Lambert's problem concerns with transferring a spacecraft from a given initial to a given terminal position within prescribed flight time via velocity control subject to a gravitational force field. We consider a probabilistic variant of the Lambert problem where the knowledge of the endpoint constraints in position vectors are replaced by the knowledge of their respective joint probability density functions. We show that the Lambert problem with endpoint joint probability density constraints is a generalized optimal mass transport (OMT) problem, thereby connecting this classical astrodynamics problem with a burgeoning area of research in modern stochastic control and stochastic machine learning. This newfound connection allows us to rigorously establish the existence and uniqueness of solution for the probabilistic Lambert problem. The same connection also helps to numerically solve the probabilistic Lambert problem via diffusion regularization, i.e., by leveraging further connection of the OMT with the Schr\"odinger bridge problem (SBP). This also shows that the probabilistic Lambert problem with additive dynamic process noise is in fact a generalized SBP, and can be solved numerically using the so-called Schr\"odinger factors, as we do in this work. We explain how the resulting analysis leads to solving a boundary-coupled system of reaction-diffusion PDEs where the nonlinear gravitational potential appears as the reaction rate. We propose novel algorithms for the same, and present illustrative numerical results. Our analysis and the algorithmic framework are nonparametric, i.e., we make neither statistical (e.g., Gaussian, first few moments, mixture or exponential family, finite dimensionality of the sufficient statistic) nor dynamical (e.g., Taylor series) approximations.
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Quality Assurance (QA) aims to prevent mistakes and defects in manufactured products and avoid problems when delivering products or services to customers. QA for AI systems, however, poses particular challenges, given their data-driven and non-deterministic nature as well as more complex architectures and algorithms. While there is growing empirical evidence about practices of machine learning in industrial contexts, little is known about the challenges and best practices of quality assurance for AI systems (QA4AI). In this paper, we report on a mixed-method study of QA4AI in industry practice from various countries and companies. Through interviews with fifteen industry practitioners and a validation survey with 50 practitioner responses, we studied the concerns as well as challenges and best practices in ensuring the QA4AI properties reported in the literature, such as correctness, fairness, interpretability and others. Our findings suggest correctness as the most important property, followed by model relevance, efficiency and deployability. In contrast, transferability (applying knowledge learned in one task to another task), security and fairness are not paid much attention by practitioners compared to other properties. Challenges and solutions are identified for each QA4AI property. For example, interviewees highlighted the trade-off challenge among latency, cost and accuracy for efficiency (latency and cost are parts of efficiency concern). Solutions like model compression are proposed. We identified 21 QA4AI practices across each stage of AI development, with 10 practices being well recognized and another 8 practices being marginally agreed by the survey practitioners.
Recent studies have uncovered intriguing phenomena in deep learning, such as grokking, double descent, and emergent abilities in large language models, which challenge human intuition and are crucial for a deeper understanding of neural models. In this paper, we present a comprehensive framework that provides a unified view of these three phenomena, focusing on the competition between memorization and generalization circuits. This approach, initially employed to explain grokking, is extended in our work to encompass a wider range of model sizes and training data volumes. Our framework delineates four distinct training dynamics, each depending on varying combinations of model size and training data quantity. Utilizing this framework, we provide a detailed analysis of the double descent phenomenon and propose two verifiable predictions regarding its occurrence, both substantiated by our experimental results. Moreover, we expand our framework to the multi-task learning paradigm, demonstrating how algorithm tasks can be turned into emergent abilities. This offers a novel perspective to understand emergent abilities in Large Language Models.
We consider gradient descent (GD) with a constant stepsize applied to logistic regression with linearly separable data, where the constant stepsize $\eta$ is so large that the loss initially oscillates. We show that GD exits this initial oscillatory phase rapidly -- in $\mathcal{O}(\eta)$ steps -- and subsequently achieves an $\tilde{\mathcal{O}}(1 / (\eta t) )$ convergence rate after $t$ additional steps. Our results imply that, given a budget of $T$ steps, GD can achieve an accelerated loss of $\tilde{\mathcal{O}}(1/T^2)$ with an aggressive stepsize $\eta:= \Theta( T)$, without any use of momentum or variable stepsize schedulers. Our proof technique is versatile and also handles general classification loss functions (where exponential tails are needed for the $\tilde{\mathcal{O}}(1/T^2)$ acceleration), nonlinear predictors in the neural tangent kernel regime, and online stochastic gradient descent (SGD) with a large stepsize, under suitable separability conditions.
Given the task of positioning a ball-like object to a goal region beyond direct reach, humans can often throw, slide, or rebound objects against the wall to attain the goal. However, enabling robots to reason similarly is non-trivial. Existing methods for physical reasoning are data-hungry and struggle with complexity and uncertainty inherent in the real world. This paper presents PhyPlan, a novel physics-informed planning framework that combines physics-informed neural networks (PINNs) with modified Monte Carlo Tree Search (MCTS) to enable embodied agents to perform dynamic physical tasks. PhyPlan leverages PINNs to simulate and predict outcomes of actions in a fast and accurate manner and uses MCTS for planning. It dynamically determines whether to consult a PINN-based simulator (coarse but fast) or engage directly with the actual environment (fine but slow) to determine optimal policy. Evaluation with robots in simulated 3D environments demonstrates the ability of our approach to solve 3D-physical reasoning tasks involving the composition of dynamic skills. Quantitatively, PhyPlan excels in several aspects: (i) it achieves lower regret when learning novel tasks compared to state-of-the-art, (ii) it expedites skill learning and enhances the speed of physical reasoning, (iii) it demonstrates higher data efficiency compared to a physics un-informed approach.
Conceptual spaces represent entities in terms of their primitive semantic features. Such representations are highly valuable but they are notoriously difficult to learn, especially when it comes to modelling perceptual and subjective features. Distilling conceptual spaces from Large Language Models (LLMs) has recently emerged as a promising strategy. However, existing work has been limited to probing pre-trained LLMs using relatively simple zero-shot strategies. We focus in particular on the task of ranking entities according to a given conceptual space dimension. Unfortunately, we cannot directly fine-tune LLMs on this task, because ground truth rankings for conceptual space dimensions are rare. We therefore use more readily available features as training data and analyse whether the ranking capabilities of the resulting models transfer to perceptual and subjective features. We find that this is indeed the case, to some extent, but having perceptual and subjective features in the training data seems essential for achieving the best results. We furthermore find that pointwise ranking strategies are competitive against pairwise approaches, in defiance of common wisdom.
Learning from noisy data has attracted much attention, where most methods focus on closed-set label noise. However, a more common scenario in the real world is the presence of both open-set and closed-set noise. Existing methods typically identify and handle these two types of label noise separately by designing a specific strategy for each type. However, in many real-world scenarios, it would be challenging to identify open-set examples, especially when the dataset has been severely corrupted. Unlike the previous works, we explore how models behave when faced with open-set examples, and find that \emph{a part of open-set examples gradually get integrated into certain known classes}, which is beneficial for the separation among known classes. Motivated by the phenomenon, we propose a novel two-step contrastive learning method CECL (Class Expansion Contrastive Learning) which aims to deal with both types of label noise by exploiting the useful information of open-set examples. Specifically, we incorporate some open-set examples into closed-set classes to enhance performance while treating others as delimiters to improve representative ability. Extensive experiments on synthetic and real-world datasets with diverse label noise demonstrate the effectiveness of CECL.
Multi-modal knowledge graph completion (MMKGC) aims to predict the missing triples in the multi-modal knowledge graphs by incorporating structural, visual, and textual information of entities into the discriminant models. The information from different modalities will work together to measure the triple plausibility. Existing MMKGC methods overlook the imbalance problem of modality information among entities, resulting in inadequate modal fusion and inefficient utilization of the raw modality information. To address the mentioned problems, we propose Adaptive Multi-modal Fusion and Modality Adversarial Training (AdaMF-MAT) to unleash the power of imbalanced modality information for MMKGC. AdaMF-MAT achieves multi-modal fusion with adaptive modality weights and further generates adversarial samples by modality-adversarial training to enhance the imbalanced modality information. Our approach is a co-design of the MMKGC model and training strategy which can outperform 19 recent MMKGC methods and achieve new state-of-the-art results on three public MMKGC benchmarks. Our code and data have been released at //github.com/zjukg/AdaMF-MAT.
Knowledge graph reasoning (KGR), aiming to deduce new facts from existing facts based on mined logic rules underlying knowledge graphs (KGs), has become a fast-growing research direction. It has been proven to significantly benefit the usage of KGs in many AI applications, such as question answering and recommendation systems, etc. According to the graph types, the existing KGR models can be roughly divided into three categories, \textit{i.e.,} static models, temporal models, and multi-modal models. The early works in this domain mainly focus on static KGR and tend to directly apply general knowledge graph embedding models to the reasoning task. However, these models are not suitable for more complex but practical tasks, such as inductive static KGR, temporal KGR, and multi-modal KGR. To this end, multiple works have been developed recently, but no survey papers and open-source repositories comprehensively summarize and discuss models in this important direction. To fill the gap, we conduct a survey for knowledge graph reasoning tracing from static to temporal and then to multi-modal KGs. Concretely, the preliminaries, summaries of KGR models, and typical datasets are introduced and discussed consequently. Moreover, we discuss the challenges and potential opportunities. The corresponding open-source repository is shared on GitHub: //github.com/LIANGKE23/Awesome-Knowledge-Graph-Reasoning.
Modeling multivariate time series has long been a subject that has attracted researchers from a diverse range of fields including economics, finance, and traffic. A basic assumption behind multivariate time series forecasting is that its variables depend on one another but, upon looking closely, it is fair to say that existing methods fail to fully exploit latent spatial dependencies between pairs of variables. In recent years, meanwhile, graph neural networks (GNNs) have shown high capability in handling relational dependencies. GNNs require well-defined graph structures for information propagation which means they cannot be applied directly for multivariate time series where the dependencies are not known in advance. In this paper, we propose a general graph neural network framework designed specifically for multivariate time series data. Our approach automatically extracts the uni-directed relations among variables through a graph learning module, into which external knowledge like variable attributes can be easily integrated. A novel mix-hop propagation layer and a dilated inception layer are further proposed to capture the spatial and temporal dependencies within the time series. The graph learning, graph convolution, and temporal convolution modules are jointly learned in an end-to-end framework. Experimental results show that our proposed model outperforms the state-of-the-art baseline methods on 3 of 4 benchmark datasets and achieves on-par performance with other approaches on two traffic datasets which provide extra structural information.
Recent years have witnessed the enormous success of low-dimensional vector space representations of knowledge graphs to predict missing facts or find erroneous ones. Currently, however, it is not yet well-understood how ontological knowledge, e.g. given as a set of (existential) rules, can be embedded in a principled way. To address this shortcoming, in this paper we introduce a framework based on convex regions, which can faithfully incorporate ontological knowledge into the vector space embedding. Our technical contribution is two-fold. First, we show that some of the most popular existing embedding approaches are not capable of modelling even very simple types of rules. Second, we show that our framework can represent ontologies that are expressed using so-called quasi-chained existential rules in an exact way, such that any set of facts which is induced using that vector space embedding is logically consistent and deductively closed with respect to the input ontology.
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