Causal Abstraction Learning based on the Semantic Embedding Principle

Thank you to the Reviewer for their effort, valuable comments, and appreciation of our work. We address below all the Reviewer’s concerns, also providing an additional theoretical contribution in [Q2]. We are happy to further discuss any additional concerns.

Claims

See [W1-W2].

Experimental Designs

[Uncertainty] See [W2].

[Brain application] See [W3].

[Fig.4] Agreed, we will report the results for our proposed methods that do not guarantee constructiveness in the Appendix.

Weaknesses

[W1] We do not assume the availability of aligned (i.e., jointly sampled) data from the SCMs. Hence, we cannot pose any regression problem.

Assuming jointly sampled data are available (but again, that’s a different work from ours), denoting by $h$ the number of nodes in the high-level model and considering full-prior knowledge, then one could solve separately $h$ linear regression problems subject to unitary $\ell_2$-norm constraints for the vectors of coefficients for complying with SEP. Thus, the individual problems are not mere regressions, rather nonconvex problems due to the constraints $\beta_i^\top \beta_i=1$, $i \in [h]$ and $\beta_i$ being the $i$-th vector of coefficients.

Thanks, we will add these details as a Remark.

[W2] We believe there is a misunderstanding, probably due to lines 369-370 where we say “by forgetting the mapping for 25%, 50%, and 75% of the variables”. This passage in the text may be ambiguous to the reader and we’ll modify it. We clarify below this point that raised a major concern.

In lines 433-435 we say: “We then express this partial information via uncertainty over B, meaning that some rows of B have more than one entry equal to one”. This is exactly what we do in the partial prior setting of the synthetic experiments in Sec.6.

Indeed, by “forgetting the mapping for 25%, 50%, and 75% of the variables” we mean that 25%, 50%, and 75% of the rows in $B$ have all entries equal to one. Thus, for each case, a specific fraction of nodes in the low-level considers all nodes in the high-level as plausible abstractions and our algorithms have to identify the correct high-level node for each low-level one.

Hence, Fig.4 already shows the results for partial prior knowledge on synthetic data.

[W3] Agreed, we will add the $B$ used in the experiments to App. L.

Comments

[C1] Thank you, we will correct the typo in Fig.5.

[C2] Thanks, we will introduce $V^\star$ before.

Questions

[Q1] In Eq.(3) we look at KL as a function of $V$. The constant term is exactly $-\log{ \det{\Sigma^h}} - h$. We will specify this in line 202 2nd col.

[Q2] Below we translate Remark 1 into a rigorous additional result we will add to the paper. It establishes a spectral characterization for linear CA and Gaussian measures, valid for any information-theoretic metric and $\phi$-divergence.

Theorem. Let $\chi^\ell \sim N(0_\ell, \Sigma^\ell)$, $\chi^h \sim N(0_h, \Sigma^h)$, where $\Sigma^\ell$ and $\Sigma^h$ are positive definite and $\ell>h$. Denote by $0<\lambda\_1\leq…\leq \lambda\_\ell$ the eigenvalues of $\Sigma^\ell$, and by $0<k\_1\leq…\leq k\_h$ those of $\Sigma^h$. If a linear CA $V \in \text{St}(\ell,h)$ complying with SEP from $\chi^\ell$ to $\chi^h$ exists, then

$$\lambda\_i \leq k\_i \leq \lambda\_{i+\ell-h}, \forall i \in [h]. \tag{1}$$

Proof. If a linear CA $\mathbf{V} \in \mathrm{St}(\ell,h)$ exists, then $\chi^h=\varphi\_\text{push}^V(\chi^\ell)$. Thus the kernel of any information-theoretic metric or $\phi$-divergence is nonempty, and $V^\top \Sigma V=\Sigma^h$ implying the eigenvalues of $V^\top \Sigma V$ are those of $\Sigma^h$. By the Ostrowski’s theorem for rectangular $V$ (cf. Th. 3.2 in Higham & Cheng, 1998) we have

$$k_i = \theta_i \mu_i, \quad i \in [h]; \tag{2}$$

where

$$\lambda\_i \leq \mu\_i \leq \lambda\_{i+\ell-h}, \tag{3}$$

and

$$\text{eigvls}(V^\top V)\_1 \leq \theta_i \leq \text{eigvls}(V^\top V)\_h . \tag{4}$$

Since $V \in \text{St}(\ell,h)$, by (4) $\theta_i=1$ for each $i \in [h]$. Substituting the latter into (2), we get $k_i=\mu_i$, thus obtaining (1) by (3).

Ref.: Higham, N. J., & Cheng, S. H. (1998). Modifying the inertia of matrices arising in optimization. Linear Algebra and its Applications, 275, 261-279.

[Q3] The matrix $S$ guarantees a constructive CA. Consider the partial prior knowledge case where $B$ has more than a single 1 per row. By disentangling the support $S$ and the coefficients $V$ we can learn and enforce the support of CA to be constructive through the second and third constraints in Eq. (5).

In the nonsmooth case we propose a simpler unconstrained formulation at the price of losing constructiveness guarantees for CA. In fact, we simply penalize entries in $V$ corresponding to zeros in $B$. Clearly, this does not guarantee the constructiveness of $V$, especially in the case of partial prior knowledge.

[Minor] Thanks, on the LHS should be $\chi^h$.

We thank the Reviewer for their effort, valuable comments, and appreciation of our work. We address below the Reviewer’s concerns in a concise manner due to text limit. We are happy to further discuss any additional concerns.

Weaknesses

[W1] The Reviewer is right, in real-world applications, systems often display nonlinear interactions. However, the weakness they point out does not apply to our work since we do not assume any linearity of the low- and high-level SCMs.

Indeed, from both the theoretical and learning perspectives, (i) the category-theoretic treatment of SCM, (ii) SEP in Def. 4.1, and (iii) the SEP-based CA learning problem in Prob.1 are general and do not make any assumptions on the functional forms of the involved SCMs.

It’s only from the application perspective that we decline Prob.1 to the case of linear CA. However, the linearity of the CA does not imply the linearity of the causal modules of the SCMs (cf. “Counterexample” in reply to Reviewer 232e).

[W2] We agree with the Reviewer on the importance of the comparison with baselines. Unfortunately, we were not able to find any baseline suitable for a fair comparison as we dropped many assumptions of existing methods, viz. (NA1)-(NA5), as the Reviewer acknowledges. Requirements:

• Zennaro et al. (2023): (NA1), (NA2)
• Felekis et al. (2024), $(\tau,\omega)$ CA setting: (NA2),(NA3)
• Massida et al. (2024): (NA4),(NA5)
• Kekic et al. (2024) perform targeted reduction of an SCM in the $(\tau,\omega)$ abstraction setting, a different task than ours. Require (NA3) (NA4)
• Dyer et al. (2024) consider a different problem within the setting of $(\tau,\omega)$-abstraction. Require (NA3)
• Geiger et al. (2021) and (2024) leverage CA formalism to rigorously analyze explainability of neural networks, a different task from ours.
  Given a neural network as a low-level SCM, the assumption is that there exists a high-level interpretable SCM. The goal is to evaluate the alignment between the two. Here, “low-level” means “black-box”; “high-level” refers to “human-understandable” SCMs built from theoretical and empirical modeling work. Differently, in our work “low-” and “high-level” mean “micro”(fine-grained) and “macro”(coarse-grained). It’s not a mere difference of terminology. Consider the SCMs $\mathcal{B}$ and $\mathcal{N}$ in Sec.3 of Geiger et al. (2024): $\mathcal{B}$ is a high-level model for $\mathcal{N}$ although both have the same structure. In our work, since there is no difference in interpretability between SCMs, the previous setting would lead to a contradiction, since it would mean considering as a (macro) abstraction of an SCM a rotation of it.
  Further, Geiger et al. (2021) and (2024) use interchange intervention training (IIT) objectives, developed for neural networks and requiring the possibility to perform interventions over both the black-box and human-interpretable models. Citing from Geiger et al. (2024): “interchange intervention (also known as activation patching), in which a neural network is provided a ‘base’ input, and sets of neurons are forced to take on the values they would have if different ‘source’ inputs were processed”. Due to (NA1)-(NA3), it is not possible to adapt IIT to our setting.

Comments

[S1] We devoted App.B to “Category theory essentials”. If the paper will be accepted, we will add a concise background to the main using the extra page.

[S2] Thanks, there is a typo in the text. The corrected version is: “... the columns of the support $(B \odot S)^\top$ must sum up to one, [...] the rows of the support $(B \odot S)^\top$ must contain at least a one.”

[S3] Thanks, we will fix the link. A non-assumption is an assumption made by existing methods that we do not make. We will specify it better in the paper.

Questions

[Q1] We work purely at the semantic (distributional) level dropping (NA1)-(NA5). Category theory (CT) is applied to isolate the distributional layer of the SCMs. This has an impact on SEP in Def. 4.1 since CT requires the involved mappings to be measurable. Finally, CT is used to generalize the existing CA $\alpha$-framework – which is posed in category-theoretic terms – into our setting.

[Q2] Please refer to [W2]. We can add the reported discussion in our manuscript to motivate the absence of a comparison with baselines. Additionally, among future works we will add the investigation of SEP in the setting of Geiger (2021, 2024). It is an intriguing research question that could lead to jointly aligning and compressing human-understandable models to AI ones in a principled manner.

[Q3] As stated in lines 194-195, “Only if we identify the true constructive abstraction, we are guaranteed interventional consistency.“. We plan to investigate in which cases interventional consistency can be guaranteed without additional assumptions (cf. Sec. 8).

References: Already in the paper. Geiger (2021) and (2024) are [1,2] of the review.

We thank the Reviewer for their effort and valuable comments. We address below the Reviewer’s major concerns on the relationship between CA learning and CRL, and we are happy to further discuss to clarify any additional points.

Claims And Evidence Causal abstraction is intuitively introduced at the beginning of Sec.1, from line 46 1st col to line 49 2nd col; and formally in Def. 2.3. However, we are happy to improve our manuscript as per Reviewer suggestion by adding the proposed text in italic reported below, within our reply to “Weaknesses”.

Theorethical Claims We do not make any claim about the identifiability of the causal abstraction for the reasons reported below in [W3.2].

Experimental Designs Or Analyses Please refer to [W1-W2] below. CA and CRL tackle different learning tasks, although a comparison between them is valuable to further improve the paper.

Relation To Broader Scientific Literature Please refer to [W1-W2] below. We believe that our work is correctly placed within the literature, although considering the comparison with CRL is valuable and planning to add it in the revised version of the manuscript.

Essential References Not Discussed Please refer to [W1-W2] below.

Weaknesses

[W1-W2] We agree with the Reviewer that an additional comparison between CA learning and CRL within Sec.1 would improve the paper, specifically aiding the reader in setting apart CA learning from CRL. We propose to add the following discussion in the revised version:

Causal abstraction (CA) learning aims at learning a mapping between two different SCMs, for instance, the architecture of a neural network and a human-interpretable causal model [2], or an SCM of brain regions of interest (ROIs) and one of brain functional activity (see Sec.7).

Within CA literature, it is usual to distinguish between low- and high-level variables (or SCMs). Although the same adjectives are usually employed also in causal representation learning (CRL, [3]) literature, they convey different meanings in the two research fields.

In CA, both low- and high-level refer to endogenous variables being causal and observed. Specifically, the latter are said to be causal since they relate to each other within the SCM, and are the relevant variables for interventions and counterfactual reasoning. Instead, in CRL, the low-level variables are observed but not causal, that is, mere mathematical functions of high-level causal but unobserved variables, where causal has the same meaning as above. As an example, the low-level variables could be the pixels of an image, whereas the high-level ones are concepts related by an SCM [4, 5]. Additionally, the high-level variables could also not be labeled [3].

Consequently, also the goal of CRL is deeply different from that of CA: Given the low-level variables, CRL algorithms aim at learning (i) the high-level variables and (ii) the causal structure underlying these variables. In brief, while CRL extracts a meaningful causal representation from non-causal data to improve model performance and interpretability [4, 5]; CA learns mappings between already meaningful representations to enable causal knowledge transfer and communication between SCMs working at different levels of abstractions.

[W3.1] Identifiability of CA is not key as identifiability of causal structures in CRL and causal discovery. In CA learning, we mainly care about (structural and distributional) interventional consistency. It is well known that there might exist multiple causal abstractions between low- and high-level SCMs (see Example 5.2 in [1] where symmetry allows for multiple interventionally consistent causal abstractions).

[W3.2] As we drop (NA1), (NA2), and (NA4), our work shows that it is possible to learn causal abstractions without assuming any knowledge of the underlying low- and high-level SCMs.

References

[1] Zennaro, F. M., Bishop, N., Dyer, J., Felekis, Y., Calinescu, A., Wooldridge, M., & Damoulas, T. (2024, September). Causally Abstracted Multi-armed Bandits. In Uncertainty in Artificial Intelligence (pp. 4109-4139). PMLR.

[2] Geiger, A., Lu, H., Icard, T., & Potts, C. (2021). Causal abstractions of neural networks. Advances in Neural Information Processing Systems, 34, 9574-9586.

[3] Schölkopf, B., Locatello, F., Bauer, S., Ke, N. R., Kalchbrenner, N., Goyal, A., & Bengio, Y. (2021). Toward causal representation learning. Proceedings of the IEEE, 109(5), 612-634.

[4] Yang, M., Liu, F., Chen, Z., Shen, X., Hao, J., & Wang, J. (2021). Causalvae: Disentangled representation learning via neural structural causal models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition (pp. 9593-9602).

[5] Komanduri, A., Wu, Y., Huang, W., Chen, F., & Wu, X. (2022, December). Scm-vae: Learning identifiable causal representations via structural knowledge. In 2022 IEEE International Conference on Big Data (Big Data) (pp. 1014-1023). IEEE.

We thank the Reviewer for their effort, valuable comments, and appreciation of our work. We address below the Reviewer’s concerns. We are happy to further discuss to dispel any additional concerns.

Claims And Evidence

[Our claims] Our claims are highlighted in the “Contributions” paragraph of Sec.1. We support them both theoretically and empirically. We are open to reducing any eventual excessive claims as per Reviewer suggestion. At the moment, it is not clear to us which claim is problematic. We kindly ask the Reviewer to specify not supported or justified claims.

[Assumptions] We believe the Reviewer’s concern is centered on the assumptions behind our work. We clarify this point below.

Regarding the SOTA, we do not believe – and do not state in the manuscript – that the assumptions of existing works are unacceptable. We say they are restrictive to tackle CA learning in real-world applications (cf. line 60 1st col, lines 398-399 2nd col). Accordingly, we make only one assumption, viz. (A1), supported by empirical evidence (cf. Sec.7).

Also leveraging (A1), we pose the general SEP-based CA learning problem (cf. Prob1). This problem does not assume any specific (i) functional form for CA, (ii) probability measures for the involved SCMs, and (iii) distance function for quantifying the misalignment (cf. lines 167 1st col - 170 2nd col). Prob.1 should be read as a learning paradigm for CA rather than a single learning problem.

As an application, we decline Prob.1 to the case of (i) linear CA, (ii) Gaussian measures for the endogenous, and (iii) KL divergence, arriving at Prob.s 2 and 3. We remark that (i)-(iii) are not assumptions within our work, rather a particular case of Prob 1.

Concerning the Reviewer’s statement, it’s unclear to us what they mean by “Gaussianity of the errors”. If error stands for exogenous variables, then we remark that we do not make any such assumption in our work. If they mean Gaussianity of the endogenous probability measures, then we provide below a simple counterexample showing that (nonlinear) functional forms other than linear for causal modules are compatible with the (i)-(iii). We will consider lognormal distributed variables as those are relevant in application domains such as quantitative finance: stock prices are considered to be log-normally distributed (Black and Scholes, 1973; Fama, 1965).

Counterexample. Denote by $U_i$ and $L_i$ the exogenous and endogenous variables of the low-level SCM. Denote by $V_j$ and $H_j$ the exogenous and endogenous of the high-level SCM. Let $T:[0,1] \rightarrow [0,1]$ be a measure preserving map (e.g., $T(x)=1- |2x-1|$), $f$ and $g$ be the quantile function and CDF of two Gaussians, $N(0, c_1^2+c_2^2+c_3^2)$ and $N(0, c_4^2)$, respectively.

Causal Abstraction complying with SEP: $H_1=1/\sqrt{2} (L_1 + L_2)$, $H_2=L_3$.

• Low-level SCM:

  • Exogenous: $U_1, U_2, U_3$; each following $\log{N(0,1)}$
  • Endogenous: $L_1=\log{U_1}; L_2=\log{U_2}; L_3= (f \circ T \circ g) (c_1 L_1 + c_2 L_2 + c_3 \log{U_3})$
  • Observational distributions for the endogenous: $L_1 \sim N(0,1), L_2 \sim N(0,1), L_3 \sim N(0,c_4^2)$.

• High-level observational distributions entailed by CA: $H_1 \sim N(0,1)$, $H_2 \sim N(0, c_4^2)$

• High-level SCM:

  • Exogenous: $V_1, V_2$; each following $\log{N(0,1)}$
  • Endogenous: $H_1 = \log{V_1}, H_2 = (f \circ T \circ g)(d_1 H_1 + d_2 \log{V_2})$; with $d_1=\sqrt{ c_1^2 + c_2^2 }$ and $d_2=c_3$.
  • Observational distributions for the endogenous: $H_1 \sim N(0,1)$, $H_2 \sim N(0, c_4^2)$

Finally, the KL divergence is used as an objective function in Prob. 2 and 3, and not limited to any specific probability measure and functional form for the SCM.

[Relations to existing methods] We deliberately selected KL divergence for its relevance in ML applications. Furthermore we consider the application of KL to probability measures of different dimensionality, differently from previous work (Kekic et al., 2023; Dyer et al., 2024). Additionally, Zennaro et al. (2023) use a regularized Jensen-Shannon divergence, Felekis et al. (2024) an $\omega$-informed cost of transport with entropic and do-intervention regularization terms; Massida et al. (2024) perform OLS estimation between the low- and high-level SCMs data. None of the works above is similar to the Riemannian optimization problems, viz. Prob.s 2 and 3, posed in our linear CA application.

Additional Ref.s To Those in The Paper

[1] Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.

[2] Fama, E. F. (1965). The behavior of stock-market prices. The Journal of Business, 38(1), 34-105.

Weaknesses

See “Claims” above.

Comments or Suggestions

Def. 4.1 and Prob. 1 are general and suitable for any CAs. SEP requires the CA having a right-inverse (cf. lines 175-176 1st col), this is the condition to be enforced during the optimization process.