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We propose a practical tool for evaluating and comparing the accuracy of FDMs for the Helmholtz equation. The tool based on Fourier analysis makes it easy to find wavenumber explicit order of convergence, and can be used for rigorous proof. It fills in the gap between the dispersion analysis and the actual error with source term.

Existing work on the alignment problem has focused mainly on (1) qualitative descriptions of the alignment problem; (2) attempting to align AI actions with human interests by focusing on value specification and learning; and/or (3) focusing on a single agent or on humanity as a monolith. Recent sociotechnical approaches highlight the need to understand complex misalignment among multiple human and AI agents. We address this gap by adapting a computational social science model of human contention to the alignment problem. Our model quantifies misalignment in large, diverse agent groups with potentially conflicting goals across various problem areas. Misalignment scores in our framework depend on the observed agent population, the domain in question, and conflict between agents' weighted preferences. Through simulations, we demonstrate how our model captures intuitive aspects of misalignment across different scenarios. We then apply our model to two case studies, including an autonomous vehicle setting, showcasing its practical utility. Our approach offers enhanced explanatory power for complex sociotechnical environments and could inform the design of more aligned AI systems in real-world applications.

Neural scaling laws are observed in a range of domains, to date with no clear understanding of why they occur. Recent theories suggest that loss power laws arise from Zipf's law, a power law observed in domains like natural language. One theory suggests that language scaling laws emerge when Zipf-distributed task quanta are learned in descending order of frequency. In this paper we examine power-law scaling in AlphaZero, a reinforcement learning algorithm, using a theory of language-model scaling. We find that game states in training and inference data scale with Zipf's law, which is known to arise from the tree structure of the environment, and examine the correlation between scaling-law and Zipf's-law exponents. In agreement with quanta scaling theory, we find that agents optimize state loss in descending order of frequency, even though this order scales inversely with modelling complexity. We also find that inverse scaling, the failure of models to improve with size, is correlated with unusual Zipf curves where end-game states are among the most frequent states. We show evidence that larger models shift their focus to these less-important states, sacrificing their understanding of important early-game states.

Population protocols are a model of distributed computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs to decide some property of their initial configuration. We investigate the behaviour of population protocols under adversarial faults that cause agents to silently crash and no longer interact with other agents. As a starting point, we consider the property ``the number of agents exceeds a given threshold $t$'', represented by the predicate $x \geq t$, and show that the standard protocol for $x \geq t$ is very fragile: one single crash in a computation with $x:=2t-1$ agents can already cause the protocol to answer incorrectly that $x \geq t$ does not hold. However, a slightly less known protocol is robust: for any number $t' \geq t$ of agents, at least $t' - t+1$ crashes must occur for the protocol to answer that the property does not hold. We formally define robustness for arbitrary population protocols, and investigate the question whether every predicate computable by population protocols has a robust protocol. Angluin et al. proved in 2007 that population protocols decide exactly the Presburger predicates, which can be represented as Boolean combinations of threshold predicates of the form $\sum_{i=1}^n a_i \cdot x_i \geq t$ for $a_1,...,a_n, t \in \mathbb{Z}$ and modulo prdicates of the form $\sum_{i=1}^n a_i \cdot x_i \bmod m \geq t $ for $a_1, \ldots, a_n, m, t \in \mathbb{N}$. We design robust protocols for all threshold and modulo predicates. We also show that, unfortunately, the techniques in the literature that construct a protocol for a Boolean combination of predicates given protocols for the conjuncts do not preserve robustness. So the question remains open.

Recent scholarly work has extensively examined the phenomenon of algorithmic collusion driven by AI-enabled pricing algorithms. However, online platforms commonly deploy recommender systems that influence how consumers discover and purchase products, thereby shaping the reward structures faced by pricing algorithms and ultimately affecting competition dynamics and equilibrium outcomes. To address this gap in the literature and elucidate the role of recommender systems, we propose a novel repeated game framework that integrates several key components. We first develop a structural search model to characterize consumers' decision-making processes in response to varying recommendation sets. This model incorporates both observable and unobservable heterogeneity in utility and search cost functions, and is estimated using real-world data. Building on the resulting consumer model, we formulate personalized recommendation algorithms designed to maximize either platform revenue or consumer utility. We further introduce pricing algorithms for sellers and integrate all these elements to facilitate comprehensive numerical experiments. Our experimental findings reveal that a revenue-maximizing recommender system intensifies algorithmic collusion, whereas a utility-maximizing recommender system encourages more competitive pricing behavior among sellers. Intriguingly, and contrary to conventional insights from the industrial organization and choice modeling literature, increasing the size of recommendation sets under a utility-maximizing regime does not consistently enhance consumer utility. Moreover, the degree of horizontal differentiation moderates this phenomenon in unexpected ways. The "more is less" effect does not arise at low levels of differentiation, but becomes increasingly pronounced as horizontal differentiation increases.

Compressing integer keys is a fundamental operation among multiple communities, such as database management (DB), information retrieval (IR), and high-performance computing (HPC). Recent advances in \emph{learned indexes} have inspired the development of \emph{learned compressors}, which leverage simple yet compact machine learning (ML) models to compress large-scale sorted keys. The core idea behind learned compressors is to \emph{losslessly} encode sorted keys by approximating them with \emph{error-bounded} ML models (e.g., piecewise linear functions) and using a \emph{residual array} to guarantee accurate key reconstruction. While the concept of learned compressors remains in its early stages of exploration, our benchmark results demonstrate that an SIMD-optimized learned compressor can significantly outperform state-of-the-art CPU-based compressors. Drawing on our preliminary experiments, this vision paper explores the potential of learned data compression to enhance critical areas in DBMS and related domains. Furthermore, we outline the key technical challenges that existing systems must address when integrating this emerging methodology.

We propose an instrumental variable framework for identifying and estimating causal effects of discrete and continuous treatments with binary instruments. The basis of our approach is a local copula representation of the joint distribution of the potential outcomes and unobservables determining treatment assignment. This representation allows us to introduce an identifying assumption, so-called copula invariance, that restricts the local dependence of the copula with respect to the treatment propensity. We show that copula invariance identifies treatment effects for the entire population and other subpopulations such as the treated. The identification results are constructive and lead to practical estimation and inference procedures based on distribution regression. An application to estimating the effect of sleep on well-being uncovers interesting patterns of heterogeneity.

Visual illusions in humans arise when interpreting out-of-distribution stimuli: if the observer is adapted to certain statistics, perception of outliers deviates from reality. Recent studies have shown that artificial neural networks (ANNs) can also be deceived by visual illusions. This revelation raises profound questions about the nature of visual information. Why are two independent systems, both human brains and ANNs, susceptible to the same illusions? Should any ANN be capable of perceiving visual illusions? Are these perceptions a feature or a flaw? In this work, we study how visual illusions are encoded in diffusion models. Remarkably, we show that they present human-like brightness/color shifts in their latent space. We use this fact to demonstrate that diffusion models can predict visual illusions. Furthermore, we also show how to generate new unseen visual illusions in realistic images using text-to-image diffusion models. We validate this ability through psychophysical experiments that show how our model-generated illusions also fool humans.

Two substantial technological advances have reshaped the public square in recent decades: first with the advent of the internet and second with the recent introduction of large language models (LLMs). LLMs offer opportunities for a paradigm shift towards more decentralized, participatory online spaces that can be used to facilitate deliberative dialogues at scale, but also create risks of exacerbating societal schisms. Here, we explore four applications of LLMs to improve digital public squares: collective dialogue systems, bridging systems, community moderation, and proof-of-humanity systems. Building on the input from over 70 civil society experts and technologists, we argue that LLMs both afford promising opportunities to shift the paradigm for conversations at scale and pose distinct risks for digital public squares. We lay out an agenda for future research and investments in AI that will strengthen digital public squares and safeguard against potential misuses of AI.

This work considers the question of how convenient access to copious data impacts our ability to learn causal effects and relations. In what ways is learning causality in the era of big data different from -- or the same as -- the traditional one? To answer this question, this survey provides a comprehensive and structured review of both traditional and frontier methods in learning causality and relations along with the connections between causality and machine learning. This work points out on a case-by-case basis how big data facilitates, complicates, or motivates each approach.