Supervised learning has seen numerous theoretical and practical advances over the last few decades. However, its basic assumption of identical train and test distributions often fails to hold in practice. One important example of this is when the training instances are subject to label noise: that is, where the observed labels do not accurately reflect the underlying ground truth. While the impact of simple noise models has been extensively studied, relatively less attention has been paid to the practically relevant setting of instance-dependent label noise. It is thus unclear whether one can learn, both in theory and in practice, good models from data subject to such noise, with no access to clean labels. We provide a theoretical analysis of this issue, with three contributions. First, we prove that for instance-dependent (but label-independent) noise, any algorithm that is consistent for classification on the noisy distribution is also consistent on the noise-free distribution. Second, we prove that consistency also holds for the area under the ROC curve, assuming the noise scales (in a precise sense) with the inherent difficulty of an instance. Third, we show that the Isotron algorithm can efficiently and provably learn from noisy samples when the noise-free distribution is a generalised linear model. We empirically confirm our theoretical findings, which we hope may stimulate further analysis of this important learning setting.

It is usual in machine learning theory to assume that the training and testing sets comprise of draws from the same distribution. This is rarely, if ever, true and one must admit the presence of corruption. There are many different types of corruption that can arise and as of yet there is no general means to compare the relative ease of learning in these settings. Such results are necessary if we are to make informed economic decisions regarding the acquisition of data. Here we begin to develop an abstract framework for tackling these problems. We present a generic method for learning from a fixed, known, reconstructible corruption, along with an analyses of its statistical properties. We demonstrate the utility of our framework via concrete novel results in solving supervised learning problems wherein the labels are corrupted, such as learning with noisy labels, semi-supervised learning and learning with partial labels.

Convex potential minimisation is the de facto approach to binary classification. However, Long and Servedio [2010] proved that under symmetric label noise (SLN), minimisation of any convex potential over a linear function class can result in classification performance equivalent to random guessing. This ostensibly shows that convex losses are not SLN-robust. In this paper, we propose a convex, classification-calibrated loss and prove that it is SLN-robust. The loss avoids the Long and Servedio [2010] result by virtue of being negatively unbounded. The loss is a modification of the hinge loss, where one does not clamp at zero; hence, we call it the unhinged loss. We show that the optimal unhinged solution is equivalent to that of a strongly regularised SVM, and is the limiting solution for any convex potential; this implies that strong l2 regularisation makes most standard learners SLN-robust. Experiments confirm the SLN-robustness of the unhinged loss.

Many classification algorithms produce a classifier that is a weighted average of kernel evaluations. When working with a high or infinite dimensional kernel, it is imperative for speed of evaluation and storage issues that as few training samples as possible are used in the kernel expansion. Popular existing approaches focus on altering standard learning algorithms, such as the Support Vector Machine, to induce sparsity, as well as post-hoc procedures for sparse approximations. Here we adopt the latter approach. We begin with a very simple classifier, given by the kernel mean $$ f(x) = \frac{1}{n} \sum\limits_{i=i}^{n} y_i K(x_i,x) $$ We then find a sparse approximation to this kernel mean via herding. The result is an accurate, easily parallelized algorithm for learning classifiers.

In supervised learning one wishes to identify a pattern present in a joint distribution $P$, of instances, label pairs, by providing a function $f$ from instances to labels that has low risk $\mathbb{E}_{P}\ell(y,f(x))$. To do so, the learner is given access to $n$ iid samples drawn from $P$. In many real world problems clean samples are not available. Rather, the learner is given access to samples from a corrupted distribution $\tilde{P}$ from which to learn, while the goal of predicting the clean pattern remains. There are many different types of corruption one can consider, and as of yet there is no general means to compare the relative ease of learning under these different corruption processes. In this paper we develop a general framework for tackling such problems as well as introducing upper and lower bounds on the risk for learning in the presence of corruption. Our ultimate goal is to be able to make informed economic decisions in regards to the acquisition of data sets. For a certain subclass of corruption processes (those that are \emph{reconstructible}) we achieve this goal in a particular sense. Our lower bounds are in terms of the coefficient of ergodicity, a simple to calculate property of stochastic matrices. Our upper bounds proceed via a generalization of the method of unbiased estimators appearing in recent work of Natarajan et al and implicit in the earlier work of Kearns.

Feature Learning aims to extract relevant information contained in data sets in an automated fashion. It is driving force behind the current deep learning trend, a set of methods that have had widespread empirical success. What is lacking is a theoretical understanding of different feature learning schemes. This work provides a theoretical framework for feature learning and then characterizes when features can be learnt in an unsupervised fashion. We also provide means to judge the quality of features via rate-distortion theory and its generalizations.

Many supervised learning problems involve learning from samples whose labels are corrupted in some way. For example, each label may be flipped with some constant probability (learning with label noise), or one may have a pool of unlabelled samples in lieu of negative samples (learning from positive and unlabelled data). This paper uses class-probability estimation to study these and other corruption processes belonging to the mutually contaminated distributions framework (Scott et al., 2013), with three conclusions. First, one can optimise balanced error and AUC without knowledge of the corruption parameters. Second, given estimates of the corruption parameters, one can minimise a range of classification risks. Third, one can estimate corruption parameters via a class-probability estimator (e.g. kernel logistic regression) trained solely on corrupted data. Experiments on label noise tasks corroborate our analysis.

"Deep Learning" methods attempt to learn generic features in an unsupervised fashion from a large unlabelled data set. These generic features should perform as well as the best hand crafted features for any learning problem that makes use of this data. We provide a definition of generic features, characterize when it is possible to learn them and provide methods closely related to the autoencoder and deep belief network of deep learning. In order to do so we use the notion of deficiency and illustrate its value in studying certain general learning problems.

Bayes theorem can be seen as the result of an optimization problem. By slightly altering this optimization problem many generalized Bayes rules can be constructed. In this work we show that a notion of a conjugate prior for non exponential family distributions can be recovered if one uses one of these generalized rules. We prove some theorems concerning this new updating rule before giving a simple example of such a generalized conjugate prior.