Interventionally Consistent Surrogates for Complex Simulation Models

Improving the discussion of results

We will use some of the additional space in the revision to provide further discussion. In particular, we will:

• state and discuss the relative computational costs of our complex simulators and our surrogates (briefly: we see that our surrogates run approximately 3-30 times faster across the experiments we consider); and
• provide a more extensive discussion of the advantages of training surrogates for interventional consistency. For example, with respect to the epidemic case study we present, we can discuss a hypothetical scenario in which a policymaker must decide whether and when to introduce a lockdown in order to minimise the average number of infections over the simulated time horizon. Please see the Global Rebuttal for further details. (Is this the kind of discussion you would like to see? We hope this provides the discussion you are looking for on the relative performance of the interventional vs observational surrogates (as in your second question); please do let us know in the discussion period if there are specific points you would like us to discuss.)

Further experiments

To address your point that "the test case is perhaps a bit simple": the reason we focus on this particular epidemic simulator is that it allowed us to clearly and transparently demonstrate that simulators can be expressed as SCMs, which is important for establishing that the theory of causal abstraction can be used to learn surrogates. To nonetheless demonstrate our framework in a further, more complex setting, we have prepared another case study in which we model an ecology of species in a predator-prey relationship and the possible effects of reintroducing an additional species into the mix. Please see the Global Rebuttal for details. We emphasise that this additional case study is indeed more complex, as requested, in that it consists of agents that move around a spatial environment and reproduce and die, and consists of stocks of natural resources that grow and are consumed over time. We hope this addresses this comment, and that it provides further evidence that our framework can be applied in various settings.

Question about alternative approaches to surrogates and RNN

We believe the approach you suggest could in principle be taken for a limited set of surrogate families. (This is not what we do in the LRNN model: as described in the Appendix, we feed $\tilde{\boldsymbol{\theta}}$ into the surrogate at each time step, such that intervening at time $t$ amounts to feeding in a modified $\tilde{\boldsymbol{\theta}}$ at that time.)

However, this will not work generally. For example, it would not work when the surrogate family is mechanistic, such as in the case of the LODE family we consider in our experiments. Thus it can't be the basis of a general framework for training interventionally consistent surrogates. In contrast, our framework is useful because it handles both cases, and is thus more general.

This approach would also have the undesirable property that an intervention at time $t$ would have a causal influence on variables "located" at times $<t$, which by definition should not be the case. This would mean that time has different meanings in the two models, which amongst other things will reduce interpretability.

Note that interventions in our surrogates are explicitly specified on abstract causal variables via $\omega$. Consider a pandemic sim where music venues can be closed. In our surrogates, $\omega$ specifies how venue closure can be interpreted as a direct intervention on, say, $\alpha$ in the LODE surrogate. If interventions aren't transformed via $\omega$ you lose this interpretability. You also lose out on scalability: $\omega$ reduces the space of possible interventions significantly.

Literature recommendations

Similarly to us [R4.1] & [R4.2] focus on improving causal reasoning methods in complex systems with causal inference. [R4.1] investigates hierarchical data settings with latent confounders, while [R4.2]'s active learning framework performs inference without assuming access to the true SCM. Instead, it assigns a Bayesian prior over the causal model class & jointly infers a posterior over both causal models (discovery) & queries of interest (inference). Both frameworks aim to handle complex causal relations which is crucial for surrogate modelling of complex systems.

Additionally, both frameworks employ GPs to model causal relationships, aligning with our objective of building surrogates when it's hard to work directly with the underlying SCM. However, they rely on several simplifying structural assumptions in order to restrict their focus to a class of causal sufficient/identifiable models. In contrast, we allow for a wider class of SCMs, resulting in a more robust framework. Further, these papers overlook the interventional consistency between the surrogate & simulator, a topic we thoroughly examine via causal abstraction. A notable distinction is our approach for addressing the computational difficulties of large-scale simulators, especially within policymaking scenarios, through a causally consistent & computationally efficient surrogate paradigm. Our goal of developing interventionally consistent surrogates aligns with the goal of accurately estimating causal effects discussed in these papers. We will update the Related Work section of the revision, & include [R4.1] & [R4.2] as examples of surrogates for causal inference.

[R4.3] is indeed relevant to our work when domain expertise is available & can be incorporated into the surrogate by enforcing monotonicity on certain features. As noted, an example could be when we identify a monotonic effect in interventions. For instance, one may enforce that longer lockdowns lead to less infections. We'll incorporate this discussion into the revision.

Refs

[R4.1] Witty et al. (2020)

[R4.2] Toth et al. (2022)

[R4.3] Riihimäki et al. (2010)

Thank you for your review and suggestions for improving our paper. We provide itemised responses to your two questions below.

a. To expand our empirical assessment of the framework we propose, and to provide practitioners with a further example of how they may use our framework, we have prepared a further case study that we will include in the revised paper. We provide details on this experiment in the Global Rebuttal above. We will also include a discussion of how causal surrogates can be benchmarked going forward, such as by assessing the degree to which the surrogates preserve the optimality or ordering of interventions in the original simulator with respect to some downstream optimisation task. Please see the Global Rebuttal for further details. We believe this would also be a useful guide to practitioners, and hope that this also addresses the reviewer's question here.

b. We will include in the revision a comparison of the runtimes of the simulators and surrogates. Briefly: in the epidemic case study presented, the LRNN/LODERNN/LODE surrogates each run approx. 3 times faster than the ABM; in the new experiment described in the previous bullet point and in the Global Rebuttal, the surrogates run 10-30 times faster. Obviously, the exact speed up will be different for different simulators. That said, there are additional considerations that motivate the use of smaller-scale surrogates with tractable likelihood functions, such as interpretability, improved communication (e.g., in our epidemic case study, we would be able to explain to a decision-maker who is familiar with the LODE surrogate but not with the complex ABM that "applying intervention X in the ABM is like applying intervention Y in the LODE"), and the ability to optimise probabilities/probability densities directly, thus even potentially obviating the need for any simulation from the surrogate in any case. We will discuss this more fully, using some of the additional space in the revision, and the appendix if necessary.

Dear authors, thank you for the rebuttal and the new experiments you mentioned in the global rebuttal.

I still think that, although the topic is relevant for the ML community, there are strong limitations in its applicability by practitioners without strong domain expertise.

Thank you again for your feedback – we hope we have been able to address your remaining concern about the limited role that domain knowledge plays in defining the variables of interest to the modeller. If you believe our improvements warrant an increase in your initial score, we would greatly appreciate seeing this be reflected in your updated score prior to the deadline in ~20 mins. Thank you again!

We thank the reviewer for their time & kind comments ("paper is a pleasure to read..very clear", "well-defined, consistent and easy to follow", "bridges the gap between.. causality.. and useful applications").

The reviewer identifies as a single weakness the novelty of our paper wrt a very recent UAI 2024 paper by Kekić et. al. [R3.1], presented at UAI less than 3 weeks ago. Despite this being categorized as a contemporaneous work by the NeurIPS Reviewer Guidelines, we are happy to detail the main differences between the two papers, highlight our novel contributions, & discuss this contemporaneous work in our paper.

The two papers share some similarities as they both start from an existing causal abstraction framework, adapt a measure of interventional consistency, and use it learn new models. However the two works differ substantially in aims, methodology & results:

• [R3.1] has an explanatory focus. They deal with targeted reduction of causal models aimed to generate a model "explaining causal influences on an observable target variable $Y$"; they describe the target variable $Y$ as a "property of interest" & the abstraction to $Y$ as a "detector" quantifying "the presence or magnitude of a phenomenon in the data". Thus, reduction works around a chosen focal point ($Y$) & the only causal dynamics of interest are the ones converging on the said variable. In contrast, our work has a simulation focus: we deal with learning surrogate models that would capture the whole causal dynamics of a system of interest at a different level of abstraction. Our work does not require the modeller to commit to any target variable: in the context of our SIRS simulation, [R3.1]'s approach would require the modeller to choose which target variable is of interest (e.g.: number of infected or number of recovered); our approach, instead, provides a simplified & accurate surrogate that simulates the dynamics of all relevant variables. To provide analogous simulation results, [R3.1]'s approach would require instantiating as many reduced models as the variables of interest
• There are also fundamental methodological differences. Since reduction is interested in a single target variable, only constructive transformations equivalent to clustering are considered & the reduced causal model is a simple collection of nodes $Z_i$ only affecting node $Y$. This trivial structure does not encode complex causal dynamics, such as influences between variables $Z_i$; it has indeed a resemblance to ICA, as explained in their discussion. On the other hand, we consider a broader class of $\tau$ maps and arbitrary surrogate models describing complex causal structures
• [R3.1] introduces further simplifications (linearity of $\tau, \omega$, affinity of the high-level mechanisms) for the sake of identifiability analysis. Experiments are run on simple physical models, as these simplifications rarely comply with real-world systems. Not requiring them, our approach is applied to actual ABMs which, as the reviewer states, "bridges the gap between the causality ivory tower and useful applications"

The most striking similitude between our works is how we adapted the existing interventional consistency loss by replacing a JS distance with a KL divergence, & substituting a max operator with an expectation. But these are common simplifications in ML adopted in causal abstraction papers predating both our works. Interestingly, from a similar loss function, we derived propositions that are relevant wrt our different aim: [R3.1] proves positivity, invariance to invertible reparametrization & zero for exactness; we prove zero for exactness and upper-bound for divergence.

We will briefly discuss this contemporaneous work in the Related Work section & add in the Appendix an exhaustive comparison along the lines above. A detailed comparison will be valuable to practitioners for choosing the approach that best suits their needs, whether modelling locally the dynamics of a single variable or whether simulating an entire system at a coarser scale.

Questions & limitations

Q1: Hard interventions are a requirement of the used abstraction framework as the loss is defined between posets of interventions which arise only in such a case. Our interventions are defined on parameters controlling the dynamics of the system; this modelling choice allows us to: disentangle the dynamics of the system & the params controlling it; rely on known ODEs for the dynamics; express parsimonious (defined by a single value) & interpretable (lockdown equal to setting the param to zero) interventions. In general, our work may be expanded to deal with soft interventions [R3.2], e.g., to model realistic uncertain/"fat-fingers" interventions by policy-makers.

Q2: Just a helpful picture! The method is general.

Q3: The \tilde{\theta}_t determine the update of the hidden state z_t of the ODE/RNN at time t, and z_t mediates the dependency of \tilde{y}_t on the \tilde{\theta}_{1:t} and I_0 (please see Appendix E). However, we have wrongly put arrows from \tilde{y}_t to \tilde{y}_{t+1}; will remove in revision.

L: It is incorrect that we do not specify "how to make the distribution of the surrogate family tractable and differentiable": we show three examples for how this can be done by combining neural networks, differentiable simulators, & standard distributions in the experimental section. We will nonetheless discuss other strategies in the revision, e.g. the use of neural generative models (e.g., normalising flows) to specify the stochasticity of surrogates, & of sample-based interventional consistency losses (e.g., MMDs) for the case of differentiable surrogates with intractable distributions.

For other limitations, please see our answer to Reviewer fR4E: specifically, on choosing interventions/finding a variable map/misspecification see Q1/L1/Q2.

Refs

[R3.2] Massidda et al. "Causal Abstraction with Soft Interventions." CLeaR (2022)

Q1 & W4: Generally, we expect $\eta$ to be informed by domain experts/policymakers based on downstream tasks, perhaps accounting for economic/political constraints. E.g. in a pandemic, economic constraints may preclude lockdowns of length >2 weeks, and political pressure may demand action soon; here, $\eta$ could be uniform over 2-week lockdowns starting within a month.

We now turn to the reviewer's question on benchmarks. Whilst numerous datasets & benchmarks have been proposed in the causality literature [R1.1-3], they are typically confined to SCMs much smaller than those of large-scale simulators. Likewise, to our best knowledge, there is no consensus in the social simulation community regarding benchmark simulators, besides simple examples (e.g. the Schelling model). Further, benchmarking surrogates requires access to a set of realistic downstream tasks. For instance, to properly assess a pandemic surrogate one must know the problems epidemiologists are interested in. Summarising, benchmarking surrogate models of complex systems requires:

• A set of large-scale standardised simulators spanning application domains.
• A set of downstream tasks for each simulator on which surrogate models can be deployed.

While we focus on presenting a new methodology for surrogate modelling grounded in causal abstraction theory, as we will discuss in revision, one way to devise a benchmark on downstream tasks is to check that interventional properties are preserved. For example, one can check that the internal ordering of interventions wrt their efficacy is preserved (see the Global Rebuttal).

Q2: As recognised by the reviewer, the surrogate family may not fully capture the behaviour of the simulator. Note that the task of learning a surrogate may be formulated as minimising $\text{KL}(P \Vert Q(\psi, \phi))$ over $\Psi \times \Phi$. Here, $P$ is a joint distribution over interventions and abstract states given by first sampling an intervention $\iota \sim \eta$ and then sampling from the base model under $\iota$, while $Q(\psi, \phi)$ is a joint distribution defined by first sampling $\iota \sim \eta$ and then sampling from the corresponding interventional distribution in the surrogate. Given the form of this problem, classical results for maximum likelihood estimation can be adapted to provide misspecification guarantees; by leveraging [R1.4], one can show that our surrogate estimation is asymptotically normal w/ sandwich covariance & mean corresponding to the surrogate model w/ minimum KL-divergence to the perfect abstract model under $\tau$.

Q3: Our method assumes no knowledge of the simulator's SCM. This ensures our method is generally applicable as it's difficult to directly characterise the SCM of large-scale simulators. It's possible that access to the base SCM/DAG may expedite abstraction by allowing us to focus on minimal intervention sets [R1.5-6], or leverage the identifiability of interventional distributions to reduce the number of simulations required from the base model [R1.7-8]. However, it's unclear that applying the do-calculus on large causal graphs is more efficient than simulating interventions directly. We will discuss these ideas in more detail in the appendices of the paper.

Q4: Many of the surrogates we consider are neural networks, thus the time complexity of sampling an interventional outcome scales linearly w/ the dimension of interventions. Moreover, our neural surrogates benefit from parallelisation within each time step, massively reducing their run-time. In contrast, social simulators often rely on pairwise interactions between potentially millions of agents that are difficult to parallelise. Thus, once trained, neural surrogates can be orders of magnitude faster to evaluate than the simulator. Precisely characterising this speed up is non-trivial & depends on the internal structure of the simulator and the surrogate. For a comparison of runtime costs in our experiments, see part (b) of our response to Reviewer fUGP.

Of course, there is a training cost associated with the surrogate, but this will be amortised over downstream tasks. That is, when the collective sample complexity of downstream tasks exceeds the sample complexity of causal abstraction, learning a surrogate is beneficial.

Furthermore, running complex simulators may require technical expertise & high performance hardware not widely available. Providing interpretable surrogates that run on commodity hardware reduces the barrier to entry in such cases. Note that this mirrors the release of low memory LLMs that run on commodity hardware commonplace today.

W1: Please see the Global Rebuttal.

W2: Our baselines are already complex wrt SOTA in social simulation (see e.g. [R1.9]). Please let us know if there are specific baselines you would like us to compare against.

W3: We agree that more complex simulators are used in practice, but we prioritised clear presentation by focusing on experiments that are easy to understand, familiar to the social simulation community, & complex enough to demonstrate our method's viability. As in our response to Q1: applying our method to more complex models is complicated by a lack of standard benchmarks in the social simulation community.

L1: We agree that selecting the appropriate granularity via $\tau$ is hard, but in many cases domain experts know what emergent properties interest them. For instance, epidemiologists/labour economists/central bankers often care about the number infected individuals/unemployment numbers/inflation rates over time. In other words, domain expertise can be leveraged to select a suitable $\tau$. We do not know of existing work that automatically selects the right granularity for abstraction; the closest we know of applies rate distortion theory to learn low dimensional representations of MAB and RL problems [R1.10-11] but do not have a causal flavour & focus on single downstream tasks.

Refs

In Official Comment. (Too few chars sorry!)

Refs

[R1.1] Geffner et al. "Deep end-to-end causal inference" NeurIPS Workshop on Causal Machine Learning for Real-World Impact (2022)

[R1.2] Melistas et al. "Benchmarking counterfactual image generation" arXiv (2022)

[R1.3] Mooij et al. "Distinguishing cause from effect using observational data: Methods and benchmarks" JMLR (2016)

[R1.4] White "Maximum likelihood estimation of misspecified models" Econometrica (1982)

[R1.5] Aglietti et al. "Causal Bayesian optimization" AISTATS (2020)

[R1.6] Lee and Bareinboim "Structural causal bandits: where to intervene?" NeurIPS (2018)

[R1.7] Lattimore et al. "Causal bandits: Learning good interventions via causal inference" NeurIPS (2016)

[R1.8] Bilodeau et al. "Adaptively exploiting d-separators with causal bandits" NeurIPS (2022)

[R1.9] Angione et al. "Using machine learning as a surrogate model for agent-based simulations" Plos one (2022)

[R1.10] Arumugam and Van Roy "Deciding what to learn: A rate-distortion approach" ICLR (2021)

[R1.11] Arumugam and Van Roy "Deciding what to model: Value-equivalent sampling for reinforcement learning" NeurIPS (2022)

Thank you all for your feedback during the review and discussion period. If there are any further points that you would like to discuss before the discussion period closes, please do let us know and we will be happy to respond.

We are particularly cognizant of the extreme discrepancy between the positive scores and feedback we have received from all reviewers (7 from fR4E, 6s from fUGP and 3hmr, positive comments from all reviewers including x5Gt) and the single outlying "Strong Reject" score from Reviewer x5Gt, whose concern is the relation of our work to a paper appearing at UAI '24 approx. 4 weeks ago (herein [R3.1]).

We have already used the rebuttal and discussion to highlight the many substantive differences between our work and [R3.1]'s. We have shown that [R3.1] differs substantially in focus from our work, and is entirely unsuitable for the settings we consider, given that it scales extremely poorly to the large-scale complex simulators of interest to us. We have also carefully consulted [R3.1]'s paper and used [R3.1]'s accompanying code to run computational experiments that verify these claims. (Please see the discussion below for details.)

While a "Strong Reject" is, therefore, inappropriately low and incompatible with Reviewer x5Gt's review and the ensuing discussion*, we would in particular like to invite any further comments from the reviewers on any concerns they may have in relation to [R3.1], in order that we may address them. And if you have no further concerns about our work, please do champion our paper!

Thank you once again for your helpful feedback.

*A "Strong Reject" is reserved for papers "with major technical flaws, and/or poor evaluation, limited impact, poor reproducibility and mostly unaddressed ethical considerations". In contrast, all reviews give "Soundness: 3: good" for our work, rather than claiming major technical flaws; "poor evaluation" is incompatible with the fact that we have provided two case studies, each studying 3 approaches to learning interventionally consistent surrogates, and conducted an experimental comparison against [R3.1]'s approach; "limited impact" is incompatible with the fact that our work currently provides the only viable (practical, scalable, and interpretable) method to learn interventionally consistent surrogates for complex simulators of the kind used to inform decision-making in complex systems; "poor reproducibility" is not raised by any reviewer; and no reviewers raise ethical concerns.