A new scalable parallel math library, dMath, is presented in this paper that demonstrates leading scaling when using intranode, or internode, hybrid-parallelism for deep-learning. dMath provides easy-to-use distributed base primitives and a variety of domain-specific algorithms. These include matrix multiplication, convolutions, and others allowing for rapid development of highly scalable applications, including Deep Neural Networks (DNN), whereas previously one was restricted to libraries that provided effective primitives for only a single GPU, like Nvidia cublas and cudnn or DNN primitives from Nervana neon framework. Development of HPC software is difficult, labor-intensive work, requiring a unique skill set. dMath allows a wide range of developers to utilize parallel and distributed hardware easily. One contribution of this approach is that data is stored persistently on the GPU hardware, avoiding costly transfers between host and device. Advanced memory management techniques are utilized, including caching of transferred data and memory reuse through pooling. A key contribution of dMath is that it delivers performance, portability, and productivity to its specific domain of support. It enables algorithm and application programmers to quickly solve problems without managing the significant complexity associated with multi-level parallelism.
Texts present coherent stories that have a particular theme or overall setting, for example science fiction or western. In this paper, we present a text generation method called {\it rewriting} that edits existing human-authored narratives to change their theme without changing the underlying story. We apply the approach to math word problems, where it might help students stay more engaged by quickly transforming all of their homework assignments to the theme of their favorite movie without changing the math concepts that are being taught. Our rewriting method uses a two-stage decoding process, which proposes new words from the target theme and scores the resulting stories according to a number of factors defining aspects of syntactic, semantic, and thematic coherence. Experiments demonstrate that the final stories typically represent the new theme well while still testing the original math concepts, outperforming a number of baselines. We also release a new dataset of human-authored rewrites of math word problems in several themes.
The paper presents a parallel math library, dMath, that demonstrates leading scaling when using intranode, internode, and hybrid-parallelism for deep learning (DL). dMath provides easy-to-use distributed primitives and a variety of domain-specific algorithms including matrix multiplication, convolutions, and others allowing for rapid development of scalable applications like deep neural networks (DNNs). Persistent data stored in GPU memory and advanced memory management techniques avoid costly transfers between host and device. dMath delivers performance, portability, and productivity to its specific domain of support.
Math word problems provide a natural abstraction to a range of natural language understanding problems that involve reasoning about quantities, such as interpreting election results, news about casualties, and the financial section of a newspaper. Units associated with the quantities often provide information that is essential to support this reasoning. This paper proposes a principled way to capture and reason about units and shows how it can benefit an arithmetic word problem solver. This paper presents the concept of Unit Dependency Graphs (UDGs), which provides a compact representation of the dependencies between units of numbers mentioned in a given problem. Inducing the UDG alleviates the brittleness of the unit extraction system and allows for a natural way to leverage domain knowledge about unit compatibility, for word problem solving. We introduce a decomposed model for inducing UDGs with minimal additional annotations, and use it to augment the expressions used in the arithmetic word problem solver of (Roy and Roth 2015) via a constrained inference framework. We show that introduction of UDGs reduces the error of the solver by over 10 %, surpassing all existing systems for solving arithmetic word problems. In addition, it also makes the system more robust to adaptation to new vocabulary and equation forms .
As data collections become larger, exploratory regression analysis becomes more important but more challenging. When observations are hierarchically clustered the problem is even more challenging because model selection with mixed effect models can produce misleading results when nonlinear effects are not included into the model (Bauer and Cai, 2009). A machine learning method called boosted decision trees (Friedman, 2001) is a good approach for exploratory regression analysis in real data sets because it can detect predictors with nonlinear and interaction effects while also accounting for missing data. We propose an extension to boosted decision decision trees called metboost for hierarchically clustered data. It works by constraining the structure of each tree to be the same across groups, but allowing the terminal node means to differ. This allows predictors and split points to lead to different predictions within each group, and approximates nonlinear group specific effects. Importantly, metboost remains computationally feasible for thousands of observations and hundreds of predictors that may contain missing values. We apply the method to predict math performance for 15,240 students from 751 schools in data collected in the Educational Longitudinal Study 2002 (Ingels et al., 2007), allowing 76 predictors to have unique effects for each school. When comparing results to boosted decision trees, metboost has 15% improved prediction performance. Results of a large simulation study show that metboost has up to 70% improved variable selection performance and up to 30% improved prediction performance compared to boosted decision trees when group sizes are small
This tutorial introduces a new and powerful set of techniques variously called "neural machine translation" or "neural sequence-to-sequence models". These techniques have been used in a number of tasks regarding the handling of human language, and can be a powerful tool in the toolbox of anyone who wants to model sequential data of some sort. The tutorial assumes that the reader knows the basics of math and programming, but does not assume any particular experience with neural networks or natural language processing. It attempts to explain the intuition behind the various methods covered, then delves into them with enough mathematical detail to understand them concretely, and culiminates with a suggestion for an implementation exercise, where readers can test that they understood the content in practice.
The monogenic signal is an image analysis methodology that was introduced by Felsberg and Sommer in 2001 and has been employed for a variety of purposes in image processing and computer vision research. In particular, it has been found to be useful in the analysis of ultrasound imagery in several research scenarios mostly in work done within the BioMedIA lab at Oxford. However, the literature on the monogenic signal can be difficult to penetrate due to the lack of a single resource to explain the various principles from basics. The purpose of this document is therefore to introduce the principles, purpose, applications, and limitations of the methodology. It assumes some background knowledge from the fields of image and signal processing, in particular a good knowledge of Fourier transforms as applied to signals and images. We will not attempt to provide a thorough math- ematical description or derivation of the monogenic signal, but rather focus on developing an intuition for understanding and using the methodology and refer the reader elsewhere for a more mathematical treatment.
Cognitive arithmetic studies the mental processes used in solving math problems. This area of research explores the retrieval mechanisms and strategies used by people during a common cognitive task. Past research has shown that human performance in arithmetic operations is correlated to the numerical size of the problem. Past research on cognitive arithmetic has pinpointed this trend to either retrieval strength, error checking, or strategy-based approaches when solving equations. This paper describes a rule-based computational model that performs the four major arithmetic operations (addition, subtraction, multiplication and division) on two operands. We then evaluated our model to probe its validity in representing the prevailing concepts observed in psychology experiments from the related works. The experiments specifically explore the problem size effect, an activation-based model for fact retrieval, backup strategies when retrieval fails, and finally optimization strategies when faced with large operands. From our experimental results, we concluded that our model's response times were comparable to results observed when people performed similar tasks during psychology experiments. The fit of our model in reproducing these results and incorporating accuracy into our model are discussed.
Turing test was long considered the measure for artificial intelligence. But with the advances in AI, it has proved to be insufficient measure. We can now aim to mea- sure machine intelligence like we measure human intelligence. One of the widely accepted measure of intelligence is standardized math and science test. In this paper, we explore the progress we have made towards the goal of making a machine smart enough to pass the standardized test. We see the challenges and opportunities posed by the domain, and note that we are quite some ways from actually making a system as smart as a even a middle school scholar.
The asynchronous nature of the state-of-the-art reinforcement learning algorithms such as the Asynchronous Advantage Actor-Critic algorithm, makes them exceptionally suitable for CPU computations. However, given the fact that deep reinforcement learning often deals with interpreting visual information, a large part of the train and inference time is spent performing convolutions. In this work we present our results on learning strategies in Atari games using a Convolutional Neural Network, the Math Kernel Library and TensorFlow 0.11rc0 machine learning framework. We also analyze effects of asynchronous computations on the convergence of reinforcement learning algorithms.
We consider the random design regression model with square loss. We propose a method that aggregates empirical minimizers (ERM) over appropriately chosen random subsets and reduces to ERM in the extreme case, and we establish sharp oracle inequalities for its risk. We show that, under the $\varepsilon^{-p}$ growth of the empirical $\varepsilon$-entropy, the excess risk of the proposed method attains the rate $n^{-2/(2+p)}$ for $p\in(0,2)$ and $n^{-1/p}$ for $p>2$ where $n$ is the sample size. Furthermore, for $p\in(0,2)$, the excess risk rate matches the behavior of the minimax risk of function estimation in regression problems under the well-specified model. This yields a conclusion that the rates of statistical estimation in well-specified models (minimax risk) and in misspecified models (minimax regret) are equivalent in the regime $p\in(0,2)$. In other words, for $p\in(0,2)$ the problem of statistical learning enjoys the same minimax rate as the problem of statistical estimation. On the contrary, for $p>2$ we show that the rates of the minimax regret are, in general, slower than for the minimax risk. Our oracle inequalities also imply the $v\log(n/v)/n$ rates for Vapnik-Chervonenkis type classes of dimension $v$ without the usual convexity assumption on the class; we show that these rates are optimal. Finally, for a slightly modified method, we derive a bound on the excess risk of $s$-sparse convex aggregation improving that of Lounici [Math. Methods Statist. 16 (2007) 246-259] and providing the optimal rate.
Deep Neural Networks (DNNs) have emerged as a core tool for machine learning. The computations performed during DNN training and inference are dominated by operations on the weight matrices describing the DNN. As DNNs incorporate more stages and more nodes per stage, these weight matrices may be required to be sparse because of memory limitations. The GraphBLAS.org math library standard was developed to provide high performance manipulation of sparse weight matrices and input/output vectors. For sufficiently sparse matrices, a sparse matrix library requires significantly less memory than the corresponding dense matrix implementation. This paper provides a brief description of the mathematics underlying the GraphBLAS. In addition, the equations of a typical DNN are rewritten in a form designed to use the GraphBLAS. An implementation of the DNN is given using a preliminary GraphBLAS C library. The performance of the GraphBLAS implementation is measured relative to a standard dense linear algebra library implementation. For various sizes of DNN weight matrices, it is shown that the GraphBLAS sparse implementation outperforms a BLAS dense implementation as the weight matrix becomes sparser.
Imagery texts are usually organized as a hierarchy of several visual elements, i.e. characters, words, text lines and text blocks. Among these elements, character is the most basic one for various languages such as Western, Chinese, Japanese, mathematical expression and etc. It is natural and convenient to construct a common text detection engine based on character detectors. However, training character detectors requires a vast of location annotated characters, which are expensive to obtain. Actually, the existing real text datasets are mostly annotated in word or line level. To remedy this dilemma, we propose a weakly supervised framework that can utilize word annotations, either in tight quadrangles or the more loose bounding boxes, for character detector training. When applied in scene text detection, we are thus able to train a robust character detector by exploiting word annotations in the rich large-scale real scene text datasets, e.g. ICDAR15 and COCO-text. The character detector acts as a key role in the pipeline of our text detection engine. It achieves the state-of-the-art performance on several challenging scene text detection benchmarks. We also demonstrate the flexibility of our pipeline by various scenarios, including deformed text detection and math expression recognition.
Science, technology, engineering, and math (STEM) fields play growing roles in national and international economies by driving innovation and generating high salary jobs. Yet, the US is lagging behind other highly industrialized nations in terms of STEM education and training. Furthermore, many economic forecasts predict a rising shortage of domestic STEM-trained professions in the US for years to come. One potential solution to this deficit is to decrease the rates at which students leave STEM-related fields in higher education, as currently over half of all students intending to graduate with a STEM degree eventually attrite. However, little quantitative research at scale has looked at causes of STEM attrition, let alone the use of machine learning to examine how well this phenomenon can be predicted. In this paper, we detail our efforts to model and predict dropout from STEM fields using one of the largest known datasets used for research on students at a traditional campus setting. Our results suggest that attrition from STEM fields can be accurately predicted with data that is routinely collected at universities using only information on students' first academic year. We also propose a method to model student STEM intentions for each academic term to better understand the timing of STEM attrition events. We believe these results show great promise in using machine learning to improve STEM retention in traditional and non-traditional campus settings.
Math word problems form a natural abstraction to a range of quantitative reasoning problems, such as understanding financial news, sports results, and casualties of war. Solving such problems requires the understanding of several mathematical concepts such as dimensional analysis, subset relationships, etc. In this paper, we develop declarative rules which govern the translation of natural language description of these concepts to math expressions. We then present a framework for incorporating such declarative knowledge into word problem solving. Our method learns to map arithmetic word problem text to math expressions, by learning to select the relevant declarative knowledge for each operation of the solution expression. This provides a way to handle multiple concepts in the same problem while, at the same time, support interpretability of the answer expression. Our method models the mapping to declarative knowledge as a latent variable, thus removing the need for expensive annotations. Experimental evaluation suggests that our domain knowledge based solver outperforms all other systems, and that it generalizes better in the realistic case where the training data it is exposed to is biased in a different way than the test data.
The Deep Learning (DL) community sees many novel topologies published each year. Achieving high performance on each new topology remains challenging, as each requires some level of manual effort. This issue is compounded by the proliferation of frameworks and hardware platforms. The current approach, which we call "direct optimization", requires deep changes within each framework to improve the training performance for each hardware backend (CPUs, GPUs, FPGAs, ASICs) and requires $\mathcal{O}(fp)$ effort; where $f$ is the number of frameworks and $p$ is the number of platforms. While optimized kernels for deep-learning primitives are provided via libraries like Intel Math Kernel Library for Deep Neural Networks (MKL-DNN), there are several compiler-inspired ways in which performance can be further optimized. Building on our experience creating neon (a fast deep learning library on GPUs), we developed Intel nGraph, a soon to be open-sourced C++ library to simplify the realization of optimized deep learning performance across frameworks and hardware platforms. Initially-supported frameworks include TensorFlow, MXNet, and Intel neon framework. Initial backends are Intel Architecture CPUs (CPU), the Intel(R) Nervana Neural Network Processor(R) (NNP), and NVIDIA GPUs. Currently supported compiler optimizations include efficient memory management and data layout abstraction. In this paper, we describe our overall architecture and its core components. In the future, we envision extending nGraph API support to a wider range of frameworks, hardware (including FPGAs and ASICs), and compiler optimizations (training versus inference optimizations, multi-node and multi-device scaling via efficient sub-graph partitioning, and HW-specific compounding of operations).
Handwritten mathematical expression recognition is a challenging problem due to the complicated two-dimensional structures, ambiguous handwriting input and variant scales of handwritten math symbols. To settle this problem, we utilize the attention based encoder-decoder model that recognizes mathematical expression images from two-dimensional layouts to one-dimensional LaTeX strings. We improve the encoder by employing densely connected convolutional networks as they can strengthen feature extraction and facilitate gradient propagation especially on a small training set. We also present a novel multi-scale attention model which is employed to deal with the recognition of math symbols in different scales and save the fine-grained details that will be dropped by pooling operations. Validated on the CROHME competition task, the proposed method significantly outperforms the state-of-the-art methods with an expression recognition accuracy of 52.8% on CROHME 2014 and 50.1% on CROHME 2016, by only using the official training dataset.
This paper is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We assume no math knowledge beyond what you learned in calculus 1, and provide links to help you refresh the necessary math where needed. Note that you do not need to understand this material before you start learning to train and use deep learning in practice; rather, this material is for those who are already familiar with the basics of neural networks, and wish to deepen their understanding of the underlying math. Don't worry if you get stuck at some point along the way---just go back and reread the previous section, and try writing down and working through some examples. And if you're still stuck, we're happy to answer your questions in the Theory category at forums.fast.ai. Note: There is a reference section at the end of the paper summarizing all the key matrix calculus rules and terminology discussed here. See related articles at http://explained.ai
We introduce MeSys, a meaning-based approach, for solving English math word problems (MWPs) via understanding and reasoning in this paper. It first analyzes the text, transforms both body and question parts into their corresponding logic forms, and then performs inference on them. The associated context of each quantity is represented with proposed role-tags (e.g., nsubj, verb, etc.), which provides the flexibility for annotating an extracted math quantity with its associated context information (i.e., the physical meaning of this quantity). Statistical models are proposed to select the operator and operands. A noisy dataset is designed to assess if a solver solves MWPs mainly via understanding or mechanical pattern matching. Experimental results show that our approach outperforms existing systems on both benchmark datasets and the noisy dataset, which demonstrates that the proposed approach understands the meaning of each quantity in the text more.
Networks capture pairwise interactions between entities and are frequently used in applications such as social networks, food networks, and protein interaction networks, to name a few. Communities, cohesive groups of nodes, often form in these applications, and identifying them gives insight into the overall organization of the network. One common quality function used to identify community structure is modularity. In Hu et al. [SIAM J. App. Math., 73(6), 2013], it was shown that modularity optimization is equivalent to minimizing a particular nonconvex total variation (TV) based functional over a discrete domain. They solve this problem, assuming the number of communities is known, using a Merriman, Bence, Osher (MBO) scheme. We show that modularity optimization is equivalent to minimizing a convex TV-based functional over a discrete domain, again, assuming the number of communities is known. Furthermore, we show that modularity has no convex relaxation satisfying certain natural conditions. We therefore, find a manageable non-convex approximation using a Ginzburg Landau functional, which provably converges to the correct energy in the limit of a certain parameter. We then derive an MBO algorithm with fewer hand-tuned parameters than in Hu et al. and which is 7 times faster at solving the associated diffusion equation due to the fact that the underlying discretization is unconditionally stable. Our numerical tests include a hyperspectral video whose associated graph has 2.9x10^7 edges, which is roughly 37 times larger than was handled in the paper of Hu et al.