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.
Citations
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Proceedings Article•
05 Dec 2016TL;DR: The authors apply this variational inference based dropout technique in LSTM and GRU models, assessing it on language modelling and sentiment analysis tasks, and to the best of their knowledge improve on the single model state-of-the-art in language modelling with the Penn Treebank (73.4 test perplexity).
Abstract: Recurrent neural networks (RNNs) stand at the forefront of many recent developments in deep learning. Yet a major difficulty with these models is their tendency to overfit, with dropout shown to fail when applied to recurrent layers. Recent results at the intersection of Bayesian modelling and deep learning offer a Bayesian interpretation of common deep learning techniques such as dropout. This grounding of dropout in approximate Bayesian inference suggests an extension of the theoretical results, offering insights into the use of dropout with RNN models. We apply this new variational inference based dropout technique in LSTM and GRU models, assessing it on language modelling and sentiment analysis tasks. The new approach outperforms existing techniques, and to the best of our knowledge improves on the single model state-of-the-art in language modelling with the Penn Treebank (73.4 test perplexity). This extends our arsenal of variational tools in deep learning.
1,557 citations
Posted Content•
TL;DR: Self-ensembling is introduced, where it is shown that this ensemble prediction can be expected to be a better predictor for the unknown labels than the output of the network at the most recent training epoch, and can thus be used as a target for training.
Abstract: In this paper, we present a simple and efficient method for training deep neural networks in a semi-supervised setting where only a small portion of training data is labeled. We introduce self-ensembling, where we form a consensus prediction of the unknown labels using the outputs of the network-in-training on different epochs, and most importantly, under different regularization and input augmentation conditions. This ensemble prediction can be expected to be a better predictor for the unknown labels than the output of the network at the most recent training epoch, and can thus be used as a target for training. Using our method, we set new records for two standard semi-supervised learning benchmarks, reducing the (non-augmented) classification error rate from 18.44% to 7.05% in SVHN with 500 labels and from 18.63% to 16.55% in CIFAR-10 with 4000 labels, and further to 5.12% and 12.16% by enabling the standard augmentations. We additionally obtain a clear improvement in CIFAR-100 classification accuracy by using random images from the Tiny Images dataset as unlabeled extra inputs during training. Finally, we demonstrate good tolerance to incorrect labels.
1,257 citations
01 Jun 2021
TL;DR: Some of the prevailing trends in embedding physics into machine learning are reviewed, some of the current capabilities and limitations are presented and diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems are discussed.
Abstract: Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Here, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems. The rapidly developing field of physics-informed learning integrates data and mathematical models seamlessly, enabling accurate inference of realistic and high-dimensional multiphysics problems. This Review discusses the methodology and provides diverse examples and an outlook for further developments.
1,114 citations
Posted Content•
TL;DR: This tutorial article aims to provide the reader with the conceptual tools needed to get started on research on offline reinforcement learning algorithms: reinforcementlearning algorithms that utilize previously collected data, without additional online data collection.
Abstract: In this tutorial article, we aim to provide the reader with the conceptual tools needed to get started on research on offline reinforcement learning algorithms: reinforcement learning algorithms that utilize previously collected data, without additional online data collection. Offline reinforcement learning algorithms hold tremendous promise for making it possible to turn large datasets into powerful decision making engines. Effective offline reinforcement learning methods would be able to extract policies with the maximum possible utility out of the available data, thereby allowing automation of a wide range of decision-making domains, from healthcare and education to robotics. However, the limitations of current algorithms make this difficult. We will aim to provide the reader with an understanding of these challenges, particularly in the context of modern deep reinforcement learning methods, and describe some potential solutions that have been explored in recent work to mitigate these challenges, along with recent applications, and a discussion of perspectives on open problems in the field.
950 citations
Cites background or methods from "Dropout as a Bayesian approximation..."
...…generalization may find use in offline RL algorithms, including techniques from causal inference (Schölkopf, 2019), uncertainty estimation (Gal and Ghahramani, 2016; Kendall and Gal, 2017), density estimation and generative modeling (Kingma et al., 2014), distributional robustness (Sinha…...
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...Moving forward, we might expect that a variety of tools developed for addressing distributional shift and facilitating generalization may find use in offline RL algorithms, including techniques from causal inference (Schölkopf, 2019), uncertainty estimation (Gal and Ghahramani, 2016; Kendall and Gal, 2017), density estimation and generative modeling (Kingma et al....
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TL;DR: This study reviews recent advances in UQ methods used in deep learning and investigates the application of these methods in reinforcement learning (RL), and outlines a few important applications of UZ methods.
Abstract: Uncertainty quantification (UQ) plays a pivotal role in reduction of uncertainties during both optimization and decision making processes. It can be applied to solve a variety of real-world applications in science and engineering. Bayesian approximation and ensemble learning techniques are two most widely-used UQ methods in the literature. In this regard, researchers have proposed different UQ methods and examined their performance in a variety of applications such as computer vision (e.g., self-driving cars and object detection), image processing (e.g., image restoration), medical image analysis (e.g., medical image classification and segmentation), natural language processing (e.g., text classification, social media texts and recidivism risk-scoring), bioinformatics, etc. This study reviews recent advances in UQ methods used in deep learning. Moreover, we also investigate the application of these methods in reinforcement learning (RL). Then, we outline a few important applications of UQ methods. Finally, we briefly highlight the fundamental research challenges faced by UQ methods and discuss the future research directions in this field.
809 citations
Cites methods from "Dropout as a Bayesian approximation..."
...To combat this, MC (MC) dropout has been introduced, which uses dropout [29] as a regularization term to compute the prediction uncertainty [30]....
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...They modeled the uncertainty by leveraging the Bayesian AL (BAL) using Dropout-sampling....
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...Monte Carlo Dropout Input (left atrium 3D MRI images) 𝑁𝑜𝑖𝑠𝑒 𝜉 𝑁𝑜𝑖𝑠𝑒 𝜉′ 𝐷𝐿 𝐷𝑈 +𝐷𝐿 ℒ𝑐 ℒ𝑠 Uncertainty map Guide estimate the distribution of the output segmentation....
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...To alleviate (a) Baseline neural network (b) Bernoulli DropConnect (c) Gaussian DropConnect (d) Bernoulli Dropout (e) Gaussian Dropout (f) Spike-and-Slab Dropout Fig....
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...Kahn et al. a [173] used both Bootstrapping and Dropout methods to estimate uncertainty in NNs and then used in UA collision prediction model....
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References
More filters
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]....
[...]
...Finally, we used mini-batches of size 32 and the Adam optimiser (Kingma & Ba, 2014)....
[...]
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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