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## **Teaching Machines to Read and Comprehend** **Karl Moritz Hermann** *[†]* **Tom´aˇs Koˇcisk´y** *[‡]* **Edward Grefenstette** *[†]* **Lasse Espeholt** *[†]* **Will Kay** *[†]* **Mustafa Suleyman** *[†]* **Phil Blunsom** *[†‡]* *†* Google DeepMind *‡* University of Oxford *{* kmh,etg,lespeholt,wkay,mustafasul,...
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### **Abstract** Teaching machines to read natural language documents remains an elusive challenge. Machine reading systems can be tested on their ability to answer questions posed on the contents of documents that they have seen, but until now large scale training and test datasets have been missing for this type of...
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### **1 Introduction** Progress on the path from shallow bag-of-words information retrieval algorithms to machines capable of reading and understanding documents has been slow. Traditional approaches to machine reading and comprehension have been based on either hand engineered grammars [1], or information extraction...
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document that it believes will help it answer a question, and also allows us to visualises its inference process. We compare these neural models to a range of baselines and heuristic benchmarks based upon a traditional frame semantic analysis provided by a state-of-the-art natural language processing 1 ----- **CN...
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# months 95 1 1 56 1 1
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{ "Header 1": "months 95 1 1 56 1 1", "Header 2": null, "Header 3": null, "Header 4": null }
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# documents 107,582 1,210 1,165 195,461 11,746 11,074
{ "id": "1506.03340", "title": "Teaching Machines to Read and Comprehend", "categories": [ "cs.CL", "cs.AI", "cs.NE" ] }
{ "Header 1": "documents 107,582 1,210 1,165 195,461 11,746 11,074", "Header 2": null, "Header 3": null, "Header 4": null }
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# queries 438,148 3,851 3,446 837,944 60,798 55,077 Max # entities 456 190 398 424 247 250 Avg # entities 29.5 32.4 30.2 41.3 44.7 45.0 Avg tokens/doc 780 809 773 1044 1061 1066 Vocab size 124,814 274,604 Table 1: Corpus statistics. Articles were collected starting in April 2007 for CNN and June 2010 for the Daily Ma...
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### **2 Supervised training data for reading comprehension** The reading comprehension task naturally lends itself to a formulation as a supervised learning problem. Specifically we seek to estimate the conditional probability *p* ( *a|c, q* ), where *c* is a context document, *q* a query relating to that document, a...
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required to solve them, sorting them into six categories. *Simple* queries can be answered without any syntactic or semantic complexity. *Lexical* queries require some form of lexical generalisation— something that comes naturally to embedding based systems, but causes difficulty for methods relying primarily on syntac...
{ "id": "1506.03340", "title": "Teaching Machines to Read and Comprehend", "categories": [ "cs.CL", "cs.AI", "cs.NE" ] }
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least two context sentences to be answered. With 30% of queries needing complex inference and another 10% ambiguous or unanswerable, it is clear that our machine reading setup is a hard task. **2.2** **Entity replacement and permutation** Note that the focus of this paper is to provide a corpus for evaluating a mod...
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### **3 Models** So far we have motivated the need for better datasets and tasks to evaluate the capabilities of machine reading models. We proceed by describing a number of baselines, benchmarks and new models to evaluate against this paradigm. We define two simple baselines, the majority baseline (maximum frequency...
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2 be.01.V match ( *p, be.01.V, y* ) ( ***x*** *, be.01.V, y* ) X is president / **Mike** is president 3 Correct frame ( *p, V, y* ) ( ***x*** *, V, z* ) X won Oscar / **Tom** won Academy Award 4 Permuted frame ( *p, V, y* ) ( *y, V,* ***x*** ) X met Suse / Suse met **Tom** 5 Matching entity ( *p, V, y* ) ( ***x*** *, Z...
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where *W* ( *a* ) indexes row *a* of weight matrix *W* and through a slight abuse of notation word types double as indexes. Note that we do not privilege entities or variables, the model must learn to differentiate these in the input sequence. The function *g* ( *d, q* ) returns a vector embedding of a document and que...
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*c* ( *t, k* ) = *f* ( *t, k* ) *c* ( *t −* 1 *, k* ) + *i* ( *t, k* ) tanh ( *W* *kxc* *x* *[′]* ( *t, k* ) + *W* *khc* *h* ( *t −* 1 *, k* ) + *b* *kc* ) *o* ( *t, k* ) = *σ* ( *W* *kxo* *x* *[′]* ( *t, k* ) + *W* *kho* *h* ( *t −* 1 *, k* ) + *W* *kco* *c* ( *t, k* ) + *b* *ko* ) *h* ( *t, k* ) = *o* ( *t, k* ) ta...
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score the embedding of the document *r* is computed as the weighted sum of the token embeddings. The model is completed with the definition of the joint document and query embedding via a nonlinear combination: *g* [AR] ( *d, q* ) = tanh ( *W* *rg* *r* + *W* *ug* *u* ) *.* **The Impatient Reader** The Attentive Rea...
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### **4 Empirical Evaluation** Having described a number of models in the previous section, we next evaluate these models on our reading comprehension corpora. Our hypothesis is that neural models should in principle be well suited for this task. However, we argued that simple recurrent models such as the LSTM probab...
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Maximum frequency 26.3 27.9 22.5 22.7 Exclusive frequency 30.8 32.6 27.3 27.7 Frame-semantic model 32.2 33.0 30.7 31.1 Word distance model 46.2 46.9 55.6 54.8 Deep LSTM Reader 49.0 49.9 57.1 57.3 Uniform attention 31.1 33.6 31.0 31.7 Attentive Reader 56.5 58.9 64.5 63.7 Impatient Reader **57.0** **60.6** **64.8**...
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*Scooter Brown* ” has the phrase “ *... turns out he is good friends with Scooter Brown, manager for* *Carly Rae Jepson* ” in the context. The word distance benchmark correctly aligns these two while the frame-semantic approach fails to pickup the friendship or management relations when parsing the query. We expect tha...
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queries (the correct answers are *ent23* and *ent63*, respectively). Both examples require significant lexical generalisation and co-reference resolution in order to be answered correctly by a given model. accuracy of close to 60% which, according to our analysis in Table 2, is the proportion of questions answerable ...
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### **5 Conclusion** The supervised paradigm for training machine reading and comprehension models provides a promising avenue for making progress on the path to building full natural language understanding systems. We have demonstrated a methodology for obtaining a large number of document-queryanswer triples and sh...
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### **References** [1] Ellen Riloff and Michael Thelen. A rule-based question answering system for reading comprehension tests. In *Proceedings of the 2000 ANLP/NAACL Workshop on Reading Compre-* *hension Tests As Evaluation for Computer-based Language Understanding Sytems - Volume 6*, ANLP/NAACL-ReadingComp ’00, pag...
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*ings of ACL*, pages 565–574, 2010. [12] Wilson L Taylor. “Cloze procedure”: a new tool for measuring readability. *Journalism Quar-* *terly*, 30:415–433, 1953. [13] Dipanjan Das, Desai Chen, Andr´e F. T. Martins, Nathan Schneider, and Noah A. Smith. Framesemantic parsing. *Computational Linguistics*, 40(1):9–56, 2...
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### **A Additional Heatmap Analysis** We expand on the analysis of the attention mechanism presented in the paper by including visualisations for additional queries from the CNN validation dataset below. We consider examples from the Attentive Reader as well as the Impatient Reader in this appendix. **A.1** **Atten...
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{ "Header 1": "queries 438,148 3,851 3,446 837,944 60,798 55,077", "Header 2": null, "Header 3": "**A Additional Heatmap Analysis**", "Header 4": null }
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## **Convolutional Networks on Graphs** **for Learning Molecular Fingerprints** **David Duvenaud** *[†]* **, Dougal Maclaurin** *[†]* **, Jorge Aguilera-Iparraguirre** **Rafael G´omez-Bombarelli, Timothy Hirzel, Al´an Aspuru-Guzik, Ryan P. Adams** Harvard University
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### **Abstract** Predicting properties of molecules requires functions that take graphs as inputs. Molecular graphs are usually preprocessed using hash-based functions to produce fixed-size fingerprint vectors, which are used as features for making predictions. We introduce a convolutional neural network that operate...
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### **1 Introduction** Recent work in materials design has applied neural networks to virtual screening, where the task is to predict the properties of novel molecules by generalizing from examples. One difficulty with this task is that the input to the predictor, a molecule, can be of arbitrary size and shape. Most ...
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representation more meaningful. *†* Equal contribution. 1 ----- ![](/content/images/1509.09292v1.pdf-1-0.jpg) ![](/content/images/1509.09292v1.pdf-1-1.jpg) Figure 1: *Left* : A visual representation of the computational graph of both standard circular fingerprints and neural graph fingerprints. First, at bo...
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### **2 Circular fingerprints** The state of the art in molecular fingerprints are extended-connectivity circular fingerprints (ECFP) [21]. Circular fingerprints [6] are a refinement of the Morgan algorithm [17], designed to identify which substructures are present in a molecule in a way that is invariant to atom-rel...
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### **3 Creating a differentiable fingerprint** The space of possible network architectures is large. In the spirit of starting from a known-good configuration, we chose an architecture analogous to existing fingerprints. This section describes our replacement of each discrete operation in circular fingerprints with ...
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12: **Return:** binary vector **f** **Al** **g** **orithm 2** Neural g ra p h fin g er p rints 1: **Input:** molecule, radius *R*, hidden weights *H* 1 [1] *[. . . H]* *R* [5] [, output weights] *[ W]* [1] *[ . . . W]* *[R]* 2: **Initialize:** fingerprint vector **f** *←* **0** *S* 3: **for** each atom *a* in molec...
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### **4 Experiments** Circular fingerprints can be interpreted as a special case of neural graph fingerprints having large random weights. This is reasonable to expect, since in the limit of large input weights, tanh nonlinearities approach step functions, which when concatenated resemble a hash function. Also, in th...
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||| Figure 3: *Left:* Comparison of pairwise distances between molecules, measured using circular fingerprints and neural graph fingerprints with large random weights. *Right* : Predictive performance of circular fingerprints (red), neural graph fingerprints with fixed large random weights (green) and neural graph fi...
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predictive of toxicity. Fragments most activated by toxicity feature on SR-MMP dataset Fragments most activated by toxicity feature on NR-AHR dataset ![](/content/images/1509.09292v1.pdf-4-0.jpg) ![](/content/images/1509.09292v1.pdf-4-1.jpg) ![](/content/images/1509.09292v1.pdf-4-2.jpg) Figure 5: Visual...
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or aromatic, whether the bond was conjugated, and whether the bond was part of a ring. **Training and Architecture** Training used batch normalization [11]. We also experimented with tanh vs relu activation functions for both the neural fingerprint network layers and the fullyconnected network layers. relu had a slig...
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as measured by [5]. *•* **Organic photovoltaic efficiency:** The Harvard Clean Energy Project [8] uses expensive DFT simulations to estimate the photovoltaic efficiency of organic molecules. We used a subset of 20,000 molecules from this dataset. **Predictive accuracy** We compared the performance of circular finge...
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### **5 Limitations** **Computational cost** Neural fingerprints have the same asymptotic complexity in the number of atoms and the depth of the network as circular fingerprints, but have additional terms due to the matrix multiplies necessary to transform the feature vector at each step. To be precise, computing the...
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### **6 Related work** This work is similar in spirit to the neural Turing machine [7], in the sense that we take an existing discrete computational architecture, and make each part differentiable in order to do gradient-based optimization. **Convolutional neural networks** Convolutional neural networks have been u...
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input is a different graph. **Neural networks on input-dependent graphs** [22] propose a neural network model for graphs having an interesting training procedure. The forward pass consists of running a message-passing scheme to equilibrium, a fact which allows the reverse-mode gradient to be computed without storing ...
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### **7 Conclusion** We generalized existing hand-crafted molecular features to allow their optimization for diverse tasks. By making each operation in the feature pipeline differentiable, we can use standard neural-network training methods to scalably optimize the parameters of these neural molecular fingerprints en...
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### **References** [1] Fr´ed´eric Bastien, Pascal Lamblin, Razvan Pascanu, James Bergstra, Ian J. Goodfellow, Arnaud Bergeron, Nicolas Bouchard, and Yoshua Bengio. Theano: new features and speed improvements. Deep Learning and Unsupervised Feature Learning NIPS 2012 Workshop, 2012. [2] Joan Bruna, Wojciech Zaremba,...
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by reducing internal covariate shift. *arXiv preprint arXiv:1502.03167*, 2015. [12] Nal Kalchbrenner, Edward Grefenstette, and Phil Blunsom. A convolutional neural network for modelling sentences. *Proceedings of the 52nd Annual Meeting of the Association for Com-* *putational Linguistics*, June 2014. [13] Diederik...
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*ceedings of the Conference on Empirical Methods in Natural Language Processing*, pages 151–161. Association for Computational Linguistics, 2011. [25] Kai Sheng Tai, Richard Socher, and Christopher D Manning. Improved semantic representations from tree-structured long short-term memory networks. *arXiv preprint* *arX...
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## A Deep Reinforced Model for Abstractive Summarization ### Romain Paulus, Caiming Xiong and Richard Socher {rpaulus,cxiong,rsocher}@salesforce.com
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### Abstract Attentional, RNN-based encoder-decoder models for abstractive summarization have achieved good performance on short input and output sequences. However, for longer documents and summaries, these models often include repetitive and incoherent phrases. We introduce a neural network model with intra-att...
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### 1 Introduction Text summarization is the process of automatically generating natural language summaries from an input document while retaining the important points. By condensing large quantities of information into short, informative summaries, summarization can aid many downstream applications such as creatin...
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intra-attention model in the decoder takes into ac count which words have already been generated by the decoder. (ii) we propose a new objective function by combining the maximum-likelihood cross-entropy loss used in prior work with rewards from policy gradient reinforcement learning to reduce exposure bias. We show th...
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### 2 Neural Intra-attention Model In this section, we present our intra-attention model based on the encoder-decoder network (Sutskever et al., 2014). In all our equations, x = {x 1, x 2, . . ., x n } represents the sequence of input (article) tokens, y = {y 1, y 2, . . ., y n ′ } the sequence of output (summary) ...
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Finally, we compute the normalized attention scores α [e] ti [across the inputs and use these weights] to obtain the input context vector c [e] t [:] e [′] ti α [e] ti [=] � nj=1 [e] [′] tj (4) 2.2 Intra-decoder attention While this intra-temporal attention function ensures that different parts of the encoded inp...
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y t, . . ., y t−1, x, even when not explicitly stated. Our token-generation layer generates the following probability distribution: p(y t |u t = 0) = (9) softmax(W out [h [d] t [∥][c] [e] t [∥][c] [d] t [] +][ b] [out] [)] On the other hand, the pointer mechanism uses the temporal attention weights α [e] ti [as t...
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never contain the same trigram twice. Based on this observation, we force our decoder to never output the same trigram more than once during testing. We do this by setting p(y t ) = 0 during beam search, when outputting y t would create a trigram that already exists in the previously decoded sequence of the current bea...
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### 3 Hybrid Learning Objective In this section, we explore different ways of training our encoder-decoder model. In particular, we propose reinforcement learning-based algorithms and their application to our summarization task. ----- 3.1 Supervised learning with teacher forcing The most widely used method to t...
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{y 1, . . ., y t−1 } and the input sequence x, we hypothesize that it can assist our policy learning algorithm to generate more natural summaries. This motivates us to define a mixed learning objective function that combines equations 14 and 15: L mixed = γL rl + (1 − γ)L ml, (16) where γ is a scaling factor accoun...
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### 4 Related Work 4.1 Neural encoder-decoder sequence models Neural encoder-decoder models are widely used in NLP applications such as machine translation (Sutskever et al., 2014), summarization (Chopra et al., 2016; Nallapati et al., 2016), and question answering (Hermann et al., 2015). These models use recurrent...
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tasks, leading to significant improvements compared to previous supervised learning methods. While their method requires an additional neural network, called a critic model, to predict the expected reward and stabilize the objective function gradients, Rennie et al. (2016) designed a selfcritical sequence training meth...
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### 5 Datasets 5.1 CNN/Daily Mail We evaluate our model on a modified version of the CNN/Daily Mail dataset (Hermann et al., 2015), following the same pre-processing steps described in Nallapati et al. (2016). We refer the reader to that paper for a detailed description. The final dataset contains 286,817 training ...
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dataset. These pre-processing steps give us an average of 549 input tokens and 40 output tokens per example, after limiting the input and output lengths to 800 and 100 tokens. Pointer supervision: We run each input and abstract sequence through the Stanford named entity recognizer (NER) (Manning et al., 2014). For al...
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### 6 Results 6.1 Experiments Setup: We evaluate the intra-decoder attention mechanism and the mixed-objective learning by running the following experiments on both datasets. We first run maximum-likelihood (ML) training with and without intra-decoder attention (removing c [d] t [from Equations][ 9][ and][ 11][ to ...
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capped a miserable weekend for the Briton; his time in Bahrain plagued by reliability issues. Button spent much of the race on Twitter delivering his verdict as the action unfolded. ’Kimi is the man to watch,’ and ’loving the sparks’, were among his pearls of wisdom, but the tweet which courted the most attention was a...
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speaks to Nico Rosberg ahead of the Bahrain Grand Prix Ground truth summary Button denied 100th race start for McLaren after ERS failure. Button then spent much of the Bahrain Grand Prix on Twitter delivering his verdict on the action as it unfolded. Lewis Hamilton has out-qualified and finished ahead of Mercedes team-...
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ROUGE-L score as a reinforcement reward. We also tried ROUGE-2 but we found that it created summaries that almost always reached the maximum length, often ending sentences abruptly. 6.2 Quantitative analysis Our results for the CNN/Daily Mail dataset are shown in Table 1, and for the NYT dataset in Ta ble 2. We...
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example of Table 3, is the presence of short and truncated sentences towards the end of sequences. This confirms that optimizing for single discrete evaluation metric such as ROUGE with RL can be detrimental to the model quality. On the other hand, our RL+ML summaries obtain the highest readability scores among our...
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### 7 Conclusion We presented a new model and training procedure that obtains state-of-the-art results in text sum marization for the CNN/Daily Mail, improves the readability of the generated summaries and is better suited to long output sequences. We also run our abstractive model on the NYT dataset for the first ...
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### References Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 2014. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473 . Jianpeng Cheng, Li Dong, and Mirella Lapata. 2016. Long short-term memory-networks for machine reading. arXiv preprint arXiv:1601.06733 . ...
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arXiv:1412.6980 . Junyi Jessy Li, Kapil Thadani, and Amanda Stent. 2016. The role of discourse units in near-extractive summarization. In 17th Annual Meeting of the Special Interest Group on Discourse and Dialogue. page 137. Chin-Yew Lin. 2004. Rouge: A package for automatic evaluation of summaries. In Text summari...
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Jeffrey Pennington, Richard Socher, and Christopher D Manning. 2014. Glove: Global vectors for word representation. In EMNLP. volume 14, pages 1532– 1543. ----- Ofir Press and Lior Wolf. 2016. Using the output embedding to improve language models. arXiv preprint arXiv:1608.05859 . Marc’Aurelio Ranzato, Sumit Chop...
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Journal of Machine Learning Research 12 (2011) 2825-2830 Submitted 3/11; Revised 8/11; Published 10/11
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## Scikit-learn: Machine Learning in Python Fabian Pedregosa fabian.pedregosa@inria.fr Ga¨el Varoquaux gael.varoquaux@normalesup.org Alexandre Gramfort alexandre.gramfort@inria.fr Vincent Michel vincent.michel@logilab.fr Bertrand Thirion bertrand.thirion@inria.fr Parietal, INRIA Saclay Neurospin, Bˆat 145, CEA ...
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### Abstract Scikit-learn is a Python module integrating a wide range of state-of-the-art machine learning algorithms for medium-scale supervised and unsupervised problems. This package focuses on bringing machine learning to non-specialists using a general-purpose high-level language. Emphasis is put on ease of use,...
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### 1. Introduction The Python programming language is establishing itself as one of the most popular languages for scientific computing. Thanks to its high-level interactive nature and its maturing ecosystem of scientific libraries, it is an appealing choice for algorithmic development and exploratory data analysis ...
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### 2. Project Vision Code quality. Rather than providing as many features as possible, the project’s goal has been to provide solid implementations. Code quality is ensured with unit tests—as of release 0.8, test coverage is 81%—and the use of static analysis tools such as pyflakes and pep8. Finally, we strive to us...
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### 3. Underlying Technologies Numpy: the base data structure used for data and model parameters. Input data is presented as numpy arrays, thus integrating seamlessly with other scientific Python libraries. Numpy’s view-based memory model limits copies, even when binding with compiled code (Van der Walt et al., 2011)...
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### 4. Code Design Objects specified by interface, not by inheritance. To facilitate the use of external objects with scikit-learn, inheritance is not enforced; instead, code conventions provide a consistent interface. The central object is an estimator, that implements a fit method, accepting as arguments an input d...
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transparently as any other estimator. Cross validation can be made more efficient for certain estimators by exploiting specific properties, such as warm restarts or regularization paths (Friedman et al., 2010). This is supported through special objects, such as the LassoCV. Finally, a Pipeline object can combine severa...
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### 5. High-level yet Efficient: Some Trade Offs While scikit-learn focuses on ease of use, and is mostly written in a high level language, care has been taken to maximize computational efficiency. In Table 1, we compare computation time for a few algorithms implemented in the major machine learning toolkits accessib...
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### 6. Conclusion Scikit-learn exposes a wide variety of machine learning algorithms, both supervised and unsupervised, using a consistent, task-oriented interface, thus enabling easy comparison of methods for a given application. Since it relies on the scientific Python ecosystem, it can easily be integrated into ap...
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### References D. Albanese, G. Merler, S.and Jurman, and R. Visintainer. MLPy: high-performance Python package for predictive modeling. In NIPS, MLOSS workshop, 2008. C.C. Chang and C.J. Lin. LIBSVM: a library for support vector machines. [http://www.csie.ntu.edu.tw/cjlin/libsvm, 2001.](http://www.csie.ntu.edu.tw/c...
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C. Gehl, and V. Franc. The SHOGUN Machine Learning Toolbox. Journal of Machine Learning Research, 11:1799–1802, 2010. S. Van der Walt, S.C Colbert, and G. Varoquaux. The NumPy array: a structure for efficient numerical computation. Computing in Science and Engineering, 11, 2011. T. Zito, N. Wilbert, L. Wiskott, and...
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## Hogwild!: A Lock-Free Approach to Parallelizing Stochastic Gradient Descent ##### Feng Niu, Benjamin Recht, Christopher R´e and Stephen J. Wright Computer Sciences Department, University of Wisconsin-Madison 1210 W Dayton St, Madison, WI 53706 June 2011 **Abstract** Stochastic Gradient Descent (SGD) is a popular...
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### **1 Introduction** With its small memory footprint, robustness against noise, and rapid learning rates, Stochastic Gradient Descent (SGD) has proved to be well suited to data-intensive machine learning tasks [3,5,27]. However, SGD’s scalability is limited by its inherently sequential nature; it is difficult to pa...
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in parallel computation to synchronization (or locking) amongst the processors [2, 13]. Thus, to enable scalable data analysis on a multicore machine, any performant solution must minimize the overhead of locking. In this work, we propose a simple strategy for eliminating the overhead associated with locking: *run SGD ...
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### **2 Sparse Separable Cost Functions** Our goal throughout is to minimize a function *f* : *X ⊆* R *[n]* *→* R of the form *f* ( *x* ) = � *f* *e* ( *x* *e* ) *.* (2.1) *e∈E* Here *e* denotes a small subset of *{* 1 *, . . ., n}* and *x* *e* denotes the values of the vector *x* on the coordinates indexed by ...
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#### (a) (b) (c) **Figure 1:** Example graphs induced by cost function. (a) A sparse SVM induces a hypergraph where each hyperedge corresponds to one example. (b) A matrix completion example induces a bipartite graph between the rows and columns with an edge between two nodes if an entry is revealed. (c) The induced ...
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where ***L*** is *n* *r* *× r*, ***R*** is *n* *c* *× r* and ***L*** *u* (resp. ***R*** *v* ) denotes the *u* th (resp. *v* th) row of ***L*** (resp. ***R*** ) [18,25,28]. To put this problem in sparse form, i.e., as (2.1), we write (2.4) as minimize ( ***L*** *,* ***R*** ) � �( ***L*** *u* ***R*** *v* *[∗]* *[−]* ...
{ "id": "1106.5730", "title": "HOGWILD!: A Lock-Free Approach to Parallelizing Stochastic Gradient\n Descent", "categories": [ "math.OC", "cs.LG" ] }
{ "Header 1": null, "Header 2": "Hogwild!: A Lock-Free Approach to Parallelizing Stochastic Gradient Descent", "Header 3": "**2 Sparse Separable Cost Functions**", "Header 4": "(a) (b) (c)" }
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*e∈E* *[|][e][|][,]* *|E|* *|E|* The quantity Ωsimply quantifies the size of the hyper edges. *ρ* determines the maximum fraction of edges that intersect any given edge. ∆determines the maximum fraction of edges that intersect any variable. *ρ* is a measure of the sparsity of the hypergraph, while ∆measures the node-...
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### **3 The Hogwild! Algorithm** Here we discuss the parallel processing setup. We assume a shared memory model with *p* processors. The decision variable *x* is accessible to all processors. Each processor can read *x*, and can contribute an update vector to *x* . The vector *x* is stored in shared memory, and we as...
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of the uniform random sampling of *e* from *E*, we have E[ *G* *e* ( *x* *e* )] *∈* *∂f* ( *x* ) *.* In Algorithm 1, each processor samples an term *e ∈* *E* uniformly at random, computes the gradient of *f* *e* at *x* *e*, and then writes *x* *v* *←* *x* *v* *−* *γb* *[T]* *v* *[G]* *[e]* [(] *[x]* [)] *[,]* for...
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### **4 Fast Rates for Lock-Free Parallelism** We now turn to our theoretical analysis of Hogwild! protocols. To make the analysis tractable, we assume that we update with the following “with replacement” procedure: each processor samples an edge *e* uniformly at random and computes a subgradient of *f* *e* at the cu...
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To state our theoretical results, we must describe several quantities that important in the analysis of our parallel stochastic gradient descent scheme. We follow the notation and assumptions of Nemirovski *et al* [24]. To simplify the analysis, we will assume that each *f* *e* in (2.1) is a convex function. We assume ...
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*n* [1] *[/]* [4], we get nearly the same recursion as in the linear rate. We prove Proposition 4.1 in two steps in the Appendix. First, we demonstrate that the sequence *a* *j* = [1] 2 [E][[] *[∥][x]* *[j]* *[ −]* *[x]* *[⋆]* *[∥]* [2] [] satisfies a recursion of the form] *[ a]* *[j]* *[ ≤]* [(1] *[ −]* *[c]* *[r]* *...
{ "id": "1106.5730", "title": "HOGWILD!: A Lock-Free Approach to Parallelizing Stochastic Gradient\n Descent", "categories": [ "math.OC", "cs.LG" ] }
{ "Header 1": null, "Header 2": "Hogwild!: A Lock-Free Approach to Parallelizing Stochastic Gradient Descent", "Header 3": "**4 Fast Rates for Lock-Free Parallelism**", "Header 4": null }
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