Tuning scientific and probabilistic machine learning models -- for example, partial differential equations, Gaussian processes, or Bayesian neural networks -- often relies on evaluating functions of matrices whose size grows with the data set or the number of parameters. While the state-of-the-art for evaluating these quantities is almost always based on Lanczos and Arnoldi iterations, the present work is the first to explain how to differentiate these workhorses of numerical linear algebra efficiently. To get there, we derive previously unknown adjoint systems for Lanczos and Arnoldi iterations, implement them in JAX, and show that the resulting code can compete with Diffrax when it comes to differentiating PDEs, GPyTorch for selecting Gaussian process models and beats standard factorisation methods for calibrating Bayesian neural networks. All this is achieved without any problem-specific code optimisation. Find the code at https://github.com/pnkraemer/experiments-lanczos-adjoints and install the library with pip install matfree.

Supervised learning approaches for causal discovery from observational data often achieve competitive performance despite seemingly avoiding explicit assumptions that traditional methods make for identifiability. In this work, we investigate CSIvA (Ke et al., 2023), a transformer-based model promising to train on synthetic data and transfer to real data. First, we bridge the gap with existing identifiability theory and show that constraints on the training data distribution implicitly define a prior on the test observations. Consistent with classical approaches, good performance is achieved when we have a good prior on the test data, and the underlying model is identifiable. At the same time, we find new trade-offs. Training on datasets generated from different classes of causal models, unambiguously identifiable in isolation, improves the test generalization. Performance is still guaranteed, as the ambiguous cases resulting from the mixture of identifiable causal models are unlikely to occur (which we formally prove). Overall, our study finds that amortized causal discovery still needs to obey identifiability theory, but it also differs from classical methods in how the assumptions are formulated, trading more reliance on assumptions on the noise type for fewer hypotheses on the mechanisms.

The ubiquity of large-scale graphs in node-classification tasks significantly hinders the real-world applications of Graph Neural Networks (GNNs). Node sampling, graph coarsening, and dataset condensation are effective strategies for enhancing data efficiency. However, owing to the interdependence of graph nodes, coreset selection, which selects subsets of the data examples, has not been successfully applied to speed up GNN training on large graphs, warranting special treatment. This paper studies graph coresets for GNNs and avoids the interdependence issue by selecting ego-graphs (i.e., neighborhood subgraphs around a node) based on their spectral embeddings. We decompose the coreset selection problem for GNNs into two phases: a coarse selection of widely spread ego graphs and a refined selection to diversify their topologies. We design a greedy algorithm that approximately optimizes both objectives. Our spectral greedy graph coreset (SGGC) scales to graphs with millions of nodes, obviates the need for model pre-training, and applies to low-homophily graphs. Extensive experiments on ten datasets demonstrate that SGGC outperforms other coreset methods by a wide margin, generalizes well across GNN architectures, and is much faster than graph condensation.

Selecting or designing an appropriate domain adaptation algorithm for a given problem remains challenging. This paper presents a Transformer model that can provably approximate and opt for domain adaptation methods for a given dataset in the in-context learning framework, where a foundation model performs new tasks without updating its parameters at test time. Specifically, we prove that Transformers can approximate instance-based and feature-based unsupervised domain adaptation algorithms and automatically select an algorithm suited for a given dataset. Numerical results indicate that in-context learning demonstrates an adaptive domain adaptation surpassing existing methods.

Transfer learning enhances prediction accuracy on a target distribution by leveraging data from a source distribution, demonstrating significant benefits in various applications. This paper introduces a novel dissimilarity measure that utilizes vicinity information, i.e., the local structure of data points, to analyze the excess error in classification under covariate shift, a transfer learning setting where marginal feature distributions differ but conditional label distributions remain the same. We characterize the excess error using the proposed measure and demonstrate faster or competitive convergence rates compared to previous techniques. Notably, our approach is effective in situations where the non-absolute continuousness assumption, which often appears in real-world applications, holds. Our theoretical analysis bridges the gap between current theoretical findings and empirical observations in transfer learning, particularly in scenarios with significant differences between source and target distributions.

However, according to the results of Kolmogorov and Vitushkin, the representation of generic smooth functions by KAN implementations using analytic functions constrained to a finite number of cutoff points cannot be exact. Hence, the convergence of KAN throughout the training process may be limited. This paper explores the relevance of smoothness in KANs, proposing that smooth, structurally informed KANs can achieve equivalence to MLPs in specific function classes. By leveraging inherent structural knowledge, KANs may reduce the data required for training and mitigate the risk of generating hallucinated predictions, thereby enhancing model reliability and performance in computational biomedicine.

Synthetic data generation holds considerable promise, offering avenues to enhance privacy, fairness, and data accessibility. Despite the availability of various methods for generating synthetic tabular data, challenges persist, particularly in specialized applications such as survival analysis. One significant obstacle in survival data generation is censoring, which manifests as not knowing the precise timing of observed (target) events for certain instances. Existing methods face difficulties in accurately reproducing the real distribution of event times for both observed (uncensored) events and censored events, i.e., the generated event-time distributions do not accurately match the underlying distributions of the real data. So motivated, we propose a simple paradigm to produce synthetic survival data by generating covariates conditioned on event times (and censoring indicators), thus allowing one to reuse existing conditional generative models for tabular data without significant computational overhead, and without making assumptions about the (usually unknown) generation mechanism underlying censoring. We evaluate this method via extensive experiments on real-world datasets. Our methodology outperforms multiple competitive baselines at generating survival data, while improving the performance of downstream survival models trained on it and tested on real data.

Machine Learning algorithms are notorious for providing point predictions but not prediction intervals. There are many applications where one requires confidence in predictions and prediction intervals. Stringing together, these intervals give rise to joint prediction regions with the desired significance level. It is an easy task to compute Joint Prediction regions (JPR) when the data is IID. However, the task becomes overly difficult when JPR is needed for time series because of the dependence between the observations. This project aims to implement Wolf and Wunderli's method for constructing JPRs and compare it with other methods (e.g. NP heuristic, Joint Marginals). The method under study is based on bootstrapping and is applied to different datasets (Min Temp, Sunspots), using different predictors (e.g. ARIMA and LSTM). One challenge of applying the method under study is to derive prediction standard errors for models, it cannot be obtained analytically. A novel method to estimate prediction standard error for different predictors is also devised. Finally, the method is applied to a synthetic dataset to find empirical averages and empirical widths and the results from the Wolf and Wunderli paper are consolidated. The experimental results show a narrowing of width with strong predictors like neural nets, widening of width with increasing forecast horizon H and decreasing significance level alpha, controlling the width with parameter k in K-FWE, and loss of information using Joint Marginals.

Offline policy evaluation (OPE) allows us to evaluate and estimate a new sequential decision-making policy's performance by leveraging historical interaction data collected from other policies. Evaluating a new policy online without a confident estimate of its performance can lead to costly, unsafe, or hazardous outcomes, especially in education and healthcare. Several OPE estimators have been proposed in the last decade, many of which have hyperparameters and require training. Unfortunately, choosing the best OPE algorithm for each task and domain is still unclear. In this paper, we propose a new algorithm that adaptively blends a set of OPE estimators given a dataset without relying on an explicit selection using a statistical procedure. We prove that our estimator is consistent and satisfies several desirable properties for policy evaluation. Additionally, we demonstrate that when compared to alternative approaches, our estimator can be used to select higher-performing policies in healthcare and robotics. Our work contributes to improving ease of use for a general-purpose, estimator-agnostic, off-policy evaluation framework for offline RL.

This paper shows that the dimensionality reduction methods, UMAP and t-SNE, can be approximately recast as MAP inference methods corresponding to a generalized Wishart-based model introduced in ProbDR. This interpretation offers deeper theoretical insights into these algorithms, while introducing tools with which similar dimensionality reduction methods can be studied.