A fundamental theoretical question in network analysis is to determine under which conditions community recovery is possible in polynomial time in the Stochastic Block Model (SBM). When the number $K$ of communities remains smaller than $\sqrt{n}$ --where $n$ denotes the number of nodes--, non-trivial community recovery is possible in polynomial time above, and only above, the Kesten--Stigum (KS) threshold, originally postulated using arguments from statistical physics. When $K \geq \sqrt{n}$, Chin, Mossel, Sohn, and Wein recently proved that, in the \emph{sparse regime}, community recovery in polynomial time is achievable below the KS threshold by counting non-backtracking paths. This finding led them to postulate a new threshold for the many-communities regime $K \geq \sqrt{n}$. Subsequently, Carpentier, Giraud, and Verzelen established the failure of low-degree polynomials below this new threshold across all density regimes, and demonstrated successful recovery above the threshold in certain moderately sparse settings. While these results provide strong evidence that, in the many community setting, the computational barrier lies at the threshold proposed in~Chin et al., the question of achieving recovery above this threshold still remains open in most density regimes. The present work is a follow-up to~Carpentier et al., in which we prove Conjecture~1.4 stated therein by: \\ 1- Constructing a family of motifs satisfying specific structural properties; and\\ 2- Proving that community recovery is possible above the proposed threshold by counting such motifs.\\ Our results complete the picture of the computational barrier for community recovery in the SBM with $K \geq \sqrt{n}$ communities. They also indicate that, in moderately sparse regimes, the optimal algorithms appear to be fundamentally different from spectral methods.

Then statement 2 follows from statement 1 and the Banach fixed point theorem. A.5 Proof of Theorem 4.2 Due to (L3), it is clear that the derivative bounds (16) hold if and only if null null null null dN We created datasets from 6 AC power network test cases. We selected 6 test cases, listed in Table 2. Fortunately, branch resistances are typically small before this modification. We then computed the resulting power flows using the runpf function and recorded the results. Geometric data object, with the following attributes: edge_index, the edge index tensor, containing the topology from the test case.

Current advances in AI and its applicability have highlighted the need to ensure its trustworthiness for legal, ethical, and even commercial reasons. Sub-symbolic machine learning algorithms, such as the LLMs, simulate reasoning but hallucinate and their decisions cannot be explained or audited (crucial aspects for trustworthiness). On the other hand, rule-based reasoners, such as Cyc, are able to provide the chain of reasoning steps but are complex and use a large number of reasoners. We propose a middle ground using s(CASP), a goal-directed constraint-based answer set programming reasoner that employs a small number of mechanisms to emulate reliable and explainable human-style commonsense reasoning. In this paper, we explain how s(CASP) supports the 16 desiderata for trustworthy AI introduced by Doug Lenat and Gary Marcus (2023), and two additional ones: inconsistency detection and the assumption of alternative worlds. To illustrate the feasibility and synergies of s(CASP), we present a range of diverse applications, including a conversational chatbot and a virtually embodied reasoner.

Then statement 2 follows from statement 1 and the Banach fixed point theorem. A.5 Proof of Theorem 4.2 Due to (L3), it is clear that the derivative bounds (16) hold if and only if null null null null dN We created datasets from 6 AC power network test cases. We selected 6 test cases, listed in Table 2. Fortunately, branch resistances are typically small before this modification. We then computed the resulting power flows using the runpf function and recorded the results. Geometric data object, with the following attributes: edge_index, the edge index tensor, containing the topology from the test case.

Commonsense temporal reasoning at scale is a core problem for cognitive systems. The correct inference of the duration for which fluents hold is required by many tasks, including natural language understanding and planning. Many AI systems have limited deductive closure because they cannot extrapolate information correctly regarding existing fluents and events. In this study, we discuss the knowledge representation and reasoning schemes required for robust temporal projection in the Cyc Knowledge Base. We discuss how events can start and end risk periods for fluents. We then use discrete survival functions, which represent knowledge of the persistence of facts, to extrapolate a given fluent. The extrapolated intervals can be truncated by temporal constraints and other types of commonsense knowledge. Finally, we present the results of experiments to demonstrate that these methods obtain significant improvements in terms of Q/A performance.

In order to train agents that can quickly adapt to new objectives or reward functions, efficient unsupervised representation learning in sequential decision-making environments can be important. Frameworks such as the Exogenous Block Markov Decision Process (Ex-BMDP) have been proposed to formalize this representation-learning problem (Efroni et al., 2022b). In the Ex-BMDP framework, the agent's high-dimensional observations of the environment have two latent factors: a controllable factor, which evolves deterministically within a small state space according to the agent's actions, and an exogenous factor, which represents time-correlated noise, and can be highly complex. The goal of the representation learning problem is to learn an encoder that maps from observations into the controllable latent space, as well as the dynamics of this space. Efroni et al. (2022b) has shown that this is possible with a sample complexity that depends only on the size of the controllable latent space, and not on the size of the noise factor. However, this prior work has focused on the episodic setting, where the controllable latent state resets to a specific start state after a finite horizon. By contrast, if the agent can only interact with the environment in a single continuous trajectory, prior works have not established sample-complexity bounds. We propose STEEL, the first provably sample-efficient algorithm for learning the controllable dynamics of an Ex-BMDP from a single trajectory, in the function approximation setting. STEEL has a sample complexity that depends only on the sizes of the controllable latent space and the encoder function class, and (at worst linearly) on the mixing time of the exogenous noise factor. We prove that STEEL is correct and sample-efficient, and demonstrate STEEL on two toy problems. Code is available at: https://github.com/midi-lab/steel.

Learning in games discusses the processes where multiple players learn their optimal strategies through the repetition of game plays. The dynamics of learning between two players in zero-sum games, such as matching pennies, where their benefits are competitive, have already been well analyzed. However, it is still unexplored and challenging to analyze the dynamics of learning among three players. In this study, we formulate a minimalistic game where three players compete to match their actions with one another. Although interaction among three players diversifies and complicates the Nash equilibria, we fully analyze the equilibria. We also discuss the dynamics of learning based on some famous algorithms categorized into Follow the Regularized Leader. From both theoretical and experimental aspects, we characterize the dynamics by categorizing three-player interactions into three forces to synchronize their actions, switch their actions rotationally, and seek competition.

Pervasive across diverse domains, stochastic systems exhibit fluctuations in processes ranging from molecular dynamics to climate phenomena. The Langevin equation has served as a common mathematical model for studying such systems, enabling predictions of their temporal evolution and analyses of thermodynamic quantities, including absorbed heat, work done on the system, and entropy production. However, inferring the Langevin equation from observed trajectories remains challenging, particularly for nonlinear and high-dimensional systems. In this study, we present a comprehensive framework that employs Bayesian neural networks for inferring Langevin equations in both overdamped and underdamped regimes. Our framework first provides the drift force and diffusion matrix separately and then combines them to construct the Langevin equation. By providing a distribution of predictions instead of a single value, our approach allows us to assess prediction uncertainties, which can prevent potential misunderstandings and erroneous decisions about the system. We demonstrate the effectiveness of our framework in inferring Langevin equations for various scenarios including a neuron model and microscopic engine, highlighting its versatility and potential impact.

Urban Air Mobility (UAM) offers a solution to current traffic congestion by providing on-demand air mobility in urban areas. Effective traffic management is crucial for efficient operation of UAM systems, especially for high-demand scenarios. In this paper, we present VertiSync, a centralized traffic management policy for on-demand UAM networks. VertiSync schedules the aircraft for either servicing trip requests or rebalancing in the network subject to aircraft safety margins and separation requirements during takeoff and landing. We characterize the system-level throughput of VertiSync, which determines the demand threshold at which travel times transition from being stabilized to being increasing over time. We show that the proposed policy is able to maximize the throughput for sufficiently large fleet sizes. We demonstrate the performance of VertiSync through a case study for the city of Los Angeles. We show that VertiSync significantly reduces travel times compared to a first-come first-serve scheduling policy.

Generative AI, the most popular current approach to AI, consists of large language models (LLMs) that are trained to produce outputs that are plausible, but not necessarily correct. Although their abilities are often uncanny, they are lacking in aspects of reasoning, leading LLMs to be less than completely trustworthy. Furthermore, their results tend to be both unpredictable and uninterpretable. We lay out 16 desiderata for future AI, and discuss an alternative approach to AI which could theoretically address many of the limitations associated with current approaches: AI educated with curated pieces of explicit knowledge and rules of thumb, enabling an inference engine to automatically deduce the logical entailments of all that knowledge. Even long arguments produced this way can be both trustworthy and interpretable, since the full step-by-step line of reasoning is always available, and for each step the provenance of the knowledge used can be documented and audited. There is however a catch: if the logical language is expressive enough to fully represent the meaning of anything we can say in English, then the inference engine runs much too slowly. That's why symbolic AI systems typically settle for some fast but much less expressive logic, such as knowledge graphs. We describe how one AI system, Cyc, has developed ways to overcome that tradeoff and is able to reason in higher order logic in real time. We suggest that any trustworthy general AI will need to hybridize the approaches, the LLM approach and more formal approach, and lay out a path to realizing that dream.