Sharpness Aware Minimization (SAM) enhances performance across various neural architectures and datasets. As models are continually scaled up to improve performance, a rigorous understanding of SAM's scaling behaviour is paramount. To this end, we study the infinite-width limit of neural networks trained with SAM, using the Tensor Programs framework. Our findings reveal that the dynamics of standard SAM effectively reduce to applying SAM solely in the last layer in wide neural networks, even with optimal hyperparameters. In contrast, we identify a stable parameterization with layerwise perturbation scaling, which we call $\textit{Maximal Update and Perturbation Parameterization}$ ($\mu$P$^2$), that ensures all layers are both feature learning and effectively perturbed in the limit. Through experiments with MLPs, ResNets and Vision Transformers, we empirically demonstrate that $\mu$P$^2$ is the first parameterization to achieve hyperparameter transfer of the joint optimum of learning rate and perturbation radius across model scales. Moreover, we provide an intuitive condition to derive $\mu$P$^2$ for other perturbation rules like Adaptive SAM and SAM-ON, also ensuring balanced perturbation effects across all layers.

Despite the growing popularity of explainable and interpretable machine learning, there is still surprisingly limited work on inherently interpretable clustering methods. Recently, there has been a surge of interest in explaining the classic k-means algorithm, leading to efficient algorithms that approximate k-means clusters using axis-aligned decision trees. However, interpretable variants of k-means have limited applicability in practice, where more flexible clustering methods are often needed to obtain useful partitions of the data. In this work, we investigate interpretable kernel clustering, and propose algorithms that construct decision trees to approximate the partitions induced by kernel k-means, a nonlinear extension of k-means. We further build on previous work on explainable k-means and demonstrate how a suitable choice of features allows preserving interpretability without sacrificing approximation guarantees on the interpretable model. The increasing predictive power of machine learning has made it a popular tool in many scientific fields. Sensitive applications such as healthcare or autonomous driving however require more than just good accuracy--it is also crucial for a model's decisions to be interpretable (Tjoa & Guan, 2020; Varshney & Alemzadeh, 2017). Unfortunately, popular machine learning models are not transparent and are often referred to as "black box" approaches. The demand for explainable machine learning has led to the development of several tools over the last few years, albeit mostly for supervised learning. Methods such as LIME or Shapley values (Ribeiro et al., 2016; Lundberg & Lee, 2017) are designed to explain the prediction of any given machine learning model.

As causal ground truth is incredibly rare, causal discovery algorithms are commonly only evaluated on simulated data. This is concerning, given that simulations reflect common preconceptions about generating processes regarding noise distributions, model classes, and more. In this work, we propose a novel method for falsifying the output of a causal discovery algorithm in the absence of ground truth. Our key insight is that while statistical learning seeks stability across subsets of data points, causal learning should seek stability across subsets of variables. Motivated by this insight, our method relies on a notion of compatibility between causal graphs learned on different subsets of variables. We prove that detecting incompatibilities can falsify wrongly inferred causal relations due to violation of assumptions or errors from finite sample effects. Although passing such compatibility tests is only a necessary criterion for good performance, we argue that it provides strong evidence for the causal models whenever compatibility entails strong implications for the joint distribution. We also demonstrate experimentally that detection of incompatibilities can aid in causal model selection.

If $X,Y,Z$ denote sets of random variables, two different data sources may contain samples from $P_{X,Y}$ and $P_{Y,Z}$, respectively. We argue that causal discovery can help inferring properties of the `unobserved joint distributions' $P_{X,Y,Z}$ or $P_{X,Z}$. The properties may be conditional independences (as in `integrative causal inference') or also quantitative statements about dependences. More generally, we define a learning scenario where the input is a subset of variables and the label is some statistical property of that subset. Sets of jointly observed variables define the training points, while unobserved sets are possible test points. To solve this learning task, we infer, as an intermediate step, a causal model from the observations that then entails properties of unobserved sets. Accordingly, we can define the VC dimension of a class of causal models and derive generalization bounds for the predictions. Here, causal discovery becomes more modest and better accessible to empirical tests than usual: rather than trying to find a causal hypothesis that is `true' a causal hypothesis is {\it useful} whenever it correctly predicts statistical properties of unobserved joint distributions. This way, a sparse causal graph that omits weak influences may be more useful than a dense one (despite being less accurate) because it is able to reconstruct the full joint distribution from marginal distributions of smaller subsets. Within such a `pragmatic' application of causal discovery, some popular heuristic approaches become justified in retrospect. It is, for instance, allowed to infer DAGs from partial correlations instead of conditional independences if the DAGs are only used to predict partial correlations.

Regression on observational data can fail to capture a causal relationship in the presence of unobserved confounding. Confounding strength measures this mismatch, but estimating it requires itself additional assumptions. A common assumption is the independence of causal mechanisms, which relies on concentration phenomena in high dimensions. While high dimensions enable the estimation of confounding strength, they also necessitate adapted estimators. In this paper, we derive the asymptotic behavior of the confounding strength estimator by Janzing and Sch\"olkopf (2018) and show that it is generally not consistent. We then use tools from random matrix theory to derive an adapted, consistent estimator.

In recent years, several results in the supervised learning setting suggested that classical statistical learning-theoretic measures, such as VC dimension, do not adequately explain the performance of deep learning models which prompted a slew of work in the infinite-width and iteration regimes. However, there is little theoretical explanation for the success of neural networks beyond the supervised setting. In this paper we argue that, under some distributional assumptions, classical learning-theoretic measures can sufficiently explain generalization for graph neural networks in the transductive setting. In particular, we provide a rigorous analysis of the performance of neural networks in the context of transductive inference, specifically by analysing the generalisation properties of graph convolutional networks for the problem of node classification. While VC Dimension does result in trivial generalisation error bounds in this setting as well, we show that transductive Rademacher complexity can explain the generalisation properties of graph convolutional networks for stochastic block models. We further use the generalisation error bounds based on transductive Rademacher complexity to demonstrate the role of graph convolutions and network architectures in achieving smaller generalisation error and provide insights into when the graph structure can help in learning.

Despite the increasing relevance of forecasting methods, the causal implications of these algorithms remain largely unexplored. This is concerning considering that, even under simplifying assumptions such as causal sufficiency, the statistical risk of a model can differ significantly from its \textit{causal risk}. Here, we study the problem of *causal generalization* -- generalizing from the observational to interventional distributions -- in forecasting. Our goal is to find answers to the question: How does the efficacy of an autoregressive (VAR) model in predicting statistical associations compare with its ability to predict under interventions? To this end, we introduce the framework of *causal learning theory* for forecasting. Using this framework, we obtain a characterization of the difference between statistical and causal risks, which helps identify sources of divergence between them. Under causal sufficiency, the problem of causal generalization amounts to learning under covariate shifts albeit with additional structure (restriction to interventional distributions). This structure allows us to obtain uniform convergence bounds on causal generalizability for the class of VAR models. To the best of our knowledge, this is the first work that provides theoretical guarantees for causal generalization in the time-series setting.

Despite the ubiquity of kernel-based clustering, surprisingly few statistical guarantees exist beyond settings that consider strong structural assumptions on the data generation process. In this work, we take a step towards bridging this gap by studying the statistical performance of kernel-based clustering algorithms under non-parametric mixture models. We provide necessary and sufficient separability conditions under which these algorithms can consistently recover the underlying true clustering. Our analysis provides guarantees for kernel clustering approaches without structural assumptions on the form of the component distributions. Additionally, we establish a key equivalence between kernel-based data-clustering and kernel density-based clustering. This enables us to provide consistency guarantees for kernel-based estimators of non-parametric mixture models. Along with theoretical implications, this connection could have practical implications, including in the systematic choice of the bandwidth of the Gaussian kernel in the context of clustering.

Given data from a general metric space, one of the standard machine learning pipelines is to first embed the data into a Euclidean space and subsequently apply out of the box machine learning algorithms to analyze the data. The quality of such an embedding is typically described in terms of a distortion measure. In this paper, we show that many of the existing distortion measures behave in an undesired way, when considered from a machine learning point of view. We investigate desirable properties of distortion measures and formally prove that most of the existing measures fail to satisfy these properties. These theoretical findings are supported by simulations, which for example demonstrate that existing distortion measures are not robust to noise or outliers and cannot serve as good indicators for classification accuracy. As an alternative, we suggest a new measure of distortion, called $\sigma$-distortion. We can show both in theory and in experiments that it satisfies all desirable properties and is a better candidate to evaluate distortion in the context of machine learning.

Given data from a general metric space, one of the standard machine learning pipelines is to first embed the data into a Euclidean space and subsequently apply out of the box machine learning algorithms to analyze the data. The quality of such an embedding is typically described in terms of a distortion measure. In this paper, we show that many of the existing distortion measures behave in an undesired way, when considered from a machine learning point of view. We investigate desirable properties of distortion measures and formally prove that most of the existing measures fail to satisfy these properties. These theoretical findings are supported by simulations, which for example demonstrate that existing distortion measures are not robust to noise or outliers and cannot serve as good indicators for classification accuracy. As an alternative, we suggest a new measure of distortion, called $\sigma$-distortion. We can show both in theory and in experiments that it satisfies all desirable properties and is a better candidate to evaluate distortion in the context of machine learning.