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We introduce Meta MLGym and MLGym-Bench, a new framework and benchmark for evaluating and developing LLM agents on AI research tasks. This is the first Gym environment for machine learning (ML) tasks, enabling research on reinforcement learning (RL) algorithms for training such agents. MLGym-bench consists of 13 diverse and open-ended AI research tasks from diverse domains such as computer vision, natural language processing, reinforcement learning, and game theory. Solving these tasks requires real-world AI research skills such as generating new ideas and hypotheses, creating and processing data, implementing ML methods, training models, running experiments, analyzing the results, and iterating through this process to improve on a given task. We evaluate a number of frontier large language models (LLMs) on our benchmarks such as Claude-3.5-Sonnet, Llama-3.1 405B, GPT-4o, o1-preview, and Gemini-1.5 Pro. Our MLGym framework makes it easy to add new tasks, integrate and evaluate models or agents, generate synthetic data at scale, as well as develop new learning algorithms for training agents on AI research tasks. We find that current frontier models can improve on the given baselines, usually by finding better hyperparameters, but do not generate novel hypotheses, algorithms, architectures, or substantial improvements. We open-source our framework and benchmark to facilitate future research in advancing the AI research capabilities of LLM agents., Comment: 35 pages, 12 figures, 10 tables

We study strong existence and pathwise uniqueness for a class of infinite-dimensional singular stochastic differential equations (SDE), with state space as the cone $\{x \in \mathbb{R}^{\mathbb{N}}: -\infty < x_1 \leq x_2 \leq \cdots\}$, referred to as an infinite system of competing Brownian particles. A `mass' parameter $p \in [0,1]$ governs the splitting proportions of the singular collision local time between adjacent state coordinates. Solutions in the case $p=1/2$ correspond to the well-studied rank-based diffusions, while the general case arises from scaling limits of interacting particle systems on the lattice with asymmetric interactions and the study of the KPZ equation. Under conditions on the initial configuration, the drift vector, and the growth of the local time terms, we establish pathwise uniqueness and strong existence of solutions to the SDE. A key observation is the connection between pathwise uniqueness and the finiteness of `chains of collisions' between adjacent particles influencing a tagged particle in the system. Ingredients in the proofs include classical comparison and monotonicity arguments for reflected Brownian motions, techniques from Brownian last-passage percolation, large deviation bounds for random matrix eigenvalues, and concentration estimates for extrema of Gaussian processes., Comment: 50 pages

We study inhomogeneous random graphs with a finite type space. For a natural generalization of the model as a dynamic network-valued process, the paper establishes the following results: (a) Functional central limit theorems for the infinite vector of microscopic type-densities and characterizations of the limits as infinite-dimensional conditionally Gaussian processes in a certain Banach space. (b) Functional (joint) central limit theorems for macroscopic observables of the giant component in the supercritical regime including size, surplus and number of vertices of various types in the giant component. As a corollary this provides central limit theorems for the size of the largest connected component, its surplus, and its type vector, for percolation on dense graphs obtained from a finite type Graphon. (c) Central limit theorem for the weight of the minimum spanning tree with random i.i.d. Exponential edge weights on dense graph sequences driven by an underlying finite type graphon., Comment: version 2, added application to weight of the random MST

Large Language Models (LLMs) and Text-To-Image (T2I) models have demonstrated the ability to generate compelling text and visual stories. However, their outputs are predominantly aligned with the sensibilities of the Global North, often resulting in an outsider's gaze on other cultures. As a result, non-Western communities have to put extra effort into generating culturally specific stories. To address this challenge, we developed a visual storytelling pipeline called KAHANI that generates culturally grounded visual stories for non-Western cultures. Our pipeline leverages off-the-shelf models GPT-4 Turbo and Stable Diffusion XL (SDXL). By using Chain of Thought (CoT) and T2I prompting techniques, we capture the cultural context from user's prompt and generate vivid descriptions of the characters and scene compositions. To evaluate the effectiveness of KAHANI, we conducted a comparative user study with ChatGPT-4 (with DALL-E3) in which participants from different regions of India compared the cultural relevance of stories generated by the two tools. Results from the qualitative and quantitative analysis performed on the user study showed that KAHANI was able to capture and incorporate more Culturally Specific Items (CSIs) compared to ChatGPT-4. In terms of both its cultural competence and visual story generation quality, our pipeline outperformed ChatGPT-4 in 27 out of the 36 comparisons., Comment: Under review

As Large Language Models (LLMs) continue to exhibit remarkable performance in natural language understanding tasks, there is a crucial need to measure their ability for human-like multi-step logical reasoning. Existing logical reasoning evaluation benchmarks often focus primarily on simplistic single-step or multi-step reasoning with a limited set of inference rules. Furthermore, the lack of datasets for evaluating non-monotonic reasoning represents a crucial gap since it aligns more closely with human-like reasoning. To address these limitations, we propose Multi-LogiEval, a comprehensive evaluation dataset encompassing multi-step logical reasoning with various inference rules and depths. Multi-LogiEval covers three logic types--propositional, first-order, and non-monotonic--consisting of more than 30 inference rules and more than 60 of their combinations with various depths. Leveraging this dataset, we conduct evaluations on a range of LLMs including GPT-4, ChatGPT, Gemini-Pro, Yi, Orca, and Mistral, employing a zero-shot chain-of-thought. Experimental results show that there is a significant drop in the performance of LLMs as the reasoning steps/depth increases (average accuracy of ~68% at depth-1 to ~43% at depth-5). We further conduct a thorough investigation of reasoning chains generated by LLMs which reveals several important findings. We believe that Multi-LogiEval facilitates future research for evaluating and enhancing the logical reasoning ability of LLMs. Data is available at https://github.com/Mihir3009/Multi-LogiEval., Comment: Accepted at EMNLP 2024 Main

We study a model for flocking given by a $n$-particle system under which each particle jumps forward by a random amount, independently sampled from a given distribution $\theta$, with rate given by a non-increasing function $w$ of its signed distance from the system center of mass. This model was introduced in Bal\'azs et. al. (2014) and some of its properties were studied for the case when $w$ is bounded. In the current work we are interested in the setting where $w$ is unbounded, and this feature results in a stochastic dynamical system for interacting particles with fast and large jumps for which little is available in the literature. We characterize the large $n$ limit (the so-called `fluid limit') of the empirical measure process associated with the system and prove a propagation of chaos result. Next, for the centered $n$-particle system, by constructing suitable Lyapunov functions, we establish existence and uniqueness of stationary distributions and study their tail properties. In the special case where $w$ is an exponential function and $\theta$ is an exponential distribution, by establishing that all stationary solutions of the McKean-Vlasov equation must be the unique fixed point of the equation, we prove a propagation of chaos result at $t=\infty$ and establish convergence of the particle system, starting from stationarity, in the large $n$ limit, to a traveling wave solution of the McKean-Vlasov equation. The proof of this result may be of interest for other interacting particle systems where convexity properties or functional inequalities generally used for establishing such a result are not available. Our work answers several open problems posed in Bal\'azs et. al., Comment: 52 pages

California schools received over $60 billion from federal and state stimulus funding to help students recover from the pandemic. This infographic summarizes the major programs districts have implemented during the 2022-23 school year. Most districts extended learning time and hired instructional supports, but few provided high-dosage tutoring. Rural districts were less likely to extend the school year, provide tutoring or provide counseling supports.

The Bou\'e-Dupuis variational formula gives a representation for log Laplace transforms of bounded measurable functions of a finite dimensional Brownian motion on a compact time interval as an infimum of a suitable cost over a collection of non-anticipative control processes. This variational formula has proved to be very useful in studying a variety of large deviation problems. In this article we collect some extensions of this basic result that have appeared in disparate venues in studying a broad range of large deviation questions. Some of these results can be found in a unified way in the recent book (Budhiraja and Dupuis(2019)), while others, to date, have been scattered at various places in the literature. The latter category includes, in particular, variational representations, when the stochastic dynamical system of interest has in addition to a driving L\'evy noise, another source of randomness, e.g. due to a random initial condition; when the functionals of interest depends on infinite-length paths of a L\'evy process; when the noise process is a Gaussian process with long-range dependence, e.g. a fractional Brownian motion, etc. The goal of this survey article is to present these diverse variational formulas in a systematic manner.

We consider a continuous-time random walk in the quarter plane for which the transition intensities are constant on each of the four faces $(0,\infty)^2$, $F_1=\{0\}\times(0,\infty)$, $F_2=(0,\infty)\times\{0\}$ and $\{(0,0)\}$. We show that when rescaled diffusively it converges in law to a Brownian motion with oblique reflection direction $d^{(i)}$ on face $F_i$, $i=1,2$, defined via the Varadhan-Williams submartingale problem. A parameter denoted by $\alpha$ was introduced in \cite{vw}, measuring the extent to which $d^{(i)}$ are inclined toward the origin. In the case of the quarter plane, $\alpha$ takes values in $(-2,2)$, and it is known that the reflected Brownian motion is a semimartingale if and only if $\alpha\in(-2,1)$. Convergence results via both the Skorohod map and the invariance principle for semimartingale reflected Brownian motion are known to hold in various settings in arbitrary dimension. In the case of the quarter plane, the invariance principle was proved for $\alpha \in (-2,1)$ whereas for tools based on the Skorohod map to be applicable it is necessary (but not sufficient) that $\alpha \in [-1,1)$. Another tool that has been used to prove convergence in general dimension is the extended Skorohod map, which in the case of the quarter plane provides convergence for $\alpha=1$. This paper focuses on the range $\alpha \in (1,2)$, where the Skorohod problem and the extended Skorohod problem do not possess a unique solution, the limit process is not a semimartingale, and convergence to reflected Brownian motion has not been shown before. The result has implications on the asymptotic analysis of two Markovian queueing models: The {\it generalized processor sharing model with parallelization slowdown}, and the {\it coupled processor model}.

We consider a family of resource sharing networks, known as bandwidth sharing models, in heavy traffic with general service and interarrival times. These networks, introduced in Massoulie and Roberts (2000) as models for internet flows, have the feature that a typical job may require simultaneous processing by multiple resources in the network. We construct simple form threshold policies that asymptotically achieve the Hierarchical Greedy Ideal (HGI) performance. This performance benchmark, which was introduced in Harrison et al. (2014), is characterized by the following two features: every resource works at full capacity whenever there is work for that resource in the system; total holding cost of jobs of each type at any instant is the minimum cost possible for the associated vector of workloads. The control policy we provide is explicit in terms of a finite collection of vectors which can be computed offline by solving a system of linear inequalities. Proof of convergence is based on path large deviation estimates for renewal processes, Lyapunov function constructions, and analyses of suitable sample path excursions., Comment: To be published in Annals of Applied Probability

Dynamic capacity allocation control for resource sharing networks (RSN) is studied when the networks are in heavy traffic. The goal is to minimize an ergodic cost with a linear holding cost function. Our main result shows that the optimal cost associated with an associated Brownian control problem provides a lower bound for the asymptotic ergodic cost in the RSN for any sequence of control policies. A similar result for an infinite horizon discounted cost has been previously shown for general resource sharing networks and for general "unitary networks." The study of an ergodic cost criterion requires different ideas as bounds on ergodic costs only yield an estimate on the controls in a time-averaged sense which makes the time-rescaling ideas used in the infinite horizon discounted cost cases hard to implement. Proofs rely on working with a weaker topology on the space of controlled processes that is more amenable to an analysis for the ergodic cost. As a corollary of the main result, we show that control policies which achieve hierarchical greedy ideal performance for general RSN are asymptotically optimal for the ergodic cost when the underlying cost per unit time has certain monotonicity properties., Comment: Submitted to special issue of QUESTA on Reflected Brownian Motion and connected subjects

We consider the uplink of a hardware-impaired intelligent reflective surfaces (IRS) aided multi-cell massive multiple-input multiple-output (mMIMO) system with mobile user equipments, whose channel age with time. For this system, we analyze the distortion due to hardware impairments, and the effect of channel aging, and propose a novel distortion-and-aging-aware MMSE (DAA-MMSE) receiver that not only provides a higher spectral efficiency (SE) than conventional maximal ratio and distortion-unaware MMSE (DU-MMSE) receivers, but also reduces the pilot overhead. By considering non-ideal hardware and spatially-correlated Rician fading channels with phase shifts, we also derive the SE lower bound for this system. We show that the SE gain of the DAA-MMSE receiver over DU-MMSE receiver increases with hardware impairments, and channel aging. Along with DAA-MMSE receiver, the IRS is also shown to reduce the pilot overhead in a mMIMO system with channel aging., Comment: This paper is accepted in WCNC 2025

Large language models (LLMs) such as ChatGPT and Google Bard have garnered significant attention in the academic community. Previous research has evaluated these LLMs for various applications such as generating programming exercises and solutions. However, these evaluations have predominantly been conducted by instructors and researchers, not considering the actual usage of LLMs by students. This study adopts a student-first approach to comprehensively understand how undergraduate computer science students utilize ChatGPT, a popular LLM, released by OpenAI. We employ a combination of student surveys and interviews to obtain valuable insights into the benefits, challenges, and suggested improvements related to ChatGPT. Our findings suggest that a majority of students (over 57%) have a convincingly positive outlook towards adopting ChatGPT as an aid in coursework-related tasks. However, our research also highlights various challenges that must be resolved for long-term acceptance of ChatGPT amongst students. The findings from this investigation have broader implications and may be applicable to other LLMs and their role in computing education., Comment: Accepted in ACE 2024: https://aceconference2024.github.io/aceconference2024/ We thank Shamik Sinha, Palak Bhardwaj, Rayyan Hussain, Kashvi Panvanda, Sidhartha Garg, Shagun Yadav, Yash Chillar, and Sourav Goyal for their assistance in the development of questionnaires, as well as their involvement in conducting selected interviews

The infinite Atlas model describes the evolution of a countable collection of Brownian particles on the real line, where the lowest particle is given a drift of $\gamma \in [0,\infty)$. We study equilibrium fluctuations for the Atlas model when the system of particles starts from an inhomogeneous stationary profile with exponentially growing density. We show that the appropriately centered and scaled occupation measure of the particle positions, with suitable translations, viewed as a space-time random field, converges to a limit given by a certain stochastic partial differential equation (SPDE). The initial condition for this equation is given by a Brownian motion, the equation is driven by an additive space-time noise that is white in time and colored in space, and the linear operator governing the evolution is the infinitesimal generator of a geometric Brownian motion. We use this SPDE to also characterize the fluctuations of the ranked particle positions with a suitable centering and scaling. Our results describe the behavior of the particles in the bulk and one finds that the Gaussian process describing the asymptotic fluctuations has the same H\"{o}lder regularity as a fractional Brownian motion with Hurst parameter $1/4$. One finds that, unlike the setting of a homogeneous profile (Dembo and Tsai (2017)), the behavior on the lower edge of the particle system is very different from the bulk behavior and in fact the variance of the Gaussian limit diverges to $\infty$ as one approaches the lower edge. Indeed, our results show that, with the gaps between particles given by one of the inhomogeneous stationary distributions, the lowest particle, started from $0$, with a linear in time translation, converges in distribution to an explicit non-Gaussian limit as $t\to \infty$., Comment: 58 pages

The rise in popularity of Large Language Models (LLMs) has prompted discussions in academic circles, with students exploring LLM-based tools for coursework inquiries and instructors exploring them for teaching and research. Even though a lot of work is underway to create LLM-based tools tailored for students and instructors, there is a lack of comprehensive user studies that capture the perspectives of students and instructors regarding LLMs. This paper addresses this gap by conducting surveys and interviews within undergraduate engineering universities in India. Using 1306 survey responses among students, 112 student interviews, and 27 instructor interviews around the academic usage of ChatGPT (a popular LLM), this paper offers insights into the current usage patterns, perceived benefits, threats, and challenges, as well as recommendations for enhancing the adoption of LLMs among students and instructors. These insights are further utilized to discuss the practical implications of LLMs in undergraduate engineering education and beyond., Comment: Under review

ChatGPT is an AI language model developed by OpenAI that can understand and generate human-like text. It can be used for a variety of use cases such as language generation, question answering, text summarization, chatbot development, language translation, sentiment analysis, content creation, personalization, text completion, and storytelling. While ChatGPT has garnered significant positive attention, it has also generated a sense of apprehension and uncertainty in academic circles. There is concern that students may leverage ChatGPT to complete take-home assignments and exams and obtain favorable grades without genuinely acquiring knowledge. This paper adopts a quantitative approach to demonstrate ChatGPT's high degree of unreliability in answering a diverse range of questions pertaining to topics in undergraduate computer science. Our analysis shows that students may risk self-sabotage by blindly depending on ChatGPT to complete assignments and exams. We build upon this analysis to provide constructive recommendations to both students and instructors., Comment: Accepted in SIGCSE TS 2024

We study the impact of channel aging on the uplink of a cell-free (CF) massive multiple-input multiple-output (mMIMO) system by considering i) spatially-correlated Rician-faded channels; ii) hardware impairments at the access points and user equipments (UEs); and iii) two-layer large-scale fading decoding (LSFD). We first derive a closed-form spectral efficiency (SE) expression for this system, and later propose two novel optimization techniques to optimize the non-convex SE metric by exploiting the minorization-maximization (MM) method. The first one requires a numerical optimization solver, and has a high computation complexity. The second one with closed-form transmit power updates, has a trivial computation complexity. We numerically show that i) the two-layer LSFD scheme effectively mitigates the interference due to channel aging for both low- and high-velocity UEs; and ii) increasing the number of AP antennas does not mitigate the SE deterioration due to channel aging. We numerically characterize the optimal pilot length required to maximize the SE for various UE speeds. We also numerically show that the proposed closed-form MM optimization yields the same SE as that of the first technique, which requires numerical solver, and that too with a much reduced time-complexity., Comment: This work has been submitted to the IEEE Transactions on Communications for possible publication, 32 pages, 14 figures

Let $\Delta^o$ be a finite set and, for each probability measure $m$ on $\Delta^o$, let $G(m)$ be a transition probability kernel on $\Delta^o$. Fix $x_0 \in \Delta^o$ and consider the chain $\{X_n, \; n \in \mathbb{N}_0\}$ of $\Delta^o$-valued random variables such that $X_0=x$, and given $X_0, \ldots , X_n$, the conditional distribution of $X_{n+1}$ is $G(L^{n+1})(X_n, \cdot)$, where $L^{n+1} = \frac{1}{n+1} \sum_{i=0}^{n} \delta_{X_i}$ is the empirical measure at instant $n$. Under conditions on $G$ we establish a large deviation principle for the empirical measure sequence $\{L^n, \; n \in \mathbb{N}\}$. As one application of this result we obtain large deviation asymptotics for the Aldous-Flannery-Palacios (1988) approximation scheme for quasistationary distributions of irreducible finite state Markov chains. The conditions on $G$ cover various other models of reinforced stochastic evolutions as well, including certain vertex reinforced and edge reinforced random walks and a variant of the PageRank algorithm. The particular case where $G(m)$ does not depend on $m$ corresponds to the classical results of Donsker and Varadhan (1975) on large deviations of empirical measures of Markov processes. However, unlike this classical setting, for the general self-interacting models considered here, the rate function takes a very different form; it is typically non-convex and is given through a dynamical variational formula with an infinite horizon discounted objective function., Comment: 54 pages

We study the impact of channel aging on the uplink of a cell-free massive multiple-input multiple-output system with hardware impairments. We consider a dynamic analog-to-digital converter architecture at the access points (APs), and low-resolution digital-to-analog converters at the user equipments (UEs). We derive a closed-form spectral efficiency expression by considering i) practical spatially-correlated Rician channels; ii) hardware impairments at the APs and the UEs; iii) channel aging; and iv) large-scale fading decoding (LSFD). We show that LSFD can effectively mitigate the detrimental effects of i) channel aging for both low and high UE velocities; and ii) inter-user interference for low-velocity UEs but not for high-velocity UEs., Comment: This paper is accepted for presentation in 2023 IEEE International Conference on Communications (ICC): Wireless Communications Symposium (IEEE ICC'23 - WC Symposium), 6 pages and 4 figures

Consider a system with $K$ parallel queues in which the server for each queue processes jobs at rate $n$ and the total arrival rate to the system is $nK-\upsilon \sqrt{n}$ where $\upsilon \in (0, \infty)$ and $n$ is large. We study rank-based routing policies in which $O(\sqrt{n})$ of the incoming jobs are routed to servers with probabilities depending on their ranked queue-length and the remaining jobs are routed uniformly at random. A particular case, referred to as the marginal join-the-shortest-queue (MJSQ) policy, is one in which the $O(\sqrt{n})$ jobs are routed using the join-the-shortest-queue (JSQ) policy. Our first result provides a heavy traffic approximation theorem for such queuing systems. It turns out that, unlike the JSQ system, there is no state space collapse in the setting of MJSQ (and for the more general rank-based routing schemes) and one obtains a novel diffusion limit which is the constrained analogue of the well studied Atlas model (and other rank-based diffusions) arising from mathematical finance. Next, we prove an interchange of limits result which shows that the steady state of the queuing system is well approximated by that of the limiting diffusion, given explicitly in terms of product laws of Exponential random variables. Using these results, we compute the time asymptotic total queue-length in the heavy traffic limit for the MJSQ system. We find the striking result that although in going from JSQ to MJSQ the communication cost is reduced by a factor of $\sqrt{n}$, the asymptotic total queue-length increases only by a constant factor which can be made arbitrarily close to one by increasing a MJSQ parameter. When the system is overloaded ($\upsilon<0$) we show that although the $K$-dimensional MJSQ system is unstable, the steady state difference between the maximum and minimum queue-lengths stays bounded in probability (in the heavy traffic parameter $n$)., Comment: 34 pages. To appear in Math. of Oper. Res

We consider a hardware-impaired multi-cell Rician faded massive multi-input multi-output (mMIMO) system with two-layer pilot decontamination precoding, also known as large-scale fading precoding (LSFP). Each BS is equipped with a flexible dynamic analog-to-digital converter (ADC)/digital-to-analog converter (DAC) architecture and the user equipments (UEs) have low-resolution ADCs. Further, both BS and UEs have hardwareimpaired radio frequency chains. The dynamic ADC/DAC architecture allows us to vary the resolution of ADC/DAC connected to each BS antenna, and suitably choose them to maximize the SE. We propose a distortion-aware minimum mean squared error (DA-MMSE) precoder and investigate its usage with two-layer LSFP and conventional single-layer precoding (SLP) for hardware-impaired mMIMO systems. We discuss the use cases of LSFP and SLP with DA-MMSE and distortion-unaware MMSE (DU-MMSE) precoders, which will provide critical insights to the system designer regarding their usage in practical systems., Comment: This paper is accepted for presentation in 2022 IEEE Global Communications Conference: Wireless Communications (Globecom 2022 WC), 7 pages and 4 figures

We consider the problem of estimating the channel in reconfigurable intelligent surface (RIS) assisted millimeter wave (mmWave) systems. We propose two variational expectation maximization (VEM) based algorithms for channel estimation in RIS-aided wireless systems. The first algorithm is a structured mean field-based sparse Bayesian learning (SM-SBL) algorithm that exploits the doubly-structured sparsity and the individual sparsity of the elements of the channel. To exploit the sparsities, we propose a column-wise coupled Gaussian prior. We next design the factorized mean field-based algorithm based on the prior we propose. This algorithm called the factorized mean field SBL (FM-SBL) algorithm, addresses the time complexities of the SM-SBL algorithm without sacrificing channel estimation accuracy. We show using extensive numerical investigations that the i) proposed SM-SBL and FM-SBL algorithms outperform several existing algorithms and ii) FM-SBL has lower time complexity than the SM-SBL algorithm.

This article presents a review of some old and new results on the long time behavior of reflected diffusions. First, we present a summary of prior results on construction, ergodicity and geometric ergodicity of reflected diffusions in the positive orthant $\mathbb{R}^d_+$, $d \in \mathbb{N}$. The geometric ergodicity results, although very general, usually give implicit convergence rates due to abstract couplings and Lyapunov functions used in obtaining them. This leads us to some recent results on an important subclass of reflected Brownian motions (RBM) (constant drift and diffusion coefficients and oblique reflection at boundaries), known as the Harrison-Reiman class, where explicit rates of convergence are obtained as functions of the system parameters and underlying dimension. In addition, sufficient conditions on system parameters of the RBM are provided under which local convergence to stationarity holds at a `dimension-free' rate, that is, for any fixed $k \in \mathbb{N}$, the rate of convergence of the $k$-marginal to equilibrium does not depend on the dimension of the whole system. Finally, we study the long time behavior of infinite dimensional rank-based diffusions, including the well-studied infinite Atlas model. The gaps between the ordered particles evolve as infinite dimensional RBM and this gap process has uncountably many explicit product form stationary distributions. Sufficient conditions for initial configurations to lie in the weak domain of attraction of the various stationary distributions are provided. Finally, it is shown that, under conditions, all of these explicit stationary distributions are extremal (equivalently, ergodic) and, in some sense, the only product form invariant probability distributions. Proof techniques involve a pathwise analysis of RBM using explicit synchronous and mirror couplings and constructing Lyapunov functions.

\noindent Consider an infinite collection of particles on the real line moving according to independent Brownian motions and such that the $i$-th particle from the left gets the drift $g_{i-1}$. The case where $g_0=1$ and $g_{i}=0$ for all $i \in \mathbb{N}$ corresponds to the well studied infinite Atlas model. Under conditions on the drift vector $\boldsymbol{g} = (g_0, g_1, \ldots)'$ it is known that the Markov process corresponding to the gap sequence of the associated ranked particles has a continuum of product form stationary distributions $\{\pi_a^{\boldsymbol{g}}, a \in S^{\boldsymbol{g}}\}$ where $S^{\boldsymbol{g}}$ is a semi-infinite interval of the real line. In this work we show that all of these stationary distributions are extremal and ergodic. We also prove that any product form stationary distribution of this Markov process that satisfies a mild integrability condition must be $\pi_a^{\boldsymbol{g}}$ for some $a \in S^{\boldsymbol{g}}$. These results are new even for the infinite Atlas model. The work makes progress on the open problem of characterizing all the invariant distributions of general competing Brownian particle systems interacting through their relative ranks. Proofs rely on synchronous and mirror coupling of Brownian particles and properties of the intersection local times of the various particles in the infinite system., Comment: 29 pages. To appear in Probability Theory and Related Fields

We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan (1976) are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the rate functions take simple and explicit forms that have striking differences from their nondegenerate counterparts., Comment: 56 pages, 1 figure

In this new digital era, social media has created a severe impact on the lives of people. In recent times, fake news content on social media has become one of the major challenging problems for society. The dissemination of fabricated and false news articles includes multimodal data in the form of text and images. The previous methods have mainly focused on unimodal analysis. Moreover, for multimodal analysis, researchers fail to keep the unique characteristics corresponding to each modality. This paper aims to overcome these limitations by proposing an Efficient Transformer based Multilevel Attention (ETMA) framework for multimodal fake news detection, which comprises the following components: visual attention-based encoder, textual attention-based encoder, and joint attention-based learning. Each component utilizes the different forms of attention mechanism and uniquely deals with multimodal data to detect fraudulent content. The efficacy of the proposed network is validated by conducting several experiments on four real-world fake news datasets: Twitter, Jruvika Fake News Dataset, Pontes Fake News Dataset, and Risdal Fake News Dataset using multiple evaluation metrics. The results show that the proposed method outperforms the baseline methods on all four datasets. Further, the computation time of the model is also lower than the state-of-the-art methods., Comment: Accepted for publication in IEEE Transactions on Computational Social Systems

Let $A$ be a transition probability kernel on a finite state space $\Delta^o =\{1, \ldots , d\}$ such that $A(x,y)>0$ for all $x,y \in \Delta^o$. Consider a reinforced chain given as a sequence $\{X_n, \; n \in \mathbb{N}_0\}$ of $\Delta^o$-valued random variables, defined recursively according to, $$L^n = \frac{1}{n}\sum_{i=0}^{n-1} \delta_{X_i}, \;\; P(X_{n+1} \in \cdot \mid X_0, \ldots, X_n) = L^n A(\cdot).$$ We establish a large deviation principle for $\{L^n\}$. The rate function takes a strikingly different form than the Donsker-Varadhan rate function associated with the empirical measure of the Markov chain with transition kernel $A$ and is described in terms of a novel deterministic infinite horizon discounted cost control problem with an associated linear controlled dynamics and a nonlinear running cost involving the relative entropy function. Proofs are based on an analysis of time-reversal of controlled dynamics in representations for log-transforms of exponential moments, and on weak convergence methods.

Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external Newtonian potential (e.g. gravitation) and decreases in proportion to the total local time of collisions with the Brownian particles. This system models a semi-permeable membrane in a fluid having microscopic impurities (Knight (2001)). We study the long-time behavior of the process $(V,\mathbf{Z})$, where $V$ is the velocity of the inert particle and $\mathbf{Z}$ is the vector of gaps between successive particles ordered by their relative positions. The system is not hypoelliptic, not reversible, and has singular form interactions. Thus the study of stability behavior of the system requires new ideas. We show that this process has a unique stationary distribution that takes an explicit product form which is Gaussian in the velocity component and Exponential in the other components. We also show that convergence in total variation distance to the stationary distribution happens at an exponential rate. We further obtain certain law of large numbers results for the particle locations and intersection local times., Comment: 55 pages, to appear in Communications in Mathematical Physics

Stochastic Heavy Ball (SHB) and Nesterov's Accelerated Stochastic Gradient (ASG) are popular momentum methods in stochastic optimization. While benefits of such acceleration ideas in deterministic settings are well understood, their advantages in stochastic optimization is still unclear. In fact, in some specific instances, it is known that momentum does not help in the sample complexity sense. Our work shows that a similar outcome actually holds for the whole of quadratic optimization. Specifically, we obtain a lower bound on the sample complexity of SHB and ASG for this family and show that the same bound can be achieved by the vanilla SGD. We note that there exist results claiming the superiority of momentum based methods in quadratic optimization, but these are based on one-sided or flawed analyses.

Graph Representation Learning (GRL) is an upcoming and promising area in recommendation systems. In this paper, we revisit the Singular Value Decomposition (SVD) of adjacency matrix for embedding generation of users and items and use a two-layer neural network on top of these embeddings to learn relevance between user-item pairs. Inspired by the success of higher-order learning in GRL, we further propose an extension of this method to include two-hop neighbors for SVD through the second order of the adjacency matrix and demonstrate improved performance compared with the simple SVD method which only uses one-hop neighbors. Empirical validation on three publicly available datasets of recommendation system demonstrates that the proposed methods, despite being simple, beat many state-of-the-art methods and for two of three datasets beats all of them up to a margin of 10%. Through our research, we want to shed light on the effectiveness of matrix factorization approaches, specifically SVD, in the deep learning era and show that these methods still contribute as important baselines in recommendation systems., Comment: 7 pages, 3 figures, and 4 tables

We consider nonlinear filters for diffusion processes when the observation and signal noises are small and of the same order. As the noise intensities approach zero, the nonlinear filter can be approximated by a certain variational problem that is closely related to Mortensen's optimization problem(1968). This approximation result can be made precise through a certain Laplace asymptotic formula. In this work we study probabilities of deviations of true filtering estimates from that obtained by solving the variational problem. Our main result gives a large deviation principle for Laplace functionals whose typical asymptotic behavior is described by Mortensen-type variational problems. Proofs rely on stochastic control representations for positive functionals of Brownian motions and Laplace asymptotics of the Kallianpur-Striebel formula., Comment: 28 pages

The infinite Atlas model describes a countable system of competing Brownian particles where the lowest particle gets a unit upward drift and the rest evolve as standard Brownian motions. The stochastic process of gaps between the particles in the infinite Atlas model does not have a unique stationary distribution and in fact for every $a \ge 0$, $\pi_a := \bigotimes_{i=1}^{\infty} \operatorname{Exp}(2 + ia)$ is a stationary measure for the gap process. We say that an initial distribution of gaps is in the weak domain of attraction of the stationary measure $\pi_a$ if the time averaged laws of the stochastic process of the gaps, when initialized using that distribution, converge to $\pi_a$ weakly in the large time limit. We provide general sufficient conditions on the initial gap distribution of the Atlas particles for it to lie in the weak domain of attraction of $\pi_a$ for each $a\ge 0$. The cases $a=0$ and $a>0$ are qualitatively different as is seen from the analysis and the sufficient conditions that we provide. Proofs are based on the analysis of synchronous couplings, namely, couplings of the ranked particle systems started from different initial configurations, but driven using the same set of Brownian motions., Comment: 37 pages. To appear in Ann. Probab

We introduce and study the basic properties of two ergodic stochastic control problems associated with the quasistationary distribution (QSD) of a diffusion process $X$ relative to a bounded domain. The two problems are in some sense dual, with one defined in terms of the generator associated with $X$ and the other in terms of its adjoint. Besides proving wellposedness of the associated Hamilton-Jacobi-Bellman equations, we describe how they can be used to characterize important properties of the QSD. Of particular note is that the QSD itself can be identified, up to normalization, in terms of the cost potential of the control problem associated with the adjoint.

This work proposes a superimposed pilot (SP)-based channel estimation and data detection framework for orthogonal time-frequency space (OTFS) scheme, wherein low-powered pilots are superimposed on to data symbols in the delay-Doppler domain. We propose two channel estimation and data detection designs for SP-OTFS systems which, unlike the existing OTFS designs, do not designate any slots for pilots, which improves their spectral efficiency (SE). The first SP design estimates channel by treating data as interference, which degrades its performance at high signal to noise ratio. The second SP design alleviates this problem by iterating between channel estimation and data detection. Both these designs detect data using message passing algorithm which exploits OTFS channel sparsity, and consequently has low computational complexity. We also derive a lower bound on the signal-to-interference-plus-noise ratio of the proposed designs, and maximize it by optimally allocating power between data and pilot symbols. We numerically validate the derived analytical results, and show that the proposed designs have superior SE than the two state-of-the-art OTFS channel estimation and data detection designs.

Cell-free (CF) massive multiple-input-multiple-output (mMIMO) deployments are usually investigated with half-duplex nodes and high-capacity fronthaul links. To leverage the possible gains in throughput and energy efficiency (EE) of full-duplex (FD) communications, we consider a FD CF mMIMO system with practical limited-capacity fronthaul links. We derive closed-form spectral efficiency (SE) lower bounds for this system with maximum-ratio combining/maximum-ratio transmission processing and optimal uniform quantization. We then optimize the weighted sum EE (WSEE) via downlink and uplink power control by using a two-layered approach: the first layer formulates the optimization as a generalized convex program, while the {second layer} solves the optimization decentrally using the alternating direction method of multipliers. We analytically show that the proposed two-layered formulation yields a Karush-Kuhn-Tucker point of the original WSEE optimization. We numerically show the influence of weights on the individual EE of the users, which demonstrates the utility of the WSEE metric to incorporate heterogeneous EE requirements of users. We show that low fronthaul capacity reduces the number of users each AP can support, and the cell-free system, consequently, becomes user-centric., Comment: Accepted for publication to the IEEE Open Journal of the Communications Society

Cell-free (CF) massive multiple-input-multiple-output (mMIMO) deployments are usually investigated with half-duplex nodes and high-capacity fronthaul links. To leverage the possible gains in throughput and energy efficiency (EE) of full-duplex (FD) communications, we consider a FD CF mMIMO system with practical limited-capacity fronthaul links. We derive closed-form spectral efficiency (SE) lower bounds for this system with maximum-ratio combining/maximum-ratio transmission processing and optimal uniform quantization. We then optimize the weighted sum EE (WSEE) via downlink and uplink power control by using a {two-layered} approach: the first layer formulates the optimization as a generalized convex program, while the second layer solves the optimization decentrally using alternating direction method of multipliers. We analytically show that the proposed two-layered formulation yields a Karush-Kuhn-Tucker point of the original WSEE optimization. We numerically show the influence of weights on the individual EE of the users, which demonstrates the utility of WSEE metric to incorporate heterogeneous EE requirements of users. We show that the low fronthaul capacity reduces the number of users each AP can support, and the cell-free system, consequently, becomes user-centric., Comment: Accepted in IEEE International Conference on Communications (ICC) 2021. arXiv admin note: substantial text overlap with arXiv:2010.14110

We propose two numerical schemes for approximating quasi-stationary distributions (QSD) of finite state Markov chains with absorbing states. Both schemes are described in terms of certain interacting chains in which the interaction is given in terms of the total time occupation measure of all particles in the system and has the impact of reinforcing transitions, in an appropriate fashion, to states where the collection of particles has spent more time. The schemes can be viewed as combining the key features of the two basic simulation-based methods for approximating QSD originating from the works of Fleming and Viot (1979) and Aldous, Flannery and Palacios (1998), respectively. The key difference between the two schemes studied here is that in the first method one starts with $a(n)$ particles at time $0$ and number of particles stays constant over time whereas in the second method we start with one particle and at most one particle is added at each time instant in such a manner that there are $a(n)$ particles at time $n$. We prove almost sure convergence to the unique QSD and establish Central Limit Theorems for the two schemes under the key assumption that $a(n)=o(n)$. When $a(n)\sim n$, the fluctuation behavior is expected to be non-standard. Some exploratory numerical results are presented to illustrate the performance of the two approximation schemes.

Orthogonal time-frequency space (OTFS) scheme, which transforms a time and frequency selective channel into an almost non-selective channel in the delay-Doppler domain, establishes reliable wireless communication for high-speed moving devices. This work designs and analyzes low-complexity zero-forcing (LZ) and minimum mean square error (LM) receivers for multiple-input multiple-output (MIMO)-OTFS systems with perfect and imperfect receive channel state information (CSI). The proposed receivers provide exactly the same solution as that of the conventional counterparts, and reduce the complexity by exploiting the doubly-circulant nature of the MIMO-OTFS channel matrix, the block-wise inverse, and Schur complement. We also derive, by exploiting the Taylor expansion and results from random matrix theory, a tight approximation of the post-processing signal-to-noise-plus-interference-ratio (SINR) expressions in closed-form for both LZ and LM receivers. We show that the derived SINR expressions, when averaged over multiple channel realizations, accurately characterize their respective bit error rate (BER) of both perfect and imperfect receive CSI. We numerically show the lower BER and lower complexity of the proposed designs over state-of-the-art exiting solutions.

One of the key steps in building deep learning systems for drug classification and generation is the choice of featurization for the molecules. Previous featurization methods have included molecular images, binary strings, graphs, and SMILES strings. This paper proposes the creation of molecular images captioned with binary vectors that encode information not contained in or easily understood from a molecular image alone. Specifically, we use Morgan fingerprints, which encode higher level structural information, and MACCS keys, which encode yes or no questions about a molecules properties and structure. We tested our method on the HIV dataset published by the Pande lab, which consists of 41,127 molecules labeled by if they inhibit the HIV virus. Our final model achieved a state of the art AUC ROC on the HIV dataset, outperforming all other methods. Moreover, the model converged significantly faster than most other methods, requiring dramatically less computational power than unaugmented images.

We study the large deviation behavior of a system of diffusing particles with a mean field interaction, described through a collection of stochastic differential equations, in which each particle is driven by a vanishing independent Brownian noise. An important object in the description of the asymptotic behavior, as the number of particles approach infinity and the noise intensity approaches zero, is the stochastic current associated with the interacting particle system in the sense of Flandoli et al. (2005). We establish a joint large deviation principle (LDP) for the path empirical measure for the particle system and the associated stochastic currents in the simultaneous large particle and small noise limit. Our work extends recent results of Orrieri (2018), in which the diffusion coefficient is taken to be identity, to a setting of a state dependent and possibly degenerate noise with the mean field interaction influencing both the drift and diffusion coefficients, and allows for a stronger topology on the space of stochastic currents in the LDP. Proof techniques differ from Orrieri (2018) and rely on methods from stochastic control, theory of weak convergence, and representation formulas for Laplace functionals of Brownian motions.

We study small noise large deviation asymptotics for stochastic differential equations with a multiplicative noise given as a fractional Brownian motion $B^H$ with Hurst parameter $H>\frac12$. The solutions of the stochastic differential equations are defined pathwise under appropriate conditions on the coefficients. The ingredients in the proof of the large deviation principle, which include a variational representation for nonnegative functionals of fractional Brownian motions and a general sufficient condition for a LDP for a collection of functionals of a fractional Brownian motions, have a broader applicability than the model considered here.

We consider the supermarket model in the usual Markovian setting where jobs arrive at rate $n \lambda_n$ for some $\lambda_n > 0$, with $n$ parallel servers each processing jobs in its queue at rate 1. An arriving job joins the shortest among $d_n \le n$ randomly selected service queues. We show that when $d_n \to \infty$ and $\lambda_n \to \lambda \in (0, \infty)$, under natural conditions on the initial queues, the state occupancy process converges in probability, in a suitable path space, to the unique solution of an infinite system of constrained ordinary differential equations parametrized by $\lambda$. Our main interest is in the study of fluctuations of the state process about its near equilibrium state in the critical regime, namely when $\lambda_n \to 1$. Previous papers have considered the regime $\frac{d_n}{\sqrt{n}\log n} \to \infty$ while the objective of the current work is to develop diffusion approximations for the state occupancy process that allow for all possible rates of growth of $d_n$. In particular we consider the three canonical regimes (a) ${d_n}/{\sqrt{n}} \to 0$; (b) ${d_n}/{\sqrt{n}} \to c\in (0,\infty)$ and, (c) ${d_n}/{\sqrt{n}} \to \infty$. In all three regimes we show, by establishing suitable functional limit theorems, that (under conditions on $\lambda_n$) fluctuations of the state process about its near equilibrium are of order $n^{-1/2}$ and are governed asymptotically by a one dimensional Brownian motion. The forms of the limit processes in the three regimes are quite different; in the first case we get a linear diffusion; in the second case we get a diffusion with an exponential drift; and in the third case we obtain a reflected diffusion in a half space. In the special case ${d_n}/({\sqrt{n}\log n}) \to \infty$ our work gives alternative proofs for the universality results established by Mukherjee et al in 2018., Comment: 45 pages with a 4 page Appendix