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

We introduce a new statistical model for time series that iteratively segments data into regimes with approximately linear dynamics and learns the parameters of each of these linear regimes. This model combines and generalizes two of the most widely used stochastic time-series models— hidden Markov models and linear dynamical systems—and is closely related to models that are widely used in the control and econometrics literatures. It can also be derived by extending the mixture of experts neural network (Jacobs, Jordan, Nowlan, & Hinton, 1991) to its fully dynamical version, in which both expert and gating networks are recurrent. Inferring the posterior probabilities of the hidden states of this model is computationally intractable, and therefore the exact expectation maximization (EM) algorithm cannot be applied. However, we present a variational approximation that maximizes a lower bound on the log-likelihood and makes use of both the forward and backward recursions for hidden Markov models and the Kalman filter recursions for linear dynamical systems. We tested the algorithm on artificial data sets and a natural data set of respiration force from a patient with sleep apnea. The results suggest that variational approximations are a viable method for inference and learning in switching state-space models.

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Variational Learning for Switching State-Space Models

This thesis develops two Bayesian learning methods relying on Gaussian processes and a rigorous statistical approach for evaluating such methods. In these experimental designs the sources of uncertainty in the estimated generalisation performances due to both variation in training and test sets are accounted for. The framework allows for estimation of generalisation performance as well as statistical tests of significance for pairwise comparisons. Two experimental designs are recommended and supported by the DELVE software environment.
Two new non-parametric Bayesian learning methods relying on Gaussian process priors over functions are developed. These priors are controlled by hyperparameters which set the characteristic length scale for each input dimension. In the simplest method, these parameters are fit from the data using optimization. In the second, fully Bayesian method, a Markov chain Monte Carlo technique is used to integrate over the hyperparameters. One advantage of these Gaussian process methods is that the priors and hyperparameters of the trained models are easy to interpret.
The Gaussian process methods are benchmarked against several other methods, on regression tasks using both real data and data generated from realistic simulations. The experiments show that small datasets are unsuitable for benchmarking purposes because the uncertainties in performance measurements are large. A second set of experiments provide strong evidence that the bagging procedure is advantageous for the Multivariate Adaptive Regression Splines (MARS) method.
The simulated datasets have controlled characteristics which make them useful for understanding the relationship between properties of the dataset and the performance of different methods. The dependency of the performance on available computation time is also investigated. It is shown that a Bayesian approach to learning in multi-layer perceptron neural networks achieves better performance than the commonly used early stopping procedure, even for reasonably short amounts of computation time. The Gaussian process methods are shown to consistently outperform the more conventional methods.

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Evaluation of gaussian processes and other methods for non-linear regression

Information extracted from aerial photographs has found applications in a wide range of areas including urban planning, crop and forest management, disaster relief, and climate modeling. At present, much of the extraction is still performed by human experts, making the process slow, costly, and error prone. The goal of this thesis is to develop methods for automatically extracting the locations of objects such as roads, buildings, and trees directly from aerial images. 
We investigate the use of machine learning methods trained on aligned aerial images and possibly outdated maps for labeling the pixels of an aerial image with semantic labels. We show how deep neural networks implemented on modern GPUs can be used to efficiently learn highly discriminative image features. We then introduce new loss functions for training neural networks that are partially robust to incomplete and poorly registered target maps. Finally, we propose two ways of improving the predictions of our system by introducing structure into the outputs of the neural networks. 
We evaluate our system on the largest and most-challenging road and building detection datasets considered in the literature and show that it works reliably under a wide variety of conditions. Furthermore, we are releasing the first large-scale road and building detection datasets to the public in order to facilitate future comparisons with other methods.

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Machine learning for aerial image labeling

We present a new learning algorithm for Boltzmann machines that contain many layers of hidden variables. Data-dependent statistics are estimated using a variational approximation that tends to focus on a single mode, and data-independent statistics are estimated using persistent Markov chains. The use of two quite different techniques for estimating the two types of statistic that enter into the gradient of the log likelihood makes it practical to learn Boltzmann machines with multiple hidden layers and millions of parameters. The learning can be made more efficient by using a layer-by-layer pretraining phase that initializes the weights sensibly. The pretraining also allows the variational inference to be initialized sensibly with a single bottom-up pass. We present results on the MNIST and NORB data sets showing that deep Boltzmann machines learn very good generative models of handwritten digits and 3D objects. We also show that the features discovered by deep Boltzmann machines are a very effective way to initialize the hidden layers of feedforward neural nets, which are then discriminatively fine-tuned.

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An efficient learning procedure for deep boltzmann machines

The Temporal Restricted Boltzmann Machine (TRBM) is a probabilistic model for sequences that is able to successfully model (ie, generate nice-looking samples of) several very high dimensional sequences, such as motion capture data and the pixels of low resolution videos of balls bouncing in a box The major disadvantage of the TRBM is that exact inference is extremely hard, since even computing a Gibbs update for a single variable of the posterior is exponentially expensive This difficulty has necessitated the use of a heuristic inference procedure, that nonetheless was accurate enough for successful learning In this paper we introduce the Recurrent TRBM, which is a very slight modification of the TRBM for which exact inference is very easy and exact gradient learning is almost tractable We demonstrate that the RTRBM is better than an analogous TRBM at generating motion capture and videos of bouncing balls

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Geoffrey E. Hinton

Papers

Variational Learning for Switching State-Space Models

Evaluation of gaussian processes and other methods for non-linear regression

Machine learning for aerial image labeling

An efficient learning procedure for deep boltzmann machines

The Recurrent Temporal Restricted Boltzmann Machine