Proceedings Article•
Dropout as a Bayesian approximation: representing model uncertainty in deep learning
19 Jun 2016-pp 1050-1059
TL;DR: A new theoretical framework is developed casting dropout training in deep neural networks (NNs) as approximate Bayesian inference in deep Gaussian processes, which mitigates the problem of representing uncertainty in deep learning without sacrificing either computational complexity or test accuracy.
Abstract: Deep learning tools have gained tremendous attention in applied machine learning. However such tools for regression and classification do not capture model uncertainty. In comparison, Bayesian models offer a mathematically grounded framework to reason about model uncertainty, but usually come with a prohibitive computational cost. In this paper we develop a new theoretical framework casting dropout training in deep neural networks (NNs) as approximate Bayesian inference in deep Gaussian processes. A direct result of this theory gives us tools to model uncertainty with dropout NNs - extracting information from existing models that has been thrown away so far. This mitigates the problem of representing uncertainty in deep learning without sacrificing either computational complexity or test accuracy. We perform an extensive study of the properties of dropout's uncertainty. Various network architectures and nonlinearities are assessed on tasks of regression and classification, using MNIST as an example. We show a considerable improvement in predictive log-likelihood and RMSE compared to existing state-of-the-art methods, and finish by using dropout's uncertainty in deep reinforcement learning.
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Posted Content•
TL;DR: In this paper, the lottery tickets hypothesis is proposed to find the subnetworks that can reach test accuracy comparable to the original network in a similar number of iterations, where the winning tickets have won the initialization lottery: their connections have initial weights that make training particularly effective.
Abstract: Neural network pruning techniques can reduce the parameter counts of trained networks by over 90%, decreasing storage requirements and improving computational performance of inference without compromising accuracy. However, contemporary experience is that the sparse architectures produced by pruning are difficult to train from the start, which would similarly improve training performance.
We find that a standard pruning technique naturally uncovers subnetworks whose initializations made them capable of training effectively. Based on these results, we articulate the "lottery ticket hypothesis:" dense, randomly-initialized, feed-forward networks contain subnetworks ("winning tickets") that - when trained in isolation - reach test accuracy comparable to the original network in a similar number of iterations. The winning tickets we find have won the initialization lottery: their connections have initial weights that make training particularly effective.
We present an algorithm to identify winning tickets and a series of experiments that support the lottery ticket hypothesis and the importance of these fortuitous initializations. We consistently find winning tickets that are less than 10-20% of the size of several fully-connected and convolutional feed-forward architectures for MNIST and CIFAR10. Above this size, the winning tickets that we find learn faster than the original network and reach higher test accuracy.
797 citations
Posted Content•
TL;DR: A large-scale benchmark of existing state-of-the-art methods on classification problems and the effect of dataset shift on accuracy and calibration is presented, finding that traditional post-hoc calibration does indeed fall short, as do several other previous methods.
Abstract: Modern machine learning methods including deep learning have achieved great success in predictive accuracy for supervised learning tasks, but may still fall short in giving useful estimates of their predictive {\em uncertainty}. Quantifying uncertainty is especially critical in real-world settings, which often involve input distributions that are shifted from the training distribution due to a variety of factors including sample bias and non-stationarity. In such settings, well calibrated uncertainty estimates convey information about when a model's output should (or should not) be trusted. Many probabilistic deep learning methods, including Bayesian-and non-Bayesian methods, have been proposed in the literature for quantifying predictive uncertainty, but to our knowledge there has not previously been a rigorous large-scale empirical comparison of these methods under dataset shift. We present a large-scale benchmark of existing state-of-the-art methods on classification problems and investigate the effect of dataset shift on accuracy and calibration. We find that traditional post-hoc calibration does indeed fall short, as do several other previous methods. However, some methods that marginalize over models give surprisingly strong results across a broad spectrum of tasks.
754 citations
Cites background or methods from "Dropout as a Bayesian approximation..."
...…Bayesian approaches include Laplace approximation (MacKay, 1992), variational inference (Graves, 2011; Blundell et al., 2015), dropout-based variational inference (Gal and Ghahramani, 2016; Kingma et al., 2015), expectation propagation and stochastic gradient MCMC (Welling and Teh, 2011)....
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...…and Gimpel, 2017) • (Temp Scaling) Post-hoc calibration by temperature scaling using a validation set (Guo et al., 2017) • (Dropout) Monte-Carlo Dropout (Gal and Ghahramani, 2016; Srivastava et al., 2015) with rate p • (Ensembles) Ensembles of M networks trained independently on the entire dataset…...
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..., 2017) • (Dropout) Monte-Carlo Dropout (Gal and Ghahramani, 2016; Srivastava et al., 2015) with rate p • (Ensembles) Ensembles of M networks trained independently on the entire dataset using random initialization (Lakshminarayanan et al....
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...• (Vanilla) Maximum softmax probability (Hendrycks and Gimpel, 2017) • (Temp Scaling) Post-hoc calibration by temperature scaling using a validation set (Guo et al., 2017) • (Dropout) Monte-Carlo Dropout (Gal and Ghahramani, 2016; Srivastava et al., 2015) with rate p • (Ensembles) Ensembles of M networks trained independently on the entire dataset using random initialization (Lakshminarayanan et al., 2017) (we set M = 10 in experiments below) • (SVI) Stochastic Variational Bayesian Inference for deep learning (Blundell et al., 2015; Graves, 2011; Louizos and Welling, 2017; 2016; Wen et al., 2018)....
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..., 2015), dropout-based variational inference (Gal and Ghahramani, 2016; Kingma et al., 2015), expectation propagation and stochastic gradient MCMC (Welling and Teh, 2011)....
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01 Apr 2017
TL;DR: This article showed that weight tying can reduce the size of neural translation models to less than half of their original size without harming their performance and proposed a new method of regularizing the output embedding.
Abstract: We study the topmost weight matrix of neural network language models. We show that this matrix constitutes a valid word embedding. When training language models, we recommend tying the input embedding and this output embedding. We analyze the resulting update rules and show that the tied embedding evolves in a more similar way to the output embedding than to the input embedding in the untied model. We also offer a new method of regularizing the output embedding. Our methods lead to a significant reduction in perplexity, as we are able to show on a variety of neural network language models. Finally, we show that weight tying can reduce the size of neural translation models to less than half of their original size without harming their performance.
730 citations
Proceedings Article•
05 Dec 2016TL;DR: Bootstrapped DQN as discussed by the authors combines deep exploration with deep neural networks for exponentially faster learning than any dithering strategy, which is a promising approach to efficient exploration with generalization.
Abstract: Efficient exploration remains a major challenge for reinforcement learning (RL). Common dithering strategies for exploration, such as '-greedy, do not carry out temporally-extended (or deep) exploration; this can lead to exponentially larger data requirements. However, most algorithms for statistically efficient RL are not computationally tractable in complex environments. Randomized value functions offer a promising approach to efficient exploration with generalization, but existing algorithms are not compatible with nonlinearly parameterized value functions. As a first step towards addressing such contexts we develop bootstrapped DQN. We demonstrate that bootstrapped DQN can combine deep exploration with deep neural networks for exponentially faster learning than any dithering strategy. In the Arcade Learning Environment bootstrapped DQN substantially improves learning speed and cumulative performance across most games.
715 citations
TL;DR: In this article, the authors analyze the challenges of using deep learning for remote sensing data analysis, review the recent advances, and provide resources to make deep learning in remote sensing ridiculously simple to start with.
Abstract: Standing at the paradigm shift towards data-intensive science, machine learning techniques are becoming increasingly important. In particular, as a major breakthrough in the field, deep learning has proven as an extremely powerful tool in many fields. Shall we embrace deep learning as the key to all? Or, should we resist a 'black-box' solution? There are controversial opinions in the remote sensing community. In this article, we analyze the challenges of using deep learning for remote sensing data analysis, review the recent advances, and provide resources to make deep learning in remote sensing ridiculously simple to start with. More importantly, we advocate remote sensing scientists to bring their expertise into deep learning, and use it as an implicit general model to tackle unprecedented large-scale influential challenges, such as climate change and urbanization.
629 citations
References
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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
"Dropout as a Bayesian approximation..." refers methods in this paper
...Finally, we used mini-batches of size 32 and the Adam optimiser [38]....
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...Finally, we used mini-batches of size 32 and the Adam optimiser (Kingma & Ba, 2014)....
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01 Jan 1998
TL;DR: In this article, a graph transformer network (GTN) is proposed for handwritten character recognition, which can be used to synthesize a complex decision surface that can classify high-dimensional patterns, such as handwritten characters.
Abstract: Multilayer neural networks trained with the back-propagation algorithm constitute the best example of a successful gradient based learning technique. Given an appropriate network architecture, gradient-based learning algorithms can be used to synthesize a complex decision surface that can classify high-dimensional patterns, such as handwritten characters, with minimal preprocessing. This paper reviews various methods applied to handwritten character recognition and compares them on a standard handwritten digit recognition task. Convolutional neural networks, which are specifically designed to deal with the variability of 2D shapes, are shown to outperform all other techniques. Real-life document recognition systems are composed of multiple modules including field extraction, segmentation recognition, and language modeling. A new learning paradigm, called graph transformer networks (GTN), allows such multimodule systems to be trained globally using gradient-based methods so as to minimize an overall performance measure. Two systems for online handwriting recognition are described. Experiments demonstrate the advantage of global training, and the flexibility of graph transformer networks. A graph transformer network for reading a bank cheque is also described. It uses convolutional neural network character recognizers combined with global training techniques to provide record accuracy on business and personal cheques. It is deployed commercially and reads several million cheques per day.
42,067 citations
Journal Article•
TL;DR: It is shown that dropout improves the performance of neural networks on supervised learning tasks in vision, speech recognition, document classification and computational biology, obtaining state-of-the-art results on many benchmark data sets.
Abstract: Deep neural nets with a large number of parameters are very powerful machine learning systems. However, overfitting is a serious problem in such networks. Large networks are also slow to use, making it difficult to deal with overfitting by combining the predictions of many different large neural nets at test time. Dropout is a technique for addressing this problem. The key idea is to randomly drop units (along with their connections) from the neural network during training. This prevents units from co-adapting too much. During training, dropout samples from an exponential number of different "thinned" networks. At test time, it is easy to approximate the effect of averaging the predictions of all these thinned networks by simply using a single unthinned network that has smaller weights. This significantly reduces overfitting and gives major improvements over other regularization methods. We show that dropout improves the performance of neural networks on supervised learning tasks in vision, speech recognition, document classification and computational biology, obtaining state-of-the-art results on many benchmark data sets.
33,597 citations
"Dropout as a Bayesian approximation..." refers background or methods in this paper
...Dropout is used in many models in deep learning as a way to avoid over-fitting (Srivastava et al., 2014), and our interpretation suggests that dropout approximately integrates over the models’ weights....
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...Furthermore, our results carry to other variants of dropout as well (such as drop-connect (Wan et al., 2013), multiplicative Gaussian noise (Srivastava et al., 2014), etc.)....
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...In this paper we give a complete theoretical treatment of the link between Gaussian processes and dropout, and develop the tools necessary to represent uncertainty in deep learning....
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TL;DR: This work bridges the divide between high-dimensional sensory inputs and actions, resulting in the first artificial agent that is capable of learning to excel at a diverse array of challenging tasks.
Abstract: The theory of reinforcement learning provides a normative account, deeply rooted in psychological and neuroscientific perspectives on animal behaviour, of how agents may optimize their control of an environment. To use reinforcement learning successfully in situations approaching real-world complexity, however, agents are confronted with a difficult task: they must derive efficient representations of the environment from high-dimensional sensory inputs, and use these to generalize past experience to new situations. Remarkably, humans and other animals seem to solve this problem through a harmonious combination of reinforcement learning and hierarchical sensory processing systems, the former evidenced by a wealth of neural data revealing notable parallels between the phasic signals emitted by dopaminergic neurons and temporal difference reinforcement learning algorithms. While reinforcement learning agents have achieved some successes in a variety of domains, their applicability has previously been limited to domains in which useful features can be handcrafted, or to domains with fully observed, low-dimensional state spaces. Here we use recent advances in training deep neural networks to develop a novel artificial agent, termed a deep Q-network, that can learn successful policies directly from high-dimensional sensory inputs using end-to-end reinforcement learning. We tested this agent on the challenging domain of classic Atari 2600 games. We demonstrate that the deep Q-network agent, receiving only the pixels and the game score as inputs, was able to surpass the performance of all previous algorithms and achieve a level comparable to that of a professional human games tester across a set of 49 games, using the same algorithm, network architecture and hyperparameters. This work bridges the divide between high-dimensional sensory inputs and actions, resulting in the first artificial agent that is capable of learning to excel at a diverse array of challenging tasks.
23,074 citations
Proceedings Article•
01 Jan 2014TL;DR: A stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case is introduced.
Abstract: How can we perform efficient inference and learning in directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce a stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case. Our contributions is two-fold. First, we show that a reparameterization of the variational lower bound yields a lower bound estimator that can be straightforwardly optimized using standard stochastic gradient methods. Second, we show that for i.i.d. datasets with continuous latent variables per datapoint, posterior inference can be made especially efficient by fitting an approximate inference model (also called a recognition model) to the intractable posterior using the proposed lower bound estimator. Theoretical advantages are reflected in experimental results.
20,769 citations
"Dropout as a Bayesian approximation..." refers background or methods in this paper
...Recent advances in variational inference introduced new techniques into the field such as sampling-based variational inference and stochastic variational inference (Blei et al., 2012; Kingma & Welling, 2013; Rezende et al., 2014; Titsias & LázaroGredilla, 2014; Hoffman et al., 2013)....
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...Recent advances in variational inference introduced new techniques such as sampling-based variational inference and stochastic variational inference [21, 22, 23, 24, 25]....
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