Niki Kilbertus

Isabel Chien, Nina Deliu, Richard E. Turner et al. · cambridge

While interest in the application of machine learning to improve healthcare has grown tremendously in recent years, a number of barriers prevent deployment in medical practice. A notable concern is the potential to exacerbate entrenched biases and existing health disparities in society. The area of fairness in machine learning seeks to address these issues of equity; however, appropriate approaches are context-dependent, necessitating domain-specific consideration. We focus on clinical trials, i.e., research studies conducted on humans to evaluate medical treatments. Clinical trials are a relatively under-explored application in machine learning for healthcare, in part due to complex ethical, legal, and regulatory requirements and high costs. Our aim is to provide a multi-disciplinary assessment of how fairness for machine learning fits into the context of clinical trials research and practice. We start by reviewing the current ethical considerations and guidelines for clinical trials and examine their relationship with common definitions of fairness in machine learning. We examine potential sources of unfairness in clinical trials, providing concrete examples, and discuss the role machine learning might play in either mitigating potential biases or exacerbating them when applied without care. Particular focus is given to adaptive clinical trials, which may employ machine learning. Finally, we highlight concepts that require further investigation and development, and emphasize new approaches to fairness that may be relevant to the design of clinical trials.

Jiri Hron, Karl Krauth, Michael I. Jordan et al.

Content creators compete for user attention. Their reach crucially depends on algorithmic choices made by developers on online platforms. To maximize exposure, many creators adapt strategically, as evidenced by examples like the sprawling search engine optimization industry. This begets competition for the finite user attention pool. We formalize these dynamics in what we call an exposure game, a model of incentives induced by algorithms, including modern factorization and (deep) two-tower architectures. We prove that seemingly innocuous algorithmic choices, e.g., non-negative vs. unconstrained factorization, significantly affect the existence and character of (Nash) equilibria in exposure games. We proffer use of creator behavior models, like exposure games, for an (ex-ante) pre-deployment audit. Such an audit can identify misalignment between desirable and incentivized content, and thus complement post-hoc measures like content filtering and moderation. To this end, we propose tools for numerically finding equilibria in exposure games, and illustrate results of an audit on the MovieLens and LastFM datasets. Among else, we find that the strategically produced content exhibits strong dependence between algorithmic exploration and content diversity, and between model expressivity and bias towards gender-based user and creator groups.

Leon Hetzel, Simon Böhm, Niki Kilbertus et al.

Single-cell transcriptomics enabled the study of cellular heterogeneity in response to perturbations at the resolution of individual cells. However, scaling high-throughput screens (HTSs) to measure cellular responses for many drugs remains a challenge due to technical limitations and, more importantly, the cost of such multiplexed experiments. Thus, transferring information from routinely performed bulk RNA HTS is required to enrich single-cell data meaningfully. We introduce chemCPA, a new encoder-decoder architecture to study the perturbational effects of unseen drugs. We combine the model with an architecture surgery for transfer learning and demonstrate how training on existing bulk RNA HTS datasets can improve generalisation performance. Better generalisation reduces the need for extensive and costly screens at single-cell resolution. We envision that our proposed method will facilitate more efficient experiment designs through its ability to generate in-silico hypotheses, ultimately accelerating drug discovery.

Luca Eyring, Dominik Klein, Théo Uscidda et al.

In optimal transport (OT), a Monge map is known as a mapping that transports a source distribution to a target distribution in the most cost-efficient way. Recently, multiple neural estimators for Monge maps have been developed and applied in diverse unpaired domain translation tasks, e.g. in single-cell biology and computer vision. However, the classic OT framework enforces mass conservation, which makes it prone to outliers and limits its applicability in real-world scenarios. The latter can be particularly harmful in OT domain translation tasks, where the relative position of a sample within a distribution is explicitly taken into account. While unbalanced OT tackles this challenge in the discrete setting, its integration into neural Monge map estimators has received limited attention. We propose a theoretically grounded method to incorporate unbalancedness into any Monge map estimator. We improve existing estimators to model cell trajectories over time and to predict cellular responses to perturbations. Moreover, our approach seamlessly integrates with the OT flow matching (OT-FM) framework. While we show that OT-FM performs competitively in image translation, we further improve performance by incorporating unbalancedness (UOT-FM), which better preserves relevant features. We hence establish UOT-FM as a principled method for unpaired image translation.

Hananeh Aliee, Till Richter, Mikhail Solonin et al.

Neural Ordinary Differential Equations (NODEs) have proven successful in learning dynamical systems in terms of accurately recovering the observed trajectories. While different types of sparsity have been proposed to improve robustness, the generalization properties of NODEs for dynamical systems beyond the observed data are underexplored. We systematically study the influence of weight and feature sparsity on forecasting as well as on identifying the underlying dynamical laws. Besides assessing existing methods, we propose a regularization technique to sparsify "input-output connections" and extract relevant features during training. Moreover, we curate real-world datasets consisting of human motion capture and human hematopoiesis single-cell RNA-seq data to realistically analyze different levels of out-of-distribution (OOD) generalization in forecasting and dynamics identification respectively. Our extensive empirical evaluation on these challenging benchmarks suggests that weight sparsity improves generalization in the presence of noise or irregular sampling. However, it does not prevent learning spurious feature dependencies in the inferred dynamics, rendering them impractical for predictions under interventions, or for inferring the true underlying dynamics. Instead, feature sparsity can indeed help with recovering sparse ground-truth dynamics compared to unregularized NODEs.

Sören Becker, Michal Klein, Alexander Neitz et al.

We develop a transformer-based sequence-to-sequence model that recovers scalar ordinary differential equations (ODEs) in symbolic form from irregularly sampled and noisy observations of a single solution trajectory. We demonstrate in extensive empirical evaluations that our model performs better or on par with existing methods in terms of accurate recovery across various settings. Moreover, our method is efficiently scalable: after one-time pretraining on a large set of ODEs, we can infer the governing law of a new observed solution in a few forward passes of the model.

Stéphane d'Ascoli, Sören Becker, Alexander Mathis et al.

We introduce ODEFormer, the first transformer able to infer multidimensional ordinary differential equation (ODE) systems in symbolic form from the observation of a single solution trajectory. We perform extensive evaluations on two datasets: (i) the existing "Strogatz" dataset featuring two-dimensional systems; (ii) ODEBench, a collection of one- to four-dimensional systems that we carefully curated from the literature to provide a more holistic benchmark. ODEFormer consistently outperforms existing methods while displaying substantially improved robustness to noisy and irregularly sampled observations, as well as faster inference. We release our code, model and benchmark dataset publicly.

Francesco Quinzan, Cecilia Casolo, Krikamol Muandet et al.

Notions of counterfactual invariance (CI) have proven essential for predictors that are fair, robust, and generalizable in the real world. We propose graphical criteria that yield a sufficient condition for a predictor to be counterfactually invariant in terms of a conditional independence in the observational distribution. In order to learn such predictors, we propose a model-agnostic framework, called Counterfactually Invariant Prediction (CIP), building on the Hilbert-Schmidt Conditional Independence Criterion (HSCIC), a kernel-based conditional dependence measure. Our experimental results demonstrate the effectiveness of CIP in enforcing counterfactual invariance across various simulated and real-world datasets including scalar and multi-variate settings.

Alistair White, Niki Kilbertus, Maximilian Gelbrecht et al.

Many successful methods to learn dynamical systems from data have recently been introduced. However, ensuring that the inferred dynamics preserve known constraints, such as conservation laws or restrictions on the allowed system states, remains challenging. We propose stabilized neural differential equations (SNDEs), a method to enforce arbitrary manifold constraints for neural differential equations. Our approach is based on a stabilization term that, when added to the original dynamics, renders the constraint manifold provably asymptotically stable. Due to its simplicity, our method is compatible with all common neural differential equation (NDE) models and broadly applicable. In extensive empirical evaluations, we demonstrate that SNDEs outperform existing methods while broadening the types of constraints that can be incorporated into NDE training.

Sören Becker, Michal Klein, Alexander Neitz et al.

Natural laws are often described through differential equations yet finding a differential equation that describes the governing law underlying observed data is a challenging and still mostly manual task. In this paper we make a step towards the automation of this process: we propose a transformer-based sequence-to-sequence model that recovers scalar autonomous ordinary differential equations (ODEs) in symbolic form from time-series data of a single observed solution of the ODE. Our method is efficiently scalable: after one-time pretraining on a large set of ODEs, we can infer the governing laws of a new observed solution in a few forward passes of the model. Then we show that our model performs better or on par with existing methods in various test cases in terms of accurate symbolic recovery of the ODE, especially for more complex expressions.

Birgit Kühbacher, Daan Crommelin, Niki Kilbertus

Weather and climate forecasts are inherently uncertain due to chaotic dynamics, imperfect initial conditions, and incomplete representation of the underlying physical processes. Operational ensemble forecasts aim to represent these uncertainties through forecast spread, yet many approaches yield underdispersive estimates, with spread that grows too slowly relative to forecast error. Using the two-scale Lorenz 1996 system as a widely used, controlled testbed, we design a systematic approach to disentangle intrinsic variability, initial-condition perturbations, and stochastic model uncertainty. We compare multiple ensemble configurations and parameterization strategies, including existing deterministic and autoregressive as well as novel Bayesian and flow-based approaches. Our results show that ensemble perturbations do not increase the system's long-term variance; rather, they regulate how rapidly trajectories decorrelate and explore the invariant measure. Stochastic parameterizations, particularly those with temporally persistent structure, enhance early spread growth and improve spread-error consistency. Overall, we bring clarity to how different sources of uncertainty interact in a chaotic system and provide guidance for the design and evaluation of stochastic parameterizations in weather and climate models.

Elisabeth Ailer, Jason Hartford, Niki Kilbertus

Instrumental variable (IV) methods are used to estimate causal effects in settings with unobserved confounding, where we cannot directly experiment on the treatment variable. Instruments are variables which only affect the outcome indirectly via the treatment variable(s). Most IV applications focus on low-dimensional treatments and crucially require at least as many instruments as treatments. This assumption is restrictive: in the natural sciences we often seek to infer causal effects of high-dimensional treatments (e.g., the effect of gene expressions or microbiota on health and disease), but can only run few experiments with a limited number of instruments (e.g., drugs or antibiotics). In such underspecified problems, the full treatment effect is not identifiable in a single experiment even in the linear case. We show that one can still reliably recover the projection of the treatment effect onto the instrumented subspace and develop techniques to consistently combine such partial estimates from different sets of instruments. We then leverage our combined estimators in an algorithm that iteratively proposes the most informative instruments at each round of experimentation to maximize the overall information about the full causal effect.

Nora Schneider, Georg Manten, Niki Kilbertus

We develop a conservative continuous-time stochastic control framework for treatment optimization from irregularly sampled patient trajectories. The unknown patient dynamics are modeled as a controlled stochastic differential equation with treatment as a continuous-time control. Naive model-based optimization can exploit model errors and propose out-of-support controls, so optimizing the estimated dynamics may not optimize the true dynamics. To limit extrapolation, we add a consistent signature-based MMD regularizer on path space that penalizes treatment plans whose induced trajectory distribution deviates from observed trajectories. The resulting objective minimizes a computable upper bound on the true cost. Experiments on benchmark datasets show improved robustness and performance compared to non-conservative baselines.

Shimeng Huang, Elisabeth Ailer, Niki Kilbertus et al.

The compositionality and sparsity of high-throughput sequencing data poses a challenge for regression and classification. However, in microbiome research in particular, conditional modeling is an essential tool to investigate relationships between phenotypes and the microbiome. Existing techniques are often inadequate: they either rely on extensions of the linear log-contrast model (which adjusts for compositionality, but is often unable to capture useful signals), or they are based on black-box machine learning methods (which may capture useful signals, but ignore compositionality in downstream analyses). We propose KernelBiome, a kernel-based nonparametric regression and classification framework for compositional data. It is tailored to sparse compositional data and is able to incorporate prior knowledge, such as phylogenetic structure. KernelBiome captures complex signals, including in the zero-structure, while automatically adapting model complexity. We demonstrate on par or improved predictive performance compared with state-of-the-art machine learning methods. Additionally, our framework provides two key advantages: (i) We propose two novel quantities to interpret contributions of individual components and prove that they consistently estimate average perturbation effects of the conditional mean, extending the interpretability of linear log-contrast models to nonparametric models. (ii) We show that the connection between kernels and distances aids interpretability and provides a data-driven embedding that can augment further analysis. Finally, we apply the KernelBiome framework to two public microbiome studies and illustrate the proposed model analysis. KernelBiome is available as an open-source Python package at https://github.com/shimenghuang/KernelBiome.

Aybüke Ulusarslan, Niki Kilbertus, Nora Schneider

Learning governing dynamics from data is a common goal across the sciences, yet it is only well-posed when the underlying mechanisms are identifiable. In practice, many data-driven methods implicitly assume identifiability; when this assumption fails, estimated models can yield spurious predictions and invalid mechanistic conclusions. Classical identifiability guarantees for controlled linear time-invariant (LTI) systems provide sufficient conditions -- controllability and persistent excitation -- but leave open whether identifiability holds when these conditions fail, and which parts of the system remain identifiable without full identifiability. We show that the experimental setup, i.e., the realized initial state and control input, dictates a fundamental limit on the information recoverable from the observed trajectory. We develop a geometric characterization of this limit and derive a closed-form description of all systems consistent with the experimental setup. Crucially, we prove that even when the full system is not identifiable, the restricted dynamics on the subspace reachable by the experiment remain uniquely determined.

Stefan Feuerriegel, Dennis Frauen, Valentyn Melnychuk et al.

Causal machine learning (ML) offers flexible, data-driven methods for predicting treatment outcomes including efficacy and toxicity, thereby supporting the assessment and safety of drugs. A key benefit of causal ML is that it allows for estimating individualized treatment effects, so that clinical decision-making can be personalized to individual patient profiles. Causal ML can be used in combination with both clinical trial data and real-world data, such as clinical registries and electronic health records, but caution is needed to avoid biased or incorrect predictions. In this Perspective, we discuss the benefits of causal ML (relative to traditional statistical or ML approaches) and outline the key components and steps. Finally, we provide recommendations for the reliable use of causal ML and effective translation into the clinic.

Jan Marco Ruiz de Vargas, Kirtan Padh, Niki Kilbertus

Finding cause-effect relationships is of key importance in science. Causal discovery aims to recover a graph from data that succinctly describes these cause-effect relationships. However, current methods face several challenges, especially when dealing with high-dimensional data and complex dependencies. Incorporating prior knowledge about the system can aid causal discovery. In this work, we leverage Cluster-DAGs as a prior knowledge framework to warm-start causal discovery. We show that Cluster-DAGs offer greater flexibility than existing approaches based on tiered background knowledge and introduce two modified constraint-based algorithms, Cluster-PC and Cluster-FCI, for causal discovery in the fully and partially observed setting, respectively. Empirical evaluation on simulated data demonstrates that Cluster-PC and Cluster-FCI outperform their respective baselines without prior knowledge.

Georg Manten, Cecilia Casolo, Emilio Ferrucci et al.

Inferring the causal structure underlying stochastic dynamical systems from observational data holds great promise in domains ranging from science and health to finance. Such processes can often be accurately modeled via stochastic differential equations (SDEs), which naturally imply causal relationships via "which variables enter the differential of which other variables". In this paper, we develop conditional independence (CI) constraints on coordinate processes over selected intervals that are Markov with respect to the acyclic dependence graph (allowing self-loops) induced by a general SDE model. We then provide a sound and complete causal discovery algorithm, capable of handling both fully and partially observed data, and uniquely recovering the underlying or induced ancestral graph by exploiting time directionality assuming a CI oracle. Finally, to make our algorithm practically usable, we also propose a flexible, consistent signature kernel-based CI test to infer these constraints from data. We extensively benchmark the CI test in isolation and as part of our causal discovery algorithms, outperforming existing approaches in SDE models and beyond.

Konstantin Hess, Dennis Frauen, Niki Kilbertus et al.

Neural operators are widely used to approximate solution maps of complex physical systems. In many applications, however, the goal is not to recover the full solution trajectory, but to summarize the solution trajectory via a scalar target quantity (e.g., a functional such as time spent in a target range, time above a threshold, accumulated cost, or total energy). In this paper, we introduce DOPE (debiased neural operator): a semiparametric estimator for such target quantities of solution trajectories obtained from neural operators. DOPE is broadly applicable to settings with both partial and irregular observations and can be combined with arbitrary neural operator architectures. We make three main contributions. (1) We show that, in contrast to DOPE, naive plug-in estimation can suffer from first-order bias. (2) To address this, we derive a novel one-step, Neyman-orthogonal estimator that treats the neural operator as a high-dimensional nuisance mapping between function spaces, and removes the leading bias term. For this, DOPE uses a weighting mechanism that simultaneously accounts for irregular observation designs and for how sensitive the target quantity is to perturbations of the underlying trajectory. (3) To learn the weights, we extend automatic debiased machine learning to operator-valued nuisances via Riesz regression. We demonstrate the benefits of DOPE across various numerical experiments.

Michelle Seng Ah Lee, Kirtan Padh, David Watson et al.

Fair machine learning (ML) methods help identify and mitigate the risk that algorithms encode or automate social injustices. Algorithmic approaches alone cannot resolve structural inequalities, but they can support socio-technical decision systems by surfacing discriminatory biases, clarifying trade-offs, and enabling governance. Although fairness is well studied in supervised learning, many real ML applications are online and sequential, with prior decisions informing future ones. Each decision is taken under uncertainty due to unobserved counterfactuals and finite samples, with dire consequences for under-represented groups, systematically under-observed due to historical exclusion and selective feedback. A bank cannot know whether a denied loan would have been repaid, and may have less data on marginalized populations. This paper introduces a taxonomy of uncertainty in sequential decision-making -- model, feedback, and prediction uncertainty -- providing shared vocabulary for assessing systems where uncertainty is unevenly distributed across groups. We formalize model and feedback uncertainty via counterfactual logic and reinforcement learning, and illustrate harms to decision makers (unrealized gains/losses) and subjects (compounding exclusion, reduced access) of policies that ignore the unobserved space. Algorithmic examples show it is possible to reduce outcome variance for disadvantaged groups while preserving institutional objectives (e.g. expected utility). Experiments on data simulated with varying bias show how unequal uncertainty and selective feedback produce disparities, and how uncertainty-aware exploration alters fairness metrics. The framework equips practitioners to diagnose, audit, and govern fairness risks. Where uncertainty drives unfairness rather than incidental noise, accounting for it is essential to fair and effective decision-making.

Cecilia Casolo, Sören Becker, Niki Kilbertus

Dynamical systems modeling is a core pillar of scientific inquiry across natural and life sciences. Increasingly, dynamical system models are learned from data, rendering identifiability a paramount concept. For systems that are not identifiable from data, no guarantees can be given about their behavior under new conditions and inputs, or about possible control mechanisms to steer the system. It is known in the community that "linear ordinary differential equations (ODE) are almost surely identifiable from a single trajectory." However, this only holds for dense matrices. The sparse regime remains underexplored, despite its practical relevance with sparsity arising naturally in many biological, social, and physical systems. In this work, we address this gap by characterizing the identifiability of sparse linear ODEs. Contrary to the dense case, we show that sparse systems are unidentifiable with a positive probability in practically relevant sparsity regimes and provide lower bounds for this probability. We further study empirically how this theoretical unidentifiability manifests in state-of-the-art methods to estimate linear ODEs from data. Our results corroborate that sparse systems are also practically unidentifiable. Theoretical limitations are not resolved through inductive biases or optimization dynamics. Our findings call for rethinking what can be expected from data-driven dynamical system modeling and allows for quantitative assessments of how much to trust a learned linear ODE.

Nora Schneider, Lars Lorch, Niki Kilbertus et al.

We consider the problem of predicting perturbation effects via causal models. In many applications, it is a priori unknown which mechanisms of a system are modified by an external perturbation, even though the features of the perturbation are available. For example, in genomics, some properties of a drug may be known, but not their causal effects on the regulatory pathways of cells. We propose a generative intervention model (GIM) that learns to map these perturbation features to distributions over atomic interventions in a jointly-estimated causal model. Contrary to prior approaches, this enables us to predict the distribution shifts of unseen perturbation features while gaining insights about their mechanistic effects in the underlying data-generating process. On synthetic data and scRNA-seq drug perturbation data, GIMs achieve robust out-of-distribution predictions on par with unstructured approaches, while effectively inferring the underlying perturbation mechanisms, often better than other causal inference methods.

Alistair White, Anna Büttner, Maximilian Gelbrecht et al.

Neural differential equations offer a powerful approach for learning dynamics from data. However, they do not impose known constraints that should be obeyed by the learned model. It is well-known that enforcing constraints in surrogate models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a new method for constraining neural differential equations based on projection of the learned vector field to the tangent space of the constraint manifold. In tests on several challenging examples, including chaotic dynamical systems and state-of-the-art power grid models, PNDEs outperform existing methods while requiring fewer hyperparameters. The proposed approach demonstrates significant potential for enhancing the modeling of constrained dynamical systems, particularly in complex domains where accuracy and reliability are essential.

Jonas Schweisthal, Dennis Frauen, Maresa Schröder et al.

Reliable estimation of treatment effects from observational data is important in many disciplines such as medicine. However, estimation is challenging when unconfoundedness as a standard assumption in the causal inference literature is violated. In this work, we leverage arbitrary (potentially high-dimensional) instruments to estimate bounds on the conditional average treatment effect (CATE). Our contributions are three-fold: (1) We propose a novel approach for partial identification through a mapping of instruments to a discrete representation space so that we yield valid bounds on the CATE. This is crucial for reliable decision-making in real-world applications. (2) We derive a two-step procedure that learns tight bounds using a tailored neural partitioning of the latent instrument space. As a result, we avoid instability issues due to numerical approximations or adversarial training. Furthermore, our procedure aims to reduce the estimation variance in finite-sample settings to yield more reliable estimates. (3) We show theoretically that our procedure obtains valid bounds while reducing estimation variance. We further perform extensive experiments to demonstrate the effectiveness across various settings. Overall, our procedure offers a novel path for practitioners to make use of potentially high-dimensional instruments (e.g., as in Mendelian randomization).

Uzair Akbar, Niki Kilbertus, Hao Shen et al.

The technique of data augmentation (DA) is often used in machine learning for regularization purposes to better generalize under i.i.d. settings. In this work, we present a unifying framework with topics in causal inference to make a case for the use of DA beyond just the i.i.d. setting, but for generalization across interventions as well. Specifically, we argue that when the outcome generating mechanism is invariant to our choice of DA, then such augmentations can effectively be thought of as interventions on the treatment generating mechanism itself. This can potentially help to reduce bias in causal effect estimation arising from hidden confounders. In the presence of such unobserved confounding we typically make use of instrumental variables (IVs) -- sources of treatment randomization that are conditionally independent of the outcome. However, IVs may not be as readily available as DA for many applications, which is the main motivation behind this work. By appropriately regularizing IV based estimators, we introduce the concept of IV-like (IVL) regression for mitigating confounding bias and improving predictive performance across interventions even when certain IV properties are relaxed. Finally, we cast parameterized DA as an IVL regression problem and show that when used in composition can simulate a worst-case application of such DA, further improving performance on causal estimation and generalization tasks beyond what simple DA may offer. This is shown both theoretically for the population case and via simulation experiments for the finite sample case using a simple linear example. We also present real data experiments to support our case.

Zhufeng Li, Niki Kilbertus

Causal discovery aims to recover graphs that represent causal relations among given variables from observations, and new methods are constantly being proposed. Increasingly, the community raises questions about how much progress is made, because properly evaluating discovered graphs remains notoriously difficult, particularly under latent confounding. We propose a graph distance measure for acyclic directed mixed graphs (ADMGs) based on the downstream task of cause-effect estimation under unobserved confounding. Our approach uses identification via fixing and a symbolic verifier to quantify how graph differences distort cause-effect estimands for different treatment-outcome pairs. We analyze the behavior of the measure under different graph perturbations and compare it against existing distance metrics.

Yakup Emre Şahin, Niki Kilbertus, Sören Becker

We introduce MIO, a transformer-based model for inferring symbolic ordinary differential equations (ODEs) from multiple observed trajectories of a dynamical system. By combining multiple instance learning with transformer-based symbolic regression, the model effectively leverages repeated observations of the same system to learn more generalizable representations of the underlying dynamics. We investigate different instance aggregation strategies and show that even simple mean aggregation can substantially boost performance. MIO is evaluated on systems ranging from one to four dimensions and under varying noise levels, consistently outperforming existing baselines.

Georg Manten, Cecilia Casolo, Søren Wengel Mogensen et al.

We develop the theory linking 'E-separation' in directed mixed graphs (DMGs) with conditional independence relations among coordinate processes in stochastic differential equations (SDEs), where causal relationships are determined by "which variables enter the governing equation of which other variables". We prove a global Markov property for cyclic SDEs, which naturally extends to partially observed cyclic SDEs, because our asymmetric independence model is closed under marginalization. We then characterize the class of graphs that encode the same set of independence relations, yielding a result analogous to the seminal 'same skeleton and v-structures' result for directed acyclic graphs (DAGs). In the fully observed case, we show that each such equivalence class of graphs has a greatest element as a parsimonious representation and develop algorithms to identify this greatest element from data. We conjecture that a greatest element also exists under partial observations, which we verify computationally for graphs with up to four nodes.

Kirtan Padh, Zhufeng Li, Cecilia Casolo et al.

Assuming a directed acyclic graph (DAG) that represents prior knowledge of causal relationships between variables is a common starting point for cause-effect estimation. Existing literature typically invokes hypothetical domain expert knowledge or causal discovery algorithms to justify this assumption. In practice, neither may propose a single DAG with high confidence. Domain experts are hesitant to rule out dependencies with certainty or have ongoing disputes about relationships; causal discovery often relies on untestable assumptions itself or only provides an equivalence class of DAGs and is commonly sensitive to hyperparameter and threshold choices. We propose an efficient, gradient-based optimization method that provides bounds for causal queries over a collection of causal graphs -- compatible with imperfect prior knowledge -- that may still be too large for exhaustive enumeration. Our bounds achieve good coverage and sharpness for causal queries such as average treatment effects in linear and non-linear synthetic settings as well as on real-world data. Our approach aims at providing an easy-to-use and widely applicable rebuttal to the valid critique of `What if your assumed DAG is wrong?'.

Thomas Schwarz, Cecilia Casolo, Niki Kilbertus

In personalized medicine, the ability to predict and optimize treatment outcomes across various time frames is essential. Additionally, the ability to select cost-effective treatments within specific budget constraints is critical. Despite recent advancements in estimating counterfactual trajectories, a direct link to optimal treatment selection based on these estimates is missing. This paper introduces a novel method integrating counterfactual estimation techniques and uncertainty quantification to recommend personalized treatment plans adhering to predefined cost constraints. Our approach is distinctive in its handling of continuous treatment variables and its incorporation of uncertainty quantification to improve prediction reliability. We validate our method using two simulated datasets, one focused on the cardiovascular system and the other on COVID-19. Our findings indicate that our method has robust performance across different counterfactual estimation baselines, showing that introducing uncertainty quantification in these settings helps the current baselines in finding more reliable and accurate treatment selection. The robustness of our method across various settings highlights its potential for broad applicability in personalized healthcare solutions.

Birgit Kühbacher, Fernando Iglesias-Suarez, Niki Kilbertus et al.

Climate models play a critical role in understanding and projecting climate change. Due to their complexity, their horizontal resolution of about 40-100 km remains too coarse to resolve processes such as clouds and convection, which need to be approximated via parameterizations. These parameterizations are a major source of systematic errors and large uncertainties in climate projections. Deep learning (DL)-based parameterizations, trained on data from computationally expensive short, high-resolution simulations, have shown great promise for improving climate models in that regard. However, their lack of interpretability and tendency to learn spurious non-physical correlations result in reduced trust in the climate simulation. We propose an efficient supervised learning framework for DL-based parameterizations that leads to physically consistent models with improved interpretability and negligible computational overhead compared to standard supervised training. First, key features determining the target physical processes are uncovered. Subsequently, the neural network is fine-tuned using only those relevant features. We show empirically that our method robustly identifies a small subset of the inputs as actual physical drivers, therefore removing spurious non-physical relationships. This results in by design physically consistent and interpretable neural networks while maintaining the predictive performance of unconstrained black-box DL-based parameterizations.

Zhufeng Li, Sandeep S Cranganore, Nicholas Youngblut et al.

Leveraging the vast genetic diversity within microbiomes offers unparalleled insights into complex phenotypes, yet the task of accurately predicting and understanding such traits from genomic data remains challenging. We propose a framework taking advantage of existing large models for gene vectorization to predict habitat specificity from entire microbial genome sequences. Based on our model, we develop attribution techniques to elucidate gene interaction effects that drive microbial adaptation to diverse environments. We train and validate our approach on a large dataset of high quality microbiome genomes from different habitats. We not only demonstrate solid predictive performance, but also how sequence-level information of entire genomes allows us to identify gene associations underlying complex phenotypes. Our attribution recovers known important interaction networks and proposes new candidates for experimental follow up.

Kirtan Padh, Jakob Zeitler, David Watson et al.

Causal effect estimation is important for many tasks in the natural and social sciences. We design algorithms for the continuous partial identification problem: bounding the effects of multivariate, continuous treatments when unmeasured confounding makes identification impossible. Specifically, we cast causal effects as objective functions within a constrained optimization problem, and minimize/maximize these functions to obtain bounds. We combine flexible learning algorithms with Monte Carlo methods to implement a family of solutions under the name of stochastic causal programming. In particular, we show how the generic framework can be efficiently formulated in settings where auxiliary variables are clustered into pre-treatment and post-treatment sets, where no fine-grained causal graph can be easily specified. In these settings, we can avoid the need for fully specifying the distribution family of hidden common causes. Monte Carlo computation is also much simplified, leading to algorithms which are more computationally stable against alternatives.

Jiri Hron, Karl Krauth, Michael I. Jordan et al.

Thanks to their scalability, two-stage recommenders are used by many of today's largest online platforms, including YouTube, LinkedIn, and Pinterest. These systems produce recommendations in two steps: (i) multiple nominators, tuned for low prediction latency, preselect a small subset of candidates from the whole item pool; (ii) a slower but more accurate ranker further narrows down the nominated items, and serves to the user. Despite their popularity, the literature on two-stage recommenders is relatively scarce, and the algorithms are often treated as mere sums of their parts. Such treatment presupposes that the two-stage performance is explained by the behavior of the individual components in isolation. This is not the case: using synthetic and real-world data, we demonstrate that interactions between the ranker and the nominators substantially affect the overall performance. Motivated by these findings, we derive a generalization lower bound which shows that independent nominator training can lead to performance on par with uniformly random recommendations. We find that careful design of item pools, each assigned to a different nominator, alleviates these issues. As manual search for a good pool allocation is difficult, we propose to learn one instead using a Mixture-of-Experts based approach. This significantly improves both precision and recall at K.

Hananeh Aliee, Fabian J. Theis, Niki Kilbertus

Spurred by tremendous success in pattern matching and prediction tasks, researchers increasingly resort to machine learning to aid original scientific discovery. Given large amounts of observational data about a system, can we uncover the rules that govern its evolution? Solving this task holds the great promise of fully understanding the causal interactions and being able to make reliable predictions about the system's behavior under interventions. We take a step towards answering this question for time-series data generated from systems of ordinary differential equations (ODEs). While the governing ODEs might not be identifiable from data alone, we show that combining simple regularization schemes with flexible neural ODEs can robustly recover the dynamics and causal structures from time-series data. Our results on a variety of (non)-linear first and second order systems as well as real data validate our method. We conclude by showing that we can also make accurate predictions under interventions on variables or the system itself.

Elisabeth Ailer, Christian L. Müller, Niki Kilbertus

Many scientific datasets are compositional in nature. Important biological examples include species abundances in ecology, cell-type compositions derived from single-cell sequencing data, and amplicon abundance data in microbiome research. Here, we provide a causal view on compositional data in an instrumental variable setting where the composition acts as the cause. First, we crisply articulate potential pitfalls for practitioners regarding the interpretation of compositional causes from the viewpoint of interventions and warn against attributing causal meaning to common summary statistics such as diversity indices in microbiome data analysis. We then advocate for and develop multivariate methods using statistical data transformations and regression techniques that take the special structure of the compositional sample space into account while still yielding scientifically interpretable results. In a comparative analysis on synthetic and real microbiome data we show the advantages and limitations of our proposal. We posit that our analysis provides a useful framework and guidance for valid and informative cause-effect estimation in the context of compositional data.

Niki Kilbertus

This thesis scrutinizes common assumptions underlying traditional machine learning approaches to fairness in consequential decision making. After challenging the validity of these assumptions in real-world applications, we propose ways to move forward when they are violated. First, we show that group fairness criteria purely based on statistical properties of observed data are fundamentally limited. Revisiting this limitation from a causal viewpoint we develop a more versatile conceptual framework, causal fairness criteria, and first algorithms to achieve them. We also provide tools to analyze how sensitive a believed-to-be causally fair algorithm is to misspecifications of the causal graph. Second, we overcome the assumption that sensitive data is readily available in practice. To this end we devise protocols based on secure multi-party computation to train, validate, and contest fair decision algorithms without requiring users to disclose their sensitive data or decision makers to disclose their models. Finally, we also accommodate the fact that outcome labels are often only observed when a certain decision has been made. We suggest a paradigm shift away from training predictive models towards directly learning decisions to relax the traditional assumption that labels can always be recorded. The main contribution of this thesis is the development of theoretically substantiated and practically feasible methods to move research on fair machine learning closer to real-world applications.

Jiri Hron, Karl Krauth, Michael I. Jordan et al.

Two-stage recommender systems are widely adopted in industry due to their scalability and maintainability. These systems produce recommendations in two steps: (i) multiple nominators preselect a small number of items from a large pool using cheap-to-compute item embeddings; (ii) with a richer set of features, a ranker rearranges the nominated items and serves them to the user. A key challenge of this setup is that optimal performance of each stage in isolation does not imply optimal global performance. In response to this issue, Ma et al. (2020) proposed a nominator training objective importance weighted by the ranker's probability of recommending each item. In this work, we focus on the complementary issue of exploration. Modeled as a contextual bandit problem, we find LinUCB (a near optimal exploration strategy for single-stage systems) may lead to linear regret when deployed in two-stage recommenders. We therefore propose a method of synchronising the exploration strategies between the ranker and the nominators. Our algorithm only relies on quantities already computed by standard LinUCB at each stage and can be implemented in three lines of additional code. We end by demonstrating the effectiveness of our algorithm experimentally.

Frederik Träuble, Elliot Creager, Niki Kilbertus et al.

The focus of disentanglement approaches has been on identifying independent factors of variation in data. However, the causal variables underlying real-world observations are often not statistically independent. In this work, we bridge the gap to real-world scenarios by analyzing the behavior of the most prominent disentanglement approaches on correlated data in a large-scale empirical study (including 4260 models). We show and quantify that systematically induced correlations in the dataset are being learned and reflected in the latent representations, which has implications for downstream applications of disentanglement such as fairness. We also demonstrate how to resolve these latent correlations, either using weak supervision during training or by post-hoc correcting a pre-trained model with a small number of labels.

Niki Kilbertus, Matt J. Kusner, Ricardo Silva

Causal treatment effect estimation is a key problem that arises in a variety of real-world settings, from personalized medicine to governmental policy making. There has been a flurry of recent work in machine learning on estimating causal effects when one has access to an instrument. However, to achieve identifiability, they in general require one-size-fits-all assumptions such as an additive error model for the outcome. An alternative is partial identification, which provides bounds on the causal effect. Little exists in terms of bounding methods that can deal with the most general case, where the treatment itself can be continuous. Moreover, bounding methods generally do not allow for a continuum of assumptions on the shape of the causal effect that can smoothly trade off stronger background knowledge for more informative bounds. In this work, we provide a method for causal effect bounding in continuous distributions, leveraging recent advances in gradient-based methods for the optimization of computationally intractable objective functions. We demonstrate on a set of synthetic and real-world data that our bounds capture the causal effect when additive methods fail, providing a useful range of answers compatible with observation as opposed to relying on unwarranted structural assumptions.

Niki Kilbertus, Philip J. Ball, Matt J. Kusner et al.

Causal approaches to fairness have seen substantial recent interest, both from the machine learning community and from wider parties interested in ethical prediction algorithms. In no small part, this has been due to the fact that causal models allow one to simultaneously leverage data and expert knowledge to remove discriminatory effects from predictions. However, one of the primary assumptions in causal modeling is that you know the causal graph. This introduces a new opportunity for bias, caused by misspecifying the causal model. One common way for misspecification to occur is via unmeasured confounding: the true causal effect between variables is partially described by unobserved quantities. In this work we design tools to assess the sensitivity of fairness measures to this confounding for the popular class of non-linear additive noise models (ANMs). Specifically, we give a procedure for computing the maximum difference between two counterfactually fair predictors, where one has become biased due to confounding. For the case of bivariate confounding our technique can be swiftly computed via a sequence of closed-form updates. For multivariate confounding we give an algorithm that can be efficiently solved via automatic differentiation. We demonstrate our new sensitivity analysis tools in real-world fairness scenarios to assess the bias arising from confounding.

Timothy D. Gebhard, Niki Kilbertus, Ian Harry et al.

In the last few years, machine learning techniques, in particular convolutional neural networks, have been investigated as a method to replace or complement traditional matched filtering techniques that are used to detect the gravitational-wave signature of merging black holes. However, to date, these methods have not yet been successfully applied to the analysis of long stretches of data recorded by the Advanced LIGO and Virgo gravitational-wave observatories. In this work, we critically examine the use of convolutional neural networks as a tool to search for merging black holes. We identify the strengths and limitations of this approach, highlight some common pitfalls in translating between machine learning and gravitational-wave astronomy, and discuss the interdisciplinary challenges. In particular, we explain in detail why convolutional neural networks alone cannot be used to claim a statistically significant gravitational-wave detection. However, we demonstrate how they can still be used to rapidly flag the times of potential signals in the data for a more detailed follow-up. Our convolutional neural network architecture as well as the proposed performance metrics are better suited for this task than a standard binary classifications scheme. A detailed evaluation of our approach on Advanced LIGO data demonstrates the potential of such systems as trigger generators. Finally, we sound a note of caution by constructing adversarial examples, which showcase interesting "failure modes" of our model, where inputs with no visible resemblance to real gravitational-wave signals are identified as such by the network with high confidence.

Niki Kilbertus, Manuel Gomez-Rodriguez, Bernhard Schölkopf et al.

Consequential decisions are increasingly informed by sophisticated data-driven predictive models. However, to consistently learn accurate predictive models, one needs access to ground truth labels. Unfortunately, in practice, labels may only exist conditional on certain decisions---if a loan is denied, there is not even an option for the individual to pay back the loan. Hence, the observed data distribution depends on how decisions are being made. In this paper, we show that in this selective labels setting, learning a predictor directly only from available labeled data is suboptimal in terms of both fairness and utility. To avoid this undesirable behavior, we propose to directly learn decision policies that maximize utility under fairness constraints and thereby take into account how decisions affect which data is observed in the future. Our results suggest the need for a paradigm shift in the context of fair machine learning from the currently prevalent idea of simply building predictive models from a single static dataset via risk minimization, to a more interactive notion of "learning to decide". In particular, such policies should not entirely neglect part of the input space, drawing connections to explore/exploit tradeoffs in reinforcement learning, data missingness, and potential outcomes in causal inference. Experiments on synthetic and real-world data illustrate the favorable properties of learning to decide in terms of utility and fairness.

Niki Kilbertus, Giambattista Parascandolo, Bernhard Schölkopf

The ability to learn and act in novel situations is still a prerogative of animate intelligence, as current machine learning methods mostly fail when moving beyond the standard i.i.d. setting. What is the reason for this discrepancy? Most machine learning tasks are anti-causal, i.e., we infer causes (labels) from effects (observations). Typically, in supervised learning we build systems that try to directly invert causal mechanisms. Instead, in this paper we argue that strong generalization capabilities crucially hinge on searching and validating meaningful hypotheses, requiring access to a causal model. In such a framework, we want to find a cause that leads to the observed effect. Anti-causal models are used to drive this search, but a causal model is required for validation. We investigate the fundamental differences between causal and anti-causal tasks, discuss implications for topics ranging from adversarial attacks to disentangling factors of variation, and provide extensive evidence from the literature to substantiate our view. We advocate for incorporating causal models in supervised learning to shift the paradigm from inference only, to search and validation.

Niki Kilbertus, Adrià Gascón, Matt J. Kusner et al.

Recent work has explored how to train machine learning models which do not discriminate against any subgroup of the population as determined by sensitive attributes such as gender or race. To avoid disparate treatment, sensitive attributes should not be considered. On the other hand, in order to avoid disparate impact, sensitive attributes must be examined, e.g., in order to learn a fair model, or to check if a given model is fair. We introduce methods from secure multi-party computation which allow us to avoid both. By encrypting sensitive attributes, we show how an outcome-based fair model may be learned, checked, or have its outputs verified and held to account, without users revealing their sensitive attributes.

Giambattista Parascandolo, Niki Kilbertus, Mateo Rojas-Carulla et al.

Statistical learning relies upon data sampled from a distribution, and we usually do not care what actually generated it in the first place. From the point of view of causal modeling, the structure of each distribution is induced by physical mechanisms that give rise to dependences between observables. Mechanisms, however, can be meaningful autonomous modules of generative models that make sense beyond a particular entailed data distribution, lending themselves to transfer between problems. We develop an algorithm to recover a set of independent (inverse) mechanisms from a set of transformed data points. The approach is unsupervised and based on a set of experts that compete for data generated by the mechanisms, driving specialization. We analyze the proposed method in a series of experiments on image data. Each expert learns to map a subset of the transformed data back to a reference distribution. The learned mechanisms generalize to novel domains. We discuss implications for transfer learning and links to recent trends in generative modeling.

Niki Kilbertus, Mateo Rojas-Carulla, Giambattista Parascandolo et al.

Recent work on fairness in machine learning has focused on various statistical discrimination criteria and how they trade off. Most of these criteria are observational: They depend only on the joint distribution of predictor, protected attribute, features, and outcome. While convenient to work with, observational criteria have severe inherent limitations that prevent them from resolving matters of fairness conclusively. Going beyond observational criteria, we frame the problem of discrimination based on protected attributes in the language of causal reasoning. This viewpoint shifts attention from "What is the right fairness criterion?" to "What do we want to assume about the causal data generating process?" Through the lens of causality, we make several contributions. First, we crisply articulate why and when observational criteria fail, thus formalizing what was before a matter of opinion. Second, our approach exposes previously ignored subtleties and why they are fundamental to the problem. Finally, we put forward natural causal non-discrimination criteria and develop algorithms that satisfy them.