A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> This paper is an attempt to motivate and justify quasi-Newton methods as useful modifications of Newton''s method for general and gradient nonlinear systems of equations. References are given to ample numerical justification; here we give an overview of many of the important theoretical results and each is accompanied by sufficient discussion to make the results and hence the methods plausible. Key Words and Phrases: unconstrained minimization, nonlinear simultaneous equations, update methods, quasi-Newton methods. <s> BIB001 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> 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. <s> BIB002 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Reinforcement learning, one of the most active research areas in artificial intelligence, is a computational approach to learning whereby an agent tries to maximize the total amount of reward it receives when interacting with a complex, uncertain environment. In Reinforcement Learning, Richard Sutton and Andrew Barto provide a clear and simple account of the key ideas and algorithms of reinforcement learning. Their discussion ranges from the history of the field's intellectual foundations to the most recent developments and applications. The only necessary mathematical background is familiarity with elementary concepts of probability. The book is divided into three parts. Part I defines the reinforcement learning problem in terms of Markov decision processes. Part II provides basic solution methods: dynamic programming, Monte Carlo methods, and temporal-difference learning. Part III presents a unified view of the solution methods and incorporates artificial neural networks, eligibility traces, and planning; the two final chapters present case studies and consider the future of reinforcement learning. <s> BIB003 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Can machine learning deliver AI? Theoretical results, inspiration from the brain and cognition, as well as machine learning experiments suggest that in order to learn the kind of complicated functions that can represent high-level abstractions (e.g. in vision, language, and other AI-level tasks), one would need deep architectures. Deep architectures are composed of multiple levels of non-linear operations, such as in neural nets with many hidden layers, graphical models with many levels of latent variables, or in complicated propositional formulae re-using many sub-formulae. Each level of the architecture represents features at a different level of abstraction, defined as a composition of lower-level features. Searching the parameter space of deep architectures is a difficult task, but new algorithms have been discovered and a new sub-area has emerged in the machine learning community since 2006, following these discoveries. Learning algorithms such as those for Deep Belief Networks and other related unsupervised learning algorithms have recently been proposed to train deep architectures, yielding exciting results and beating the state-of-the-art in certain areas. Learning Deep Architectures for AI discusses the motivations for and principles of learning algorithms for deep architectures. By analyzing and comparing recent results with different learning algorithms for deep architectures, explanations for their success are proposed and discussed, highlighting challenges and suggesting avenues for future explorations in this area. <s> BIB004 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> We develop a 2nd-order optimization method based on the "Hessian-free" approach, and apply it to training deep auto-encoders. Without using pre-training, we obtain results superior to those reported by Hinton & Salakhutdinov (2006) on the same tasks they considered. Our method is practical, easy to use, scales nicely to very large datasets, and isn't limited in applicability to auto-encoders, or any specific model class. We also discuss the issue of "pathological curvature" as a possible explanation for the difficulty of deep-learning and how 2nd-order optimization, and our method in particular, effectively deals with it. <s> BIB005 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> 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. <s> BIB006 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> 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. <s> BIB007 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> We propose a new stochastic gradient method for optimizing the sum of a finite set of smooth functions, where the sum is strongly convex. While standard stochastic gradient methods converge at sublinear rates for this problem, the proposed method incorporates a memory of previous gradient values in order to achieve a linear convergence rate. In a machine learning context, numerical experiments indicate that the new algorithm can dramatically outperform standard algorithms, both in terms of optimizing the training error and reducing the test error quickly. <s> BIB008 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> For the challenging pole balancing task we propose a system which uses raw visual input data for reinforcement learning to evolve a control strategy. Therefore we use a neural network --- a deep autoencoder --- to encode the camera images and thus the system states in a low dimensional feature space. The system is compared to controllers that work directly on the motor sensor data. We show that the performances of both systems are settled in the same order of magnitude. <s> BIB009 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> We develop stochastic variational inference, a scalable algorithm for approximating posterior distributions. We develop this technique for a large class of probabilistic models and we demonstrate it with two probabilistic topic models, latent Dirichlet allocation and the hierarchical Dirichlet process topic model. Using stochastic variational inference, we analyze several large collections of documents: 300K articles from Nature, 1.8M articles from The New York Times, and 3.8M articles from Wikipedia. Stochastic inference can easily handle data sets of this size and outperforms traditional variational inference, which can only handle a smaller subset. (We also show that the Bayesian nonparametric topic model outperforms its parametric counterpart.) Stochastic variational inference lets us apply complex Bayesian models to massive data sets. <s> BIB010 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> We present an integrated framework for using Convolutional Networks for classification, localization and detection. We show how a multiscale and sliding window approach can be efficiently implemented within a ConvNet. We also introduce a novel deep learning approach to localization by learning to predict object boundaries. Bounding boxes are then accumulated rather than suppressed in order to increase detection confidence. We show that different tasks can be learned simultaneously using a single shared network. This integrated framework is the winner of the localization task of the ImageNet Large Scale Visual Recognition Challenge 2013 (ILSVRC2013) and obtained very competitive results for the detection and classifications tasks. In post-competition work, we establish a new state of the art for the detection task. Finally, we release a feature extractor from our best model called OverFeat. <s> BIB011 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> We consider the automated recognition of human actions in surveillance videos. Most current methods build classifiers based on complex handcrafted features computed from the raw inputs. Convolutional neural networks (CNNs) are a type of deep model that can act directly on the raw inputs. However, such models are currently limited to handling 2D inputs. In this paper, we develop a novel 3D CNN model for action recognition. This model extracts features from both the spatial and the temporal dimensions by performing 3D convolutions, thereby capturing the motion information encoded in multiple adjacent frames. The developed model generates multiple channels of information from the input frames, and the final feature representation combines information from all channels. To further boost the performance, we propose regularizing the outputs with high-level features and combining the predictions of a variety of different models. We apply the developed models to recognize human actions in the real-world environment of airport surveillance videos, and they achieve superior performance in comparison to baseline methods. <s> BIB012 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Recurrent neural networks (RNNs) are a powerful model for sequential data. End-to-end training methods such as Connectionist Temporal Classification make it possible to train RNNs for sequence labelling problems where the input-output alignment is unknown. The combination of these methods with the Long Short-term Memory RNN architecture has proved particularly fruitful, delivering state-of-the-art results in cursive handwriting recognition. However RNN performance in speech recognition has so far been disappointing, with better results returned by deep feedforward networks. This paper investigates deep recurrent neural networks, which combine the multiple levels of representation that have proved so effective in deep networks with the flexible use of long range context that empowers RNNs. When trained end-to-end with suitable regularisation, we find that deep Long Short-term Memory RNNs achieve a test set error of 17.7% on the TIMIT phoneme recognition benchmark, which to our knowledge is the best recorded score. <s> BIB013 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Stochastic gradient descent is popular for large scale optimization but has slow convergence asymptotically due to the inherent variance. To remedy this problem, we introduce an explicit variance reduction method for stochastic gradient descent which we call stochastic variance reduced gradient (SVRG). For smooth and strongly convex functions, we prove that this method enjoys the same fast convergence rate as those of stochastic dual coordinate ascent (SDCA) and Stochastic Average Gradient (SAG). However, our analysis is significantly simpler and more intuitive. Moreover, unlike SDCA or SAG, our method does not require the storage of gradients, and thus is more easily applicable to complex problems such as some structured prediction problems and neural network learning. <s> BIB014 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> We present the first deep learning model to successfully learn control policies directly from high-dimensional sensory input using reinforcement learning. The model is a convolutional neural network, trained with a variant of Q-learning, whose input is raw pixels and whose output is a value function estimating future rewards. We apply our method to seven Atari 2600 games from the Arcade Learning Environment, with no adjustment of the architecture or learning algorithm. We find that it outperforms all previous approaches on six of the games and surpasses a human expert on three of them. <s> BIB015 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Convolutional Neural Networks (CNNs) have been established as a powerful class of models for image recognition problems. Encouraged by these results, we provide an extensive empirical evaluation of CNNs on large-scale video classification using a new dataset of 1 million YouTube videos belonging to 487 classes. We study multiple approaches for extending the connectivity of a CNN in time domain to take advantage of local spatio-temporal information and suggest a multiresolution, foveated architecture as a promising way of speeding up the training. Our best spatio-temporal networks display significant performance improvements compared to strong feature-based baselines (55.3% to 63.9%), but only a surprisingly modest improvement compared to single-frame models (59.3% to 60.9%). We further study the generalization performance of our best model by retraining the top layers on the UCF-101 Action Recognition dataset and observe significant performance improvements compared to the UCF-101 baseline model (63.3% up from 43.9%). <s> BIB016 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> 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. <s> BIB017 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> 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. <s> BIB018 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Abstract: We describe a new class of learning models called memory networks. Memory networks reason with inference components combined with a long-term memory component; they learn how to use these jointly. The long-term memory can be read and written to, with the goal of using it for prediction. We investigate these models in the context of question answering (QA) where the long-term memory effectively acts as a (dynamic) knowledge base, and the output is a textual response. We evaluate them on a large-scale QA task, and a smaller, but more complex, toy task generated from a simulated world. In the latter, we show the reasoning power of such models by chaining multiple supporting sentences to answer questions that require understanding the intension of verbs. <s> BIB019 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for defining distant proposals with high acceptance probabilities in a Metropolis-Hastings framework, enabling more efficient exploration of the state space than standard random-walk proposals. The popularity of such methods has grown significantly in recent years. However, a limitation of HMC methods is the required gradient computation for simulation of the Hamiltonian dynamical system-such computation is infeasible in problems involving a large sample size or streaming data. Instead, we must rely on a noisy gradient estimate computed from a subset of the data. In this paper, we explore the properties of such a stochastic gradient HMC approach. Surprisingly, the natural implementation of the stochastic approximation can be arbitrarily bad. To address this problem we introduce a variant that uses second-order Langevin dynamics with a friction term that counteracts the effects of the noisy gradient, maintaining the desired target distribution as the invariant distribution. Results on simulated data validate our theory. We also provide an application of our methods to a classification task using neural networks and to online Bayesian matrix factorization. <s> BIB020 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> 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. <s> BIB021 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Text classification is a foundational task in many NLP applications. Traditional text classifiers often rely on many human-designed features, such as dictionaries, knowledge bases and special tree kernels. In contrast to traditional methods, we introduce a recurrent convolutional neural network for text classification without human-designed features. In our model, we apply a recurrent structure to capture contextual information as far as possible when learning word representations, which may introduce considerably less noise compared to traditional window-based neural networks. We also employ a max-pooling layer that automatically judges which words play key roles in text classification to capture the key components in texts. We conduct experiments on four commonly used datasets. The experimental results show that the proposed method outperforms the state-of-the-art methods on several datasets, particularly on document-level datasets. <s> BIB022 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> This paper introduces the Deep Recurrent Attentive Writer (DRAW) neural network architecture for image generation. DRAW networks combine a novel spatial attention mechanism that mimics the foveation of the human eye, with a sequential variational auto-encoding framework that allows for the iterative construction of complex images. The system substantially improves on the state of the art for generative models on MNIST, and, when trained on the Street View House Numbers dataset, it generates images that cannot be distinguished from real data with the naked eye. <s> BIB023 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> In recent years, supervised learning with convolutional networks (CNNs) has seen huge adoption in computer vision applications. Comparatively, unsupervised learning with CNNs has received less attention. In this work we hope to help bridge the gap between the success of CNNs for supervised learning and unsupervised learning. We introduce a class of CNNs called deep convolutional generative adversarial networks (DCGANs), that have certain architectural constraints, and demonstrate that they are a strong candidate for unsupervised learning. Training on various image datasets, we show convincing evidence that our deep convolutional adversarial pair learns a hierarchy of representations from object parts to scenes in both the generator and discriminator. Additionally, we use the learned features for novel tasks - demonstrating their applicability as general image representations. <s> BIB024 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> 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. <s> BIB025 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Many data-fitting applications require the solution of an optimization problem involving a sum of large number of functions of high dimensional parameter. Here, we consider the problem of minimizing a sum of $n$ functions over a convex constraint set $\mathcal{X} \subseteq \mathbb{R}^{p}$ where both $n$ and $p$ are large. In such problems, sub-sampling as a way to reduce $n$ can offer great amount of computational efficiency. Within the context of second order methods, we first give quantitative local convergence results for variants of Newton's method where the Hessian is uniformly sub-sampled. Using random matrix concentration inequalities, one can sub-sample in a way that the curvature information is preserved. Using such sub-sampling strategy, we establish locally Q-linear and Q-superlinear convergence rates. We also give additional convergence results for when the sub-sampled Hessian is regularized by modifying its spectrum or Levenberg-type regularization. Finally, in addition to Hessian sub-sampling, we consider sub-sampling the gradient as way to further reduce the computational complexity per iteration. We use approximate matrix multiplication results from randomized numerical linear algebra (RandNLA) to obtain the proper sampling strategy and we establish locally R-linear convergence rates. In such a setting, we also show that a very aggressive sample size increase results in a R-superlinearly convergent algorithm. While the sample size depends on the condition number of the problem, our convergence rates are problem-independent, i.e., they do not depend on the quantities related to the problem. Hence, our analysis here can be used to complement the results of our basic framework from the companion paper, [38], by exploring algorithmic trade-offs that are important in practice. <s> BIB026 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> The paper studies the solution of stochastic optimization problems in which approximations to the gradient and Hessian are obtained through subsampling. We first consider Newton-like methods that employ these approximations and discuss how to coordinate the accuracy in the gradient and Hessian to yield a superlinear rate of convergence in expectation. The second part of the paper analyzes an inexact Newton method that solves linear systems approximately using the conjugate gradient (CG) method, and that samples the Hessian and not the gradient (the gradient is assumed to be exact). We provide a complexity analysis for this method based on the properties of the CG iteration and the quality of the Hessian approximation, and compare it with a method that employs a stochastic gradient iteration instead of the CG method. We report preliminary numerical results that illustrate the performance of inexact subsampled Newton methods on machine learning applications based on logistic regression. <s> BIB027 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Neural network based methods have obtained great progress on a variety of natural language processing tasks. However, in most previous works, the models are learned based on single-task supervised objectives, which often suffer from insufficient training data. In this paper, we use the multi-task learning framework to jointly learn across multiple related tasks. Based on recurrent neural network, we propose three different mechanisms of sharing information to model text with task-specific and shared layers. The entire network is trained jointly on all these tasks. Experiments on four benchmark text classification tasks show that our proposed models can improve the performance of a task with the help of other related tasks. <s> BIB028 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Modeling the distribution of natural images is a landmark problem in unsupervised learning. This task requires an image model that is at once expressive, tractable and scalable. We present a deep neural network that sequentially predicts the pixels in an image along the two spatial dimensions. Our method models the discrete probability of the raw pixel values and encodes the complete set of dependencies in the image. Architectural novelties include fast two-dimensional recurrent layers and an effective use of residual connections in deep recurrent networks. We achieve log-likelihood scores on natural images that are considerably better than the previous state of the art. Our main results also provide benchmarks on the diverse ImageNet dataset. Samples generated from the model appear crisp, varied and globally coherent. <s> BIB029 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Learning from a few examples remains a key challenge in machine learning. Despite recent advances in important domains such as vision and language, the standard supervised deep learning paradigm does not offer a satisfactory solution for learning new concepts rapidly from little data. In this work, we employ ideas from metric learning based on deep neural features and from recent advances that augment neural networks with external memories. Our framework learns a network that maps a small labelled support set and an unlabelled example to its label, obviating the need for fine-tuning to adapt to new class types. We then define one-shot learning problems on vision (using Omniglot, ImageNet) and language tasks. Our algorithm improves one-shot accuracy on ImageNet from 87.6% to 93.2% and from 88.0% to 93.8% on Omniglot compared to competing approaches. We also demonstrate the usefulness of the same model on language modeling by introducing a one-shot task on the Penn Treebank. <s> BIB030 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Despite recent breakthroughs in the applications of deep neural networks, one setting that presents a persistent challenge is that of "one-shot learning." Traditional gradient-based networks require a lot of data to learn, often through extensive iterative training. When new data is encountered, the models must inefficiently relearn their parameters to adequately incorporate the new information without catastrophic interference. Architectures with augmented memory capacities, such as Neural Turing Machines (NTMs), offer the ability to quickly encode and retrieve new information, and hence can potentially obviate the downsides of conventional models. Here, we demonstrate the ability of a memory-augmented neural network to rapidly assimilate new data, and leverage this data to make accurate predictions after only a few samples. We also introduce a new method for accessing an external memory that focuses on memory content, unlike previous methods that additionally use memory location-based focusing mechanisms. <s> BIB031 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> The move from hand-designed features to learned features in machine learning has been wildly successful. In spite of this, optimization algorithms are still designed by hand. In this paper we show how the design of an optimization algorithm can be cast as a learning problem, allowing the algorithm to learn to exploit structure in the problems of interest in an automatic way. Our learned algorithms, implemented by LSTMs, outperform generic, hand-designed competitors on the tasks for which they are trained, and also generalize well to new tasks with similar structure. We demonstrate this on a number of tasks, including simple convex problems, training neural networks, and styling images with neural art. <s> BIB032 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> 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%, respectively, 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 overfitting 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. <s> BIB033 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Despite superior training outcomes, adaptive optimization methods such as Adam, Adagrad or RMSprop have been found to generalize poorly compared to Stochastic gradient descent (SGD). These methods tend to perform well in the initial portion of training but are outperformed by SGD at later stages of training. We investigate a hybrid strategy that begins training with an adaptive method and switches to SGD when appropriate. Concretely, we propose SWATS, a simple strategy which switches from Adam to SGD when a triggering condition is satisfied. The condition we propose relates to the projection of Adam steps on the gradient subspace. By design, the monitoring process for this condition adds very little overhead and does not increase the number of hyperparameters in the optimizer. We report experiments on several standard benchmarks such as: ResNet, SENet, DenseNet and PyramidNet for the CIFAR-10 and CIFAR-100 data sets, ResNet on the tiny-ImageNet data set and language modeling with recurrent networks on the PTB and WT2 data sets. The results show that our strategy is capable of closing the generalization gap between SGD and Adam on a majority of the tasks. <s> BIB034 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> We propose an algorithm for meta-learning that is model-agnostic, in the sense that it is compatible with any model trained with gradient descent and applicable to a variety of different learning problems, including classification, regression, and reinforcement learning. The goal of meta-learning is to train a model on a variety of learning tasks, such that it can solve new learning tasks using only a small number of training samples. In our approach, the parameters of the model are explicitly trained such that a small number of gradient steps with a small amount of training data from a new task will produce good generalization performance on that task. In effect, our method trains the model to be easy to fine-tune. We demonstrate that this approach leads to state-of-the-art performance on two few-shot image classification benchmarks, produces good results on few-shot regression, and accelerates fine-tuning for policy gradient reinforcement learning with neural network policies. <s> BIB035 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Though deep neural networks have shown great success in the large data domain, they generally perform poorly on few-shot learning tasks, where a model has to quickly generalize after seeing very few examples from each class. The general belief is that gradient-based optimization in high capacity models requires many iterative steps over many examples to perform well. Here, we propose an LSTM-based meta-learner model to learn the exact optimization algorithm used to train another learner neural network in the few-shot regime. The parametrization of our model allows it to learn appropriate parameter updates specifically for the scenario where a set amount of updates will be made, while also learning a general initialization of the learner network that allows for quick convergence of training. We demonstrate that this meta-learning model is competitive with deep metric-learning techniques for few-shot learning. <s> BIB036 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Recurrent neural network (RNN) and long short-term memory (LSTM) have achieved great success in processing sequential multimedia data and yielded the state-of-the-art results in speech recognition, digital signal processing, video processing, and text data analysis. In this paper, we propose a novel action recognition method by processing the video data using convolutional neural network (CNN) and deep bidirectional LSTM (DB-LSTM) network. First, deep features are extracted from every sixth frame of the videos, which helps reduce the redundancy and complexity. Next, the sequential information among frame features is learnt using DB-LSTM network, where multiple layers are stacked together in both forward pass and backward pass of DB-LSTM to increase its depth. The proposed method is capable of learning long term sequences and can process lengthy videos by analyzing features for a certain time interval. Experimental results show significant improvements in action recognition using the proposed method on three benchmark data sets including UCF-101, YouTube 11 Actions, and HMDB51 compared with the state-of-the-art action recognition methods. <s> BIB037 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> In recent years, a specific machine learning method called deep learning has gained huge attraction, as it has obtained astonishing results in broad applications such as pattern recognition, speech recognition, computer vision, and natural language processing. Recent research has also been shown that deep learning techniques can be combined with reinforcement learning methods to learn useful representations for the problems with high dimensional raw data input. This chapter reviews the recent advances in deep reinforcement learning with a focus on the most used deep architectures such as autoencoders, convolutional neural networks and recurrent neural networks which have successfully been come together with the reinforcement learning framework. <s> BIB038 </s> A Survey of Optimization Methods from a Machine Learning Perspective <s> I. INTRODUCTION <s> Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that these algorithms fail to converge to an optimal solution (or a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit example of a simple convex optimization setting where Adam does not converge to the optimal solution, and describe the precise problems with the previous analysis of Adam algorithm. Our analysis suggests that the convergence issues can be fixed by endowing such algorithms with `long-term memory' of past gradients, and propose new variants of the Adam algorithm which not only fix the convergence issues but often also lead to improved empirical performance. <s> BIB039
R ECENTLY, machine learning has grown at a remarkable rate, attracting a great number of researchers and practitioners. It has become one of the most popular research directions and plays a significant role in many fields, such as machine translation, speech recognition, image recognition, recommendation system, etc. Optimization is one of the core components of machine learning. The essence of most machine learning algorithms is to build an optimization model and learn the parameters in the objective function from the given data. In the era of immense data, the effectiveness and efficiency of the numerical optimization algorithms dramatically influence the popularization and application of the machine learning models. In order to promote the development of machine learning, a series of effective optimization methods were put forward, which have improved the performance and efficiency of machine learning methods. From the perspective of the gradient information in optimization, popular optimization methods can be divided into three categories: first-order optimization methods, which are represented by the widely used stochastic gradient methods; high-order optimization methods, in which Newton's method is a typical example; and heuristic derivative-free optimization methods, in which coordinate descent method is a representative. As the representative of first-order optimization methods, the stochastic gradient descent method, as well as its variants, has been widely used in recent years and is evolving at a high speed. However, many users pay little attention to the characteristics or application scope of these methods. They often adopt them as black box optimizers, which may limit the functionality of the optimization methods. In this paper, we comprehensively introduce the fundamental optimization methods. Particularly, we systematically explain their advantages and disadvantages, their application scope, and the characteristics of their parameters. We hope that the targeted introduction will help users to choose the first-order optimization methods more conveniently and make parameter adjustment more reasonable in the learning process. Compared with first-order optimization methods, high-order methods converge at a faster speed in which the curvature information makes the search direction more effective. Highorder optimizations attract widespread attention but face more challenges. The difficulty in high-order methods lies in the operation and storage of the inverse matrix of the Hessian matrix. To solve this problem, many variants based on Newton's method have been developed, most of which try to approximate the Hessian matrix through some techniques BIB001 , BIB005 . In subsequent studies, the stochastic quasi-Newton method and its variants are introduced to extend high-order methods to large-scale data BIB026 ,  , BIB027 . Derivative-free optimization methods are mainly used in the case that the derivative of the objective function may not exist or be difficult to calculate. There are two main ideas in derivative-free optimization methods. One is adopting a heuristic search based on empirical rules, and the other is fitting the objective function with samples. Derivativefree optimization methods can also work in conjunction with gradient-based methods. Most machine learning problems, once formulated, can be solved as optimization problems. Optimization in the fields of deep neural network, reinforcement learning, meta learning, variational inference and Markov chain Monte Carlo encounters different difficulties and challenges. The optimization methods developed in the specific machine learning fields are different, which can be inspiring to the development of general optimization methods. Deep neural networks (DNN) have shown great success in pattern recognition and machine learning. There are two very popular NNs, i.e., convolutional neural networks (CNNs) BIB002 and recurrent neural networks (RNNs), which play important roles in various fields of machine learning. CNNs are feedforward neural networks with convolution calculation. CNNs have been successfully used in many fields such as image processing BIB033 , BIB011 , video processing BIB016 and natural language processing (NLP)  , BIB012 . RNNs are a kind of sequential model and very active in NLP BIB022 , BIB017 , BIB028 , BIB013 . Besides, RNNs are also popular in the fields of image processing BIB023 , BIB029 and video processing BIB037 . Stochastic gradient-based algorithms are widely used in deep neural networks BIB006 , BIB007 ,  , BIB018 . However, various problems are emerging when employing stochastic gradientbased algorithms. For example, the learning rate will be oscillating in the later training stage of some adaptive methods BIB039 , BIB024 , which may lead to the problem of non-converging. Thus, further optimization algorithms based on variance reduction were proposed to improve the convergence rate BIB008 , BIB014 . Moreover, combining the stochastic gradient descent and the characteristics of its variants is a possible direction to improve the optimization. Especially, switching an adaptive algorithm to the stochastic gradient descent method can improve the accuracy and convergence speed of the algorithm BIB034 . Reinforcement learning (RL) is a branch of machine learning, for which an agent interacts with the environment by trialand-error mechanism and learns an optimal policy by maximizing cumulative rewards BIB003 . Deep reinforcement learning combines the RL and deep learning techniques, and enables the RL agent to have a good perception of its environment. Recent research has shown that deep learning can be applied to learn a useful representation for reinforcement learning problems BIB009 , BIB015 , BIB025 , BIB004 , BIB038 . Stochastic optimization algorithms are commonly used in RL and deep RL models. Meta learning  ,  has recently become very popular in the field of machine learning. The goal of meta learning is to design a model that can efficiently adapt to the new environment with as few samples as possible. The application of meta learning in supervised learning can solve the few-shot learning problems BIB035 . In general, meta learning methods can be summarized into the following three types  : metricbased methods  ,  , BIB030 ,  , model-based methods BIB031 , BIB019 and optimization-based methods BIB032 , BIB036 , BIB035 . We will describe the details of optimization-based meta learning methods in the subsequent sections. Variational inference is a useful approximation method which aims to approximate the posterior distributions in Bayesian machine learning. It can be considered as an optimization problem. For example, mean-field variational inference uses coordinate ascent to solve this optimization problem  . As the amount of data increases continuously, it is not friendly to use the traditional optimization method to handle the variational inference. Thus, the stochastic variational inference was proposed, which introduces natural gradients and extends the variational inference to large-scale data BIB010 . The stochastic optimization method can also be applied to Markov chain Monte Carlo (MCMC) sampling to improve efficiency. In this kind of application, stochastic gradient Hamiltonian Monte Carlo (HMC) is a representative method BIB020 where the stochastic gradient accelerates the step of gradient update when handling large-scale samples. The noise introduced by the stochastic gradient can be characterized by introducing Gaussian noise and friction terms. Additionally, the deviation caused by HMC discretization can be eliminated by the friction term, and thus the Metropolis-Hasting step can be omitted. The hyper-parameter settings in the HMC will affect the performance of the model. There are some efficient ways to automatically adjust the hyperparameters and improve the performance of the sampler. The development of optimization brings a lot of contributions to the progress of machine learning. However, there are still many challenges and open problems for optimization problems in machine learning. 1) How to improve optimization performance with insufficient data in deep neural networks is a tricky problem. If there are not enough samples in the training of deep neural network, it is prone to cause the problem of high variances and overfitting BIB021 . In addition, non-convex optimization has been one of the difficulties in deep neural networks, which makes the optimization tend to get a locally optimal solution rather than the global optimal solution. 2) For sequential models, the samples are often truncated by batches when the sequence is too long, which will cause deviation. How to analyze the deviation of stochastic optimization in this case and correct it is vital. 3) The stochastic variational inference is graceful and practical, and it is probably a good choice to develop methods of applying high-order gradient information to stochastic variational inference. 4) It may be a great idea to introduce the stochastic technique to the conjugate gradient method to obtain an elegant and powerful optimization algorithm. The detailed techniques to make improvements in the stochastic conjugate gradient is an interesting and challenging problem. The purpose of this paper is to summarize and analyze classical and modern optimization methods from a machine learning perspective. The remainder of this paper is organized as follows. Section II summarizes the machine learning problems from the perspective of optimization. Section III discusses the classical optimization algorithms and their latest developments in machine learning. Particularly, the recent popular optimization methods including the first and second order optimization algorithms are emphatically introduced. Section IV describes the developments and applications of optimization methods in some specific machine learning fields. Section V presents the challenges and open problems in the optimization methods. Finally, we conclude the whole paper.