Can classical consensus models predict the group behavior of large language models (LLMs)? We examine multi-round interactions among LLM agents through the DeGroot framework, where agents exchange text-based messages over diverse communication graphs. To track opinion evolution, we map each message to an opinion score via sentiment analysis. We find that agents typically reach consensus and the disagreement between the agents decays exponentially. However, the limiting opinion departs from DeGroot's network-centrality-weighted forecast. The consensus between LLM agents turns out to be largely insensitive to initial conditions and instead depends strongly on the discussion subject and inherent biases. Nevertheless, transient dynamics align with classical graph theory and the convergence rate of opinions is closely related to the second-largest eigenvalue of the graph's combination matrix. Together, these findings can be useful for LLM-driven social-network simulations and the design of resource-efficient multi-agent LLM applications.

Algorithmic stability is an established tool for analyzing generalization. While adversarial training enhances model robustness, it often suffers from robust overfitting and an enlarged generalization gap. Recent work has established the convergence of adversarial training in decentralized networks, but its generalization properties remain unexplored. This work presents a stability-based generalization analysis of adversarial training under the diffusion strategy for convex losses. We derive a bound showing that the generalization error grows with both the adversarial perturbation strength and the number of training steps, a finding consistent with the single-agent case but novel for decentralized settings. Numerical experiments on logistic regression validate these theoretical predictions.

Under distributed decision-making (a.k.a. social learning), several agents cooperate according to a communication graph to solve a classification task. A traditional assumption in social learning is that the spatial dependence across the agents’ data is not available, while each individual agent is assumed to know a local model encoding the marginal distribution of its personal data. Recent work shows that, when some joint parametric model is available to describe the spatial dependence, the theory of copulas can be exploited to learn the joint distribution of the data and improve the decision-making performance. However, in most cases a model for the joint distribution is not available. This work addresses this challenging scenario. We propose a fully decentralized strategy where the network of agents reproduces a multilayer perceptron to build a discriminative model capturing the spatial dependence. It is shown that this strategy sensibly outperforms the existing strategies.

This paper studies a stochastic dynamic game between two competing teams, each consisting of a network of collaborating agents. Unlike fully cooperative settings, where all agents share a common objective, each team in this game aims to minimize its own distinct objective. In the adversarial setting, their objectives could be conflicting as in zero-sum games. Throughout the competition, agents share strategic information within their own team while simultaneously inferring and adapting to the strategies of the opposing team. We propose diffusion learning algorithms to address two important classes of this network game: i) a zero-sum game characterized by weak cross-team subgraph interactions, and ii) a general non-zero-sum game exhibiting strong cross-team subgraph interactions. We analyze the stability performance of the proposed algorithms under reasonable assumptions and illustrate the theoretical results through experiments on Cournot team competition and decentralized GAN training.

This study proposes the use of a social learning method to estimate a global state within a multi-agent off-policy actor-critic algorithm for reinforcement learning (RL) operating in a partially observable environment. We assume that the network of agents operates in a fully-decentralized manner, possessing the capability to exchange variables with their immediate neighbors. The proposed design methodology is supported by an analysis demonstrating that the difference between final outcomes, obtained when the global state is fully observed versus estimated through the social learning method, is ε-bounded when an appropriate number of iterations of social learning updates are implemented. Unlike many existing dec-POMDP-based RL approaches, the proposed algorithm is suitable for model-free multi-agent reinforcement learning as it does not require knowledge of a transition model. Furthermore, experimental results illustrate the efficacy of the algorithm and demonstrate its superiority over the current state-of-the-art methods.

Distributed adaptation and learning over directed, unbalanced graphs poses unique challenges due to asymmetric communication and heterogeneous data across nodes. In this work, we introduce a novel class of first-order primal–dual stochastic gradient algorithms for such graphs. Our flagship algorithm, called primal-dual pull diffusion stochastic gradient, is designed to update both the decision variables (primal) and the associated multipliers (dual) using two left-stochastic combination matrices. This design maintains data privacy while ensuring that the estimates remain accurate and unbiased. Building on this, we develop pull-based exact diffusion and pull–push variants that reduce communication costs or eliminate the need for prior knowledge of Perron vector. We also provide a mean-square stability analysis for the primal-dual pull diffusion method, demonstrating the steady-state error proportional to the stepsize. Finally, simulation results on randomly generated directed graphs validate the efficiency of the proposed algorithms and show faster convergence or lower steady-state error compared to existing gradient-tracking-type methods.

The study of tail behaviour of SGD-induced processes has been attracting a lot of interest, due to offering strong guarantees with respect to individual runs of an algorithm. While many works provide high-probability guarantees, quantifying the error rate for a fixed probability threshold, there is a lack of work directly studying the probability of failure, i.e., quantifying the tail decay rate for a fixed error threshold. Moreover, existing results are of finite-time nature, limiting their ability to capture the true long-term tail decay which is more informative for modern learning models, typically trained for millions of iterations. Our work closes these gaps, by studying the long-term tail decay of SGD-based methods through the lens of large deviations theory, establishing several strong results in the process. First, we provide an upper bound on the tails of the gradient norm-squared of the best iterate produced by (vanilla) SGD, for non-convex costs and bounded noise, with long-term decay at rate e^-t/log(t). Next, we relax the noise assumption by considering clipped SGD (c-SGD) under heavy-tailed noise with bounded moment of order p ∈ (1,2], showing an upper bound with long-term decay at rate e^-t^β_p/log(t), where β_p = 4(p-1)/3p-2 for p ∈ (1,2) and e^-t/log^2(t) for p = 2. Finally, we provide lower bounds on the tail decay, at rate e^-t, showing that our rates for both SGD and c-SGD are tight, up to poly-logarithmic factors. Notably, our results demonstrate an order of magnitude faster long-term tail decay compared to existing work based on finite-time bounds, which show rates e^-√(t) and e^-t^β_p/2, p ∈ (1,2], for SGD and c-SGD, respectively. As such, we uncover regimes where the tails decay much faster than previously known, providing stronger long-term guarantees for individual runs.

We propose ZOMA, a unified Zeroth-Order decentralized accelerated MinimAx framework for multi-agent nonconvex Polyak–Łojasiewicz minimax optimization. The proposed framework only requires evaluating the function value and, as such, is tailored to gradient-free environments, where exact gradient information is either unavailable or computationally prohibitive to obtain. A central contribution of our ZOMA framework is a multi-level unification, along the following directions: (i) estimator - our framework adopts a hybrid zeroth-order estimator, which accommodates, among others, both coordinate-wise and randomized uniform smoothing estimators; (ii) bias correction - our framework subsumes a wide range of bias-correction strategies, including gradient tracking (GT), exact diffusion (ED), and EXTRA and (iii) acceleration - our framework facilitates a broad class of acceleration techniques, including zeroth-order versions of STORM, PAGE, and L2S. The general nature of ZOMA leads to many novel decentralized zeroth-order minimax methods and allows us to establish unified convergence guarantees, matching the performance of state-of-the-art centralized zeroth-order minimax methods, while providing benefits, such as linear speed-up in the number of users. The unified framework also provides a systematic way to assess algorithmic suitability by specializing the convergence rates to specific problem structures and method designs. We validate the performance of the proposed algorithms via numerical simulations.

Gradient-free (zeroth-order) optimization is well-suited for black-box optimization and memory-efficient learning when gradient information is unavailable or costly to obtain. In this paper, we study decentralized stochastic and finite-sum nonconvex optimization using only function information from a stochastic zeroth-order oracle. We develop coordinate-wise zeroth-order algorithms built on a first-order decentralized learning framework that subsumes a broad class of decentralized strategies, including adapt-then-combine gradient-tracking, exact diffusion, and EXTRA. To further improve the zeroth-order gradient estimator, we incorporate a probabilistic variance reduction strategy based on the GRACE acceleration strategy. We establish convergence guarantees for the proposed methods over sparsely connected networks and identify parameter regimes that achieve linear speedup with respect to the number of agents K in stochastic settings. In particular, the zeroth-order exact diffusion method attains a favorable per-agent function query complexity of $\mathcal{O}\left( {\frac{{d{\varepsilon ^{ - 3}}}}{K} + \frac{{d{\varepsilon ^{ - 2}}}}{{{{(1 - \lambda )}^2}}}} \right)$ in the stochastic regime, and $\mathcal{O}\left( {\frac{{d\sqrt N {\varepsilon ^{ - 2}}}}{{\sqrt K {{(1 - \lambda )}^2}}}} \right)$ in the finite-sum regime, among others, for finding a ε-stationary point, where λ denotes the second largest eigenvalue in magnitude of the communication matrix, N is the local sample size, and d is the problem dimension. Numerical experiments illustrate the effectiveness of the proposed decentralized gradient-free method.

The increasing heterogeneity of multi-agent systems poses significant challenges for jointly training a global model across agents. At the same time, cooperative inference between agents has long been recognized as a powerful mechanism for distributed decision making over networks. Motivated by these observations, we propose a collaboration framework for distributed binary classification over multi-agent networks, where a set of independently trained agents, potentially differing in architecture, feature space, or modality, coordinate their actions during test time to form collective predictions. This coordination is achieved by exchanging local decision statistics through a distributed learning protocol. We develop a theoretical and experimental study of this independent training and cooperative inference paradigm, and examine its performance under different communication budgets and distributed learning rules. We establish classification error guarantees under sufficient, finite-round, and finite-precision communication, together with PAC-style generalization bounds. These results capture the influence of model heterogeneity, network topology, combination policy, and communication constraints on prediction accuracy. Taken together with the experimental results, they reveal both the price of independent training and the benefit of collective prediction for the proposed distributed decision making framework with models learned from data.

Convergence in high-probability (HP) has attracted increasing interest, due to implying exponentially decaying tail bounds and strong guarantees for individual runs of an algorithm. While many works study HP guarantees in centralized settings, much less is understood in the decentralized setup, where existing works require strong assumptions, like uniformly bounded gradients, or asymptotically vanishing noise. This results in a significant gap between the assumptions used to establish convergence in the HP and the mean-squared error (MSE) sense, and is also contrary to centralized settings, where it is known that 𝚂𝙶𝙳 converges in HP under the same conditions on the cost function as needed for MSE convergence. Motivated by these observations, we study the HP convergence of Decentralized 𝚂𝙶𝙳 (𝙳𝚂𝙶𝙳) in the presence of light-tailed noise, providing several strong results. First, we show that 𝙳𝚂𝙶𝙳 converges in HP under the same conditions on the cost as in the MSE sense, removing the restrictive assumptions used in prior works. Second, our sharp analysis yields order-optimal rates for both non-convex and strongly convex costs. Third, we establish a linear speed-up in the number of users, leading to matching or strictly better transient times than those obtained from MSE results, further underlining the tightness of our analysis. To the best of our knowledge, this is the first work that shows 𝙳𝚂𝙶𝙳 achieves a linear speed-up in the HP sense. Our relaxed assumptions and sharp rates stem from several technical results of independent interest, including a result on the variance-reduction effect of decentralized methods in the HP sense, as well as a novel bound on the moment-generating function of strongly convex costs, of interest even in centralized settings. Numerical experiments validate our theory.

This work addresses stochastic multi-network competing problems. Unlike prior studies that focus exclusively on the two-team (or two-network) case, we consider the more general setting with T ≥ 2 teams. In the multi-team setting, each team consists of a set of partially connected cooperative agents, and there exists at least one link between any two teams to enable information exchange. We propose a new algorithm based on a diffusion learning strategy, which subsumes the existing two-team competing algorithm as a special case. We show that the state vectors generated by the learning algorithm asymptotically converge to a neighborhood of the Nash equilibrium in the sense of first-, second-, and fourth-order error moments. The theoretical claims are supported by simulations on a quadratic game.

Non-Bayesian social learning (NBSL) is a framework in distributed inference that describes how agents in a network combine local observations and their neighbors’ beliefs to infer a hidden state. While the framework has been extensively analyzed in theory, its role as a model of belief formation in realistic multi-agent systems remains under-explored. In this work, we investigate whether NBSL can capture the belief dynamics of interacting large language model (LLM) agents. We design controlled experiments in which LLM agents revise their beliefs sequentially based on local evidence and exchanges with their neighbors. Then, we compare their belief trajectories to those predicted by NBSL. Our results provide the first empirical evaluations of NBSL on modern AI collectives, and show that LLM networks can exhibit belief evolution patterns that closely follow NBSL dynamics.

Lower-bound analyses for nonconvex strongly-concave minimax optimization problems have shown that stochastic first-order algorithms require at least 𝒪(ε^-4) sample complexity to find an ε-stationary point. Some works indicate that this complexity can be improved to 𝒪(ε^-3) when the stochastic loss gradient is Lipschitz continuous. The question of achieving enhanced convergence rates under distinct conditions, remains open. In this work, we address this question for optimization problems that are nonconvex in the minimization variable and strongly concave or Polyak-Lojasiewicz (PL) in the maximization variable. We introduce novel bias-corrected momentum algorithms utilizing efficient Hessian-vector products. We establish convergence conditions and demonstrate a lower iteration complexity of 𝒪(ε^-3) for the proposed algorithms. The effectiveness of the proposed method is validated through applications to robust logistic regression and robust adaptive cruise control.

Social Learning is a decentralized decision-making paradigm in which spatially dispersed agents collect streaming observations regulated by one of a finite number of models (the hypotheses). The agents are interested in assigning probability scores (the beliefs) to the possible hypotheses. To this end, the agents exchange their beliefs according to a certain communication graph. It has been shown that, under reasonable conditions on the identifiability of the decision model and the network connectivity, each agent ultimately places all the belief mass on the true hypothesis governing the data. However, several questions remain unanswered regarding the evaluation of the social learning performance. One recently adopted performance metric is the rejection rate, i.e., the rate at which the beliefs about the erroneous hypotheses vanish. One contribution of this work is to establish that the rejection rate leads to several paradoxes, which make it unsuitable as a valid performance measure. We then focus on studying the error probability measure. For a binary Gaussian problem, we derive an analytical formula characterizing the ratio between the individual agents' probabilities and the optimal Bayesian probability. The formula shows that this ratio is expressed by the product of two terms quantifying the effect of the network connectivity and the role of the prior information. As a result, an irreducible gap emerges between the decentralized and the centralized error probabilities, which is agent-dependent and does not disappear asymptotically.

This work leverages adaptive social learning to estimate partially observable global states in multi-agent reinforcement learning (MARL) problems. Unlike existing methods, the proposed approach enables the concurrent operation of social learning and reinforcement learning. Specifically, it alternates between a single step of social learning and a single step of MARL, eliminating the need for the time- and computation-intensive two-timescale learning frameworks. Theoretical guarantees are provided to support the effectiveness of the proposed method. Simulation results verify that the performance of the proposed methodology can approach that of reinforcement learning when the true state is known.

In social learning, a network of agents assigns probability scores (beliefs) to some hypotheses of interest, which rule the generation of local streaming data observed by each agent. Belief formation takes place by means of an iterative two-step procedure where: i) the agents update locally their beliefs by using some likelihood models; and ii) the updated beliefs are combined with the beliefs of the neighboring agents, using a pooling rule. This procedure can fail to perform well in the presence of dynamic drifts, leading the agents to incorrect decision making. Here, we focus on the fully online setting where both the true hypothesis and the likelihood models can change over time. We propose the doubly adaptive social learning (A2 SL) strategy, which infuses social learning with the necessary adaptation capabilities. This goal is achieved by exploiting two adaptation stages: i) a stochastic gradient descent update to learn and track the drifts in the decision model; ii) and an adaptive belief update to track the true hypothesis changing over time. These stages are controlled by two adaptation parameters that govern the evolution of the error probability for each agent. We show that all agents learn consistently for sufficiently small adaptation parameters, in the sense that they ultimately place all their belief mass on the true hypothesis. In particular, the probability of choosing the wrong hypothesis converges to values on the order of the adaptation parameters. The theoretical analysis is illustrated both on synthetic data and by applying the A2 SL strategy to a social learning problem in the online setting using real data.

Distributed decision-making over graphs involves a group of agents that collaboratively work toward a common objective. In the social learning framework, the agents are tasked to infer an unknown state from a finite set by using a stream of local observations. The probability of decision errors for each agent asymptotically converges to zero at an exponential rate, characterized by the error exponent, which depends on the combination policy employed by the network. This work addresses the challenge of identifying optimal combination policies to maximize the error exponent for the true state while ensuring the errors for all other states converge to zero as well. We derive an upper bound on the achievable error exponent under the social learning rule, and then establish conditions for the combination policy to reach this upper bound. Moreover, we examine the performance loss scenarios when the combination policy is chosen inappropriately. From a geometric perspective, each combination policy induces a weighted nearest neighbor classifier where the weights correspond to the agents' Perron centralities. By implementing an optimized combination policy, we enhance the error exponent, leading to improved accuracy and efficiency in the distributed decision-making process.

This paper proposes a theoretical framework to evaluate and compare the performance of stochastic gradient algorithms for distributed learning in relation to their behavior around local minima in nonconvex environments. Previous works have noticed that convergence toward flat local minima tend to enhance the generalization ability of learning algorithms. This work discovers three interesting results. First, it shows that decentralized learning strategies are able to escape faster away from local minima and favor convergence toward flatter minima relative to the centralized solution. Second, in decentralized methods, the consensus strategy has a worse excess-risk performance than diffusion, giving it a better chance of escaping from local minima and favoring flatter minima. Third, and importantly, the ultimate classification accuracy is not solely dependent on the flatness of the local minimum but also on how well a learning algorithm can approach that minimum. In other words, the classification accuracy is a function of both flatness and optimization performance. In this regard, since diffusion has a lower excess-risk than consensus, when both algorithms are trained starting from random initial points, diffusion enhances the classification accuracy. The paper examines the interplay between the two measures of flatness and optimization error closely. One important conclusion is that decentralized strategies deliver in general enhanced classification accuracy because they strike a more favorable balance between flatness and optimization performance compared to the centralized solution.

Diffusion learning is a framework that endows edge devices with advanced intelligence. By processing and analyzing data locally and allowing each agent to communicate with its immediate neighbors, diffusion effectively protects the privacy of edge devices, enables real-time response, and reduces reliance on central servers. However, traditional diffusion learning relies on communication at every iteration, leading to communication overhead, especially with large learning models. Furthermore, the inherent volatility of edge devices, stemming from power outages or signal loss, poses challenges to reliable communication between neighboring agents. To mitigate these issues, this paper investigates an enhanced diffusion learning approach incorporating local updates and partial agent participation. Local updates will curtail communication frequency, while partial agent participation will allow for the inclusion of agents based on their availability. We prove that the resulting algorithm is stable in the mean-square error sense and provide a tight analysis of its Mean-Square-Deviation (MSD) performance. Various numerical experiments are conducted to illustrate our theoretical findings.