First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper addresses the problem of learning with label proportions, wherein one only knows the proportions of the labels of bags of samples that are positive (or more generally that belong to one class). It makes very substantive contributions by: a. analysis that shows that a general class of loss functions allow efficient learning via the mean operator without requiring homogeneity (as was the case with previous literature) b. Developing fast learning algorithms for estimating the mean operator c. allow the use of standard binary classifier learning algorithms to solve the LLP problem via reduction. It is also very well written, though it is quite dense because of the sheer volume of novel material that the authors try to cover in a short NIPS paper.

In Learning with Label Proportions (LLP), the objective is to learn a supervised classifier when, instead of labels, only label proportions for bags of observations are known. This setting has broad practical relevance, in particular for privacy preserving data processing. We first show that the mean operator, a statistic which aggregates all labels, is minimally sufficient for the minimization of many proper scoring losses with linear (or kernelized) classifiers without using labels. We provide a fast learning algorithm that estimates the mean operator via a manifold regularizer with guaranteed approximation bounds. Then, we present an iterative learning algorithm that uses this as initialization. We ground this algorithm in Rademacher-style generalization bounds that fit the LLP setting, introducing a generalization of Rademacher complexity and a Label Proportion Complexity measure.

In Learning with Label Proportions (LLP), the objective is to learn a supervised classifier when, instead of labels, only label proportions for bags of observations are known. This setting has broad practical relevance, in particular for privacy preserving data processing. We first show that the mean operator, a statistic which aggregates all labels, is minimally sufficient for the minimization of many proper scoring losses with linear (or kernelized) classifiers without using labels. We provide a fast learning algorithm that estimates the mean operator via a manifold regularizer with guaranteed approximation bounds. Then, we present an iterative learning algorithm that uses this as initialization. We ground this algorithm in Rademacher-style generalization bounds that fit the LLP setting, introducing a generalization of Rademacher complexity and a Label Proportion Complexity measure.

In Learning with Label Proportions (LLP), the objective is to learn a supervised classifier when, instead of labels, only label proportions for bags of observations are known. This setting has broad practical relevance, in particular for privacy preserving data processing. We first show that the mean operator, a statistic which aggregates all labels, is minimally sufficient for the minimization of many proper scoring losses with linear (or kernelized) classifiers without using labels. We provide a fast learning algorithm that estimates the mean operator via a manifold regularizer with guaranteed approximation bounds. Then, we present an iterative learning algorithm that uses this as initialization. We ground this algorithm in Rademacher-style generalization bounds that fit the LLP setting, introducing a generalization of Rademacher complexity and a Label Proportion Complexity measure.

K-Nearest Neighbors algorithm is one of the most used classifiers in terms of simplicity and performance. Although, when a dataset has many outliers or when it is small or unbalanced, KNN doesn't work well. This paper aims to propose a novel classifier, based on K-Nearest Neighbors which calculates the local means of every class using the Power Muirhead Mean operator to overcome alluded issues. We called our new algorithm Power Muirhead Mean K-Nearest Neighbors (PMM-KNN). Eventually, we used five well-known datasets to assess PMM-KNN performance. The research results demonstrate that the PMM-KNN has outperformed three state-of-the-art classification methods in all experiments.

In Learning with Label Proportions (LLP), the objective is to learn a supervised classifier when, instead of labels, only label proportions for bags of observations are known. This setting has broad practical relevance, in particular for privacy preserving data processing. We first show that the mean operator, a statistic which aggregates all labels, is minimally sufficient for the minimization of many proper scoring losses with linear (or kernelized) classifiers without using labels. We provide a fast learning algorithm that estimates the mean operator via a manifold regularizer with guaranteed approximation bounds. Then, we present an iterative learning algorithm that uses this as initialization.

We prove that the empirical risk of most well-known loss functions factors into a linear term aggregating all labels with a term that is label free, and can further be expressed by sums of the loss. This holds true even for non-smooth, non-convex losses and in any RKHS. The first term is a (kernel) mean operator --the focal quantity of this work-- which we characterize as the sufficient statistic for the labels. The result tightens known generalization bounds and sheds new light on their interpretation. Factorization has a direct application on weakly supervised learning. In particular, we demonstrate that algorithms like SGD and proximal methods can be adapted with minimal effort to handle weak supervision, once the mean operator has been estimated. We apply this idea to learning with asymmetric noisy labels, connecting and extending prior work. Furthermore, we show that most losses enjoy a data-dependent (by the mean operator) form of noise robustness, in contrast with known negative results.

In Learning with Label Proportions (LLP), the objective is to learn a supervised classifier when, instead of labels, only label proportions for bags of observations are known. This setting has broad practical relevance, in particular for privacy preserving data processing. We first show that the mean operator, a statistic which aggregates all labels, is minimally sufficient for the minimization of many proper scoring losses with linear (or kernelized) classifiers without using labels. We provide a fast learning algorithm that estimates the mean operator via a manifold regularizer with guaranteed approximation bounds. Then, we present an iterative learning algorithm that uses this as initialization. We ground this algorithm in Rademacher-style generalization bounds that fit the LLP setting, introducing a generalization of Rademacher complexity and a Label Proportion Complexity measure. This latter algorithm optimizes tractable bounds for the corresponding bag-empirical risk. Experiments are provided on fourteen domains, whose size ranges up to 300K observations. They display that our algorithms are scalable and tend to consistently outperform the state of the art in LLP. Moreover, in many cases, our algorithms compete with or are just percents of AUC away from the Oracle that learns knowing all labels. On the largest domains, half a dozen proportions can suffice, i.e. roughly 40K times less than the total number of labels.