Early stopping of iterative algorithms is a widely-used form of regularization in statistical learning, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including $L^2$-boost, LogitBoost and AdaBoost, among others), we connect the performance of a stopped iterate to the localized Rademacher/Gaussian complexity of the associated function class. This connection allows us to show that local fixed point analysis, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes, and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

Early stopping of iterative algorithms is a widely-used form of regularization in statistical learning, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including $L 2$-boost, LogitBoost and AdaBoost, among others), we connect the performance of a stopped iterate to the localized Rademacher/Gaussian complexity of the associated function class. This connection allows us to show that local fixed point analysis, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes, and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

In-vivo examination of the physical connectivity of axonal projections through the white matter of the human brain is made possible by diffusion weighted magnetic resonance imaging (dMRI) Analysis of dMRI commonly considers derived scalar metrics such as fractional anisotrophy as proxies for "white matter integrity," and differences of such measures have been observed as significantly correlating with various neurological diagnosis and clinical measures such as executive function, presence of multiple sclerosis, and genetic similarity. The analysis of such voxel measures is confounded in areas of more complicated fiber wiring due to crossing, kissing, and dispersing fibers. Recently, Volz et al. introduced a simple probabilistic measure of the count of distinct fiber populations within a voxel, which was shown to reduce variance in group comparisons. We propose a complementary measure that considers the complexity of a voxel in context of its local region, with an aim to quantify the localized wiring complexity of every part of white matter. This allows, for example, identification of particularly ambiguous regions of the brain for tractographic approaches of modeling global wiring connectivity. Our method builds on recent advances in image inpainting, in which the task is to plausibly fill in a missing region of an image. Our proposed method builds on a Bayesian estimate of heteroscedastic aleatoric uncertainty of a region of white matter by inpainting it from its context. We define the localized wiring complexity of white matter as how accurately and confidently a well-trained model can predict the missing patch. In our results, we observe low aleatoric uncertainty along major neuronal pathways which increases at junctions and towards cortex boundaries. This directly quantifies the difficulty of lesion inpainting of dMRI images at all parts of white matter.

We show two novel concentration inequalities for suprema of empirical processes when sampling without replacement, which both take the variance of the functions into account. While these inequalities may potentially have broad applications in learning theory in general, we exemplify their significance by studying the transductive setting of learning theory. For which we provide the first excess risk bounds based on the localized complexity of the hypothesis class, which can yield fast rates of convergence also in the transductive learning setting. We give a preliminary analysis of the localized complexities for the prominent case of kernel classes.