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.

/pdf/deep-residual-learning-for-image-recognition-b75jg61qrq.pdf

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

Linear relational embedding (LRE) was introduced previously by the authors (1999) as a means of extracting a distributed representation of concepts from relational data. The original formulation cannot use negative information and cannot properly handle data in which there are multiple correct answers. In this paper we propose an extended formulation of LRE that solves both these problems. We present results in two simple domains, which show that learning leads to good generalization.

/pdf/extracting-distributed-representations-of-concepts-and-26yzua45b4.pdf

Extracting distributed representations of concepts and relations from positive and negative propositions

http://www.cs.toronto.edu/~hinton/absps/ncrg.pdf

Instantiating deformable models with a neural net

In models that define probabilities via energies, maximum likelihood learning typically involves using Markov Chain Monte Carlo to sample from the model's distribution. If the Markov chain is started at the data distribution, learning often works well even if the chain is only run for a few time steps [3]. But if the data distribution contains modes separated by regions of very low density, brief MCMC will not ensure that different modes have the correct relative energies because it cannot move particles from one mode to another. We show how to improve brief MCMC by allowing long-range moves that are suggested by the data distribution. If the model is approximately correct, these long-range moves have a reasonable acceptance rate.

/pdf/wormholes-improve-contrastive-divergence-13zrl29vyr.pdf

Wormholes Improve Contrastive Divergence

A system and method for generating training images. An existing training image is associated with a classification. The system includes an image processing module that performs color-space deformation on each pixel of the existing training image and then associates the classification to the color-space deformed training image. The technique may be applied to increase the size of a training set for training a neural network.

System and method for generating training cases for image classification

In this thesis I develop a method for recognizing isolated handprinted digits using trainable deformable models. Each digit is modelled by a cubic B-spline whose basic shape is defined by the "home" positions of the control points. A Gaussian distribution over displacements of the control points away from their home locations defines a probability distribution over shapes. The quality of the match of a spline model to an image is calculated as the likelihood of the data under a mixture of Gaussian "ink generators" placed along the length of the spline. Each spline model is adjusted to minimize an energy function that includes both the deformation energy of the model and the likelihood of the data, using a elastic matching procedure which is a generalization of the Expectation Maximization (EM) algorithm. I show that the matching procedure can be significantly speeded up by using a neural net to provide better starting points for the search.
The use of deformable models has a number of advantages. (1) After identifying the model most likely to have generated the data, the system not only produces a classification of the digit but also a rich description of the instantiation parameters. I have shown that these can be used to detect writing style consistency within a string of digits. (2) During the process of explaining the image, generative models can perform recognition-driven segmentation. (3) Unlike many other recognition schemes the method does not rely on some form of pre-normalization of input images, but can handle arbitrary scalings, translations and a limited degree of image rotation. The main disadvantage of the method is it requires much more computation than more standard optical character recognition techniques.

Geoffrey E. Hinton

Papers

Extracting distributed representations of concepts and relations from positive and negative propositions

Instantiating deformable models with a neural net

Wormholes Improve Contrastive Divergence

System and method for generating training cases for image classification

Combining deformable models and neural networks for handprinted digit recognition