Adjustment for Confounding using Pre-Trained Representations

There is growing interest in extending average treatment effect (ATE) estimation to incorporate non-tabular data, such as images and text, which may act as sources of confounding. Neglecting these effects risks biased results and flawed scientific conclusions. However, incorporating non-tabular data necessitates sophisticated feature extractors, often in combination with ideas of transfer learning. In this work, we investigate how latent features from pre-trained neural networks can be leveraged to adjust for sources of confounding. We formalize conditions under which these latent features enable valid adjustment and statistical inference in ATE estimation, demonstrating results along the example of double machine learning. We discuss critical challenges inherent to latent feature learning and downstream parameter estimation arising from the high dimensionality and non-identifiability of representations. Common structural assumptions for obtaining fast convergence rates with additive or sparse linear models are shown to be unrealistic for latent features. We argue, however, that neural networks are largely insensitive to these issues. In particular, we show that neural networks can achieve fast convergence rates by adapting to intrinsic notions of sparsity and dimension of the learning problem.

Vine Copulas as Differentiable Computational Graphs

Vine copulas are sophisticated models for multivariate distributions and are increasingly used in machine learning. To facilitate their integration into modern ML pipelines, we introduce the vine computational graph, a DAG that abstracts the multilevel vine structure and associated computations. On this foundation, we devise new algorithms for conditional sampling, efficient sampling-order scheduling, and constructing vine structures for customized conditioning variables. We implement these ideas in torchvinecopulib, a GPU-accelerated Python library built upon PyTorch, delivering improved scalability for fitting, sampling, and density evaluation. Our experiments illustrate how gradient flowing through the vine can improve Vine Copula Autoencoders and that incorporating vines for uncertainty quantification in deep learning can outperform MC-dropout, deep ensembles, and Bayesian Neural Networks in sharpness, calibration, and runtime. By recasting vine copula models as computational graphs, our work connects classical dependence modeling with modern deep-learning toolchains and facilitates the integration of state-of-the-art copula methods in modern machine learning pipelines.

Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginals

Density regression models allow a comprehensive understanding of data by modeling the complete conditional probability distribution. While flexible estimation approaches such as normalizing flows (NF) work particularly well in multiple dimensions, interpreting the input-output relationship of such models is often difficult, due to the black-box character of deep learning models. In contrast, existing statistical methods for multivariate outcomes such as multivariate conditional transformation models (MCTM) are restricted in flexibility and are often not expressive enough to represent complex multivariate probability distributions. In this paper, we combine MCTM with state-of-the-art and autoregressive NF to leverage the transparency of MCTM for modeling interpretable feature effects on the marginal distributions in the first step and the flexibility of neural-network-based NF techniques to account for complex and non-linear relationships in the joint data distribution. We demonstrate our method's versatility in various numerical experiments and compare it with MCTM and other NF models on both simulated and real-world data.

Uncertainty Quantification for Prior-Data Fitted Networks using Martingale Posteriors

Prior-data fitted networks (PFNs) have emerged as promising foundation models for prediction from tabular data sets, achieving state-of-the-art performance on small to moderate data sizes without tuning. While PFNs are motivated by Bayesian ideas, they do not provide any uncertainty quantification for predictive means, quantiles, or similar quantities. We propose a principled and efficient sampling procedure to construct Bayesian posteriors for such estimates based on Martingale posteriors, and prove its convergence. Several simulated and real-world data examples showcase the uncertainty quantification of our method in inference applications.

Constructing Confidence Intervals for 'the' Generalization Error -- a Comprehensive Benchmark Study

When assessing the quality of prediction models in machine learning, confidence intervals (CIs) for the generalization error, which measures predictive performance, are a crucial tool. Luckily, there exist many methods for computing such CIs and new promising approaches are continuously being proposed. Typically, these methods combine various resampling procedures, most popular among them cross-validation and bootstrapping, with different variance estimation techniques. Unfortunately, however, there is currently no consensus on when any of these combinations may be most reliably employed and how they generally compare. In this work, we conduct the first large-scale study comparing CIs for the generalization error - empirically evaluating 13 different methods on a total of 18 tabular regression and classification problems, using four different inducers and a total of eight loss functions. We give an overview of the methodological foundations and inherent challenges of constructing CIs for the generalization error and provide a concise review of all 13 methods in a unified framework. Finally, the CI methods are evaluated in terms of their relative coverage frequency, width, and runtime. Based on these findings, we are able to identify a subset of methods that we would recommend. We also publish the datasets as a benchmarking suite on OpenML and our code on GitHub to serve as a basis for further studies.

Label-wise Aleatoric and Epistemic Uncertainty Quantification

We present a novel approach to uncertainty quantification in classification tasks based on label-wise decomposition of uncertainty measures. This label-wise perspective allows uncertainty to be quantified at the individual class level, thereby improving cost-sensitive decision-making and helping understand the sources of uncertainty. Furthermore, it allows to define total, aleatoric, and epistemic uncertainty on the basis of non-categorical measures such as variance, going beyond common entropy-based measures. In particular, variance-based measures address some of the limitations associated with established methods that have recently been discussed in the literature. We show that our proposed measures adhere to a number of desirable properties. Through empirical evaluation on a variety of benchmark data sets -- including applications in the medical domain where accurate uncertainty quantification is crucial -- we establish the effectiveness of label-wise uncertainty quantification.

Reshuffling Resampling Splits Can Improve Generalization of Hyperparameter Optimization

Hyperparameter optimization is crucial for obtaining peak performance of machine learning models. The standard protocol evaluates various hyperparameter configurations using a resampling estimate of the generalization error to guide optimization and select a final hyperparameter configuration. Without much evidence, paired resampling splits, i.e., either a fixed train-validation split or a fixed cross-validation scheme, are often recommended. We show that, surprisingly, reshuffling the splits for every configuration often improves the final model's generalization performance on unseen data. Our theoretical analysis explains how reshuffling affects the asymptotic behavior of the validation loss surface and provides a bound on the expected regret in the limiting regime. This bound connects the potential benefits of reshuffling to the signal and noise characteristics of the underlying optimization problem. We confirm our theoretical results in a controlled simulation study and demonstrate the practical usefulness of reshuffling in a large-scale, realistic hyperparameter optimization experiment. While reshuffling leads to test performances that are competitive with using fixed splits, it drastically improves results for a single train-validation holdout protocol and can often make holdout become competitive with standard CV while being computationally cheaper.

Generalizing Orthogonalization for Models with Non-linearities

The complexity of black-box algorithms can lead to various challenges, including the introduction of biases. These biases present immediate risks in the algorithms' application. It was, for instance, shown that neural networks can deduce racial information solely from a patient's X-ray scan, a task beyond the capability of medical experts. If this fact is not known to the medical expert, automatic decision-making based on this algorithm could lead to prescribing a treatment (purely) based on racial information. While current methodologies allow for the "orthogonalization" or "normalization" of neural networks with respect to such information, existing approaches are grounded in linear models. Our paper advances the discourse by introducing corrections for non-linearities such as ReLU activations. Our approach also encompasses scalar and tensor-valued predictions, facilitating its integration into neural network architectures. Through extensive experiments, we validate our method's effectiveness in safeguarding sensitive data in generalized linear models, normalizing convolutional neural networks for metadata, and rectifying pre-existing embeddings for undesired attributes.

Interpretable Machine Learning for TabPFN

The recently developed Prior-Data Fitted Networks (PFNs) have shown very promising results for applications in low-data regimes. The TabPFN model, a special case of PFNs for tabular data, is able to achieve state-of-the-art performance on a variety of classification tasks while producing posterior predictive distributions in mere seconds by in-context learning without the need for learning parameters or hyperparameter tuning. This makes TabPFN a very attractive option for a wide range of domain applications. However, a major drawback of the method is its lack of interpretability. Therefore, we propose several adaptations of popular interpretability methods that we specifically design for TabPFN. By taking advantage of the unique properties of the model, our adaptations allow for more efficient computations than existing implementations. In particular, we show how in-context learning facilitates the estimation of Shapley values by avoiding approximate retraining and enables the use of Leave-One-Covariate-Out (LOCO) even when working with large-scale Transformers. In addition, we demonstrate how data valuation methods can be used to address scalability challenges of TabPFN. Our proposed methods are implemented in a package tabpfn_iml and made available at https://github.com/david-rundel/tabpfn_iml.

Second-Order Uncertainty Quantification: Variance-Based Measures

Uncertainty quantification is a critical aspect of machine learning models, providing important insights into the reliability of predictions and aiding the decision-making process in real-world applications. This paper proposes a novel way to use variance-based measures to quantify uncertainty on the basis of second-order distributions in classification problems. A distinctive feature of the measures is the ability to reason about uncertainties on a class-based level, which is useful in situations where nuanced decision-making is required. Recalling some properties from the literature, we highlight that the variance-based measures satisfy important (axiomatic) properties. In addition to this axiomatic approach, we present empirical results showing the measures to be effective and competitive to commonly used entropy-based measures.