scispace - formally typeset
Search or ask a question
Journal Article

Adaptive Subgradient Methods for Online Learning and Stochastic Optimization

TL;DR: This work describes 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 functions that can be chosen in hindsight.
Abstract: We present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradient-based learning. Metaphorically, the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features. Our paradigm stems from recent advances in stochastic optimization and online learning which employ proximal functions to control the gradient steps of the algorithm. We describe and analyze an apparatus for adaptively modifying the proximal function, which significantly simplifies setting a learning rate and results in regret guarantees that are provably as good as the best proximal function that can be chosen in hindsight. We give several efficient algorithms for empirical risk minimization problems with common and important regularization functions and domain constraints. We experimentally study our theoretical analysis and show that adaptive subgradient methods outperform state-of-the-art, yet non-adaptive, subgradient algorithms.

Content maybe subject to copyright    Report

Citations
More filters
Proceedings Article
Sergey Ioffe1, Christian Szegedy1
06 Jul 2015
TL;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


Cites methods from "Adaptive Subgradient Methods for On..."

  • ...A model employing Batch Normalization can be trained using batch gradient descent, or Stochastic Gradient Descent with a mini-batch sizem > 1, or with any of its variants such as Adagrad (Duchi et al., 2011)....

    [...]

  • ..., 2013) and Adagrad (Duchi et al., 2011) have been used to achieve state of the art performance....

    [...]

  • ...Stochastic gradient descent (SGD) has proved to be an effective way of training deep networks, and SGD variants such as momentum (Sutskever et al., 2013) and Adagrad (Duchi et al., 2011) have been used to achieve state of the art performance....

    [...]

  • ...A model employing Batch Normalization can be trained using batch gradient descent, or Stochastic Gradient Descent with a mini-batch size m > 1, or with any of its variants such as Adagrad (Duchi et al., 2011)....

    [...]

Proceedings ArticleDOI
01 Oct 2014
TL;DR: A new global logbilinear regression model that combines the advantages of the two major model families in the literature: global matrix factorization and local context window methods and produces a vector space with meaningful substructure.
Abstract: Recent methods for learning vector space representations of words have succeeded in capturing fine-grained semantic and syntactic regularities using vector arithmetic, but the origin of these regularities has remained opaque. We analyze and make explicit the model properties needed for such regularities to emerge in word vectors. The result is a new global logbilinear regression model that combines the advantages of the two major model families in the literature: global matrix factorization and local context window methods. Our model efficiently leverages statistical information by training only on the nonzero elements in a word-word cooccurrence matrix, rather than on the entire sparse matrix or on individual context windows in a large corpus. The model produces a vector space with meaningful substructure, as evidenced by its performance of 75% on a recent word analogy task. It also outperforms related models on similarity tasks and named entity recognition.

30,558 citations


Cites methods from "Adaptive Subgradient Methods for On..."

  • ...For all our experiments, we set xmax = 100, α = 3/4, and train the model using AdaGrad (Duchi et al., 2011), stochastically sampling nonzero elements from X , with initial learning rate of 0....

    [...]

  • ...For all our experiments, we set xmax = 100, α = 3/4, and train the model using AdaGrad (Duchi et al., 2011), stochastically sampling nonzero elements from X , with initial learning rate of 0.05....

    [...]

Posted Content
TL;DR: In this article, the adaptive estimates of lower-order moments are used for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimate of lowerorder moments.
Abstract: We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.

23,486 citations

Posted Content
TL;DR: This paper proposed two novel model architectures for computing continuous vector representations of words from very large data sets, and the quality of these representations is measured in a word similarity task and the results are compared to the previously best performing techniques based on different types of neural networks.
Abstract: We propose two novel model architectures for computing continuous vector representations of words from very large data sets. The quality of these representations is measured in a word similarity task, and the results are compared to the previously best performing techniques based on different types of neural networks. We observe large improvements in accuracy at much lower computational cost, i.e. it takes less than a day to learn high quality word vectors from a 1.6 billion words data set. Furthermore, we show that these vectors provide state-of-the-art performance on our test set for measuring syntactic and semantic word similarities.

20,077 citations

Book ChapterDOI
08 Oct 2016
TL;DR: The approach, named SSD, discretizes the output space of bounding boxes into a set of default boxes over different aspect ratios and scales per feature map location, which makes SSD easy to train and straightforward to integrate into systems that require a detection component.
Abstract: We present a method for detecting objects in images using a single deep neural network. Our approach, named SSD, discretizes the output space of bounding boxes into a set of default boxes over different aspect ratios and scales per feature map location. At prediction time, the network generates scores for the presence of each object category in each default box and produces adjustments to the box to better match the object shape. Additionally, the network combines predictions from multiple feature maps with different resolutions to naturally handle objects of various sizes. SSD is simple relative to methods that require object proposals because it completely eliminates proposal generation and subsequent pixel or feature resampling stages and encapsulates all computation in a single network. This makes SSD easy to train and straightforward to integrate into systems that require a detection component. Experimental results on the PASCAL VOC, COCO, and ILSVRC datasets confirm that SSD has competitive accuracy to methods that utilize an additional object proposal step and is much faster, while providing a unified framework for both training and inference. For \(300 \times 300\) input, SSD achieves 74.3 % mAP on VOC2007 test at 59 FPS on a Nvidia Titan X and for \(512 \times 512\) input, SSD achieves 76.9 % mAP, outperforming a comparable state of the art Faster R-CNN model. Compared to other single stage methods, SSD has much better accuracy even with a smaller input image size. Code is available at https://github.com/weiliu89/caffe/tree/ssd.

19,543 citations


Cites methods from "Adaptive Subgradient Methods for On..."

  • ...4% top-1 accuracy on the ILSVRC CLS-LOC val dataset, and fine-tune it for SSD using batch-normalization [11] and Adagrad [1] with initial learning rate 0....

    [...]

References
More filters
Proceedings ArticleDOI
Jia Deng1, Wei Dong1, Richard Socher1, Li-Jia Li1, Kai Li1, Li Fei-Fei1 
20 Jun 2009
TL;DR: A new database called “ImageNet” is introduced, a large-scale ontology of images built upon the backbone of the WordNet structure, much larger in scale and diversity and much more accurate than the current image datasets.
Abstract: The explosion of image data on the Internet has the potential to foster more sophisticated and robust models and algorithms to index, retrieve, organize and interact with images and multimedia data. But exactly how such data can be harnessed and organized remains a critical problem. We introduce here a new database called “ImageNet”, a large-scale ontology of images built upon the backbone of the WordNet structure. ImageNet aims to populate the majority of the 80,000 synsets of WordNet with an average of 500-1000 clean and full resolution images. This will result in tens of millions of annotated images organized by the semantic hierarchy of WordNet. This paper offers a detailed analysis of ImageNet in its current state: 12 subtrees with 5247 synsets and 3.2 million images in total. We show that ImageNet is much larger in scale and diversity and much more accurate than the current image datasets. Constructing such a large-scale database is a challenging task. We describe the data collection scheme with Amazon Mechanical Turk. Lastly, we illustrate the usefulness of ImageNet through three simple applications in object recognition, image classification and automatic object clustering. We hope that the scale, accuracy, diversity and hierarchical structure of ImageNet can offer unparalleled opportunities to researchers in the computer vision community and beyond.

49,639 citations


"Adaptive Subgradient Methods for On..." refers background or methods in this paper

  • ...2 Image Ranking ImageNet (Deng et al., 2009) consists of images organized according to the nouns in the WordNet hierarchy, where each noun is associated on average with more than 500 images collected from the web. We selected 15,000 important nouns from the hierarchy and conducted a large scale image ranking task for each noun. This approach is identical to the task tackled by Grangier and Bengio (2008) using the Passive-Aggressive algorithm....

    [...]

  • ...2 Image Ranking ImageNet (Deng et al., 2009) consists of images organized according to the nouns in the WordNet hierarchy, where each noun is associated on average with more than 500 images collected from the web....

    [...]

  • ...Experiments We performed experiments with several real world datasets with different characteristics: the ImageNet image database (Deng et al., 2009), the Reuters RCV1 text classification dataset (Lewis et al....

    [...]

Book
01 Mar 2004
TL;DR: In this article, the focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them, and a comprehensive introduction to the subject is given. But the focus of this book is not on the optimization problem itself, but on the problem of finding the appropriate technique to solve it.
Abstract: Convex optimization problems arise frequently in many different fields. A comprehensive introduction to the subject, this book shows in detail how such problems can be solved numerically with great efficiency. The focus is on recognizing convex optimization problems and then finding the most appropriate technique for solving them. The text contains many worked examples and homework exercises and will appeal to students, researchers and practitioners in fields such as engineering, computer science, mathematics, statistics, finance, and economics.

33,341 citations

Book
01 Jan 1985
TL;DR: In this article, the authors present results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrate their importance in a variety of applications, such as linear algebra and matrix theory.
Abstract: Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.

23,986 citations

01 Jan 2007

17,341 citations

Book
01 Jan 1995

12,671 citations


Additional excerpts

  • ...(3) implies for all x ∈ X and φ′(xt+1) ∈ ∂φ(xt+1) (Bertsekas, 1999) 〈x− xt+1, ηf (xt) +∇ψt(xt+1)−∇ψt(xt) + ηφ(xt+1)〉 ≥ 0....

    [...]