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.
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 .
@cite_7 @cite_21 @cite_6 @cite_19 @cite_25
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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. We study the potential benefits of unlabeled data to classification prediction to the learner. We compare learning in the semi-supervised model to the standard, supervised PAC (distribution free) model, considering both the realizable and the unrealizable (agnostic) settings. Roughly speaking, our conclusion is that access to unlabeled samples cannot provide sample size guarantees that are better than those obtainable without access to unlabeled data, unless one postulates very strong assumptions about the distribution of the labels. In particular, we prove that for basic hypothesis classes over the real line, if the distribution of unlabeled data is smooth, knowledge of that distribution cannot improve the labeled sample complexity by more than a constant factor (e.g., 2). We conjecture that a similar phenomena holds for any hypothesis class and any unlabeled data distribution. We also discuss the utility of semi-supervised learning under the common cluster assumption concerning the distribution of labels, and show that even in the most accommodating cases, where data is generated by two uni-modal label-homogeneous distributions, common SSL paradigms may be misleading and inflict poor prediction performance. The insufficiency of labeled training data is a major obstacle in automatic image annotation. To tackle this problem, we propose a semi-supervised manifold kernel density estimation (SSMKDE) approach based on a recently proposed manifold KDE method. Our contributions are twofold. First, SSMKDE leverages both labeled and unlabeled samples and formulates all data in a manifold structure, which enables a more accurate label prediction. Second, the relationship between KDE-based methods and graph-based semi-supervised learning (SSL) methods is analyzed, which helps to better understand graph-based SSL methods. Extensive experiments demonstrate the superiority of SSMKDE over existing KDE-based and graph-based SSL methods. a b s t r a c t Constructing an informative and discriminative graph plays an important role in various pattern recogni- tion tasks such as clustering and classification. Among the existing graph-based learning models, low-rank representation (LRR) is a very competitive one, which has been extensively employed in spectral cluster- ing and semi-supervised learning (SSL). In SSL, the graph is composed of both labeled and unlabeled sam- ples, where the edge weights are calculated based on the LRR coefficients. However, most of existing LRR related approaches fail to consider the geometrical structure of data, which has been shown beneficial for discriminative tasks. In this paper, we propose an enhanced LRR via sparse manifold adaption, termed manifold low-rank representation (MLRR), to learn low-rank data representation. MLRR can explicitly take the data local manifold structure into consideration, which can be identified by the geometric sparsity idea; specifically, the local tangent space of each data point was sought by solving a sparse representation objective. Therefore, the graph to depict the relationship of data points can be built once the manifold information is obtained. We incorporate a regularizer into LRR to make the learned coefficients preserve the geometric constraints revealed in the data space. As a result, MLRR combines both the global informa- tion emphasized by low-rank property and the local information emphasized by the identified manifold structure. Extensive experimental results on semi-supervised classification tasks demonstrate that MLRR is an excellent method in comparison with several state-of-the-art graph construction approaches. 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.