Nicolai Meinshausen

Andrew Jesson, Alyson Douglas, Peter Manshausen et al.

Estimating the effects of continuous-valued interventions from observational data is a critically important task for climate science, healthcare, and economics. Recent work focuses on designing neural network architectures and regularization functions to allow for scalable estimation of average and individual-level dose-response curves from high-dimensional, large-sample data. Such methodologies assume ignorability (observation of all confounding variables) and positivity (observation of all treatment levels for every covariate value describing a set of units), assumptions problematic in the continuous treatment regime. Scalable sensitivity and uncertainty analyses to understand the ignorance induced in causal estimates when these assumptions are relaxed are less studied. Here, we develop a continuous treatment-effect marginal sensitivity model (CMSM) and derive bounds that agree with the observed data and a researcher-defined level of hidden confounding. We introduce a scalable algorithm and uncertainty-aware deep models to derive and estimate these bounds for high-dimensional, large-sample observational data. We work in concert with climate scientists interested in the climatological impacts of human emissions on cloud properties using satellite observations from the past 15 years. This problem is known to be complicated by many unobserved confounders.

Xinwei Shen, Nicolai Meinshausen

Distributional regression aims to estimate the full conditional distribution of a target variable, given covariates. Popular methods include linear and tree-ensemble based quantile regression. We propose a neural network-based distributional regression methodology called `engression'. An engression model is generative in the sense that we can sample from the fitted conditional distribution and is also suitable for high-dimensional outcomes. Furthermore, we find that modelling the conditional distribution on training data can constrain the fitted function outside of the training support, which offers a new perspective to the challenging extrapolation problem in nonlinear regression. In particular, for `pre-additive noise' models, where noise is added to the covariates before applying a nonlinear transformation, we show that engression can successfully perform extrapolation under some assumptions such as monotonicity, whereas traditional regression approaches such as least-squares or quantile regression fall short under the same assumptions. Our empirical results, from both simulated and real data, validate the effectiveness of the engression method and indicate that the pre-additive noise model is typically suitable for many real-world scenarios. The software implementations of engression are available in both R and Python.

Sorawit Saengkyongam, Juan L. Gamella, Andrew C. Miller et al. · eth-zurich

The problem of domain generalization concerns learning predictive models that are robust to distribution shifts when deployed in new, previously unseen environments. Existing methods typically require labeled data from multiple training environments, limiting their applicability when labeled data are scarce. In this work, we study domain generalization in an anti-causal setting, where the outcome causes the observed covariates. Under this structure, environment perturbations that affect the covariates do not propagate to the outcome, which motivates regularizing the model's sensitivity to these perturbations. Crucially, estimating these perturbation directions does not require labels, enabling us to leverage unlabeled data from multiple environments. We propose two methods that penalize the model's sensitivity to variations in the mean and covariance of the covariates across environments, respectively, and prove that these methods have worst-case optimality guarantees under certain classes of environments. Finally, we demonstrate the empirical performance of our approach on a controlled physical system and a physiological signal dataset.

Enikő Székely, Sebastian Sippel, Nicolai Meinshausen et al.

Fingerprints are key tools in climate change detection and attribution (D&A) that are used to determine whether changes in observations are different from internal climate variability (detection), and whether observed changes can be assigned to specific external drivers (attribution). We propose a direct D&A approach based on supervised learning to extract fingerprints that lead to robust predictions under relevant interventions on exogenous variables, i.e., climate drivers other than the target. We employ anchor regression, a distributionally-robust statistical learning method inspired by causal inference that extrapolates well to perturbed data under the interventions considered. The residuals from the prediction achieve either uncorrelatedness or mean independence with the exogenous variables, thus guaranteeing robustness. We define D&A as a unified hypothesis testing framework that relies on the same statistical model but uses different targets and test statistics. In the experiments, we first show that the CO2 forcing can be robustly predicted from temperature spatial patterns under strong interventions on the solar forcing. Second, we illustrate attribution to the greenhouse gases and aerosols while protecting against interventions on the aerosols and CO2 forcing, respectively. Our study shows that incorporating robustness constraints against relevant interventions may significantly benefit detection and attribution of climate change.

Domagoj Ćevid, Loris Michel, Jeffrey Näf et al.

Random Forest (Breiman, 2001) is a successful and widely used regression and classification algorithm. Part of its appeal and reason for its versatility is its (implicit) construction of a kernel-type weighting function on training data, which can also be used for targets other than the original mean estimation. We propose a novel forest construction for multivariate responses based on their joint conditional distribution, independent of the estimation target and the data model. It uses a new splitting criterion based on the MMD distributional metric, which is suitable for detecting heterogeneity in multivariate distributions. The induced weights define an estimate of the full conditional distribution, which in turn can be used for arbitrary and potentially complicated targets of interest. The method is very versatile and convenient to use, as we illustrate on a wide range of examples. The code is available as Python and R packages drf.

Xinwei Shen, Nicolai Meinshausen, Tong Zhang

Learning complex distributions is a fundamental challenge in contemporary applications. Shen and Meinshausen (2024) introduced engression, a generative approach based on scoring rules that maps noise (and covariates, if available) directly to data. While effective, engression can struggle with highly complex distributions, such as those encountered in image data. In this work, we propose reverse Markov learning (RML), a framework that defines a general forward process transitioning from the target distribution to a known distribution (e.g., Gaussian) and then learns a reverse Markov process using multiple engression models. This reverse process reconstructs the target distribution step by step. This framework accommodates general forward processes, allows for dimension reduction, and naturally discretizes the generative process. In the special case of diffusion-based forward processes, RML provides an efficient discretization strategy for both training and inference in diffusion models. We further introduce an alternating sampling scheme to enhance post-training performance. Our statistical analysis establishes error bounds for RML and elucidates its advantages in estimation efficiency and flexibility in forward process design. Empirical results on simulated and climate data corroborate the theoretical findings, demonstrating the effectiveness of RML in capturing complex distributions.

Xinwei Shen, Nicolai Meinshausen

Dimension reduction techniques usually lose information in the sense that reconstructed data are not identical to the original data. However, we argue that it is possible to have reconstructed data identically distributed as the original data, irrespective of the retained dimension or the specific mapping. This can be achieved by learning a distributional model that matches the conditional distribution of data given its low-dimensional latent variables. Motivated by this, we propose Distributional Principal Autoencoder (DPA) that consists of an encoder that maps high-dimensional data to low-dimensional latent variables and a decoder that maps the latent variables back to the data space. For reducing the dimension, the DPA encoder aims to minimise the unexplained variability of the data with an adaptive choice of the latent dimension. For reconstructing data, the DPA decoder aims to match the conditional distribution of all data that are mapped to a certain latent value, thus ensuring that the reconstructed data retains the original data distribution. Our numerical results on climate data, single-cell data, and image benchmarks demonstrate the practical feasibility and success of the approach in reconstructing the original distribution of the data. DPA embeddings are shown to preserve meaningful structures of data such as the seasonal cycle for precipitations and cell types for gene expression.

Julius von Kügelgen, Jakob Ketterer, Xinwei Shen et al.

We consider the problem of modelling the effects of unseen perturbations such as gene knockdowns or drug combinations on low-level measurements such as RNA sequencing data. Specifically, given data collected under some perturbations, we aim to predict the distribution of measurements for new perturbations. To address this challenging extrapolation task, we posit that perturbations act additively in a suitable, unknown embedding space. More precisely, we formulate the generative process underlying the observed data as a latent variable model, in which perturbations amount to mean shifts in latent space and can be combined additively. Unlike previous work, we prove that, given sufficiently diverse training perturbations, the representation and perturbation effects are identifiable up to affine transformation, and use this to characterize the class of unseen perturbations for which we obtain extrapolation guarantees. To estimate the model from data, we propose a new method, the perturbation distribution autoencoder (PDAE), which is trained by maximising the distributional similarity between true and predicted perturbation distributions. The trained model can then be used to predict previously unseen perturbation distributions. Empirical evidence suggests that PDAE compares favourably to existing methods and baselines at predicting the effects of unseen perturbations.

Drago Plečko, Nicolas Bennett, Nicolai Meinshausen

Machine learning algorithms are useful for various predictions tasks, but they can also learn how to discriminate, based on gender, race or other sensitive attributes. This realization gave rise to the field of fair machine learning, which aims to measure and mitigate such algorithmic bias. This manuscript describes the R-package fairadapt, which implements a causal inference pre-processing method. By making use of a causal graphical model and the observed data, the method can be used to address hypothetical questions of the form "What would my salary have been, had I been of a different gender/race?". Such individual level counterfactual reasoning can help eliminate discrimination and help justify fair decisions. We also discuss appropriate relaxations which assume certain causal pathways from the sensitive attribute to the outcome are not discriminatory.

Michael Moor, Nicolas Bennet, Drago Plecko et al.

Despite decades of clinical research, sepsis remains a global public health crisis with high mortality, and morbidity. Currently, when sepsis is detected and the underlying pathogen is identified, organ damage may have already progressed to irreversible stages. Effective sepsis management is therefore highly time-sensitive. By systematically analysing trends in the plethora of clinical data available in the intensive care unit (ICU), an early prediction of sepsis could lead to earlier pathogen identification, resistance testing, and effective antibiotic and supportive treatment, and thereby become a life-saving measure. Here, we developed and validated a machine learning (ML) system for the prediction of sepsis in the ICU. Our analysis represents the largest multi-national, multi-centre in-ICU study for sepsis prediction using ML to date. Our dataset contains $156,309$ unique ICU admissions, which represent a refined and harmonised subset of five large ICU databases originating from three countries. Using the international consensus definition Sepsis-3, we derived hourly-resolved sepsis label annotations, amounting to $26,734$ ($17.1\%$) septic stays. We compared our approach, a deep self-attention model, to several clinical baselines as well as ML baselines and performed an extensive internal and external validation within and across databases. On average, our model was able to predict sepsis with an AUROC of $0.847 \pm 0.050$ (internal out-of sample validation) and $0.761 \pm 0.052$ (external validation). For a harmonised prevalence of $17\%$, at $80\%$ recall our model detects septic patients with $39\%$ precision 3.7 hours in advance.

Drago Plečko, Nicolai Meinshausen

Fairness of classification and regression has received much attention recently and various, partially non-compatible, criteria have been proposed. The fairness criteria can be enforced for a given classifier or, alternatively, the data can be adapated to ensure that every classifier trained on the data will adhere to desired fairness criteria. We present a practical data adaption method based on quantile preservation in causal structural equation models. The data adaptation is based on a presumed counterfactual model for the data. While the counterfactual model itself cannot be verified experimentally, we show that certain population notions of fairness are still guaranteed even if the counterfactual model is misspecified. The precise nature of the fulfilled non-causal fairness notion (such as demographic parity, separation or sufficiency) depends on the structure of the underlying causal model and the choice of resolving variables. We describe an implementation of the proposed data adaptation procedure based on Random Forests and demonstrate its practical use on simulated and real-world data.

Christina Heinze-Deml, Nicolai Meinshausen

When training a deep neural network for image classification, one can broadly distinguish between two types of latent features of images that will drive the classification. We can divide latent features into (i) "core" or "conditionally invariant" features $X^\text{core}$ whose distribution $X^\text{core}\vert Y$, conditional on the class $Y$, does not change substantially across domains and (ii) "style" features $X^{\text{style}}$ whose distribution $X^{\text{style}} \vert Y$ can change substantially across domains. Examples for style features include position, rotation, image quality or brightness but also more complex ones like hair color, image quality or posture for images of persons. Our goal is to minimize a loss that is robust under changes in the distribution of these style features. In contrast to previous work, we assume that the domain itself is not observed and hence a latent variable. We do assume that we can sometimes observe a typically discrete identifier or "$\mathrm{ID}$ variable". In some applications we know, for example, that two images show the same person, and $\mathrm{ID}$ then refers to the identity of the person. The proposed method requires only a small fraction of images to have $\mathrm{ID}$ information. We group observations if they share the same class and identifier $(Y,\mathrm{ID})=(y,\mathrm{id})$ and penalize the conditional variance of the prediction or the loss if we condition on $(Y,\mathrm{ID})$. Using a causal framework, this conditional variance regularization (CoRe) is shown to protect asymptotically against shifts in the distribution of the style variables. Empirically, we show that the CoRe penalty improves predictive accuracy substantially in settings where domain changes occur in terms of image quality, brightness and color while we also look at more complex changes such as changes in movement and posture.

Christina Heinze-Deml, Brian McWilliams, Nicolai Meinshausen

Privacy is crucial in many applications of machine learning. Legal, ethical and societal issues restrict the sharing of sensitive data making it difficult to learn from datasets that are partitioned between many parties. One important instance of such a distributed setting arises when information about each record in the dataset is held by different data owners (the design matrix is "vertically-partitioned"). In this setting few approaches exist for private data sharing for the purposes of statistical estimation and the classical setup of differential privacy with a "trusted curator" preparing the data does not apply. We work with the notion of $(ε,δ)$-distributed differential privacy which extends single-party differential privacy to the distributed, vertically-partitioned case. We propose PriDE, a scalable framework for distributed estimation where each party communicates perturbed random projections of their locally held features ensuring $(ε,δ)$-distributed differential privacy is preserved. For $\ell_2$-penalized supervised learning problems PriDE has bounded estimation error compared with the optimal estimates obtained without privacy constraints in the non-distributed setting. We confirm this empirically on real world and synthetic datasets.

Gabriel Krummenacher, Brian McWilliams, Yannic Kilcher et al.

Adaptive stochastic gradient methods such as AdaGrad have gained popularity in particular for training deep neural networks. The most commonly used and studied variant maintains a diagonal matrix approximation to second order information by accumulating past gradients which are used to tune the step size adaptively. In certain situations the full-matrix variant of AdaGrad is expected to attain better performance, however in high dimensions it is computationally impractical. We present Ada-LR and RadaGrad two computationally efficient approximations to full-matrix AdaGrad based on randomized dimensionality reduction. They are able to capture dependencies between features and achieve similar performance to full-matrix AdaGrad but at a much smaller computational cost. We show that the regret of Ada-LR is close to the regret of full-matrix AdaGrad which can have an up-to exponentially smaller dependence on the dimension than the diagonal variant. Empirically, we show that Ada-LR and RadaGrad perform similarly to full-matrix AdaGrad. On the task of training convolutional neural networks as well as recurrent neural networks, RadaGrad achieves faster convergence than diagonal AdaGrad.

Gian-Andrea Thanei, Nicolai Meinshausen, Rajen D. Shah

When performing regression on a dataset with $p$ variables, it is often of interest to go beyond using main linear effects and include interactions as products between individual variables. For small-scale problems, these interactions can be computed explicitly but this leads to a computational complexity of at least $\mathcal{O}(p^2)$ if done naively. This cost can be prohibitive if $p$ is very large. We introduce a new randomised algorithm that is able to discover interactions with high probability and under mild conditions has a runtime that is subquadratic in $p$. We show that strong interactions can be discovered in almost linear time, whilst finding weaker interactions requires $\mathcal{O}(p^α)$ operations for $1 < α< 2$ depending on their strength. The underlying idea is to transform interaction search into a closestpair problem which can be solved efficiently in subquadratic time. The algorithm is called $\mathit{xyz}$ and is implemented in the language R. We demonstrate its efficiency for application to genome-wide association studies, where more than $10^{11}$ interactions can be screened in under $280$ seconds with a single-core $1.2$ GHz CPU.

Christina Heinze, Brian McWilliams, Nicolai Meinshausen

We present DUAL-LOCO, a communication-efficient algorithm for distributed statistical estimation. DUAL-LOCO assumes that the data is distributed according to the features rather than the samples. It requires only a single round of communication where low-dimensional random projections are used to approximate the dependences between features available to different workers. We show that DUAL-LOCO has bounded approximation error which only depends weakly on the number of workers. We compare DUAL-LOCO against a state-of-the-art distributed optimization method on a variety of real world datasets and show that it obtains better speedups while retaining good accuracy.

Dominik Rothenhäusler, Christina Heinze, Jonas Peters et al.

We propose a simple method to learn linear causal cyclic models in the presence of latent variables. The method relies on equilibrium data of the model recorded under a specific kind of interventions ("shift interventions"). The location and strength of these interventions do not have to be known and can be estimated from the data. Our method, called backShift, only uses second moments of the data and performs simple joint matrix diagonalization, applied to differences between covariance matrices. We give a sufficient and necessary condition for identifiability of the system, which is fulfilled almost surely under some quite general assumptions if and only if there are at least three distinct experimental settings, one of which can be pure observational data. We demonstrate the performance on some simulated data and applications in flow cytometry and financial time series. The code is made available as R-package backShift.

Jonas Peters, Peter Bühlmann, Nicolai Meinshausen

What is the difference of a prediction that is made with a causal model and a non-causal model? Suppose we intervene on the predictor variables or change the whole environment. The predictions from a causal model will in general work as well under interventions as for observational data. In contrast, predictions from a non-causal model can potentially be very wrong if we actively intervene on variables. Here, we propose to exploit this invariance of a prediction under a causal model for causal inference: given different experimental settings (for example various interventions) we collect all models that do show invariance in their predictive accuracy across settings and interventions. The causal model will be a member of this set of models with high probability. This approach yields valid confidence intervals for the causal relationships in quite general scenarios. We examine the example of structural equation models in more detail and provide sufficient assumptions under which the set of causal predictors becomes identifiable. We further investigate robustness properties of our approach under model misspecification and discuss possible extensions. The empirical properties are studied for various data sets, including large-scale gene perturbation experiments.

Christina Heinze, Brian McWilliams, Nicolai Meinshausen et al.

We propose LOCO, an algorithm for large-scale ridge regression which distributes the features across workers on a cluster. Important dependencies between variables are preserved using structured random projections which are cheap to compute and must only be communicated once. We show that LOCO obtains a solution which is close to the exact ridge regression solution in the fixed design setting. We verify this experimentally in a simulation study as well as an application to climate prediction. Furthermore, we show that LOCO achieves significant speedups compared with a state-of-the-art distributed algorithm on a large-scale regression problem.

Rajen D. Shah, Nicolai Meinshausen

Large-scale regression problems where both the number of variables, $p$, and the number of observations, $n$, may be large and in the order of millions or more, are becoming increasingly more common. Typically the data are sparse: only a fraction of a percent of the entries in the design matrix are non-zero. Nevertheless, often the only computationally feasible approach is to perform dimension reduction to obtain a new design matrix with far fewer columns and then work with this compressed data. $b$-bit min-wise hashing (Li and Konig, 2011) is a promising dimension reduction scheme for sparse matrices which produces a set of random features such that regression on the resulting design matrix approximates a kernel regression with the resemblance kernel. In this work, we derive bounds on the prediction error of such regressions. For both linear and logistic models we show that the average prediction error vanishes asymptotically as long as $q \|β^*\|_2^2 /n \rightarrow 0$, where $q$ is the average number of non-zero entries in each row of the design matrix and $β^*$ is the coefficient of the linear predictor. We also show that ordinary least squares or ridge regression applied to the reduced data can in fact allow us fit more flexible models. We obtain non-asymptotic prediction error bounds for interaction models and for models where an unknown row normalisation must be applied in order for the signal to be linear in the predictors.

Aurélie C. Lozano, Nicolai Meinshausen

We propose a minimum distance estimation method for robust regression in sparse high-dimensional settings. The traditional likelihood-based estimators lack resilience against outliers, a critical issue when dealing with high-dimensional noisy data. Our method, Minimum Distance Lasso (MD-Lasso), combines minimum distance functionals, customarily used in nonparametric estimation for their robustness, with l1-regularization for high-dimensional regression. The geometry of MD-Lasso is key to its consistency and robustness. The estimator is governed by a scaling parameter that caps the influence of outliers: the loss per observation is locally convex and close to quadratic for small squared residuals, and flattens for squared residuals larger than the scaling parameter. As the parameter approaches infinity, the estimator becomes equivalent to least-squares Lasso. MD-Lasso enjoys fast convergence rates under mild conditions on the model error distribution, which hold for any of the solutions in a convexity region around the true parameter and in certain cases for every solution. Remarkably, a first-order optimization method is able to produce iterates very close to the consistent solutions, with geometric convergence and regardless of the initialization. A connection is established with re-weighted least-squares that intuitively explains MD-Lasso robustness. The merits of our method are demonstrated through simulation and eQTL data analysis.

Rajen Dinesh Shah, Nicolai Meinshausen

Finding interactions between variables in large and high-dimensional datasets is often a serious computational challenge. Most approaches build up interaction sets incrementally, adding variables in a greedy fashion. The drawback is that potentially informative high-order interactions may be overlooked. Here, we propose at an alternative approach for classification problems with binary predictor variables, called Random Intersection Trees. It works by starting with a maximal interaction that includes all variables, and then gradually removing variables if they fail to appear in randomly chosen observations of a class of interest. We show that informative interactions are retained with high probability, and the computational complexity of our procedure is of order $p^κ$ for a value of $κ$ that can reach values as low as 1 for very sparse data; in many more general settings, it will still beat the exponent $s$ obtained when using a brute force search constrained to order $s$ interactions. In addition, by using some new ideas based on min-wise hash schemes, we are able to further reduce the computational cost. Interactions found by our algorithm can be used for predictive modelling in various forms, but they are also often of interest in their own right as useful characterisations of what distinguishes a certain class from others.