We consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle; we call this (random) time step the dispersion time. In this work we study the case where $G$ is the complete graph on $n$ vertices and the number of particles is $M=n/2+\alpha n^{1/2} + o(n^{1/2})$, $\alpha\in \mathbb{R}$. This choice of $M$ corresponds to the critical window of the process, with respect to the dispersion time. We show that the dispersion time, if rescaled by $n^{-1/2}$, converges in $p$-th mean, as $n\rightarrow \infty$ and for any $p \in \mathbb{R}$, to a continuous and almost surely positive random variable $T_\alpha$. We find that $T_\alpha$ is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that $\mathbb{E}[T_0] = \pi^{3/2}/\sqrt{7}$, and furthermore we formulate explicit asymptotics when $|\alpha|$ gets large that quantify the transition into and out of the critical window. We also study the random variable counting the total number of jumps that are performed by the particles until the dispersion time is reached and prove that, if rescaled by $n\ln n$, it converges to $2/7$ in probability.
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We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.
This paper presents an $O^{*}(1.42^{n})$ time algorithm for the Maximum Cut problem on split graphs, along with a subexponential time algorithm for its decision variant.
In Path Set Packing, the input is an undirected graph $G$, a collection $\calp$ of simple paths in $G$, and a positive integer $k$. The problem is to decide whether there exist $k$ edge-disjoint paths in $\calp$. We study the parameterized complexity of Path Set Packing with respect to both natural and structural parameters. We show that the problem is $W[1]$-hard with respect to vertex cover number, and $W[1]$-hard respect to pathwidth plus maximum degree plus solution size. These results answer an open question raised in COCOON 2018. On the positive side, we present an FPT algorithm parameterized by feedback vertex number plus maximum degree, and present an FPT algorithm parameterized by treewidth plus maximum degree plus maximum length of a path in $\calp$. These positive results complement the hardness of Path Set Packing with respect to any subset of the parameters used in the FPT algorithms. We also give a $4$-approximation algorithm for maximum path set packing problem which runs in FPT time when parameterized by feedback edge number.
We study reinforcement learning with multinomial logistic (MNL) function approximation where the underlying transition probability kernel of the Markov decision processes (MDPs) is parametrized by an unknown transition core with features of state and action. For the finite horizon episodic setting with inhomogeneous state transitions, we propose provably efficient algorithms with randomized exploration having frequentist regret guarantees. For our first algorithm, $\texttt{RRL-MNL}$, we adapt optimistic sampling to ensure the optimism of the estimated value function with sufficient frequency and establish that $\texttt{RRL-MNL}$ is both statistically and computationally efficient, achieving a $\tilde{O}(\kappa^{-1} d^{\frac{3}{2}} H^{\frac{3}{2}} \sqrt{T})$ frequentist regret bound with constant-time computational cost per episode. Here, $d$ is the dimension of the transition core, $H$ is the horizon length, $T$ is the total number of steps, and $\kappa$ is a problem-dependent constant. Despite the simplicity and practicality of $\texttt{RRL-MNL}$, its regret bound scales with $\kappa^{-1}$, which is potentially large in the worst case. To improve the dependence on $\kappa^{-1}$, we propose $\texttt{ORRL-MNL}$, which estimates the value function using local gradient information of the MNL transition model. We show that its frequentist regret bound is $\tilde{O}(d^{\frac{3}{2}} H^{\frac{3}{2}} \sqrt{T} + \kappa^{-1} d^2 H^2)$. To the best of our knowledge, these are the first randomized RL algorithms for the MNL transition model that achieve both computational and statistical efficiency. Numerical experiments demonstrate the superior performance of the proposed algorithms.
In offline reinforcement learning, the challenge of out-of-distribution (OOD) is pronounced. To address this, existing methods often constrain the learned policy through policy regularization. However, these methods often suffer from the issue of unnecessary conservativeness, hampering policy improvement. This occurs due to the indiscriminate use of all actions from the behavior policy that generates the offline dataset as constraints. The problem becomes particularly noticeable when the quality of the dataset is suboptimal. Thus, we propose Adaptive Advantage-guided Policy Regularization (A2PR), obtaining high-advantage actions from an augmented behavior policy combined with VAE to guide the learned policy. A2PR can select high-advantage actions that differ from those present in the dataset, while still effectively maintaining conservatism from OOD actions. This is achieved by harnessing the VAE capacity to generate samples matching the distribution of the data points. We theoretically prove that the improvement of the behavior policy is guaranteed. Besides, it effectively mitigates value overestimation with a bounded performance gap. Empirically, we conduct a series of experiments on the D4RL benchmark, where A2PR demonstrates state-of-the-art performance. Furthermore, experimental results on additional suboptimal mixed datasets reveal that A2PR exhibits superior performance. Code is available at //github.com/ltlhuuu/A2PR.
We prove bounds on the variance of a function $f$ under the empirical measure of the samples obtained by the Sequential Monte Carlo (SMC) algorithm, with time complexity depending on local rather than global Markov chain mixing dynamics. SMC is a Markov Chain Monte Carlo (MCMC) method, which starts by drawing $N$ particles from a known distribution, and then, through a sequence of distributions, re-weights and re-samples the particles, at each instance applying a Markov chain for smoothing. In principle, SMC tries to alleviate problems from multi-modality. However, most theoretical guarantees for SMC are obtained by assuming global mixing time bounds, which are only efficient in the uni-modal setting. We show that bounds can be obtained in the truly multi-modal setting, with mixing times that depend only on local MCMC dynamics.
Retrieval-Augmented Generation (RAG) merges retrieval methods with deep learning advancements to address the static limitations of large language models (LLMs) by enabling the dynamic integration of up-to-date external information. This methodology, focusing primarily on the text domain, provides a cost-effective solution to the generation of plausible but incorrect responses by LLMs, thereby enhancing the accuracy and reliability of their outputs through the use of real-world data. As RAG grows in complexity and incorporates multiple concepts that can influence its performance, this paper organizes the RAG paradigm into four categories: pre-retrieval, retrieval, post-retrieval, and generation, offering a detailed perspective from the retrieval viewpoint. It outlines RAG's evolution and discusses the field's progression through the analysis of significant studies. Additionally, the paper introduces evaluation methods for RAG, addressing the challenges faced and proposing future research directions. By offering an organized framework and categorization, the study aims to consolidate existing research on RAG, clarify its technological underpinnings, and highlight its potential to broaden the adaptability and applications of LLMs.
The success of AI models relies on the availability of large, diverse, and high-quality datasets, which can be challenging to obtain due to data scarcity, privacy concerns, and high costs. Synthetic data has emerged as a promising solution by generating artificial data that mimics real-world patterns. This paper provides an overview of synthetic data research, discussing its applications, challenges, and future directions. We present empirical evidence from prior art to demonstrate its effectiveness and highlight the importance of ensuring its factuality, fidelity, and unbiasedness. We emphasize the need for responsible use of synthetic data to build more powerful, inclusive, and trustworthy language models.
Data augmentation, the artificial creation of training data for machine learning by transformations, is a widely studied research field across machine learning disciplines. While it is useful for increasing the generalization capabilities of a model, it can also address many other challenges and problems, from overcoming a limited amount of training data over regularizing the objective to limiting the amount data used to protect privacy. Based on a precise description of the goals and applications of data augmentation (C1) and a taxonomy for existing works (C2), this survey is concerned with data augmentation methods for textual classification and aims to achieve a concise and comprehensive overview for researchers and practitioners (C3). Derived from the taxonomy, we divided more than 100 methods into 12 different groupings and provide state-of-the-art references expounding which methods are highly promising (C4). Finally, research perspectives that may constitute a building block for future work are given (C5).
Named entity recognition (NER) is the task to identify text spans that mention named entities, and to classify them into predefined categories such as person, location, organization etc. NER serves as the basis for a variety of natural language applications such as question answering, text summarization, and machine translation. Although early NER systems are successful in producing decent recognition accuracy, they often require much human effort in carefully designing rules or features. In recent years, deep learning, empowered by continuous real-valued vector representations and semantic composition through nonlinear processing, has been employed in NER systems, yielding stat-of-the-art performance. In this paper, we provide a comprehensive review on existing deep learning techniques for NER. We first introduce NER resources, including tagged NER corpora and off-the-shelf NER tools. Then, we systematically categorize existing works based on a taxonomy along three axes: distributed representations for input, context encoder, and tag decoder. Next, we survey the most representative methods for recent applied techniques of deep learning in new NER problem settings and applications. Finally, we present readers with the challenges faced by NER systems and outline future directions in this area.
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