Abstract-Recently, a curse-of-dimensionality-free method was developed for solution of Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) for nonlinear control problems, using semiconvex duality and max-plus analysis. The curse-of-dimensionality-free method may be applied to HJB PDEs where the Hamiltonian is given as (or well-approximated by) a pointwise maximum of quadratic forms. Such HJB PDEs also arise in certain switched linear systems. The method constructs the correct solution of an HJB PDE from a maxplus linear combination of quadratics. The method completely avoids the curse-of-dimensionality, and is subject to cubic computational growth as a function of space dimension. However, it is subject to a curse-of-complexity. In particular, the number of quadratics in the approximation grows exponentially with the number of iterations. Efficacy of such a method depends on the pruning of quadratics to keep the complexity growth at a reasonable level. Here we apply a pruning algorithm based on semidefinite programming. Computational speeds are exceptional, with an example HJB PDE in six-dimensional Euclidean space solved to the indicated quality in approximately 30 minutes on a typical desktop machine.

Deep Learning has managed to push boundaries in a wide variety of tasks. One area of interest is to tackle problems in reasoning and understanding, with an aim to emulate human intelligence. In this work, we describe a deep learning model that addresses the reasoning task of question-answering on categorical plots. We introduce a novel architecture FigureNet, that learns to identify various plot elements, quantify the represented values and determine a relative ordering of these statistical values. We test our model on the FigureQA dataset which provides images and accompanying questions for scientific plots like bar graphs and pie charts, augmented with rich annotations. Our approach outperforms the state-of-the-art Relation Networks baseline by approximately 7% on this dataset, with a training time that is over an order of magnitude lesser.

In this paper, we propose a simple and effective technique to allow for efficient selfsupervised learning with bi-directional Transformers. Our approach is motivated by recent studies demonstrating that self-attention patterns in trained models contain a majority of non-linguistic regularities. We propose a computationally efficient auxiliary loss function to guide attention heads to conform to such patterns. Our method is agnostic to the actual pretraining objective and results in faster convergence of models as well as better performance on downstream tasks compared to the baselines, achieving state of the art results in lowresource settings. Surprisingly, we also find that linguistic properties of attention heads are not necessarily correlated with language modeling performance. 1

No abstract

While recent work on multilingual language models has demonstrated their capacity for cross-lingual zero-shot transfer, there is a lack of consensus in the community as to what shared properties between languages enable transfer on downstream tasks. Analyses involving pairs of natural languages are often inconclusive and contradictory since languages simultaneously differ in many linguistic aspects. In this paper, we perform a large-scale empirical study to isolate the effects of various linguistic properties by measuring zeroshot transfer between four diverse natural languages and their counterparts constructed by modifying aspects such as the script, word order, and syntax. Among other things, our experiments show that the absence of sub-word overlap significantly affects zero-shot transfer when languages differ in their word order, and there is a strong correlation between transfer performance and word embedding alignment between languages (e.g., ρ s = 0.94 on the task of NLI). Our results call for focus in multilingual models on explicitly improving word embedding alignment between languages rather than relying on its implicit emergence. 1