Gaussian processes (GPs) are used to make medical and scientific decisions, including in cardiac care and monitoring of carbon dioxide emissions. 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 formulate this sensitivity analysis as 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 substantial sensitivity to kernel choice, even when prior draws are qualitatively interchangeable to a user.

Models like LASSO and ridge regression are extensively used in practice due to their interpretability, ease of use, and strong theoretical guarantees. Cross-validation (CV) is widely used for hyperparameter tuning in these models, but do practical methods minimize the true out-of-sample loss? A recent line of research promises to show that the optimum of the CV loss matches the optimum of the out-of-sample loss (possibly after simple corrections). It remains to show how tractable it is to minimize the CV loss.In the present paper, we show that, in the case of ridge regression, the CV loss may fail to be quasiconvex and thus may have multiple local optima. We can guarantee that the CV loss is quasiconvex in at least one case: when the spectrum of the covariate matrix is nearly flat and the noise in the observed responses is not too high. More generally, we show that quasiconvexity status is independent of many properties of the observed data (response norm, covariate-matrix right singular vectors and singular-value scaling) and has a complex dependence on the few that remain. We empirically confirm our theory using simulated experiments.

Leave-one-out cross-validation (LOOCV) can be particularly accurate among cross-validation (CV) variants for machine learning assessment tasks – e.g., assessing methods’ error or variability. But it is expensive to re-fit a model $N$ times for a dataset of size $N$. Previous work has shown that approximations to LOOCV can be both fast and accurate – when the unknown parameter is of small, fixed dimension. But these approximations incur a running time roughly cubic in dimension – and we show that, besides computational issues, their accuracy dramatically deteriorates in high dimensions. Authors have suggested many potential and seemingly intuitive solutions, but these methods have not yet been systematically evaluated or compared. We find that all but one perform so poorly as to be unusable for approximating LOOCV. Crucially, though, we are able to show, both empirically and theoretically, that one approximation can perform well in high dimensions – in cases where the high-dimensional parameter exhibits sparsity. Under interpretable assumptions, our theory demonstrates that the problem can be reduced to working within an empirically recovered (small) support. This procedure is straightforward to implement, and we prove that its running time and error depend on the (small) support size even when the full parameter dimension is large.

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.

Many recent advances in machine learning are driven by a challenging trifecta: large data size $N$, high dimensions, and expensive algorithms. In this setting, cross-validation (CV) serves as an important tool for model assessment. Recent advances in approximate cross validation (ACV) provide accurate approximations to CV with only a single model fit, avoiding traditional CV's requirement for repeated runs of expensive algorithms. Unfortunately, these ACV methods can lose both speed and accuracy in high dimensions --- unless sparsity structure is present in the data. Fortunately, there is an alternative type of simplifying structure that is present in most data: approximate low rank (ALR). Guided by this observation, we develop a new algorithm for ACV that is fast and accurate in the presence of ALR data. Our first key insight is that the Hessian matrix --- whose inverse forms the computational bottleneck of existing ACV methods --- is ALR. We show that, despite our use of the \emph{inverse} Hessian, a low-rank approximation using the largest (rather than the smallest) matrix eigenvalues enables fast, reliable ACV. Our second key insight is that, in the presence of ALR data, error in existing ACV methods roughly grows with the (approximate, low) rank rather than with the (full, high) dimension. These insights allow us to prove theoretical guarantees on the quality of our proposed algorithm --- along with fast-to-compute upper bounds on its error. We demonstrate the speed and accuracy of our method, as well as the usefulness of our bounds, on a range of real and simulated data sets.

The error or variability of machine learning algorithms is often assessed by repeatedly refitting a model with different weighted versions of the observed data. The ubiquitous tools of cross-validation (CV) and the bootstrap are examples of this technique. These methods are powerful in large part due to their model agnosticism but can be slow to run on modern, large data sets due to the need to repeatedly re-fit the model. In this work, we use a linear approximation to the dependence of the fitting procedure on the weights, producing results that can be faster than repeated re-fitting by an order of magnitude. This linear approximation is sometimes known as the "infinitesimal jackknife" in the statistics literature, where it is mostly used as a theoretical tool to prove asymptotic results. We provide explicit finite-sample error bounds for the infinitesimal jackknife in terms of a small number of simple, verifiable assumptions. Our results apply whether the weights and data are stochastic or deterministic, and so can be used as a tool for proving the accuracy of the infinitesimal jackknife on a wide variety of problems. As a corollary, we state mild regularity conditions under which our approximation consistently estimates true leave k-out cross-validation for any fixed k. These theoretical results, together with modern automatic differentiation software, support the application of the infinitesimal jackknife to a wide variety of practical problems in machine learning, providing a "Swiss Army infinitesimal jackknife." We demonstrate the accuracy of our methods on a range of simulated and real datasets.

Bayesian nonparametric hidden Markov models are typically learned via fixed truncations of the infinite state space or local Monte Carlo proposals that make small changes to the state space. We develop an inference algorithm for the sticky hierarchical Dirichlet process hidden Markov model that scales to big datasets by processing a few sequences at a time yet allows rapid adaptation of the state space cardinality. Unlike previous point-estimate methods, our novel variational bound penalizes redundant or irrelevant states and thus enables optimization of the state space. Our birth proposals use observed data statistics to create useful new states that escape local optima. Merge and delete proposals remove ineffective states to yield simpler models with more affordable future computations. Experiments on speaker diarization, motion capture, and epigenetic chromatin datasets discover models that are more compact, more interpretable, and better aligned to ground truth segmentations than competitors. We have released an open-source Python implementation which can parallelize local inference steps across sequences.