When learning to walk, infants seem to address a coarse version of the problem first - stay upright, reach the caregiver - and refine it only when further practice at that resolution stops paying off. Reinforcement learning offers multiple techniques for building simple versions of complex tasks, but lacks general principles for how to dynamically adjust the granularity of these abstractions during learning. This paper proposes one such principle: refine the abstraction as soon as the learning error within it becomes comparable to the error induced by the abstraction itself. Here, we investigate one way of formalising this principle via a performance certificate that decomposes value error into two terms: a learning error bound captured by a Bellman residual, and an abstraction error bound given by a bisimulation metric. The resulting switching strategy is implemented by soft state-action abstractions built from rate-distortion principles, whose resolution along state and action axes can be continuously adjusted. We validate this construction in a range of tabular settings, showing that near-optimal performance can be achieved under substantial lossy compression of state and action information.

Agent-based models (ABMs) are a powerful tool for modelling complex decision-making systems across application domains, including the social sciences (Baptista et al., 2016), epidemiology (Kerr et al., 2021), and finance (Cont, 2007). Such models provide high-fidelity and granular representations of intricate systems of autonomous, interacting, and decision-making agents by modelling the system under consideration at the level of its individual constituent actors. In this way, ABMs enable decision-makers to experiment with, and understand the potential consequences of, policy interventions of interest, thereby allowing for more effective control of the potentially deleterious effects that arise from the endogenous dynamics of the real-world system. In economic systems, for example, such policy interventions may take the form of imposed limits on loan-to-value ratios in housing markets as a means for attenuating housing price cycles (Baptista et al., 2016), while in epidemiology, such interventions may take the form of (non-)pharmaceutical interventions to inhibit the transmission of a disease (Kerr et al., 2021). Whilst ABMs promise many benefits, their complexity generally necessitates the use of simulation studies to understand their behaviours, and their granularity can result in large computational costs even for single forward simulations. In many cases, such costs can be prohibitively large, presenting a barrier to their use as synthetic test environments for potential policy interventions in practice. Moreover, the high-fidelity data generated by ABMs can be difficult for policymakers to interpret and relate to policy interventions that act system-wide (Haldane and Turrell, 2018).

Causal abstraction (CA) theory establishes formal criteria for relating multiple structural causal models (SCMs) at different levels of granularity by defining maps between them. These maps have significant relevance for real-world challenges such as synthesizing causal evidence from multiple experimental environments, learning causally consistent representations at different resolutions, and linking interventions across multiple SCMs. In this work, we propose COTA, the first method to learn abstraction maps from observational and interventional data without assuming complete knowledge of the underlying SCMs. In particular, we introduce a multi-marginal Optimal Transport (OT) formulation that enforces do-calculus causal constraints, together with a cost function that relies on interventional information. We extensively evaluate COTA on synthetic and real world problems, and showcase its advantages over non-causal, independent and aggregated COTA formulations. Finally, we demonstrate the efficiency of our method as a data augmentation tool by comparing it against the state-of-the-art CA learning framework, which assumes fully specified SCMs, on a real-world downstream task.

Interventional causal models describe joint distributions over some variables used to describe a system, one for each intervention setting. They provide a formal recipe for how to move between joint distributions and make predictions about the variables upon intervening on the system. Yet, it is difficult to formalise how we may change the underlying variables used to describe the system, say from fine-grained to coarse-grained variables. Here, we argue that compositionality is a desideratum for model transformations and the associated errors. We develop a framework for model transformations and abstractions with a notion of error that is compositional: when abstracting a reference model M modularly, first obtaining M' and then further simplifying that to obtain M'', then the composite transformation from M to M'' exists and its error can be bounded by the errors incurred by each individual transformation step. Category theory, the study of mathematical objects via the compositional transformations between them, offers a natural language for developing our framework. We introduce a category of finite interventional causal models and, leveraging theory of enriched categories, prove that our framework enjoys the desired compositionality properties.

There is thus a great need to define inputs, outputs, and their interactive relationships clearly. Inevitably, technologists would code fairness as a narrowly defined modular property of the machine learning system. However, fairness is not a well defined nor universally applicable concept, to begin with as it has to be understood amidst a particular social context. Abstracting away this context is thus an abstraction error. With the presence of this error, AI would have an ineffective, inaccurate and misguided interpretation and thus, quantification of fairness when it is introduced to varying societal systems.