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.

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Deep Residual Learning for Image Recognition

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.

Adam: A Method for Stochastic Optimization

We trained a large, deep convolutional neural network to classify the 1.2 million high-resolution images in the ImageNet LSVRC-2010 contest into the 1000 different classes. On the test data, we achieved top-1 and top-5 error rates of 37.5% and 17.0% which is considerably better than the previous state-of-the-art. The neural network, which has 60 million parameters and 650,000 neurons, consists of five convolutional layers, some of which are followed by max-pooling layers, and three fully-connected layers with a final 1000-way softmax. To make training faster, we used non-saturating neurons and a very efficient GPU implementation of the convolution operation. To reduce overriding in the fully-connected layers we employed a recently-developed regularization method called "dropout" that proved to be very effective. We also entered a variant of this model in the ILSVRC-2012 competition and achieved a winning top-5 test error rate of 15.3%, compared to 26.2% achieved by the second-best entry.

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ImageNet Classification with Deep Convolutional Neural Networks

Learning to store information over extended time intervals by recurrent backpropagation takes a very long time, mostly because of insufficient, decaying error backflow. We briefly review Hochreiter's (1991) analysis of this problem, then address it by introducing a novel, efficient, gradient based method called long short-term memory (LSTM). Truncating the gradient where this does not do harm, LSTM can learn to bridge minimal time lags in excess of 1000 discrete-time steps by enforcing constant error flow through constant error carousels within special units. Multiplicative gate units learn to open and close access to the constant error flow. LSTM is local in space and time; its computational complexity per time step and weight is O. 1. Our experiments with artificial data involve local, distributed, real-valued, and noisy pattern representations. In comparisons with real-time recurrent learning, back propagation through time, recurrent cascade correlation, Elman nets, and neural sequence chunking, LSTM leads to many more successful runs, and learns much faster. LSTM also solves complex, artificial long-time-lag tasks that have never been solved by previous recurrent network algorithms.

Long short-term memory

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.

Very Deep Convolutional Networks for Large-Scale Image Recognition

To illustrate the potential of multilayer neural networks for adaptive interfaces, a VPL Data-Glove connected to a DECtalk speech synthesizer via five neural networks was used to implement a hand-gesture to speech system. Using minor variations of the standard backpropagation learning procedure, the complex mapping of hand movements to speech is learned using data obtained from a single 'speaker' in a simple training phase. With a 203 gesture-to-word vocabulary, the wrong word is produced less than 1% of the time, and no word is produced about 5% of the time. Adaptive control of the speaking rate and word stress is also available. The training times and final performance speed are improved by using small, separate networks for each naturally defined subtask. The system demonstrates that neural networks can be used to develop the complex mappings required in a high bandwidth interface that adapts to the individual user. >

Glove-Talk: a neural network interface between a data-glove and a speech synthesizer

This paper describes two new methods for modeling the manifolds of digitized images of handwritten digits. The models allow a priori information about the structure of the manifolds to be combined with empirical data. Accurate modeling of the manifolds allows digits to be discriminated using the relative probability densities under the alternative models. One of the methods is grounded in principal components analysis, the other in factor analysis. Both methods are based on locally linear low-dimensional approximations to the underlying data manifold. Links with other methods that model the manifold are discussed.

Modeling the manifolds of images of handwritten digits

It is possible to combine multiple probabilistic models of the same data by multiplying the probabilities together and then renormalizing. This is a very efficient way to model high-dimensional data which simultaneously satisfies many different low dimensional constraints. Each individual expert model can focus on giving high probability to data vectors that satisfy just one of the constraints. Data vectors that satisfy this one constraint but violate other constraints will be ruled out by their low probability under the other expert models. Training a product of models appears difficult because, in addition to maximizing the probabilities that the individual models assign to the observed data, it is necessary to make the models disagree on unobserved regions of the data space. However, if the individual models are tractable there is a fairly efficient way to train a product of models. This training algorithm suggests a biologically plausible way of learning neural population codes.

Products of experts

This paper reports our recent exploration of the layer-by-layer learning strategy for training a multi-layer generative model of patches of speech spectrograms. The top layer of the generative model learns binary codes that can be used for efficient compression of speech and could also be used for scalable speech recognition or rapid speech content retrieval. Each layer of the generative model is fully connected to the layer below and the weights on these connections are pretrained efficiently by using the contrastive divergence approximation to the log likelihood gradient. After layer-bylayer pre-training we “unroll” the generative model to form a deep auto-encoder, whose parameters are then fine-tuned using back-propagation. To reconstruct the full-length speech spectrogram, individual spectrogram segments predicted by their respective binary codes are combined using an overlapand-add method. Experimental results on speech spectrogram coding demonstrate that the binary codes produce a logspectral distortion that is approximately 2 dB lower than a subband vector quantization technique over the entire frequency range of wide-band speech. Index Terms: deep learning, speech feature extraction, neural networks, auto-encoder, binary codes, Boltzmann machine

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Binary Coding of Speech Spectrograms Using a Deep Auto-encoder

: Rumelhart, Hinton and Williams (Rumelhart 86) describe a learning procedure for layered networks of deterministic, neuron-like units. This paper describes further research on the learning procedure. We start by describing the units, the way they are connected, the learning procedure, and the extension to iterative nets. We then give an example in which a network learns a set of filters that enable it to discriminate formant-like patterns in the presence of noise. The speed of learning is strongly dependent on the shape of the surface formed by the error measure in weight space . We give examples of the shape of the error surface for a typical task and illustrate how an acceleration method speeds up descent in weight space. The main drawback of the learning procedure is the way it scales as the size of the task and the network increases. We give some preliminary results on scaling and show how the magnitude of the optimal weight changes depends on the fan-in of the units. Additional results illustrate the effects on learning speed of the amount of interaction between the weights. A variation of the learning procedure that back-propagates desired state information rather than error gradients is developed and compared with the standard procedure. Finally, we discuss the relationship between our iterative networks and the analog networks described by Hopefield and Tank (Hopfield 85). The learning procedure can discover appropriate weights in their kind of network, as well as determine an optimal schedule for varying the nonlinearity of the units during a search.

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Geoffrey E. Hinton

Papers

Glove-Talk: a neural network interface between a data-glove and a speech synthesizer

Modeling the manifolds of images of handwritten digits

Products of experts

Binary Coding of Speech Spectrograms Using a Deep Auto-encoder

Experiments on Learning by Back Propagation.