In this work we study the semi-supervised framework of confidence set classification with controlled expected size in minimax settings. We obtain semi-supervised minimax rates of convergence under the margin assumption and a H o lder condition on the regression function. Besides, we show that if no further assumptions are made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work. We establish that the best achievable rate for any supervised method is n^ --1 2 , even if the margin assumption is extremely favorable. On the contrary, semi-supervised estimators can achieve faster rates of convergence provided that sufficiently many unlabeled samples are available. We additionally perform numerical evaluation of the proposed algorithms empirically confirming our theoretical findings.
Confidence set approach for classification was pioneered by @cite_9 @cite_13 @cite_8 by the means of conformal prediction theory. They rely on non-conformity measures which are based on some pattern recognition methods, and develop an asymptotic theory. In this work, we consider a statistical perspective of confidence set classification and put our focus on non-asymptotic minimax theory.
@cite_9 @cite_13 @cite_8
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Confident prediction is highly relevant in machine learning; for example, in applications such as medical diagnoses, wrong prediction can be fatal. For classification, there already exist procedures that allow to not classify data when the confidence in their prediction is weak. This approach is known as classification with reject option. In the present paper, we provide new methodology for this approach. Predicting a new instance via a confidence set, we ensure an exact control of the probability of classification. Moreover, we show that this methodology is easily implementable and entails attractive theoretical and numerical properties. We propose a new method for pointwise estimation of monotone, uni- modal and U-shaped failure rates, under a right-censoring mechanism, using non- parametric likelihood ratios. The asymptotic distribution of the likelihood ratio is pivotal, though non-standard, and can therefore be used to construct asymptotic confidence intervals for the failure rate at a point of interest, via inversion. Major advantages of the new method lie in the facts that it completely avoids estima- tion of nuisance parameters, or the choice of a bandwidth tuning parameter, and is extremely easy to implement. The new method is shown to perform competi- tively in simulations, and is illustrated on a data set involving time to diagnosis of schizophrenia in the Jerusalem Perinatal Cohort Schizophrenia Study. A framework for classification is developed with a notion of confidence. In this framework, a classifier consists of two tolerance regions in the predictor space, with a specified coverage level for each class. The classifier also produces an ambiguous region where the classification needs further investigation. Theoretical analysis reveals interesting structures of the confidence-ambiguity trade-off, and the optimal solution is characterized by extending the Neyman–Pearson lemma. We provide general estimating procedures, along with rates of convergence, based on estimates of the conditional probabilities. The method can be easily implemented with good robustness, as illustrated through theory, simulation and a data example. We consider the estimation of an i.i.d. random vector observed through a linear transform followed by a componentwise, probabilistic (possibly nonlinear) measurement channel. A novel algorithm, called generalized approximate message passing (GAMP), is presented that provides computationally efficient approximate implementations of max-sum and sum-problem loopy belief propagation for such problems. The algorithm extends earlier approximate message passing methods to incorporate arbitrary distributions on both the input and output of the transform and can be applied to a wide range of problems in nonlinear compressed sensing and learning. Extending an analysis by Bayati and Montanari, we argue that the asymptotic componentwise behavior of the GAMP method under large, i.i.d. Gaussian transforms is described by a simple set of state evolution (SE) equations. From the SE equations, one can predict the asymptotic value of virtually any componentwise performance metric including mean-squared error or detection accuracy. Moreover, the analysis is valid for arbitrary input and output distributions, even when the corresponding optimization problems are non-convex. The results match predictions by Guo and Wang for relaxed belief propagation on large sparse matrices and, in certain instances, also agree with the optimal performance predicted by the replica method. The GAMP methodology thus provides a computationally efficient methodology, applicable to a large class of non-Gaussian estimation problems with precise asymptotic performance guarantees.