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. High demand for computation resources severely hinders deployment of large-scale Deep Neural Networks (DNN) in resource constrained devices. In this work, we propose a Structured Sparsity Learning (SSL) method to regularize the structures (i.e., filters, channels, filter shapes, and layer depth) of DNNs. SSL can: (1) learn a compact structure from a bigger DNN to reduce computation cost; (2) obtain a hardware-friendly structured sparsity of DNN to efficiently accelerate the DNN's evaluation. Experimental results show that SSL achieves on average 5.1 × and 3.1 × speedups of convolutional layer computation of AlexNet against CPU and GPU, respectively, with off-the-shelf libraries. These speedups are about twice speedups of non-structured sparsity; (3) regularize the DNN structure to improve classification accuracy. The results show that for CIFAR-10, regularization on layer depth reduces a 20-layer Deep Residual Network (ResNet) to 18 layers while improves the accuracy from 91.25 to 92.60 , which is still higher than that of original ResNet with 32 layers. For AlexNet, SSL reduces the error by 1 . High demand for computation resources severely hinders deployment of large-scale Deep Neural Networks (DNN) in resource constrained devices. In this work, we propose a Structured Sparsity Learning (SSL) method to regularize the structures (i.e., filters, channels, filter shapes, and layer depth) of DNNs. SSL can: (1) learn a compact structure from a bigger DNN to reduce computation cost; (2) obtain a hardware-friendly structured sparsity of DNN to efficiently accelerate the DNNs evaluation. Experimental results show that SSL achieves on average 5.1x and 3.1x speedups of convolutional layer computation of AlexNet against CPU and GPU, respectively, with off-the-shelf libraries. These speedups are about twice speedups of non-structured sparsity; (3) regularize the DNN structure to improve classification accuracy. The results show that for CIFAR-10, regularization on layer depth can reduce 20 layers of a Deep Residual Network (ResNet) to 18 layers while improve the accuracy from 91.25 to 92.60 , which is still slightly higher than that of original ResNet with 32 layers. For AlexNet, structure regularization by SSL also reduces the error by around 1 . Open source code is in this https URL The nearestor near-neighbor query problems arise in a large variety of database applications, usually in the context of similarity searching. Of late, there has been increasing interest in building search index structures for performing similarity search over high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. Unfortunately, all known techniques for solving this problem fall prey to the of dimensionality." That is, the data structures scale poorly with data dimensionality; in fact, if the number of dimensions exceeds 10 to 20, searching in k-d trees and related structures involves the inspection of a large fraction of the database, thereby doing no better than brute-force linear search. It has been suggested that since the selection of features and the choice of a distance metric in typical applications is rather heuristic, determining an approximate nearest neighbor should su ce for most practical purposes. In this paper, we examine a novel scheme for approximate similarity search based on hashing. The basic idea is to hash the points Supported by NAVY N00014-96-1-1221 grant and NSF Grant IIS-9811904. Supported by Stanford Graduate Fellowship and NSF NYI Award CCR-9357849. Supported by ARO MURI Grant DAAH04-96-1-0007, NSF Grant IIS-9811904, and NSF Young Investigator Award CCR9357849, with matching funds from IBM, Mitsubishi, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the VLDB copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Very Large Data Base Endowment. To copy otherwise, or to republish, requires a fee and or special permission from the Endowment. Proceedings of the 25th VLDB Conference, Edinburgh, Scotland, 1999. from the database so as to ensure that the probability of collision is much higher for objects that are close to each other than for those that are far apart. We provide experimental evidence that our method gives signi cant improvement in running time over other methods for searching in highdimensional spaces based on hierarchical tree decomposition. Experimental results also indicate that our scheme scales well even for a relatively large number of dimensions (more than 50).