We present new deterministic algorithms for computing distributed weighted minimum weight cycle (MWC) in undirected and directed graphs and distributed weighted all nodes shortest cycle (ANSC) in directed graphs. Our algorithms for these problems run in $\tilde{O}(n)$ rounds in the CONGEST model on graphs with arbitrary non-negative edge weights, matching the lower bound up to polylogarithmic factors. Before our work, no near linear rounds deterministic algorithms were known for these problems. The previous best bound for solving these problems deterministically requires an initial computation of all pairs shortest paths (APSP) on the given graph, followed by post-processing of $O(n)$ rounds, and in total takes $\tilde{O}(n^{4/3})$ rounds, using deterministic APSP~\cite{AR-SPAA20}. The main component of our new $\tilde{O}(n)$ rounds algorithms is a deterministic technique for constructing a sequence of successive blocker sets. These blocker sets are then treated as source nodes to compute $h$-hop shortest paths, which can then be used to compute candidate shortest cycles whose hop length lies in a particular range. The shortest cycles can then be obtained by selecting the cycle with the minimum weight from all these candidate cycles. Additionally using the above blocker set sequence technique, we also obtain $\tilde{O}(n)$ rounds deterministic algorithm for the multi-source shortest paths problem (MSSP) for both directed and undirected graphs, given that the size of the source set is at most $\sqrt{n}$. This new result for MSSP can be a step towards obtaining a $o(n^{4/3})$ rounds algorithm for deterministic APSP. We also believe that our new blocker set sequence technique may have potential applications for other distributed algorithms.
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In recent years, strong expectations have been raised for the possible power of quantum computing for solving difficult optimization problems, based on theoretical, asymptotic worst-case bounds. Can we expect this to have consequences for Linear and Integer Programming when solving instances of practically relevant size, a fundamental goal of Mathematical Programming, Operations Research and Algorithm Engineering? Answering this question faces a crucial impediment: The lack of sufficiently large quantum platforms prevents performing real-world tests for comparison with classical methods. In this paper, we present a quantum analog for classical runtime analysis when solving real-world instances of important optimization problems. To this end, we measure the expected practical performance of quantum computers by analyzing the expected gate complexity of a quantum algorithm. The lack of practical quantum platforms for experimental comparison is addressed by hybrid benchmarking, in which the algorithm is performed on a classical system, logging the expected cost of the various subroutines that are employed by the quantum versions. In particular, we provide an analysis of quantum methods for Linear Programming, for which recent work has provided asymptotic speedup through quantum subroutines for the Simplex method. We show that a practical quantum advantage for realistic problem sizes would require quantum gate operation times that are considerably below current physical limitations.
We propose an efficient semi-Lagrangian characteristic mapping method for solving the one+one-dimensional Vlasov-Poisson equations with high precision on a coarse grid. The flow map is evolved numerically and exponential resolution in linear time is obtained. Global third-order convergence in space and time is shown and conservation properties are assessed. For benchmarking, we consider linear and nonlinear Landau damping and the two-stream instability. We compare the results with a Fourier pseudo-spectral method. The extreme fine-scale resolution features are illustrated showing the method's capabilities to efficiently treat filamentation in fusion plasma simulations.
Structural Priming Demonstrates Abstract Grammatical Representations in Multilingual Language Models
Abstract grammatical knowledge - of parts of speech and grammatical patterns - is key to the capacity for linguistic generalization in humans. But how abstract is grammatical knowledge in large language models? In the human literature, compelling evidence for grammatical abstraction comes from structural priming. A sentence that shares the same grammatical structure as a preceding sentence is processed and produced more readily. Because confounds exist when using stimuli in a single language, evidence of abstraction is even more compelling from crosslingual structural priming, where use of a syntactic structure in one language primes an analogous structure in another language. We measure crosslingual structural priming in large language models, comparing model behavior to human experimental results from eight crosslingual experiments covering six languages, and four monolingual structural priming experiments in three non-English languages. We find evidence for abstract monolingual and crosslingual grammatical representations in the models that function similarly to those found in humans. These results demonstrate that grammatical representations in multilingual language models are not only similar across languages, but they can causally influence text produced in different languages.
We give a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/nonlinear constraints. Instead of taking a full stepsize, DPALM adopts a damped dual stepsize to ensure the boundedness of dual iterates. We show that DPALM can produce a (near) $\vareps$-KKT point within $O(\vareps^{-2})$ outer iterations if each DPALM subproblem is solved to a proper accuracy. In addition, we establish overall iteration complexity of DPALM when the objective is either a regularized smooth function or in a regularized compositional form. For the former case, DPALM achieves the complexity of $\widetilde{\mathcal{O}}\left(\varepsilon^{-2.5} \right)$ to produce an $\varepsilon$-KKT point by applying an accelerated proximal gradient (APG) method to each DPALM subproblem. For the latter case, the complexity of DPALM is $\widetilde{\mathcal{O}}\left(\varepsilon^{-3} \right)$ to produce a near $\varepsilon$-KKT point by using an APG to solve a Moreau-envelope smoothed version of each subproblem. Our outer iteration complexity and the overall complexity either generalize existing best ones from unconstrained or linear-constrained problems to convex-constrained ones, or improve over the best-known results on solving the same-structured problems. Furthermore, numerical experiments on linearly/quadratically constrained non-convex quadratic programs and linear-constrained robust nonlinear least squares are conducted to demonstrate the empirical efficiency of the proposed DPALM over several state-of-the art methods.
Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in a high condition number of the underlying solution matrix, posing a major challenge for iterative linear solvers. This paper introduces a two-level domain decomposition preconditioner that addresses this issue for the linear Schr\"odinger eigenvalue problem, even in the presence of a vanishing eigenvalue gap in non-uniform, expanding domains. Since the quasi-optimal shift, which is already available as the solution to a spectral cell problem, is required for the eigenvalue solver, it is logical to also use its associated eigenfunction as a generator to construct a coarse space. We analyze the resulting two-level additive Schwarz preconditioner and obtain a condition number bound that is independent of the domain's anisotropy, despite the need for only one basis function per subdomain for the coarse solver. Several numerical examples are presented to illustrate its flexibility and efficiency.
Probabilistic couplings are the foundation for many probabilistic relational program logics and arise when relating random sampling statements across two programs. In relational program logics, this manifests as dedicated coupling rules that, e.g., say we may reason as if two sampling statements return the same value. However, this approach fundamentally requires aligning or "synchronizing" the sampling statements of the two programs which is not always possible. In this paper, we develop Clutch, a higher-order probabilistic relational separation logic that addresses this issue by supporting asynchronous probabilistic couplings. We use Clutch to develop a logical step-indexed logical relational to reason about contextual refinement and equivalence of higher-order programs written in a rich language with higher-order local state and impredicative polymorphism. Finally, we demonstrate the usefulness of our approach on a number of case studies. All the results that appear in the paper have been formalized in the Coq proof assistant using the Coquelicot library and the Iris separation logic framework.
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.
Deep reinforcement learning algorithms can perform poorly in real-world tasks due to the discrepancy between source and target environments. This discrepancy is commonly viewed as the disturbance in transition dynamics. Many existing algorithms learn robust policies by modeling the disturbance and applying it to source environments during training, which usually requires prior knowledge about the disturbance and control of simulators. However, these algorithms can fail in scenarios where the disturbance from target environments is unknown or is intractable to model in simulators. To tackle this problem, we propose a novel model-free actor-critic algorithm -- namely, state-conservative policy optimization (SCPO) -- to learn robust policies without modeling the disturbance in advance. Specifically, SCPO reduces the disturbance in transition dynamics to that in state space and then approximates it by a simple gradient-based regularizer. The appealing features of SCPO include that it is simple to implement and does not require additional knowledge about the disturbance or specially designed simulators. Experiments in several robot control tasks demonstrate that SCPO learns robust policies against the disturbance in transition dynamics.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.
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