Power ISA(TM) Version 3.1 has introduced a new family of matrix math instructions, collectively known as the Matrix-Multiply Assist (MMA) facility. The instructions in this facility implement numerical linear algebra operations on small matrices and are meant to accelerate computation-intensive kernels, such as matrix multiplication, convolution and discrete Fourier transform. These instructions have led to a power- and area-efficient implementation of a high throughput math engine in the future POWER10 processor. Performance per core is 4 times better, at constant frequency, than the previous generation POWER9 processor. We also advocate the use of compiler built-ins as the preferred way of leveraging these instructions, which we illustrate through case studies covering matrix multiplication and convolution.
As GPUs scale their low precision matrix math throughput to boost deep learning (DL) performance, they upset the balance between math throughput and memory system capabilities. We demonstrate that converged GPU design trying to address diverging architectural requirements between FP32 (or larger) based HPC and FP16 (or smaller) based DL workloads results in sub-optimal configuration for either of the application domains. We argue that a Composable On-PAckage GPU (COPAGPU) architecture to provide domain-specialized GPU products is the most practical solution to these diverging requirements. A COPA-GPU leverages multi-chip-module disaggregation to support maximal design reuse, along with memory system specialization per application domain. We show how a COPA-GPU enables DL-specialized products by modular augmentation of the baseline GPU architecture with up to 4x higher off-die bandwidth, 32x larger on-package cache, 2.3x higher DRAM bandwidth and capacity, while conveniently supporting scaled-down HPC-oriented designs. This work explores the microarchitectural design necessary to enable composable GPUs and evaluates the benefits composability can provide to HPC, DL training, and DL inference. We show that when compared to a converged GPU design, a DL-optimized COPA-GPU featuring a combination of 16x larger cache capacity and 1.6x higher DRAM bandwidth scales per-GPU training and inference performance by 31% and 35% respectively and reduces the number of GPU instances by 50% in scale-out training scenarios.
Current CoVID-19 pandemic is spreading rapidly worldwide, and it may become one of the largest pandemic events in modern history if out of control. It appears most of the SARS-CoV2 virus infection resulted deaths are mainly due to dysfunctions or failures of the lung or multiple organs that could be attributed to hosts immunodysfunctions particularly hyperinflammatory type disorders. In this brief review and study, a math model is proposed to correlate the Pathogen Infection Recovery Probability (PIRP) versus Proinflammatory Anti-Pathogen Species (PIAPS) levels within a host unit, where a maximum PIRP is exhibited when the PIAPS levels are equal to or around PIAPS equilibrium levels at the pathogen elimination or clearance onset. Based on this model, rational or effective therapeutic strategies at right stages or timing, with right type of agents (immuno-stimulators or immuno-suppressors), and right dosages, may be designed and implemented that are expected to effectively achieve maximum PIRP or reduce the mortality.
Many schools in large urban districts have more applicants than seats. Centralized school assignment algorithms ration seats at over-subscribed schools using randomly assigned lottery numbers, non-lottery tie-breakers like test scores, or both. The New York City public high school match illustrates the latter, using test scores and other criteria to rank applicants at ``screened'' schools, combined with lottery tie-breaking at unscreened ``lottery'' schools. We show how to identify causal effects of school attendance in such settings. Our approach generalizes regression discontinuity methods to allow for multiple treatments and multiple running variables, some of which are randomly assigned. The key to this generalization is a local propensity score that quantifies the school assignment probabilities induced by lottery and non-lottery tie-breakers. The local propensity score is applied in an empirical assessment of the predictive value of New York City's school report cards. Schools that receive a high grade indeed improve SAT math scores and increase graduation rates, though by much less than OLS estimates suggest. Selection bias in OLS estimates is egregious for screened schools.
In problem solving, understanding the problem that one seeks to solve is an essential initial step. In this paper, we propose computational methods for facilitating problem understanding through the task of recognizing the unknown in specifications of long Math problems. We focus on the topic of Probability. Our experimental results show that learning models yield strong results on the task, a promising first step towards human interpretable, modular approaches to understanding long Math problems.
The problem of designing NLP solvers for math word problems (MWP) has seen sustained research activity and steady gains in the test accuracy. Since existing solvers achieve high performance on the benchmark datasets for elementary level MWPs containing one-unknown arithmetic word problems, such problems are often considered "solved" with the bulk of research attention moving to more complex MWPs. In this paper, we restrict our attention to English MWPs taught in grades four and lower. We provide strong evidence that the existing MWP solvers rely on shallow heuristics to achieve high performance on the benchmark datasets. To this end, we show that MWP solvers that do not have access to the question asked in the MWP can still solve a large fraction of MWPs. Similarly, models that treat MWPs as bag-of-words can also achieve surprisingly high accuracy. Further, we introduce a challenge dataset, SVAMP, created by applying carefully chosen variations over examples sampled from existing datasets. The best accuracy achieved by state-of-the-art models is substantially lower on SVAMP, thus showing that much remains to be done even for the simplest of the MWPs.
We present a Neural Network based Handwritten Text Recognition (HTR) model architecture that can be trained to recognize full pages of handwritten or printed text without image segmentation. Being based on Image to Sequence architecture, it can extract text present in an image and then sequence it correctly without imposing any constraints regarding orientation, layout and size of text and non-text. Further, it can also be trained to generate auxiliary markup related to formatting, layout and content. We use character level vocabulary, thereby enabling language and terminology of any subject. The model achieves a new state-of-art in paragraph level recognition on the IAM dataset. When evaluated on scans of real world handwritten free form test answers - beset with curved and slanted lines, drawings, tables, math, chemistry and other symbols - it performs better than all commercially available HTR cloud APIs. It is deployed in production as part of a commercial web application.
In the Priority $k$-Center problem, the input consists of a metric space $(X,d)$, an integer $k$, and for each point $v \in X$ a priority radius $r(v)$. The goal is to choose $k$-centers $S \subseteq X$ to minimize $\max_{v \in X} \frac{1}{r(v)} d(v,S)$. If all $r(v)$'s are uniform, one obtains the $k$-Center problem. Plesn\'ik [Plesn\'ik, Disc. Appl. Math. 1987] introduced the Priority $k$-Center problem and gave a $2$-approximation algorithm matching the best possible algorithm for $k$-Center. We show how the problem is related to two different notions of fair clustering [Harris et al., NeurIPS 2018; Jung et al., FORC 2020]. Motivated by these developments we revisit the problem and, in our main technical contribution, develop a framework that yields constant factor approximation algorithms for Priority $k$-Center with outliers. Our framework extends to generalizations of Priority $k$-Center to matroid and knapsack constraints, and as a corollary, also yields algorithms with fairness guarantees in the lottery model of Harris et al [Harris et al, JMLR 2019].
Many intellectual endeavors require mathematical problem solving, but this skill remains beyond the capabilities of computers. To measure this ability in machine learning models, we introduce MATH, a new dataset of 12,500 challenging competition mathematics problems. Each problem in MATH has a full step-by-step solution which can be used to teach models to generate answer derivations and explanations. To facilitate future research and increase accuracy on MATH, we also contribute a large auxiliary pretraining dataset which helps teach models the fundamentals of mathematics. Even though we are able to increase accuracy on MATH, our results show that accuracy remains relatively low, even with enormous Transformer models. Moreover, we find that simply increasing budgets and model parameter counts will be impractical for achieving strong mathematical reasoning if scaling trends continue. While scaling Transformers is automatically solving most other text-based tasks, scaling is not currently solving MATH. To have more traction on mathematical problem solving we will likely need new algorithmic advancements from the broader research community.
Traditional statistics forbids use of test data (a.k.a. holdout data) during training. Dwork et al. 2015 pointed out that current practices in machine learning, whereby researchers build upon each other's models, copying hyperparameters and even computer code -- amounts to implicitly training on the test set. Thus error rate on test data may not reflect the true population error. This observation initiated {\em adaptive data analysis}, which provides evaluation mechanisms with guaranteed upper bounds on this difference. With statistical query (i.e. test accuracy) feedbacks, the best upper bound is fairly pessimistic: the deviation can hit a practically vacuous value if the number of models tested is quadratic in the size of the test set. In this work, we present a simple new estimate, {\em Rip van Winkle's Razor}. It relies upon a new notion of \textquotedblleft information content\textquotedblright\ of a model: the amount of information that would have to be provided to an expert referee who is intimately familiar with the field and relevant science/math, and who has been just been woken up after falling asleep at the moment of the creation of the test data (like \textquotedblleft Rip van Winkle\textquotedblright\ of the famous fairy tale). This notion of information content is used to provide an estimate of the above deviation which is shown to be non-vacuous in many modern settings.
Quantization enables efficient acceleration of deep neural networks by reducing model memory footprint and exploiting low-cost integer math hardware units. Quantization maps floating-point weights and activations in a trained model to low-bitwidth integer values using scale factors. Excessive quantization, reducing precision too aggressively, results in accuracy degradation. When scale factors are shared at a coarse granularity across many dimensions of each tensor, effective precision of individual elements within the tensor are limited. To reduce quantization-related accuracy loss, we propose using a separate scale factor for each small vector of ($\approx$16-64) elements within a single dimension of a tensor. To achieve an efficient hardware implementation, the per-vector scale factors can be implemented with low-bitwidth integers when calibrated using a two-level quantization scheme. We find that per-vector scaling consistently achieves better inference accuracy at low precision compared to conventional scaling techniques for popular neural networks without requiring retraining. We also modify a deep learning accelerator hardware design to study the area and energy overheads of per-vector scaling support. Our evaluation demonstrates that per-vector scaled quantization with 4-bit weights and activations achieves 37% area saving and 24% energy saving while maintaining over 75% accuracy for ResNet50 on ImageNet. 4-bit weights and 8-bit activations achieve near-full-precision accuracy for both BERT-base and BERT-large on SQuAD while reducing area by 26% compared to an 8-bit baseline.
Imported COVID-19 cases, if unchecked, can jeopardize the effort of domestic containment. We aim to find out what sustainable border control options for different entities (e.g., countries, states) exist during the reopening phases, given their own choice of domestic control measures and new technologies such as contact tracing. We propose a SUIHR model, which represents an extension to the discrete time SIR models. The model focuses on studying the spreading of virus predominantly by asymptomatic and pre-symptomatic patients. Imported risk and (1-tier) contact tracing are both built into the model. Under plausible parameter assumptions, we seek sustainable border control policies, in combination with sufficient internal measures, which allow entities to confine the virus without the need to revert back to more restrictive life styles or to rely on herd immunity. When the base reproduction number of COVID-19 exceeds 2.5, even 100% effective contact tracing alone is not enough to contain the spreading. For an entity that has completely eliminated the virus domestically, and resumes "normal", very strict pre-departure screening and test and isolation upon arrival combined with effective contact tracing can only delay another outbreak by 6 months. However, if the total net imported cases are non-increasing, and the entity employs a confining domestic control policy, then the total new cases can be contained even without border control.
Problem-Based Learning (PBL) is a popular approach to instruction that supports students to get hands-on training by solving problems. Question Pool websites (QPs) such as LeetCode, Code Chef, and Math Playground help PBL by supplying authentic, diverse, and contextualized questions to students. Nonetheless, empirical findings suggest that 40% to 80% of students registered in QPs drop out in less than two months. This research is the first attempt to understand and predict student dropouts from QPs via exploiting students' engagement moods. Adopting a data-driven approach, we identify five different engagement moods for QP students, which are namely challenge-seeker, subject-seeker, interest-seeker, joy-seeker, and non-seeker. We find that students have collective preferences for answering questions in each engagement mood, and deviation from those preferences increases their probability of dropping out significantly. Last but not least, this paper contributes by introducing a new hybrid machine learning model (we call Dropout-Plus) for predicting student dropouts in QPs. The test results on a popular QP in China, with nearly 10K students, show that Dropout-Plus can exceed the rival algorithms' dropout prediction performance in terms of accuracy, F1-measure, and AUC. We wrap up our work by giving some design suggestions to QP managers and online learning professionals to reduce their student dropouts.
Differential evolution (DE) is an effective global evolutionary optimization algorithm using to solve global optimization problems mainly in a continuous domain. In this field, researchers pay more attention to improving the capability of DE to find better global solutions, however, the computational performance of DE is also a very interesting aspect especially when the problem scale is quite large. Firstly, this paper analyzes the design of parallel computation of DE which can easily be executed in Math Kernel Library (MKL) and Compute Unified Device Architecture (CUDA). Then the essence of the exponential crossover operator is described and we point out that it cannot be used for better parallel computation. Later, we propose a new exponential crossover operator (NEC) that can be executed parallelly with MKL/CUDA. Next, the extended experiments show that the new crossover operator can speed up DE greatly. In the end, we test the new parallel DE structure, illustrating that the former is much faster.
Gulisashvili et al. [Quant. Finance, 2018, 18(10), 1753-1765] provide a small-time asymptotics for the mass at zero under the uncorrelated stochastic-alpha-beta-rho (SABR) model by approximating the integrated variance with a moment-matched lognormal distribution. We improve the accuracy of the numerical integration by using the Gauss--Hermite quadrature. We further obtain the option price by integrating the constant elasticity of variance (CEV) option prices in the same manner without resorting to the small-strike volatility smile asymptotics of De Marco et al. [SIAM J. Financ. Math., 2017, 8(1), 709-737]. For the uncorrelated SABR model, the new option pricing method is accurate and arbitrage-free across all strike prices.
Solving algebra story problems remains a challenging task in artificial intelligence, which requires a detailed understanding of real-world situations and a strong mathematical reasoning capability. Previous neural solvers of math word problems directly translate problem texts into equations, lacking an explicit interpretation of the situations, and often fail to handle more sophisticated situations. To address such limits of neural solvers, we introduce the concept of a \emph{situation model}, which originates from psychology studies to represent the mental states of humans in problem-solving, and propose \emph{SMART}, which adopts attributed grammar as the representation of situation models for algebra story problems. Specifically, we first train an information extraction module to extract nodes, attributes, and relations from problem texts and then generate a parse graph based on a pre-defined attributed grammar. An iterative learning strategy is also proposed to improve the performance of SMART further. To rigorously study this task, we carefully curate a new dataset named \emph{ASP6.6k}. Experimental results on ASP6.6k show that the proposed model outperforms all previous neural solvers by a large margin while preserving much better interpretability. To test these models' generalization capability, we also design an out-of-distribution (OOD) evaluation, in which problems are more complex than those in the training set. Our model exceeds state-of-the-art models by 17\% in the OOD evaluation, demonstrating its superior generalization ability.
Previous neural solvers of math word problems (MWPs) are learned with full supervision and fail to generate diverse solutions. In this paper, we address this issue by introducing a \textit{weakly-supervised} paradigm for learning MWPs. Our method only requires the annotations of the final answers and can generate various solutions for a single problem. To boost weakly-supervised learning, we propose a novel \textit{learning-by-fixing} (LBF) framework, which corrects the misperceptions of the neural network via symbolic reasoning. Specifically, for an incorrect solution tree generated by the neural network, the \textit{fixing} mechanism propagates the error from the root node to the leaf nodes and infers the most probable fix that can be executed to get the desired answer. To generate more diverse solutions, \textit{tree regularization} is applied to guide the efficient shrinkage and exploration of the solution space, and a \textit{memory buffer} is designed to track and save the discovered various fixes for each problem. Experimental results on the Math23K dataset show the proposed LBF framework significantly outperforms reinforcement learning baselines in weakly-supervised learning. Furthermore, it achieves comparable top-1 and much better top-3/5 answer accuracies than fully-supervised methods, demonstrating its strength in producing diverse solutions.
We present a computational model of mathematical reasoning according to which mathematics is a fundamentally stochastic process. That is, on our model, whether or not a given formula is deemed a theorem in some axiomatic system is not a matter of certainty, but is instead governed by a probability distribution. We then show that this framework gives a compelling account of several aspects of mathematical practice. These include: 1) the way in which mathematicians generate research programs, 2) the applicability of Bayesian models of mathematical heuristics, 3) the role of abductive reasoning in mathematics, 4) the way in which multiple proofs of a proposition can strengthen our degree of belief in that proposition, and 5) the nature of the hypothesis that there are multiple formal systems that are isomorphic to physically possible universes. Thus, by embracing a model of mathematics as not perfectly predictable, we generate a new and fruitful perspective on the epistemology and practice of mathematics.
Recent years have seen significant advancement in text generation tasks with the help of neural language models. However, there exists a challenging task: generating math problem text based on mathematical equations, which has made little progress so far. In this paper, we present a novel equation-to-problem text generation model. In our model, 1) we propose a flexible scheme to effectively encode math equations, we then enhance the equation encoder by a Varitional Autoen-coder (VAE) 2) given a math equation, we perform topic selection, followed by which a dynamic topic memory mechanism is introduced to restrict the topic distribution of the generator 3) to avoid commonsense violation in traditional generation model, we pretrain word embedding with background knowledge graph (KG), and we link decoded words to related words in KG, targeted at injecting background knowledge into our model. We evaluate our model through both automatic metrices and human evaluation, experiments demonstrate our model outperforms baseline and previous models in both accuracy and richness of generated problem text.
Recent work has proven the existence of extreme inbreeding in a European ancestry sample taken from the contemporary UK population \cite{nature_01}. This result brings our attention again to a math problem related to inbreeding family trees and diversity. Groups with a finite number of individuals could give a variety of genetic relationships. { In previous works \cite{PhysRevE.92.052132, PhysRevE.90.022125, JARNE20191}, we have addressed the issue of building inbreeding trees for biparental reproduction using Markovian models. Here, we extend these studies by presenting an algorithm to generate and represent inbreeding trees with no overlapping generations. We explicitly assume a two-gender reproductory scheme, and we pay particular attention to the links between nodes. We show that even for a simple case with a relatively small number of nodes in the tree, there are a large number of possible ways to rearrange the links between generations. We present an open-source python code to generate the tree graph, the adjacency matrix, and the histogram of the links for each different tree representation. We show how this mapping reflects the difference between tree realizations, and how valuable information may be extracted upon inspection of these matrices. The algorithm includes a feature to average several tree realizations, obtain the connectivity distribution, and calculate the average and mean value. We used this feature to compare trees with a different number of generations and nodes. The code presented here, available in Git-Hub, may be easily modified to be applied to other areas of interest involving connections between individuals, extend the study to add more characteristics of the different nodes, etc.