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ADADELTA: An Adaptive Learning Rate Method
TL;DR: A novel per-dimension learning rate method for gradient descent called ADADELTA that dynamically adapts over time using only first order information and has minimal computational overhead beyond vanilla stochastic gradient descent is presented.
Abstract: We present a novel per-dimension learning rate method for gradient descent called ADADELTA. The method dynamically adapts over time using only first order information and has minimal computational overhead beyond vanilla stochastic gradient descent. The method requires no manual tuning of a learning rate and appears robust to noisy gradient information, different model architecture choices, various data modalities and selection of hyperparameters. We show promising results compared to other methods on the MNIST digit classification task using a single machine and on a large scale voice dataset in a distributed cluster environment.
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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
Cites methods from "ADADELTA: An Adaptive Learning Rate..."
...Other stochastic optimization methods include vSGD (Schaul et al., 2012), AdaDelta (Zeiler, 2012) and the natural Newton method from Roux & Fitzgibbon (2010), all setting stepsizes by estimating curvature from first-order information....
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Posted Content•
TL;DR: In this article, the adaptive estimates of lower-order moments are used for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimate of lowerorder moments.
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.
23,486 citations
Proceedings Article•
01 Jan 2015TL;DR: It is conjecture that the use of a fixed-length vector is a bottleneck in improving the performance of this basic encoder-decoder architecture, and it is proposed to extend this by allowing a model to automatically (soft-)search for parts of a source sentence that are relevant to predicting a target word, without having to form these parts as a hard segment explicitly.
Abstract: Neural machine translation is a recently proposed approach to machine translation. Unlike the traditional statistical machine translation, the neural machine translation aims at building a single neural network that can be jointly tuned to maximize the translation performance. The models proposed recently for neural machine translation often belong to a family of encoder-decoders and consists of an encoder that encodes a source sentence into a fixed-length vector from which a decoder generates a translation. In this paper, we conjecture that the use of a fixed-length vector is a bottleneck in improving the performance of this basic encoder-decoder architecture, and propose to extend this by allowing a model to automatically (soft-)search for parts of a source sentence that are relevant to predicting a target word, without having to form these parts as a hard segment explicitly. With this new approach, we achieve a translation performance comparable to the existing state-of-the-art phrase-based system on the task of English-to-French translation. Furthermore, qualitative analysis reveals that the (soft-)alignments found by the model agree well with our intuition.
20,027 citations
Cites methods from "ADADELTA: An Adaptive Learning Rate..."
...We use a minibatch stochastic gradient descent (SGD) algorithm together with Adadelta (Zeiler, 2012) to train each model....
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...Adadelta (Zeiler, 2012) was used to automatically adapt the learning rate of each parameter ( = 10−6 and ρ = 0.95)....
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01 Jan 2014
TL;DR: In this paper, the encoder and decoder of the RNN Encoder-Decoder model are jointly trained to maximize the conditional probability of a target sequence given a source sequence.
Abstract: In this paper, we propose a novel neural network model called RNN Encoder‐ Decoder that consists of two recurrent neural networks (RNN). One RNN encodes a sequence of symbols into a fixedlength vector representation, and the other decodes the representation into another sequence of symbols. The encoder and decoder of the proposed model are jointly trained to maximize the conditional probability of a target sequence given a source sequence. The performance of a statistical machine translation system is empirically found to improve by using the conditional probabilities of phrase pairs computed by the RNN Encoder‐Decoder as an additional feature in the existing log-linear model. Qualitatively, we show that the proposed model learns a semantically and syntactically meaningful representation of linguistic phrases.
19,998 citations
TL;DR: This historical survey compactly summarizes relevant work, much of it from the previous millennium, review deep supervised learning, unsupervised learning, reinforcement learning & evolutionary computation, and indirect search for short programs encoding deep and large networks.
14,635 citations
Additional excerpts
...RmsProp (Schaul, Zhang, & LeCun, 2013; Tieleman & Hinton, 2012) can speed up first order gradient descent methods (Sections 5.5, 5.6.2); compare vario-η (Neuneier & Zimmermann, 1996), Adagrad (Duchi, Hazan, & Singer, 2011) and Adadelta (Zeiler, 2012)....
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References
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TL;DR: Back-propagation repeatedly adjusts the weights of the connections in the network so as to minimize a measure of the difference between the actual output vector of the net and the desired output vector, which helps to represent important features of the task domain.
Abstract: We describe a new learning procedure, back-propagation, for networks of neurone-like units. The procedure repeatedly adjusts the weights of the connections in the network so as to minimize a measure of the difference between the actual output vector of the net and the desired output vector. As a result of the weight adjustments, internal ‘hidden’ units which are not part of the input or output come to represent important features of the task domain, and the regularities in the task are captured by the interactions of these units. The ability to create useful new features distinguishes back-propagation from earlier, simpler methods such as the perceptron-convergence procedure1.
23,814 citations
"ADADELTA: An Adaptive Learning Rate..." refers methods in this paper
...One method of speeding up training per-dimension is the momentum method [2]....
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TL;DR: In this article, a method for making successive experiments at levels x1, x2, ··· in such a way that xn will tend to θ in probability is presented.
Abstract: Let M(x) denote the expected value at level x of the response to a certain experiment. M(x) is assumed to be a monotone function of x but is unknown to the experimenter, and it is desired to find the solution x = θ of the equation M(x) = α, where a is a given constant. We give a method for making successive experiments at levels x1, x2, ··· in such a way that xn will tend to θ in probability.
9,312 citations
Proceedings Article•
01 Jan 2010TL;DR: Adaptive subgradient methods as discussed by the authors dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient-based learning, which allows us to find needles in haystacks in the form of very predictive but rarely seen features.
Abstract: We present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient-based learning. Metaphorically, the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features. Our paradigm stems from recent advances in stochastic optimization and online learning which employ proximal functions to control the gradient steps of the algorithm. We describe and analyze an apparatus for adaptively modifying the proximal function, which significantly simplifies setting a learning rate and results in regret guarantees that are provably as good as the best proximal function that can be chosen in hindsight. We give several efficient algorithms for empirical risk minimization problems with common and important regularization functions and domain constraints. We experimentally study our theoretical analysis and show that adaptive subgradient methods outperform state-of-the-art, yet non-adaptive, subgradient algorithms.
7,244 citations
Proceedings Article•
03 Dec 2012TL;DR: This paper considers the problem of training a deep network with billions of parameters using tens of thousands of CPU cores and develops two algorithms for large-scale distributed training, Downpour SGD and Sandblaster L-BFGS, which increase the scale and speed of deep network training.
Abstract: Recent work in unsupervised feature learning and deep learning has shown that being able to train large models can dramatically improve performance. In this paper, we consider the problem of training a deep network with billions of parameters using tens of thousands of CPU cores. We have developed a software framework called DistBelief that can utilize computing clusters with thousands of machines to train large models. Within this framework, we have developed two algorithms for large-scale distributed training: (i) Downpour SGD, an asynchronous stochastic gradient descent procedure supporting a large number of model replicas, and (ii) Sandblaster, a framework that supports a variety of distributed batch optimization procedures, including a distributed implementation of L-BFGS. Downpour SGD and Sandblaster L-BFGS both increase the scale and speed of deep network training. We have successfully used our system to train a deep network 30x larger than previously reported in the literature, and achieves state-of-the-art performance on ImageNet, a visual object recognition task with 16 million images and 21k categories. We show that these same techniques dramatically accelerate the training of a more modestly- sized deep network for a commercial speech recognition service. Although we focus on and report performance of these methods as applied to training large neural networks, the underlying algorithms are applicable to any gradient-based machine learning algorithm.
3,475 citations
"ADADELTA: An Adaptive Learning Rate..." refers methods in this paper
...A recent first order method called ADAGRAD [3] has shown remarkably good results on large scale learning tasks in a distributed environment [4]....
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...The network was trained using the distributed system of [4] in which a centralized parameter server accumulates the gradient information reported back from several replicas of the neural network....
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TL;DR: A pre-trained deep neural network hidden Markov model (DNN-HMM) hybrid architecture that trains the DNN to produce a distribution over senones (tied triphone states) as its output that can significantly outperform the conventional context-dependent Gaussian mixture model (GMM)-HMMs.
Abstract: We propose a novel context-dependent (CD) model for large-vocabulary speech recognition (LVSR) that leverages recent advances in using deep belief networks for phone recognition. We describe a pre-trained deep neural network hidden Markov model (DNN-HMM) hybrid architecture that trains the DNN to produce a distribution over senones (tied triphone states) as its output. The deep belief network pre-training algorithm is a robust and often helpful way to initialize deep neural networks generatively that can aid in optimization and reduce generalization error. We illustrate the key components of our model, describe the procedure for applying CD-DNN-HMMs to LVSR, and analyze the effects of various modeling choices on performance. Experiments on a challenging business search dataset demonstrate that CD-DNN-HMMs can significantly outperform the conventional context-dependent Gaussian mixture model (GMM)-HMMs, with an absolute sentence accuracy improvement of 5.8% and 9.2% (or relative error reduction of 16.0% and 23.2%) over the CD-GMM-HMMs trained using the minimum phone error rate (MPE) and maximum-likelihood (ML) criteria, respectively.
3,120 citations