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Detecting Modularity in Deep Neural Networks.
TL;DR: In this article, the authors consider the problem of assessing the modularity exhibited by a partitioning of a network's neurons and propose two proxies for this: importance, which reflects how crucial sets of neurons are to network performance, and coherence, which consistently their neurons associate with features of the inputs.
Abstract: A neural network is modular to the extent that parts of its computational graph (i.e. structure) can be represented as performing some comprehensible subtask relevant to the overall task (i.e. functionality). Are modern deep neural networks modular? How can this be quantified? In this paper, we consider the problem of assessing the modularity exhibited by a partitioning of a network's neurons. We propose two proxies for this: importance, which reflects how crucial sets of neurons are to network performance; and coherence, which reflects how consistently their neurons associate with features of the inputs. To measure these proxies, we develop a set of statistical methods based on techniques conventionally used to interpret individual neurons. We apply the proxies to partitionings generated by spectrally clustering a graph representation of the network's neurons with edges determined either by network weights or correlations of activations. We show that these partitionings, even ones based only on weights (i.e. strictly from non-runtime analysis), reveal groups of neurons that are important and coherent. These results suggest that graph-based partitioning can reveal modularity and help us understand how deep neural networks function.
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TL;DR: In this paper, a layer-wise pruning method based on modularity has been proposed to identify and locate redundant layers in deep neural networks, which provides theoretical guidance for layer pruning.
Abstract: There are good arguments to support the claim that feature representations
eventually transition from general to specific in deep neural networks (DNNs),
but this transition remains relatively underexplored. In this work, we move a
tiny step towards understanding the transition of feature representations. We
first characterize this transition by analyzing the class separation in
intermediate layers, and next model the process of class separation as
community evolution in dynamic graphs. Then, we introduce modularity, a common
metric in graph theory, to quantify the evolution of communities. We find that
modularity tends to rise as the layer goes deeper, but descends or reaches a
plateau at particular layers. Through an asymptotic analysis, we show that
modularity can provide quantitative analysis of the transition of the feature
representations. With the insight on feature representations, we demonstrate
that modularity can also be used to identify and locate redundant layers in
DNNs, which provides theoretical guidance for layer pruning. Based on this
inspiring finding, we propose a layer-wise pruning method based on modularity.
Further experiments show that our method can prune redundant layers with
minimal impact on performance. The codes are available at
https://github.com/yaolu-zjut/Dynamic-Graphs-Construction.
References
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27 Jun 2016
TL;DR: In this article, the authors proposed a residual learning framework to ease the training of networks that are substantially deeper than those used previously, which won the 1st place on the ILSVRC 2015 classification task.
Abstract: Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers—8× deeper than VGG nets [40] but still having lower complexity. An ensemble of these residual nets achieves 3.57% error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28% relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions1, where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation.
123,388 citations
Proceedings Article•
01 Jan 2015TL;DR: This work introduces Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments, and provides a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework.
Abstract: We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.
111,197 citations
TL;DR: In this paper, a different approach to problems of multiple significance testing is presented, which calls for controlling the expected proportion of falsely rejected hypotheses -the false discovery rate, which is equivalent to the FWER when all hypotheses are true but is smaller otherwise.
Abstract: SUMMARY The common approach to the multiplicity problem calls for controlling the familywise error rate (FWER). This approach, though, has faults, and we point out a few. A different approach to problems of multiple significance testing is presented. It calls for controlling the expected proportion of falsely rejected hypotheses -the false discovery rate. This error rate is equivalent to the FWER when all hypotheses are true but is smaller otherwise. Therefore, in problems where the control of the false discovery rate rather than that of the FWER is desired, there is potential for a gain in power. A simple sequential Bonferronitype procedure is proved to control the false discovery rate for independent test statistics, and a simulation study shows that the gain in power is substantial. The use of the new procedure and the appropriateness of the criterion are illustrated with examples.
83,420 citations
Proceedings Article•
04 Sep 2014TL;DR: This work investigates the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting using an architecture with very small convolution filters, which shows that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 weight layers.
Abstract: In this work we investigate the effect of the convolutional network depth on its accuracy in the large-scale image recognition setting. Our main contribution is a thorough evaluation of networks of increasing depth using an architecture with very small (3x3) convolution filters, which shows that a significant improvement on the prior-art configurations can be achieved by pushing the depth to 16-19 weight layers. These findings were the basis of our ImageNet Challenge 2014 submission, where our team secured the first and the second places in the localisation and classification tracks respectively. We also show that our representations generalise well to other datasets, where they achieve state-of-the-art results. We have made our two best-performing ConvNet models publicly available to facilitate further research on the use of deep visual representations in computer vision.
55,235 citations
Journal Article•
TL;DR: Scikit-learn is a Python module integrating a wide range of state-of-the-art machine learning algorithms for medium-scale supervised and unsupervised problems, focusing on bringing machine learning to non-specialists using a general-purpose high-level language.
Abstract: Scikit-learn is a Python module integrating a wide range of state-of-the-art machine learning algorithms for medium-scale supervised and unsupervised problems. This package focuses on bringing machine learning to non-specialists using a general-purpose high-level language. Emphasis is put on ease of use, performance, documentation, and API consistency. It has minimal dependencies and is distributed under the simplified BSD license, encouraging its use in both academic and commercial settings. Source code, binaries, and documentation can be downloaded from http://scikit-learn.sourceforge.net.
47,974 citations