This paper introduces e-fold cross-validation, an energy-efficient alternative to k-fold cross-validation. It dynamically adjusts the number of folds based on a stopping criterion. The criterion checks after each fold whether the standard deviation of the evaluated folds has consistently decreased or remained stable. Once met, the process stops early. We tested e-fold cross-validation on 15 datasets and 10 machine-learning algorithms. On average, it required 4 fewer folds than 10-fold cross-validation, reducing evaluation time, computational resources, and energy use by about 40%. Performance differences between e-fold and 10-fold cross-validation were less than 2% for larger datasets. More complex models showed even smaller discrepancies. In 96% of iterations, the results were within the confidence interval, confirming statistical significance. E-fold cross-validation offers a reliable and efficient alternative to k-fold, reducing computational costs while maintaining comparable accuracy.

Careful tuning of a regularization parameter is indispensable in many machine learning tasks because it has a significant impact on generalization performances.Nevertheless, current practice of regularization parameter tuning is more of an art than a science, e.g., it is hard to tell how many grid-points would be needed in cross-validation (CV) for obtaining a solution with sufficiently small CV error.In this paper we propose a novel framework for computing a lower bound of the CV errors as a function of the regularization parameter, which we call regularization path of CV error lower bounds.The proposed framework can be used for providing a theoretical approximation guarantee on a set of solutions in the sense that how far the CV error of the current best solution could be away from best possible CV error in the entire range of the regularization parameters.We demonstrate through numerical experiments that a theoretically guaranteed a choice of regularization parameter in the above sense is possible with reasonable computational costs.

We present a novel method for tuning the regularization hyper-parameter, \lambda, of a ridge regression that is faster to compute than leave-one-out cross-validation (LOOCV) while yielding estimates of the regression parameters of equal, or particularly in the setting of sparse covariates, superior quality to those obtained by minimising the LOOCV risk. The LOOCV risk can suffer from multiple and bad local minima for finite n and thus requires the specification of a set of candidate \lambda, which can fail to provide good solutions. In contrast, we show that the proposed method is guaranteed to find a unique optimal solution for large enough n, under relatively mild conditions, without requiring the specification of any difficult to determine hyper-parameters. This is based on a Bayesian formulation of ridge regression that we prove to have a unimodal posterior for large enough n, allowing for both the optimal \lambda and the regression coefficients to be jointly learned within an iterative expectation maximization (EM) procedure. Importantly, we show that by utilizing an appropriate preprocessing step, a single iteration of the main EM loop can be implemented in O(\min(n, p)) operations, for input data with n rows and p columns.

This work develops central limit theorems for cross-validation and consistent estimators of the asymptotic variance under weak stability conditions on the learning algorithm. Together, these results provide practical, asymptotically-exact confidence intervals for k-fold test error and valid, powerful hypothesis tests of whether one learning algorithm has smaller k-fold test error than another. These results are also the first of their kind for the popular choice of leave-one-out cross-validation. In our experiments with diverse learning algorithms, the resulting intervals and tests outperform the most popular alternative methods from the literature.

In many real-world applications of machine learning, we are interested to know if it is possible to train on the data that we have gathered so far, and obtain accurate predictions on a new test data subset that is qualitatively different in some respect (time period, geographic region, etc). Another question is whether data subsets are similar enough so that it is beneficial to combine subsets during model training. We propose SOAK, Same/Other/All K-fold cross-validation, a new method which can be used to answer both questions. SOAK systematically compares models which are trained on different subsets of data, and then used for prediction on a fixed test subset, to estimate the similarity of learnable/predictable patterns in data subsets. We show results of using SOAK on six new real data sets (with geographic/temporal subsets, to check if predictions are accurate on new subsets), 3 image pair data sets (subsets are different image types, to check that we get smaller prediction error on similar images), and 11 benchmark data sets with predefined train/test splits (to check similarity of predefined splits).

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).

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.

The evolving field of mobile robotics has indeed increased the demand for simultaneous localization and mapping (SLAM) systems. To augment the localization accuracy and mapping efficacy of SLAM, we refined the core module of the SLAM system. Within the feature matching phase, we introduced cross-validation matching to filter out mismatches. In the keyframe selection strategy, an exponential threshold function is constructed to quantify the keyframe selection process. Compared with a single robot, the multi-robot collaborative SLAM (CSLAM) system substantially improves task execution efficiency and robustness. By employing a centralized structure, we formulate a multi-robot SLAM system and design a coarse-to-fine matching approach for multi-map point cloud registration. Our system, built upon ORB-SLAM3, underwent extensive evaluation utilizing the TUM RGB-D, EuRoC MAV, and TUM_VI datasets. The experimental results demonstrate a significant improvement in the positioning accuracy and mapping quality of our enhanced algorithm compared to those of ORB-SLAM3, with a 12.90% reduction in the absolute trajectory error.

Emulating the mapping between quantities of interest and their control parameters using surrogate models finds widespread application in engineering design, including in numerical optimization and uncertainty quantification. Gaussian process models can serve as a probabilistic surrogate model of unknown functions, thereby making them highly suitable for engineering design and decision-making in the presence of uncertainty. In this work, we are interested in emulating quantities of interest observed from models of a system at multiple fidelities, which trade accuracy for computational efficiency. Using multifidelity Gaussian process models, to efficiently fuse models at multiple fidelities, we propose a novel method to actively learn the surrogate model via leave-one-out cross-validation (LOO-CV). Our proposed multifidelity cross-validation (\texttt{MFCV}) approach develops an adaptive approach to reduce the LOO-CV error at the target (highest) fidelity, by learning the correlations between the LOO-CV at all fidelities. \texttt{MFCV} develops a two-step lookahead policy to select optimal input-fidelity pairs, both in sequence and in batches, both for continuous and discrete fidelity spaces. We demonstrate the utility of our method on several synthetic test problems as well as on the thermal stress analysis of a gas turbine blade.

Two key tasks in high-dimensional regularized regression are tuning the regularization strength for good predictions and estimating the out-of-sample risk. It is known that the standard approach -- $k$-fold cross-validation -- is inconsistent in modern high-dimensional settings. While leave-one-out and generalized cross-validation remain consistent in some high-dimensional cases, they become inconsistent when samples are dependent or contain heavy-tailed covariates. To model structured sample dependence and heavy tails, we use right-rotationally invariant covariate distributions - a crucial concept from compressed sensing. In the common modern proportional asymptotics regime where the number of features and samples grow comparably, we introduce a new framework, ROTI-GCV, for reliably performing cross-validation. Along the way, we propose new estimators for the signal-to-noise ratio and noise variance under these challenging conditions. We conduct extensive experiments that demonstrate the power of our approach and its superiority over existing methods.