Gaussian processes (GPs) are commonly used for geospatial analysis, but they suffer from high computational complexity when dealing with massive data. For instance, the log-likelihood function required in estimating the statistical model parameters for geospatial data is a computationally intensive procedure that involves computing the inverse of a covariance matrix with size n X n, where n represents the number of geographical locations. As a result, in the literature, studies have shifted towards approximation methods to handle larger values of n effectively while maintaining high accuracy. These methods encompass a range of techniques, including low-rank and sparse approximations. Vecchia approximation is one of the most promising methods to speed up evaluating the log-likelihood function. This study presents a parallel implementation of the Vecchia approximation, utilizing batched matrix computations on contemporary GPUs. The proposed implementation relies on batched linear algebra routines to efficiently execute individual conditional distributions in the Vecchia algorithm. We rely on the KBLAS linear algebra library to perform batched linear algebra operations, reducing the time to solution compared to the state-of-the-art parallel implementation of the likelihood estimation operation in the ExaGeoStat software by up to 700X, 833X, 1380X on 32GB GV100, 80GB A100, and 80GB H100 GPUs, respectively. We also successfully manage larger problem sizes on a single NVIDIA GPU, accommodating up to 1M locations with 80GB A100 and H100 GPUs while maintaining the necessary application accuracy. We further assess the accuracy performance of the implemented algorithm, identifying the optimal settings for the Vecchia approximation algorithm to preserve accuracy on two real geospatial datasets: soil moisture data in the Mississippi Basin area and wind speed data in the Middle East.
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Optimizing static risk-averse objectives in Markov decision processes is difficult because they do not admit standard dynamic programming equations common in Reinforcement Learning (RL) algorithms. Dynamic programming decompositions that augment the state space with discrete risk levels have recently gained popularity in the RL community. Prior work has shown that these decompositions are optimal when the risk level is discretized sufficiently. However, we show that these popular decompositions for Conditional-Value-at-Risk (CVaR) and Entropic-Value-at-Risk (EVaR) are inherently suboptimal regardless of the discretization level. In particular, we show that a saddle point property assumed to hold in prior literature may be violated. However, a decomposition does hold for Value-at-Risk and our proof demonstrates how this risk measure differs from CVaR and EVaR. Our findings are significant because risk-averse algorithms are used in high-stake environments, making their correctness much more critical.
We develop the notion of a locally homomorphic channel and prove an approximate equivalence between those and codes for computing functions. Further, we derive decomposition properties of locally homomorphic channels which we use to analyze and construct codes where two messages must be encoded independently. This leads to new results for identification and K-identification when all messages are sent over multiple-access channels, which yield surprising rate improvements compared to naive code constructions. In particular, we demonstrate that for the example of identification with deterministic encoders, both encoders can be constructed independently.
Decision-making processes often involve dealing with uncertainty, which is traditionally addressed through probabilistic models. However, in practical scenarios, assessing probabilities reliably can be challenging, compounded by diverse perceptions of probabilistic information among decision makers. To address this variability and accommodate diverse preferences regarding uncertainty, we introduce the Probabilistic Abstract Decision Framework (PADF). PADF offers a structured approach for reasoning across different decision criteria, encompassing the optimistic, pessimistic, and Laplace perspectives, each tailored to distinct perceptions of uncertainty. We illustrate how PADF facilitates the computation of optimal decisions aligned with these criteria by leveraging probabilistic rules. Furthermore, we present strategies for optimizing the computational efficiency of these rules, leveraging appropriate independence assumptions to navigate the extensive search space inherent in PADF. Through these contributions, our framework provides a robust and adaptable tool for effectively navigating the complexities of decision-making under uncertainty.
Many stochastic processes in the physical and biological sciences can be modelled as Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.
We propose and study a realistic Continual Learning (CL) setting where learning algorithms are granted a restricted computational budget per time step while training. We apply this setting to large-scale semi-supervised Continual Learning scenarios with sparse label rates. Previous proficient CL methods perform very poorly in this challenging setting. Overfitting to the sparse labeled data and insufficient computational budget are the two main culprits for such a poor performance. Our new setting encourages learning methods to effectively and efficiently utilize the unlabeled data during training. To that end, we propose a simple but highly effective baseline, DietCL, which utilizes both unlabeled and labeled data jointly. DietCL meticulously allocates computational budget for both types of data. We validate our baseline, at scale, on several datasets, e.g., CLOC, ImageNet10K, and CGLM, under constraint budget setups. DietCL outperforms, by a large margin, all existing supervised CL algorithms as well as more recent continual semi-supervised methods. Our extensive analysis and ablations demonstrate that DietCL is stable under a full spectrum of label sparsity, computational budget, and various other ablations.
Control barrier functions (CBFs) have recently been introduced as a systematic tool to ensure safety by establishing set invariance. When combined with a control Lyapunov function (CLF), they form a safety-critical control mechanism. However, the effectiveness of CBFs and CLFs is closely tied to the system model. In practice, model uncertainty can jeopardize safety and stability guarantees and may lead to undesirable performance. In this paper, we develop a safe learning-based control strategy for switching systems in the face of uncertainty. We focus on the case that a nominal model is available for a true underlying switching system. This uncertainty results in piecewise residuals for each switching surface, impacting the CLF and CBF constraints. We introduce a batch multi-output Gaussian process (MOGP) framework to approximate these piecewise residuals, thereby mitigating the adverse effects of uncertainty. A particular structure of the covariance function enables us to convert the MOGP-based chance constraints CLF and CBF into second-order cone constraints, which leads to a convex optimization. We analyze the feasibility of the resulting optimization and provide the necessary and sufficient conditions for feasibility. The effectiveness of the proposed strategy is validated through a simulation of a switching adaptive cruise control system.
To comprehensively gauge the capacity of current models for complex reasoning, it is crucial to assess their step-by-step reasoning in a scalable manner. Established reference-based evaluation metrics rely on human-annotated reasoning chains as references to assess the model-derived chains. However, such "gold-standard" human-written reasoning chains may not be unique and their acquisition is often labor-intensive. Existing reference-free reasoning evaluation metrics, while eliminating the need for human-crafted reasoning chains as references, often require fine-tuning with human-derived chains before evaluation, complicating the process and questioning their adaptability to other datasets. To address these challenges, we harness GPT-4 to automatically evaluate reasoning chain quality, thereby removing the dependency on human-written reasoning chains for both model fine-tuning and evaluative purposes. Leveraging the Socratic method, we develop SocREval ({\bf Soc}ratic Method-Inspired {\bf R}easoning {\bf Eval}uation), a novel approach for prompt design in reference-free reasoning evaluation. Empirical results from four human annotated datasets reveal that SocREval significantly improves GPT-4's performance, surpassing existing reference-free and reference-based reasoning evaluation metrics. Beyond its demonstrated efficacy, SocREval, proves to be both cost-efficient and robust to prompt writing and example selection, as substantiated by our in-depth analysis.
Code-switching (CSW) is a common phenomenon among multilingual speakers where multiple languages are used in a single discourse or utterance. Mixed language utterances may still contain grammatical errors however, yet most existing Grammar Error Correction (GEC) systems have been trained on monolingual data and not developed with CSW in mind. In this work, we conduct the first exploration into the use of GEC systems on CSW text. Through this exploration, we propose a novel method of generating synthetic CSW GEC datasets by translating different spans of text within existing GEC corpora. We then investigate different methods of selecting these spans based on CSW ratio, switch-point factor and linguistic constraints, and identify how they affect the performance of GEC systems on CSW text. Our best model achieves an average increase of 1.57 $F_{0.5}$ across 3 CSW test sets (English-Chinese, English-Korean and English-Japanese) without affecting the model's performance on a monolingual dataset. We furthermore discovered that models trained on one CSW language generalise relatively well to other typologically similar CSW languages.
As artificial intelligence (AI) models continue to scale up, they are becoming more capable and integrated into various forms of decision-making systems. For models involved in moral decision-making, also known as artificial moral agents (AMA), interpretability provides a way to trust and understand the agent's internal reasoning mechanisms for effective use and error correction. In this paper, we provide an overview of this rapidly-evolving sub-field of AI interpretability, introduce the concept of the Minimum Level of Interpretability (MLI) and recommend an MLI for various types of agents, to aid their safe deployment in real-world settings.
The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.
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