Most existing neural network-based approaches for solving stochastic optimal control problems using the associated backward dynamic programming principle rely on the ability to simulate the underlying state variables. However, in some problems, this simulation is infeasible, leading to the discretization of state variable space and the need to train one neural network for each data point. This approach becomes computationally inefficient when dealing with large state variable spaces. In this paper, we consider a class of this type of stochastic optimal control problems and introduce an effective solution employing multitask neural networks. To train our multitask neural network, we introduce a novel scheme that dynamically balances the learning across tasks. Through numerical experiments on real-world derivatives pricing problems, we prove that our method outperforms state-of-the-art approaches.
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神經網絡（Neural Networks）是世界上三個最古老的神經建模學會的檔案期刊:國際神經網絡學會(INNS)、歐洲神經網絡學會(ENNS)和日本神經網絡學會(JNNS)。神經網絡提供了一個論壇，以發展和培育一個國際社會的學者和實踐者感興趣的所有方面的神經網絡和相關方法的計算智能。神經網絡歡迎高質量論文的提交，有助于全面的神經網絡研究，從行為和大腦建模，學習算法，通過數學和計算分析，系統的工程和技術應用，大量使用神經網絡的概念和技術。這一獨特而廣泛的范圍促進了生物和技術研究之間的思想交流，并有助于促進對生物啟發的計算智能感興趣的跨學科社區的發展。因此，神經網絡編委會代表的專家領域包括心理學，神經生物學，計算機科學，工程，數學，物理。該雜志發表文章、信件和評論以及給編輯的信件、社論、時事、軟件調查和專利信息。文章發表在五個部分之一:認知科學，神經科學，學習系統，數學和計算分析、工程和應用。
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The consistency of the maximum likelihood estimator for mixtures of elliptically-symmetric distributions for estimating its population version is shown, where the underlying distribution $P$ is nonparametric and does not necessarily belong to the class of mixtures on which the estimator is based. In a situation where $P$ is a mixture of well enough separated but nonparametric distributions it is shown that the components of the population version of the estimator correspond to the well separated components of $P$. This provides some theoretical justification for the use of such estimators for cluster analysis in case that $P$ has well separated subpopulations even if these subpopulations differ from what the mixture model assumes.
Charts, figures, and text derived from data play an important role in decision making, from data-driven policy development to day-to-day choices informed by online articles. Making sense of, or fact-checking, outputs means understanding how they relate to the underlying data. Even for domain experts with access to the source code and data sets, this poses a significant challenge. In this paper we introduce a new program analysis framework which supports interactive exploration of fine-grained I/O relationships directly through computed outputs, making use of dynamic dependence graphs. Our main contribution is a novel notion in data provenance which we call related inputs, a relation of mutual relevance or "cognacy" which arises between inputs when they contribute to common features of the output. Queries of this form allow readers to ask questions like "What outputs use this data element, and what other data elements are used along with it?". We show how Jonsson and Tarski's concept of conjugate operators on Boolean algebras appropriately characterises the notion of cognacy in a dependence graph, and give a procedure for computing related inputs over such a graph.
In many application settings, the data have missing entries which make analysis challenging. An abundant literature addresses missing values in an inferential framework: estimating parameters and their variance from incomplete tables. Here, we consider supervised-learning settings: predicting a target when missing values appear in both training and testing data. We show the consistency of two approaches in prediction. A striking result is that the widely-used method of imputing with a constant, such as the mean prior to learning is consistent when missing values are not informative. This contrasts with inferential settings where mean imputation is pointed at for distorting the distribution of the data. That such a simple approach can be consistent is important in practice. We also show that a predictor suited for complete observations can predict optimally on incomplete data,through multiple imputation.Finally, to compare imputation with learning directly with a model that accounts for missing values, we analyze further decision trees. These can naturally tackle empirical risk minimization with missing values, due to their ability to handle the half-discrete nature of incomplete variables. After comparing theoretically and empirically different missing values strategies in trees, we recommend using the "missing incorporated in attribute" method as it can handle both non-informative and informative missing values.
Robust Markov Decision Processes (RMDPs) are a widely used framework for sequential decision-making under parameter uncertainty. RMDPs have been extensively studied when the objective is to maximize the discounted return, but little is known for average optimality (optimizing the long-run average of the rewards obtained over time) and Blackwell optimality (remaining discount optimal for all discount factors sufficiently close to 1). In this paper, we prove several foundational results for RMDPs beyond the discounted return. We show that average optimal policies can be chosen stationary and deterministic for sa-rectangular RMDPs but, perhaps surprisingly, that history-dependent (Markovian) policies strictly outperform stationary policies for average optimality in s-rectangular RMDPs. We also study Blackwell optimality for sa-rectangular RMDPs, where we show that {\em approximate} Blackwell optimal policies always exist, although Blackwell optimal policies may not exist. We also provide a sufficient condition for their existence, which encompasses virtually any examples from the literature. We then discuss the connection between average and Blackwell optimality, and we describe several algorithms to compute the optimal average return. Interestingly, our approach leverages the connections between RMDPs and stochastic games.
One of the main challenges for interpreting black-box models is the ability to uniquely decompose square-integrable functions of non-independent random inputs into a sum of functions of every possible subset of variables. However, dealing with dependencies among inputs can be complicated. We propose a novel framework to study this problem, linking three domains of mathematics: probability theory, functional analysis, and combinatorics. We show that, under two reasonable assumptions on the inputs (non-perfect functional dependence and non-degenerate stochastic dependence), it is always possible to decompose such a function uniquely. This generalizes the well-known Hoeffding decomposition. The elements of this decomposition can be expressed using oblique projections and allow for novel interpretability indices for evaluation and variance decomposition purposes. The properties of these novel indices are studied and discussed. This generalization offers a path towards a more precise uncertainty quantification, which can benefit sensitivity analysis and interpretability studies whenever the inputs are dependent. This decomposition is illustrated analytically, and the challenges for adopting these results in practice are discussed.
Several mixed-effects models for longitudinal data have been proposed to accommodate the non-linearity of late-life cognitive trajectories and assess the putative influence of covariates on it. No prior research provides a side-by-side examination of these models to offer guidance on their proper application and interpretation. In this work, we examined five statistical approaches previously used to answer research questions related to non-linear changes in cognitive aging: the linear mixed model (LMM) with a quadratic term, LMM with splines, the functional mixed model, the piecewise linear mixed model, and the sigmoidal mixed model. We first theoretically describe the models. Next, using data from two prospective cohorts with annual cognitive testing, we compared the interpretation of the models by investigating associations of education on cognitive change before death. Lastly, we performed a simulation study to empirically evaluate the models and provide practical recommendations. Except for the LMM-quadratic, the fit of all models was generally adequate to capture non-linearity of cognitive change and models were relatively robust. Although spline-based models have no interpretable nonlinearity parameters, their convergence was easier to achieve, and they allow graphical interpretation. In contrast, piecewise and sigmoidal models, with interpretable non-linear parameters, may require more data to achieve convergence.
With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting.
Multiphysics simulations frequently require transferring solution fields between subproblems with non-matching spatial discretizations, typically using interpolation techniques. Standard methods are usually based on measuring the closeness between points by means of the Euclidean distance, which does not account for curvature, cuts, cavities or other non-trivial geometrical or topological features of the domain. This may lead to spurious oscillations in the interpolant in proximity to these features. To overcome this issue, we propose a modification to rescaled localized radial basis function (RL-RBF) interpolation to account for the geometry of the interpolation domain, by yielding conformity and fidelity to geometrical and topological features. The proposed method, referred to as RL-RBF-G, relies on measuring the geodesic distance between data points. RL-RBF-G removes spurious oscillations appearing in the RL-RBF interpolant, resulting in increased accuracy in domains with complex geometries. We demonstrate the effectiveness of RL-RBF-G interpolation through a convergence study in an idealized setting. Furthermore, we discuss the algorithmic aspects and the implementation of RL-RBF-G interpolation in a distributed-memory parallel framework, and present the results of a strong scalability test yielding nearly ideal results. Finally, we show the effectiveness of RL-RBF-G interpolation in multiphysics simulations by considering an application to a whole-heart cardiac electromecanics model.
A component-splitting method is proposed to improve convergence characteristics for implicit time integration of compressible multicomponent reactive flows. The characteristic decomposition of flux jacobian of multicomponent Navier-Stokes equations yields a large sparse eigensystem, presenting challenges of slow convergence and high computational costs for implicit methods. To addresses this issue, the component-splitting method segregates the implicit operator into two parts: one for the flow equations (density/momentum/energy) and the other for the component equations. Each part's implicit operator employs flux-vector splitting based on their respective spectral radii to achieve accelerated convergence. This approach improves the computational efficiency of implicit iteration, mitigating the quadratic increase in time cost with the number of species. Two consistence corrections are developed to reduce the introduced component-splitting error and ensure the numerical consistency of mass fraction. Importantly, the impact of component-splitting method on accuracy is minimal as the residual approaches convergence. The accuracy, efficiency, and robustness of component-splitting method are thoroughly investigated and compared with the coupled implicit scheme through several numerical cases involving thermo-chemical nonequilibrium hypersonic flows. The results demonstrate that the component-splitting method decreases the required number of iteration steps for convergence of residual and wall heat flux, decreases the computation time per iteration step, and diminishes the residual to lower magnitude. The acceleration efficiency is enhanced with increases in CFL number and number of species.
Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.
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