A Swiss Army Infinitesimal Jackknife

Abstract

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.

Artificial Intelligence for Earth System Predictability (AI4ESP) (2021 Workshop Report)

“…• Scalable Bayesian Inference: Methods inspired from "Infinitesimal Jackknife" (Giordano et al 2019) and "Deep Ensembles" (D'Angelo and Fortuin 2021).…”

Model-specific Data Subsampling with Influence Functions

“…For example, Hampel et al [12] use influence functions to study the impact of data contamination on statistical procedures. In machine learning, influence functions have been used by [4] to reason about the stability of kernel regression, by [16] to study perturbations of neural networks, and by [11] to obtain inexpensive approximations of resampling procedures like cross-validation and the bootstrap. Ting and Brochu [25] consider single-shot sampling, as compared to our iterative approach, using influence functions, motivating their strategy as minimizing the variance of an asymptotically linear estimator.…”

“…Our assumptions are similar to the ones made in [11]. We write the Hessian of the empirical risk minimization problem over the empirical distribution P M at parameter θ as…”

Learning Bounds for Greedy Approximation with Explicit Feature Maps from Multiple Kernels

“…Of course, optimizing over σ (e.g. using cross-validation, jackknife, or their approximate surrogates [54,55,56]) may provide better results. For our method as well as GLK, we do not optimize over the convex combination weights (uniform weights are assigned).…”