We propose a method to assess the sensitivity of econometric analyses to the removal of a small fraction of the sample. Analyzing all possible data subsets of a certain size is computationally prohibitive, so we provide a finite-sample metric to approximately compute the number (or fraction) of observations that has the greatest influence on a given result when dropped. We call our resulting metric the Approximate Maximum Influence Perturbation. Our approximation is automatically computable and works for common estimators (including OLS, IV, GMM, MLE, and variational Bayes). We provide explicit finite-sample error bounds on our approximation for linear and instrumental variables regressions. At minimal computational cost, our metric provides an exact finite-sample lower bound on sensitivity for any estimator, so any non-robustness our metric finds is conclusive. We demonstrate that the Approximate Maximum Influence Perturbation is driven by a low signal-to-noise ratio in the inference problem, is not reflected in standard errors, does not disappear asymptotically, and is not a product of misspecification. Several empirical applications show that even 2-parameter linear regression analyses of randomized trials can be highly sensitive. While we find some applications are robust, in others the sign of a treatment effect can be changed by dropping less than 1% of the sample even when standard errors are small.

Cross validation (CV) and the bootstrap are ubiquitous model-agnostic tools for assessing the error or variability of machine learning and statistical estimators. However, these methods require repeatedly re-fitting the model with different weighted versions of the original dataset, which can be prohibitively time-consuming. For sufficiently regular optimization problems the optimum depends smoothly on the data weights, and so the process of repeatedly re-fitting can be approximated with a Taylor series that can be often evaluated relatively quickly. The first-order approximation is known as the "infinitesimal jackknife" in the statistics literature and has been the subject of recent interest in machine learning for approximate CV. In this work, we consider high-order approximations, which we call the "higherorder infinitesimal jackknife" (HOIJ). Under mild regularity conditions, we provide a simple recursive procedure to compute approximations of all orders with finite-sample accuracy bounds. Additionally, we show that the HOIJ can be efficiently computed even in high dimensions using forwardmode automatic differentiation. We show that a linear approximation with bootstrap weights approximation is equivalent to those provided by asymptotic normal approximations. Consequently, the HOIJ opens up the possibility of enjoying higher-order accuracy properties of the bootstrap using local approximations. Consistency of the HOIJ for leave-one-out CV under different asymptotic regimes follows as corollaries from our finitesample bounds under additional regularity assumptions. The generality of the computation and bounds motivate the name "higher-order Swiss Army infinitesimal jackknife."

Astronomical catalogs derived from wide-field imaging surveys are an important tool for understanding the Universe. We construct an astronomical catalog from 55 TB of imaging data using Celeste, a Bayesian variational inference code written entirely in the high-productivity programming language Julia. Using over 1.3 million threads on 650,000 Intel Xeon Phi cores of the Cori Phase II supercomputer, Celeste achieves a peak rate of 1.54 DP PFLOP/s. Celeste is able to jointly optimize parameters for 188M stars and galaxies, loading and processing 178 TB across 8192 nodes in 14.6 minutes. To achieve this, Celeste exploits parallelism at multiple levels (cluster, node, and thread) and accelerates I/O through Cori's Burst Buffer. Julia's native performance enables Celeste to employ high-level constructs without resorting to handwritten or generated low-level code (C/C++/Fortran), and yet achieve petascale performance.

Astronomical catalogs derived from wide-field imaging surveys are an important tool for understanding the Universe. We construct an astronomical catalog from 55 TB of imaging data using Celeste, a Bayesian variational inference code written entirely in the high-productivity programming language Julia. Using over 1.3 million threads on 650,000 Intel Xeon Phi cores of the Cori Phase II supercomputer, Celeste achieves a peak rate of 1.54 DP PFLOP/s. Celeste is able to jointly optimize parameters for 188M stars and galaxies, loading and processing 178 TB across 8192 nodes in 14.6 minutes. To achieve this, Celeste exploits parallelism at multiple levels (cluster, node, and thread) and accelerates I/O through Cori's Burst Buffer. Julia's native performance enables Celeste to employ high-level constructs without resorting to handwritten or generated low-level code (C/C++/Fortran), and yet achieve petascale performance.

The error or variability of machine learning algorithms is often assessed by repeatedly re-fitting 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 leavek-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.