A data analyst might worry about generalization if dropping a very small fraction of data points from a study could change its substantive conclusions. Checking this non-robustness directly poses a combinatorial optimization problem and is intractable even for simple models and moderate data sizes. Recently various authors have proposed a diverse set of approximations to detect this non-robustness. In the present work, we show that, even in a setting as simple as ordinary least squares (OLS) linear regression, many of these approximations can fail to detect (true) non-robustness in realistic data arrangements. We focus on OLS in the present work due its widespread use and since some approximations work only for OLS. Across our synthetic and real-world data sets, we find that a simple recursive greedy algorithm is the sole algorithm that does not fail any of our tests and also that it can be orders of magnitude faster to run than some competitors.

If the conclusion of a data analysis is sensitive to dropping very few data points, that conclusion might hinge on the particular data at hand rather than representing a more broadly applicable truth. How could we check whether this sensitivity holds? One idea is to consider every small subset of data, drop it from the dataset, and re-run our analysis. But running MCMC to approximate a Bayesian posterior is already very expensive; running multiple times is prohibitive, and the number of re-runs needed here is combinatorially large. Recent work proposes a fast and accurate approximation to find the worst-case dropped data subset, but that work was developed for problems based on estimating equations -- and does not directly handle Bayesian posterior approximations using MCMC. We make two principal contributions in the present work. We adapt the existing data-dropping approximation to estimators computed via MCMC. Observing that Monte Carlo errors induce variability in the approximation, we use a variant of the bootstrap to quantify this uncertainty. We demonstrate how to use our approximation in practice to determine whether there is non-robustness in a problem. Empirically, our method is accurate in simple models, such as linear regression. In models with complicated structure, such as hierarchical models, the performance of our method is mixed.

Completely random measures (CRMs) and their normalizations (NCRMs) offer flexible models in Bayesian nonparametrics. But their infinite dimensionality presents challenges for inference. Two popular finite approximations are truncated finite approximations (TFAs) and independent finite approximations (IFAs). While the former have been well-studied, IFAs lack similarly general bounds on approximation error, and there has been no systematic comparison between the two options. In the present work, we propose a general recipe to construct practical finite-dimensional approximations for homogeneous CRMs and NCRMs, in the presence or absence of power laws. We call our construction the automated independent finite approximation (AIFA). Relative to TFAs, we show that AIFAs facilitate more straightforward derivations and use of parallel computing in approximate inference. We upper bound the approximation error of AIFAs for a wide class of common CRMs and NCRMs -- and thereby develop guidelines for choosing the approximation level. Our lower bounds in key cases suggest that our upper bounds are tight. We prove that, for worst-case choices of observation likelihoods, TFAs are more efficient than AIFAs. Conversely, we find that in real-data experiments with standard likelihoods, AIFAs and TFAs perform similarly. Moreover, we demonstrate that AIFAs can be used for hyperparameter estimation even when other potential IFA options struggle or do not apply.

Test log-likelihood is commonly used to compare different models of the same data or different approximate inference algorithms for fitting the same probabilistic model. We present simple examples demonstrating how comparisons based on test log-likelihood can contradict comparisons according to other objectives. Specifically, our examples show that (i) approximate Bayesian inference algorithms that attain higher test log-likelihoods need not also yield more accurate posterior approximations and (ii) conclusions about forecast accuracy based on test log-likelihood comparisons may not agree with conclusions based on root mean squared error.

Prior knowledge and symbolic rules in machine learning are often expressed in the form of label constraints, especially in structured prediction problems. In this work, we compare two common strategies for encoding label constraints in a machine learning pipeline, regularization with constraints and constrained inference, by quantifying their impact on model performance. For regularization, we show that it narrows the generalization gap by precluding models that are inconsistent with the constraints. However, its preference for small violations introduces a bias toward a suboptimal model. For constrained inference, we show that it reduces the population risk by correcting a model's violation, and hence turns the violation into an advantage. Given these differences, we further explore the use of two approaches together and propose conditions for constrained inference to compensate for the bias introduced by regularization, aiming to improve both the model complexity and optimal risk.

Gaussian processes (GPs) are used to make medical and scientific decisions, including in cardiac care and monitoring of atmospheric carbon dioxide levels. Notably, the choice of GP kernel is often somewhat arbitrary. In particular, uncountably many kernels typically align with qualitative prior knowledge (e.g. function smoothness or stationarity). But in practice, data analysts choose among a handful of convenient standard kernels (e.g. squared exponential). In the present work, we ask: Would decisions made with a GP differ under other, qualitatively interchangeable kernels? We show how to answer this question by solving a constrained optimization problem over a finite-dimensional space. We can then use standard optimizers to identify substantive changes in relevant decisions made with a GP. We demonstrate in both synthetic and real-world examples that decisions made with a GP can exhibit non-robustness to kernel choice, even when prior draws are qualitatively interchangeable to a user.

Markov chain Monte Carlo (MCMC) methods are often used in clustering since they guarantee asymptotically exact expectations in the infinite-time limit. In finite time, though, slow mixing often leads to poor performance. Modern computing environments offer massive parallelism, but naive implementations of parallel MCMC can exhibit substantial bias. In MCMC samplers of continuous random variables, Markov chain couplings can overcome bias. But these approaches depend crucially on paired chains meetings after a small number of transitions. We show that straightforward applications of existing coupling ideas to discrete clustering variables fail to meet quickly. This failure arises from the "label-switching problem": semantically equivalent cluster relabelings impede fast meeting of coupled chains. We instead consider chains as exploring the space of partitions rather than partitions' (arbitrary) labelings. Using a metric on the partition space, we formulate a practical algorithm using optimal transport couplings. Our theory confirms our method is accurate and efficient. In experiments ranging from clustering of genes or seeds to graph colorings, we show the benefits of our coupling in the highly parallel, time-limited regime.

Computational couplings of Markov chains provide a practical route to unbiased Monte Carlo estimation that can utilize parallel computation. However, these approaches depend crucially on chains meeting after a small number of transitions. For models that assign data into groups, e.g. mixture models, the obvious approaches to couple Gibbs samplers fail to meet quickly. This failure owes to the so-called "label-switching" problem; semantically equivalent relabelings of the groups contribute well-separated posterior modes that impede fast mixing and cause large meeting times. We here demonstrate how to avoid label switching by considering chains as exploring the space of partitions rather than labelings. Using a metric on this space, we employ an optimal transport coupling of the Gibbs conditionals. This coupling outperforms alternative couplings that rely on labelings and, on a real dataset, provides estimates more precise than usual ergodic averages in the limited time regime. Code is available at github.com/tinnguyen96/coupling-Gibbs-partition.

Many modern data analyses benefit from explicitly modeling dependence structure in data -- such as measurements across time or space, ordered words in a sentence, or genes in a genome. A gold standard evaluation technique is structured cross-validation (CV), which leaves out some data subset (such as data within a time interval or data in a geographic region) in each fold. But CV here can be prohibitively slow due to the need to re-run already-expensive learning algorithms many times. Previous work has shown approximate cross-validation (ACV) methods provide a fast and provably accurate alternative in the setting of empirical risk minimization. But this existing ACV work is restricted to simpler models by the assumptions that (i) data across CV folds are independent and (ii) an exact initial model fit is available. In structured data analyses, both these assumptions are often untrue. In the present work, we address (i) by extending ACV to CV schemes with dependence structure between the folds. To address (ii), we verify -- both theoretically and empirically -- that ACV quality deteriorates smoothly with noise in the initial fit. We demonstrate the accuracy and computational benefits of our proposed methods on a diverse set of real-world applications.

Bayesian nonparametric models based on completely random measures (CRMs) offer flexibility when the number of clusters or latent components in a data set is unknown. However, managing the infinite dimensionality of CRMs often leads to slow computation during inference. Practical inference typically relies on either integrating out the infinite-dimensional parameter or using a finite approximation: a truncated finite approximation (TFA) or an independent finite approximation (IFA). The atom weights of TFAs are constructed sequentially, while the atoms of IFAs are independent, which facilitates more convenient inference schemes. While the approximation error of TFA has been systematically addressed, there has not yet been a similar study of IFA. We quantify the approximation error between IFAs and two common target nonparametric priors (beta-Bernoulli process and Dirichlet process mixture model) and prove that, in the worst-case, TFAs provide more component-efficient approximations than IFAs. However, in experiments on image denoising and topic modeling tasks with real data, we find that the error of Bayesian approximation methods overwhelms any finite approximation error, and IFAs perform very similarly to TFAs.

We present a weighted-majority classification approach over subtrees of a fixed tree, which provably achieves excess-risk of the same order as the best tree-pruning. Furthermore, the computational efficiency of pruning is maintained at both training and testing time despite having to aggregate over an exponential number of subtrees. We believe this is the first subtree aggregation approach with such guarantees. The guarantees are obtained via a simple combination of insights from PAC-Bayes theory, which we believe should be of independent interest, as it generically implies consistency for weighted-voting classifiers w.r.t. Bayes - while, in contrast, usual PAC-bayes approaches only establish consistency of Gibbs classifiers.