text: Autoencoders are widely used for unsupervised learning and as a regularization scheme in semi-supervised learning. However, theoretical understanding of their generalization properties and of the manner in which they can assist supervised learning has been lacking. We utilize recent advances in the theory of deep learning generalization, together with a novel reconstruction loss, to provide generalization bounds for autoencoders. To the best of our knowledge, this is the first such bound. We further show that, under appropriate assumptions, an autoencoder with good generalization properties can improve any semi-supervised learning scheme. We support our theoretical results with empirical demonstrations.
  @cite_7: Manifold regularization (, 2006) is a geometrically motivated framework for machine learning within which several semi-supervised algorithms have been constructed. Here we try to provide some theoretical understanding of this approach. Our main result is to expose the natural structure of a class of problems on which manifold regularization methods are helpful. We show that for such problems, no supervised learner can learn effectively. On the other hand, a manifold based learner (that knows the manifold or "learns" it from unlabeled examples) can learn with relatively few labeled examples. Our analysis follows a minimax style with an emphasis on finite sample results (in terms of n: the number of labeled examples). These results allow us to properly interpret manifold regularization and related spectral and geometric algorithms in terms of their potential use in semi-supervised learning.
  @cite_21: In this paper, on the one hand, we aim to give a review on literature dealing with the problem of supervised learning aided by additional unlabeled data. On the other hand, being a part of the author's first year PhD report, the paper serves as a frame to bundle related work by the author as well as numerous suggestions for potential future work. Therefore, this work contains more speculative and partly subjective material than the reader might expect from a literature review. We give a rigorous definition of the problem and relate it to supervised and unsupervised learning. The crucial role of prior knowledge is put forward, and we discuss the important notion of input-dependent regularization. We postulate a number of baseline methods, being algorithms or algorithmic schemes which can more or less straightforwardly be applied to the problem, without the need for genuinely new concepts. However, some of them might serve as basis for a genuine method. In the literature review, we try to cover the wide variety of (recent) work and to classify this work into meaningful categories. We also mention work done on related problems and suggest some ideas towards synthesis. Finally, we discuss some caveats and tradeoffs of central importance to the problem.
  @cite_6: 
  @cite_19: Empirical evidence shows that in favorable situations semi-supervised learning (SSL) algorithms can capitalize on the abundance of unlabeled training data to improve the performance of a learning task, in the sense that fewer labeled training data are needed to achieve a target error bound. However, in other situations unlabeled data do not seem to help. Recent attempts at theoretically characterizing SSL gains only provide a partial and sometimes apparently conflicting explanations of whether, and to what extent, unlabeled data can help. In this paper, we attempt to bridge the gap between the practice and theory of semi-supervised learning. We develop a finite sample analysis that characterizes the value of un-labeled data and quantifies the performance improvement of SSL compared to supervised learning. We show that there are large classes of problems for which SSL can significantly outperform supervised learning, in finite sample regimes and sometimes also in terms of error convergence rates.
  @cite_25: Semi-supervised methods use unlabeled data in addition to labeled data to construct predictors. While existing semi-supervised methods have shown some promising empirical performance, their development has been based largely based on heuristics. In this paper we study semi-supervised learning from the viewpoint of minimax theory. Our first result shows that some common methods based on regularization using graph Laplacians do not lead to faster minimax rates of convergence. Thus, the estimators that use the unlabeled data do not have smaller risk than the estimators that use only labeled data. We then develop several new approaches that provably lead to improved performance. The statistical tools of minimax analysis are thus used to offer some new perspective on the problem of semi-supervised learning.
summary: The value of SSL has been subject to much debate. @cite_6 provide a mathematical framework for the intuitive of @cite_21 , and show that for unlabeled data to be beneficial, some clustering criterion is required (specifically, that the data consists of separated clusters with identical labels within each cluster). Based on a density level set approach, they prove fast rates of convergence in the SSL setting. @cite_25 and @cite_7 ) study SSL within a minimax framework, the former work shows that, under the so-called manifold assumption, optimal minimax rates of convergence may be achieved, while the second work demonstrates a separation between two classes of problems. When the structure of the data manifold is known, fast rates can be achieved, while without such knowledge, convergence cannot be guaranteed. More directly related to our work, and building on the clustering assumption, @cite_19 identify situations in which semi-supervised learning can improve upon supervised learning. Unfortunately, their SSL bound suffers from the curse of dimensionality, and so depends exponentially on the dimension. Our work can be seen as allowing a trade-off between the clustering separation and the dimension, suggesting how to improve the bounds in @cite_19 .