Proceedings Article•
Snapshot Ensembles: Train 1, Get M for Free
01 Apr 2017-
TL;DR: This paper proposes a method to obtain the seemingly contradictory goal of ensembling multiple neural networks at no additional training cost by training a single neural network, converging to several local minima along its optimization path and saving the model parameters.
Abstract: Ensembles of neural networks are known to be much more robust and accurate than individual networks. However, training multiple deep networks for model averaging is computationally expensive. In this paper, we propose a method to obtain the seemingly contradictory goal of ensembling multiple neural networks at no additional training cost. We achieve this goal by training a single neural network, converging to several local minima along its optimization path and saving the model parameters. To obtain repeated rapid convergence, we leverage recent work on cyclic learning rate schedules. The resulting technique, which we refer to as Snapshot Ensembling, is simple, yet surprisingly effective. We show in a series of experiments that our approach is compatible with diverse network architectures and learning tasks. It consistently yields lower error rates than state-of-the-art single models at no additional training cost, and compares favorably with traditional network ensembles. On CIFAR-10 and CIFAR-100 our DenseNet Snapshot Ensembles obtain error rates of 3.4% and 17.4% respectively.
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Posted Content•
TL;DR: This work proposes a simple modification to recover the original formulation of weight decay regularization by decoupling the weight decay from the optimization steps taken w.r.t. the loss function, and provides empirical evidence that this modification substantially improves Adam's generalization performance.
Abstract: L$_2$ regularization and weight decay regularization are equivalent for standard stochastic gradient descent (when rescaled by the learning rate), but as we demonstrate this is \emph{not} the case for adaptive gradient algorithms, such as Adam. While common implementations of these algorithms employ L$_2$ regularization (often calling it "weight decay" in what may be misleading due to the inequivalence we expose), we propose a simple modification to recover the original formulation of weight decay regularization by \emph{decoupling} the weight decay from the optimization steps taken w.r.t. the loss function. We provide empirical evidence that our proposed modification (i) decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam and (ii) substantially improves Adam's generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperformed by the latter). Our proposed decoupled weight decay has already been adopted by many researchers, and the community has implemented it in TensorFlow and PyTorch; the complete source code for our experiments is available at this https URL
6,909 citations
Cites methods from "Snapshot Ensembles: Train 1, Get M ..."
...SGDR has been successfully adopted to lead to new state-of-the-art results for popular image classification benchmarks (Huang et al., 2017; Gastaldi, 2017; Zoph et al., 2017), and we therefore tried extending it to Adam....
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Posted Content•
TL;DR: In this paper, a simple warm restart technique for stochastic gradient descent was proposed to improve its anytime performance when training deep neural networks, which achieved state-of-the-art results on both the CIFAR-10 and CifAR-100 datasets.
Abstract: Restart techniques are common in gradient-free optimization to deal with multimodal functions. Partial warm restarts are also gaining popularity in gradient-based optimization to improve the rate of convergence in accelerated gradient schemes to deal with ill-conditioned functions. In this paper, we propose a simple warm restart technique for stochastic gradient descent to improve its anytime performance when training deep neural networks. We empirically study its performance on the CIFAR-10 and CIFAR-100 datasets, where we demonstrate new state-of-the-art results at 3.14% and 16.21%, respectively. We also demonstrate its advantages on a dataset of EEG recordings and on a downsampled version of the ImageNet dataset. Our source code is available at this https URL
3,497 citations
Proceedings Article•
13 Aug 2016TL;DR: This paper proposes a simple warm restart technique for stochastic gradient descent to improve its anytime performance when training deep neural networks and empirically studies its performance on the CIFAR-10 and CIFARS datasets.
Abstract: Restart techniques are common in gradient-free optimization to deal with multimodal functions. Partial warm restarts are also gaining popularity in gradient-based optimization to improve the rate of convergence in accelerated gradient schemes to deal with ill-conditioned functions. In this paper, we propose a simple warm restart technique for stochastic gradient descent to improve its anytime performance when training deep neural networks. We empirically study its performance on the CIFAR-10 and CIFAR-100 datasets, where we demonstrate new state-of-the-art results at 3.14% and 16.21%, respectively. We also demonstrate its advantages on a dataset of EEG recordings and on a downsampled version of the ImageNet dataset. Our source code is available at this https URL
1,656 citations
Cites background or methods or result from "Snapshot Ensembles: Train 1, Get M ..."
...Our results reproduce the finding by Huang et al. (2016a) that intermediate models generated by SGDR can be used to build efficient ensembles at no cost....
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...Alternative network structures should be also considered; e.g., soon after our initial arXiv report (Loshchilov & Hutter, 2016), Zhang et al. (2016); Huang et al. (2016b); Han et al. (2016) reported that WRNs models can be replaced by more memory-efficient models....
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...…state-of-the-art results on CIFAR-10, CIFAR-100, SVHN, ImageNet, PASCAL VOC and MS COCO datasets were obtained by Residual Neural Networks (He et al., 2015; Huang et al., 2016c; He et al., 2016; Zagoruyko & Komodakis, 2016) trained without the use of advanced methods such as AdaDelta and Adam....
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...Three runs (N = 3) of SGDR with M = 3 snapshots per run are sufficient to greatly improve the results to 3.25% on CIFAR-10 and 16.64% on CIFAR-100 outperforming the results of Huang et al. (2016a)....
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...Our initial arXiv report on SGDR (Loshchilov & Hutter, 2016) inspired a follow-up study by Huang et al. (2016a) in which the authors suggest to takeM snapshots of the models obtained by SGDR (in their paper referred to as cyclical learning rate schedule and cosine annealing cycles) right before M…...
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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
Posted Content•
15 Feb 2018TL;DR: This work decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam and substantially improves Adam's generalization performance, allowing it to compete with SGD with momentum on image classification datasets.
Abstract: We note that common implementations of adaptive gradient algorithms, such as Adam, limit the potential benefit of weight decay regularization, because the weights do not decay multiplicatively (as would be expected for standard weight decay) but by an additive constant factor. We propose a simple way to resolve this issue by decoupling weight decay and the optimization steps taken w.r.t. the loss function. We provide empirical evidence that our proposed modification (i) decouples the optimal choice of weight decay factor from the setting of the learning rate for both standard SGD and Adam, and (ii) substantially improves Adam's generalization performance, allowing it to compete with SGD with momentum on image classification datasets (on which it was previously typically outperformed by the latter). We also demonstrate that longer optimization runs require smaller weight decay values for optimal results and introduce a normalized variant of weight decay to reduce this dependence. Finally, we propose a version of Adam with warm restarts (AdamWR) that has strong anytime performance while achieving state-of-the-art results on CIFAR-10 and ImageNet32x32. Our source code will become available after the review process.
1,173 citations
Cites background or methods from "Snapshot Ensembles: Train 1, Get M ..."
...Nevertheless, state-of-the-art results for popular image classification datasets, such as CIFAR-10 and CIFAR-100 Krizhevsky (2009), are still obtained by applying SGD with momentum (Huang et al., 2016; 2017; Loshchilov & Hutter, 2016; Gastaldi, 2017)....
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...SGDR has been successfully adopted to lead to new state-of-the-art results for popular image classification benchmarks (Huang et al., 2017; Gastaldi, 2017), and we therefore tried extending it to Adam shortly afterwards....
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References
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27 Jun 2016
TL;DR: In this article, the authors proposed a residual learning framework to ease the training of networks that are substantially deeper than those used previously, which won the 1st place on the ILSVRC 2015 classification task.
Abstract: 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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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.
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"Snapshot Ensembles: Train 1, Get M ..." refers background in this paper
...Stochastic Gradient Descent (SGD) (Bottou, 2010) and its accelerated variants (Kingma & Ba, 2014; Duchi et al., 2011) have become the de-facto approaches for optimizing deep neural networks....
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Proceedings Article•
03 Dec 2012TL;DR: The state-of-the-art performance of CNNs was achieved by Deep Convolutional Neural Networks (DCNNs) as discussed by the authors, which 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.
Abstract: 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.
73,978 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
Proceedings Article•
06 Jul 2015TL;DR: Applied to a state-of-the-art image classification model, Batch Normalization achieves the same accuracy with 14 times fewer training steps, and beats the original model by a significant margin.
Abstract: Training Deep Neural Networks is complicated by the fact that the distribution of each layer's inputs changes during training, as the parameters of the previous layers change. This slows down the training by requiring lower learning rates and careful parameter initialization, and makes it notoriously hard to train models with saturating nonlinearities. We refer to this phenomenon as internal covariate shift, and address the problem by normalizing layer inputs. Our method draws its strength from making normalization a part of the model architecture and performing the normalization for each training mini-batch. Batch Normalization allows us to use much higher learning rates and be less careful about initialization, and in some cases eliminates the need for Dropout. Applied to a state-of-the-art image classification model, Batch Normalization achieves the same accuracy with 14 times fewer training steps, and beats the original model by a significant margin. Using an ensemble of batch-normalized networks, we improve upon the best published result on ImageNet classification: reaching 4.82% top-5 test error, exceeding the accuracy of human raters.
30,843 citations
"Snapshot Ensembles: Train 1, Get M ..." refers methods in this paper
...Our approach is naturally compatible with other methods to improve the accuracy, such as data augmentation, stochastic depth (Huang et al., 2016b), or batch normalization (Ioffe & Szegedy, 2015)....
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