Ontheonehand, thediversedata processing steps lead tothelack oflarge-scale7 benchmarks and therefore decelerate iterative methodology improvement on a8 single problem.

In addition, the underlying signalxisassumed to lie in the range of anL-Lipschitz continuous generativemodel with boundedkdimensionalinputs.Weproposeatwo-stepapproach,forwhichthefirststepplays the role ofspectral initialization and the second step refines the estimated vector produced by the first step iteratively. We show that both steps enjoy a statistical rate oforder p (klogL) (logm)/mundersuitable conditions.

Moreover,ourtheoretical analysis reliesonstandard assumptions only,works inthedistributed heterogeneous data setting, and leads tobetter and more meaningful rates.

We study three notions of uncertainty quantification--calibration, confidence intervals and prediction sets--for binary classification in the distribution-free setting, that is without making any distributional assumptions on the data. With a focus towards calibration, we establish a'tripod' of theorems that connect these three notions for score-based classifiers. A direct implication is that distributionfree calibration is only possible, even asymptotically, using a scoring function whose level sets partition the feature space into at most countably many sets. Parametric calibration schemes such as variants of Platt scaling do not satisfy this requirement, while nonparametric schemes based on binning do.