We study the problem of combining neural networks with symbolic reasoning. Recently introduced frameworks for Probabilistic Neurosymbolic Learning (PNL), such as DeepProbLog, perform exponential-time exact inference, limiting the scalability of PNL solutions. We introduce Approximate Neurosymbolic Inference (A-NeSI): a new framework for PNL that uses neural networks for scalable approximate inference. A-NeSI 1) performs approximate inference in polynomial time without changing the semantics of probabilistic logics; 2) is trained using data generated by the background knowledge; 3) can generate symbolic explanations of predictions; and 4) can guarantee the satisfaction of logical constraints at test time, which is vital in safety-critical applications. Our experiments show that A-NeSI is the first end-to-end method to solve three neurosymbolic tasks with exponential combinatorial scaling. Finally, our experiments show that A-NeSI achieves explainability and safety without a penalty in performance.
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Given a limited labeling budget, active learning (AL) aims to sample the most informative instances from an unlabeled pool to acquire labels for subsequent model training. To achieve this, AL typically measures the informativeness of unlabeled instances based on uncertainty and diversity. However, it does not consider erroneous instances with their neighborhood error density, which have great potential to improve the model performance. To address this limitation, we propose $REAL$, a novel approach to select data instances with $\underline{R}$epresentative $\underline{E}$rrors for $\underline{A}$ctive $\underline{L}$earning. It identifies minority predictions as \emph{pseudo errors} within a cluster and allocates an adaptive sampling budget for the cluster based on estimated error density. Extensive experiments on five text classification datasets demonstrate that $REAL$ consistently outperforms all best-performing baselines regarding accuracy and F1-macro scores across a wide range of hyperparameter settings. Our analysis also shows that $REAL$ selects the most representative pseudo errors that match the distribution of ground-truth errors along the decision boundary. Our code is publicly available at //github.com/withchencheng/ECML_PKDD_23_Real.
In this paper, we study the partial pole assignment problem in symmetric quadratic pencil with time delay. A novel multi-step method is proposed to solve this problem, resulting in the undesired eigenvalues being moved to desired values, and the remaining eigenvalues unchanged. By establishing a new matrix equality relation and using a multi-step method, the problem is transformed into solving linear systems with low order. Specifically, assuming that there are $p$ undesired eigenvalues requiring reassigned, the size of the linear system we finally solved is $p^2$. Notably, the method demonstrates high efficiency for large systems with only a few poles requiring reassigned. Numerical examples are provided to illustrate the effectiveness of the proposed method
A series of experiments in stationary and moving passenger rail cars were conducted to measure removal rates of particles in the size ranges of SARS-CoV-2 viral aerosols, and the air changes per hour provided by existing and modified air handling systems. Such methods for exposure assessments are customarily based on mechanistic models derived from physical laws of particle movement that are deterministic and do not account for measurement errors inherent in data collection. The resulting analysis compromises on reliably learning about mechanistic factors such as ventilation rates, aerosol generation rates and filtration efficiencies from field measurements. This manuscript develops a Bayesian state space modeling framework that synthesizes information from the mechanistic system as well as the field data. We derive a stochastic model from finite difference approximations of differential equations explaining particle concentrations. Our inferential framework trains the mechanistic system using the field measurements from the chamber experiments and delivers reliable estimates of the underlying physical process with fully model-based uncertainty quantification. Our application falls within the realm of Bayesian ``melding'' of mechanistic and statistical models and is of significant relevance to environmental hygienists and public health researchers working on assessing performance of aerosol removal rates for rail car fleets.
The solution set of a system of polynomial equations typically contains ill-behaved, singular points. Resolution is a fundamental process in geometry in which we replace singular points with smooth points, while keeping the rest of the solution set unchanged. Resolutions are not unique: the usual way to describe them involves repeatedly performing a fundamental operation known as "blowing-up", and the complexity of the resolution highly depends on certain choices. The process can be translated into various versions of a 2-player game, the so-called Hironaka game, and a winning strategy for the first player provides a solution to the resolution problem. In this paper we introduce a new approach to the Hironaka game that uses reinforcement learning agents to find optimal resolutions of singularities. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.
This paper focuses on developing a reduction-based algebraic multigrid method that is suitable for solving general (non)symmetric linear systems and is naturally robust from pure advection to pure diffusion. Initial motivation comes from a new reduction-based algebraic multigrid (AMG) approach, $\ell$AIR (local approximate ideal restriction), that was developed for solving advection-dominated problems. Though this new solver is very effective in the advection dominated regime, its performance degrades in cases where diffusion becomes dominant. This is consistent with the fact that in general, reduction-based AMG methods tend to suffer from growth in complexity and/or convergence rates as the problem size is increased, especially for diffusion dominated problems in two or three dimensions. Motivated by the success of $\ell$AIR in the advective regime, our aim in this paper is to generalize the AIR framework with the goal of improving the performance of the solver in diffusion dominated regimes. To do so, we propose a novel way to combine mode constraints as used commonly in energy minimization AMG methods with the local approximation of ideal operators used in $\ell$AIR. The resulting constrained $\ell$AIR (C$\ell$AIR) algorithm is able to achieve fast scalable convergence on advective and diffusive problems. In addition, it is able to achieve standard low complexity hierarchies in the diffusive regime through aggressive coarsening, something that has been previously difficult for reduction-based methods.
We address the computational efficiency in solving the A-optimal Bayesian design of experiments problems for which the observational model is based on partial differential equations and, consequently, is computationally expensive to evaluate. A-optimality is a widely used and easy-to-interpret criterion for the Bayesian design of experiments. The criterion seeks the optimal experiment design by minimizing the expected conditional variance, also known as the expected posterior variance. This work presents a novel likelihood-free method for seeking the A-optimal design of experiments without sampling or integrating the Bayesian posterior distribution. In our approach, the expected conditional variance is obtained via the variance of the conditional expectation using the law of total variance, while we take advantage of the orthogonal projection property to approximate the conditional expectation. Through an asymptotic error estimation, we show that the intractability of the posterior does not affect the performance of our approach. We use an artificial neural network (ANN) to approximate the nonlinear conditional expectation to implement our method. For dealing with continuous experimental design parameters, we integrate the training process of the ANN into minimizing the expected conditional variance. Specifically, we propose a non-local approximation of the conditional expectation and apply transfer learning to reduce the number of evaluations of the observation model. Through numerical experiments, we demonstrate that our method significantly reduces the number of observational model evaluations compared with common importance sampling-based approaches. This reduction is crucial, considering the computationally expensive nature of these models.
Univariate and multivariate normal probability distributions are widely used when modeling decisions under uncertainty. Computing the performance of such models requires integrating these distributions over specific domains, which can vary widely across models. Besides some special cases, there exist no general analytical expressions, standard numerical methods or software for these integrals. Here we present mathematical results and open-source software that provide (i) the probability in any domain of a normal in any dimensions with any parameters, (ii) the probability density, cumulative distribution, and inverse cumulative distribution of any function of a normal vector, (iii) the classification errors among any number of normal distributions, the Bayes-optimal discriminability index and relation to the operating characteristic, (iv) dimension reduction and visualizations for such problems, and (v) tests for how reliably these methods may be used on given data. We demonstrate these tools with vision research applications of detecting occluding objects in natural scenes, and detecting camouflage.
Parallel software codes in high performance computing (HPC) continue to grow in complexity and scale as we enter the exascale era. A diverse set of emerging hardware and programming paradigms make developing, optimizing, and maintaining parallel software burdensome for developers. One way to alleviate some of these burdens is with automated development and analysis tools. Such tools can perform complex and/or remedial tasks for developers that increase their productivity and decrease the chance for error. So far, such tools for code development and performance analysis have been limited in the complexity of tasks they can perform. However, with recent advancements in language modeling, and the wealth of code related data that is now available online, these tools have started to utilize predictive language models to automate more complex tasks. In this paper, we show how large language models (LLMs) can be applied to tasks specific to high performance and scientific codes. We train LLMs using code and performance data that is specific to parallel codes. We compare several recent LLMs on HPC related tasks and introduce a new model, HPC-Coder, trained on parallel code. In our experiments we show that this model can auto-complete HPC functions where general models cannot, decorate for loops with OpenMP pragmas, and model performance changes in two scientific application repositories.
Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference -- other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient, nonlinear additive noise models, which we model using Gaussian processes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, we demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.
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