learning search, which matters when each score evaluation is expensive.
  • On dense 30-node and 100-node linear-Gaussian graphs, DrBO reaches structural Hamming distance near 1–2 where baselines remain above 100 SHD at the same evaluation budget.
  • The same method transfers to nonlinear additive-noise data, binary and discrete data, BGe scores, and standardized data, with SHD often close to zero.
  • Because the surrogate scales with the number of evaluations and with dimensionality, DrBO can run into tens of thousands of trials on 100-node graphs, where GP-based Bayesian optimization would be impractical.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The authors leave implicit that the node-wise surrogate trick only requires the score to decompose additively over nodes; any consistent score with a local decomposition can drop into DrBO unchanged, making it a template for score-family-specific search.
  • Their ablation shows lower ranks give more unique DAGs per random batch and faster convergence, so rank $k$ could be annealed upward during search as a built-in curriculum, a variant the paper does not test.
  • The representability condition is graph-side, not data-side: checking whether a candidate DAG is in the image of $\tau$ at fixed $k$ is independent of sample size, so the feasibility question the paper leaves open is directly testable on its own benchmarks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper proposes DrBO, a Bayesian optimization (BO) framework for score-based causal discovery from observational data. DrBO maps DAGs to a low-rank continuous representation tau(p,R)=H(grad(p)) odot H(R R^T) with default rank k=8, then uses batch Thompson sampling with dropout neural networks as scalable surrogates. The surrogates predict node-wise local score components (e.g., log-MSE), which are combined to form an acquisition function; candidate DAGs are proposed in a trust region around the current best representation, evaluated with the true BIC score, and the final output is the highest-scoring evaluated DAG, optionally pruned. The paper reports extensive experiments on synthetic linear-Gaussian, nonlinear GP, discrete, and real-world networks, showing that DrBO reaches low SHD with far fewer DAG evaluations than gradient-based (NOTEARS+TMPI, DAGMA, COSMO, GOLEM) and RL-based (CORL, ALIAS) baselines, especially on dense 30-node and 100-node graphs. Ablations examine rank, surrogate model choice, indirect local-score modeling, continual training, and hyperparameters. The authors provide code and release the implementation.

Significance. If the empirical findings hold, DrBO is a substantial practical advance: it demonstrates that BO with scalable neural surrogates can be competitive for high-dimensional discrete structure search, achieving near-perfect recovery on dense graphs where baselines remain far from the truth. The paper is notable for the breadth of settings (linear, nonlinear, discrete, standardized, large-scale), the per-component ablations, and the fact that the evaluation protocol is not circular: the final DAG is selected by the true BIC among evaluated candidates, while the dropout surrogates only guide candidate generation. The low-rank parameterization is a clever way to reduce the search dimension, and the paper honestly acknowledges the representational caveat for k<d. The main limitation is that the central claim that the accurate DAG is found rests on an empirical, not proven, sufficiency of the rank-8 family.

major comments (2)
  1. [Section 4.1 (Eq. (4)), Figure 3(a), Section 4.6] The search-space map tau(p,R)=H(grad(p)) odot H(R R^T) with default rank k=8 is the central device that turns Eq. (1) into an unconstrained problem, and the paper's own text concedes that for k<d this family may not contain every DAG. This is load-bearing for the abstract claim that DrBO 'can find the accurate DAG': on dense 30ER8 graphs the reported near-zero SHD (Figure 1(a), final SHD ~1.6) is computed after the pruning step described in Section 4.6, so a representable supergraph plus pruning could produce the same result even if the ground-truth DAG is not in the image of tau(.,.) for k=8. The rank ablation in Figure 3(a) only shows that lower ranks reach low SHD faster; it does not report whether the k=8 curve converges to SHD=0 with more evaluations, nor whether the full-rank representation (which can represent any DAG) eventually reaches SHD=0 on the same 30ER8 instances. Please add a direct test of representational sufficiency: for example, run k=d on 30ER8 with a large budget and report the unpruned best-SHD, report the SHD of the best unpruned DAG for k=8, or check membership of the ground-truth DAGs in the image of tau for k=8. Without such evidence, the dense-graph results remain conditional on an unverified assumption.
  2. [Abstract, Section 5.1, Table 7] The claim that DrBO finds the accurate DAG in 'less time' is not supported by the per-evaluation runtime data in Table 7: at 50,000 evaluations DrBO takes 22.9 minutes on 30ER8, while ALIAS, CORL, and COSMO take 2.4, 5.2, and 2.8 minutes respectively, and DAGMA takes <0.1 minutes. The 'less time' statement is only true in the sense of time to reach a target SHD (e.g., Figure 1(a), fourth column). Please either qualify the abstract and Section 5 to state the time-to-accuracy interpretation, or provide a comparison in which DrBO is faster in wall-clock for the same evaluation budget; as written, the abstract's 'less time' is contradicted by the authors' own table.
minor comments (5)
  1. [Appendix A.1] In the proof of Lemma 1, 'multiplying this adjacency matrix with the second term' should read 'taking the Hadamard product with the second term'; the current wording suggests matrix multiplication.
  2. [Figure 1] The first-column bar panels report TPR, Precision, Recall, F1, and FDR without numeric labels on the bars; providing the values (or a companion table) would make the aggregate comparison reproducible without reading the text.
  3. [Table 7] The DAGMA runtime entries are reported as 0.0 minutes; given that these values are used to support runtime claims, reporting more significant digits would be more informative.
  4. [Section 5.1] The phrase 'absolute overall performance' is vague; it should be stated explicitly that DrBO achieves the best value on all five metrics simultaneously.
  5. [Appendix D.3 / Section 5.2] The BnLearn experiments generate continuous data from the real network structures; stating this in Section 5.2 (rather than only in the appendix) would prevent a misreading that discrete real data are being modeled.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: DrBO selects by true BIC and the low-rank representability caveat is an explicit empirical limitation, not a definitional shortcut.

full rationale

The paper's central claim is empirical and self-contained: Algorithm 1 evaluates the true BIC for each proposed DAG (step 7) and returns the candidate with the highest true BIC (step 10). The dropout surrogate only proposes candidates; it never defines the target, so there is no fitted-input-called-prediction step. The low-rank map tau(p,R) is defined in Eq. (4) independently of the data, and Lemmas 1 and 2 are proved in the appendix rather than imported from prior work. The paper explicitly concedes that k<d may not cover all DAGs (Section 4.1: "while this may not hold for k < d, our empirical evaluations reveal that this representation suffices"), so the dense-graph near-zero SHD results are an empirical claim about this restricted family, not a derivation that assumes the conclusion. The citation to Duong et al. (2024) for full-rank representability is not load-bearing because the default rank is k=8<d and the main results do not depend on that theorem; it is a normal prior-work reference. Hyperparameters are fixed and presented with ablations. No circular step reduces the prediction to its inputs.

Assumptions & free parameters 9 free parameters · 5 assumptions · 0 invented entities

No new physical or statistical entities are introduced. The low-rank embedding is a reparametrization of DAGs, and dropout networks are standard surrogates. The main burden is the set of hand-chosen hyperparameters and the empirical low-rank sufficiency assumption.

free parameters (9)
  • DAG rank k = 8 default; ablations use 2,4,8,12,32 and full rank
    Fixed by hand across experiments; smaller k improves sample efficiency but restricts which DAGs are representable (Section 4.1 and Figure 3a).
  • Batch size B = 64
    Ablation shows a trade-off between SHD and runtime (Figure 9).
  • Number of preliminary candidates C = 100,000
    Ablation finds 10,000 optimal, but 100,000 is used for stability (Figure 10).
  • Learning rate = 0.1
    Adam learning rate, fixed; ablation shows a weak effect below 1 (Figure 13).
  • Hidden units h = 64
    Surrogate capacity; 32 also achieves vanishing SHD in ablation (Figure 14).
  • Dropout rate p = 0.1
    Higher dropout improves performance in ablation (Figure 15).
  • Replay buffer size n_replay = 1024
    Larger values reduce SHD without much runtime increase (Figure 12).
  • Training steps per iteration n_grads = 10
    Middle values are best; small values underfit and large values overfit (Figure 11).
  • Trust region counters n_succ and n_fail = 3 and 5
    Fixed following TuRBO practice (Appendix C.1).
assumptions (5)
  • domain assumption Causal sufficiency, causal minimality, and identifiability of the true DAG
    Section 3.2 assumes no unobserved confounders, no constant mechanisms, and identifiable causal models; otherwise score maximization may not recover ground truth.
  • standard math BIC is a consistent scoring function
    Section 3.3 relies on consistency of BIC so that the global maximizer of the score corresponds to the true DAG.
  • ad hoc to paper Low-rank DAG family suffices for k much smaller than d
    Section 4.1 admits this is not guaranteed for k<d and supports it only empirically; if false, the search domain excludes the optimum.
  • domain assumption Dropout neural network stochastic forward passes provide a useful approximate posterior
    Section 4.3 relies on Gal and Ghahramani's dropout approximation to guide Thompson sampling; the surrogate must rank candidates well enough for BO to help.
  • domain assumption Trust-region candidate generation covers promising regions of z-space
    Section 4.2 and Appendix C.1 assume that candidate sets near the current best point contain high-scoring DAGs; if the trust region collapses prematurely, exploration fails.

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Cite this review

Pith. "Pith review of Causal Discovery via Bayesian Optimization." pith.science (2026). https://pith.science/paper/K76TUSKB

@misc{pith2026250114997,
  author       = {Pith},
  title        = {Pith review of: Causal Discovery via Bayesian Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K76TUSKB}},
  note         = {Machine review of arXiv:2501.14997}
}
read the original abstract

Existing score-based methods for directed acyclic graph (DAG) learning from observational data struggle to recover the causal graph accurately and sample-efficiently. To overcome this, in this study, we propose DrBO (DAG recovery via Bayesian Optimization)-a novel DAG learning framework leveraging Bayesian optimization (BO) to find high-scoring DAGs. We show that, by sophisticatedly choosing the promising DAGs to explore, we can find higher-scoring ones much more efficiently. To address the scalability issues of conventional BO in DAG learning, we replace Gaussian Processes commonly employed in BO with dropout neural networks, trained in a continual manner, which allows for (i) flexibly modeling the DAG scores without overfitting, (ii) incorporation of uncertainty into the estimated scores, and (iii) scaling with the number of evaluations. As a result, DrBO is computationally efficient and can find the accurate DAG in fewer trials and less time than existing state-of-the-art methods. This is demonstrated through an extensive set of empirical evaluations on many challenging settings with both synthetic and real data. Our implementation is available at https://github.com/baosws/DrBO.

Figures

Figures reproduced from arXiv: 2501.14997 by the authors.

Figure 1
Figure 1. DAG learning results on Synthetic data. First column: overall performance in terms of True Positive Rate (TPR, higher is better), Precision, Recall, and F1 score (higher is better), as well as False Discovery Rate (FDR, lower is better). Second column: we track the best Bayesian Information Criterion (BIC, higher is better) so far at every optimization step. Third and Fourth columns: we monitor the Structural Hammin… view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Ablating our design choices. All configurations are evaluated on 5 linear-Gaussian datasets of 1,000 samples on 30ER8 graphs. Shaded areas indicate 95% confidence intervals. against data standardization for both linear and nonlinear data. Large-scale Nonlinear Data: Fig￾ure 8 demonstrates DrBO’s competitive performance and efficiency for nonlinear data on 50- and 100-node graphs. Additional baselines. In Appendix G,… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Causal Discovery Performance with Varying Sample Sizes. We apply our DrBO method on linear-Gaussian data with 20ER4 graphs. Shaded areas represent 95% confidence interval over 5 runs. F.1.1 DIFFERENT GRAPH TYPES We evaluate our method on different graph types, namely E…
Figure 5
Figure 5. Figure 5: BGe for Markov Equivalence Class Discovery. We compare the BGe score of ground truth DAGs and the estimations from DrBO with two popular baselines PC (Spirtes et al., 2000) and GES (Chickering, 2002). Each point corresponds to one of 50 random datasets with linear Gaus…
Figure 6
Figure 6. Figure 6: Causal Discovery performance on Binary Data. We compare our DrBO method using the BIC score for the logistic model with DAGMA (Bello et al., 2022) using the logistic loss. Each point corresponds to one of 50 random datasets with logistic data on ER graphs of 5 nodes an…
Figure 7
Figure 7. Figure 7: Causal Discovery Performance on Standardized Data. Performance metrics are Struc￾tural Hamming Distance (SHD), number of Missing, Extra, and Reverse edges. Lower values are more preferable. Error bars indicate 95% confidence intervals over 5 simulations. F.1.6 WHY LOW-…
Figure 8
Figure 8. Figure 8: DAG learning results on large-scale nonlinear data. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]
Figure 9
Figure 9. Figure 9: Effect of Evaluation Batch Size B. We evaluate our method on linear-Gaussian data with 20ER4 graphs and 1,000 observations. Error bars indicate 95% confidence intervals over 5 runs. The number of evaluations is limited to 20,000. F.2.4 EFFECT OF NUMBER OF PRELIMINARY C…
Figure 10
Figure 10. Figure 10: Effect of Number of Preliminary Candidates C. We evaluate our method on linear￾Gaussian data with 20ER4 graphs and 1,000 observations. Error bars indicate 95% confidence intervals over 5 runs. The number of evaluations is limited to 20,000. F.2.5 EFFECT OF NUMBER OF T…
Figure 11
Figure 11. Figure 11: Effect of Number of Training Steps per BO Iteration ngrads. We evaluate our method on linear-Gaussian data with 20ER4 graphs and 1,000 observations. Error bars indicate 95% confi￾dence intervals over 5 runs. The number of evaluations is limited to 20,000. F.2.6 EFFECT…
Figure 12
Figure 12. Figure 12: Effect of Replay Buffer Size nreplay. We evaluate our method on linear-Gaussian data with 20ER4 graphs and 1,000 observations. Error bars indicate 95% confidence intervals over 5 runs. The number of evaluations is limited to 20,000. F.2.7 EFFECT OF LEARNING RATE [PIT…
Figure 13
Figure 13. Figure 13: depicts that the learning rate has a weak effect on the performance and scalability of our method, where any value below 1 can achieve the same level of SHD and runtime. The SHD only becomes large for a high learning rate of 1. 0 1 2 3 4 0.0001 0.001 0.01 0.1 1.0 Lear…
Figure 14
Figure 14. Figure 14: Effect of Number of Hidden Units h. We evaluate our method on linear-Gaussian data with 20ER4 graphs and 1,000 observations. Error bars indicate 95% confidence intervals over 5 runs. The number of evaluations is limited to 20,000. F.2.9 EFFECT OF NUMBER OF DROPOUT RAT…
Figure 15
Figure 15. Figure 15: suggests that the performance of our method improves with higher dropout rates, while the runtime does not vary significantly. 0 1 2 3 4 0.01 0.05 0.1 0.2 0.3 Dropout rate ( p) Structural Hamming Distance 8.42 8.44 8.46 8.48 8.50 Minutes [PITH_FULL_IMAGE:figures/full…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.