Ensembling has a long history in statistical data analysis, with many impactful applications. However, in many modern machine learning settings, the benefits of ensembling are less ubiquitous and less obvious.

Ensembling has a long history in statistical data analysis, with many impactful applications. However, in many modern machine learning settings, the benefits of ensembling are less ubiquitous and less obvious. We study, both theoretically and empirically, the fundamental question of when ensembling yields significant performance improvements in classification tasks. Theoretically, we prove new results relating the \emph{ensemble improvement rate} (a measure of how much ensembling decreases the error rate versus a single model, on a relative scale) to the \emph{disagreement-error ratio}. We show that ensembling improves performance significantly whenever the disagreement rate is large relative to the average error rate; and that, conversely, one classifier is often enough whenever the disagreement rate is low relative to the average error rate. On the way to proving these results, we derive, under a mild condition called \emph{competence}, improved upper and lower bounds on the average test error rate of the majority vote classifier.To complement this theory, we study ensembling empirically in a variety of settings, verifying the predictions made by our theory, and identifying practical scenarios where ensembling does and does not result in large performance improvements. Perhaps most notably, we demonstrate a distinct difference in behavior between interpolating models (popular in current practice) and non-interpolating models (such as tree-based methods, where ensembling is popular), demonstrating that ensembling helps considerably more in the latter case than in the former.

In this section, we provide proofs for our main results. We first state and prove two lemmas that will be used in the proof of Theorem 1 . P ( X x) d x concludes the proof. With these two lemmas, we now provide the proof of Theorem 1 . Now we will derive a lower bound of the second term.

We address these questions in the following ways.

In this section, we provide proofs for our main results. We first state and prove two lemmas that will be used in the proof of Theorem 1 . P ( X x) d x concludes the proof. With these two lemmas, we now provide the proof of Theorem 1 . Now we will derive a lower bound of the second term.

Long-term test-time adaptation (TTA) is a challenging task due to error accumulation. Recent approaches tackle this issue by actively labeling a small proportion of samples in each batch, yet the annotation burden quickly grows as the batch number increases. In this paper, we investigate how to achieve effortless active labeling so that a maximum of one sample is selected for annotation in each batch. First, we annotate the most valuable sample in each batch based on the single-step optimization perspective in the TTA context. In this scenario, the samples that border between the source- and target-domain data distributions are considered the most feasible for the model to learn in one iteration. Then, we introduce an efficient strategy to identify these samples using feature perturbation. Second, we discover that the gradient magnitudes produced by the annotated and unannotated samples have significant variations. Therefore, we propose balancing their impact on model optimization using two dynamic weights. Extensive experiments on the popular ImageNet-C, -R, -K, -A and PACS databases demonstrate that our approach consistently outperforms state-of-the-art methods with significantly lower annotation costs.

As performance gains through scaling data and/or model size experience diminishing returns, it is becoming increasingly popular to turn to ensembling, where the predictions of multiple models are combined to improve accuracy. In this paper, we provide a detailed analysis of how the disagreement and the polarization (a notion we introduce and define in this paper) among classifiers relate to the performance gain achieved by aggregating individual classifiers, for majority vote strategies in classification tasks. We address these questions in the following ways. (1) An upper bound for polarization is derived, and we propose what we call a neural polarization law: most interpolating neural network models are 4/3-polarized. Our empirical results not only support this conjecture but also show that polarization is nearly constant for a dataset, regardless of hyperparameters or architectures of classifiers. (2) The error of the majority vote classifier is considered under restricted entropy conditions, and we present a tight upper bound that indicates that the disagreement is linearly correlated with the target, and that the slope is linear in the polarization. (3) We prove results for the asymptotic behavior of the disagreement in terms of the number of classifiers, which we show can help in predicting the performance for a larger number of classifiers from that of a smaller number. Our theories and claims are supported by empirical results on several image classification tasks with various types of neural networks.

Ensembling has a long history in statistical data analysis, with many impactful applications. However, in many modern machine learning settings, the benefits of ensembling are less ubiquitous and less obvious. We study, both theoretically and empirically, the fundamental question of when ensembling yields significant performance improvements in classification tasks. Theoretically, we prove new results relating the \emph{ensemble improvement rate} (a measure of how much ensembling decreases the error rate versus a single model, on a relative scale) to the \emph{disagreement-error ratio}. We show that ensembling improves performance significantly whenever the disagreement rate is large relative to the average error rate; and that, conversely, one classifier is often enough whenever the disagreement rate is low relative to the average error rate.