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Vikram Krishnamurthy

Publications and source records attributed to Vikram Krishnamurthy.

At least 37 records · Page 2Linked to original sources
arXiv2025

Interacting Large Language Model Agents. Interpretable Models and Social Learning

This paper discusses the theory and algorithms for interacting large language model agents (LLMAs) using methods from statistical signal processing and microeconomics. While both fields are mature, their application to decision-making involving interacting LLMAs remains unexplored. Motivated by Bayesian sentiment analysis on online platforms, we construct interpretable models and algorithms that enable LLMAs to interact and perform Bayesian inference. Because interacting LLMAs learn from both prior decisions and external inputs, they can exhibit bias and herding behavior. Thus, developing interpretable models and stochastic control algorithms is essential to understand and mitigate these behaviors. This paper has three main results. First, we show using Bayesian revealed preferences from microeconomics that an individual LLMA satisfies the necessary and sufficient conditions for rationally inattentive (bounded rationality) Bayesian utility maximization and, given an observation, the LLMA chooses an action that maximizes a regularized utility. Second, we utilize Bayesian social learning to construct interpretable models for LLMAs that interact sequentially with each other and the environment while performing Bayesian inference. Our proposed models capture the herding behavior exhibited by interacting LLMAs. Third, we propose a stochastic control framework to delay herding and improve state estimation accuracy under 2 settings: (a) centrally controlled LLMAs (b) autonomous LLMAs with incentives. We demonstrate the effectiveness of our methods on real datasets for hate speech classification and product quality assessment, using open-source models like LLaMA and closed-source models like ChatGPT. The main takeaway of this paper, based on empirical analysis and mathematical formalism, is that LLMAs act as rationally bounded Bayesian agents that exhibit social learning when interacting.

cs.LG↗
arXiv2025

Approximate MLE of High-Dimensional STAP Covariance Matrices with Banded & Spiked Structure -- A Convex Relaxation Approach

Estimating the clutter-plus-noise covariance matrix in high-dimensional STAP is challenging in the presence of Internal Clutter Motion (ICM) and a high noise floor. The problem becomes more difficult in low-sample regimes, where the Sample Covariance Matrix (SCM) becomes ill-conditioned. To capture the ICM and high noise floor, we model the covariance matrix using a ``Banded+Spiked'' structure. Since the Maximum Likelihood Estimation (MLE) for this model is non-convex, we propose a convex relaxation which is formulated as a Frobenius norm minimization with non-smooth convex constraints enforcing banded sparsity. This relaxation serves as a provable upper bound for the non-convex likelihood maximization and extends to cases where the covariance matrix dimension exceeds the number of samples. We derive a variational inequality-based bound to assess its quality. We introduce a novel algorithm to jointly estimate the banded clutter covariance and noise power. Additionally, we establish conditions ensuring the estimated covariance matrix remains positive definite and the bandsize is accurately recovered. Numerical results using the high-fidelity RFView radar simulation environment demonstrate that our algorithm achieves a higher Signal-to-Clutter-plus-Noise Ratio (SCNR) than state-of-the-art methods, including TABASCO, Spiked Covariance Stein Shrinkage, and Diagonal Loading, particularly when the covariance matrix dimension exceeds the number of samples.

eess.SP↗
arXiv2025

Information Diffusion and Preferential Attachment in a Network of Large Language Models

This paper models information diffusion in a network of Large Language Models (LLMs) that is designed to answer queries from distributed datasets, where the LLMs can hallucinate the answer. We introduce a two-time-scale dynamical model for the centrally administered network, where opinions evolve faster while the network's degree distribution changes more slowly. Using a mean-field approximation, we establish conditions for a locally asymptotically stable equilibrium where all LLMs remain truthful. We provide approximation guarantees for the mean-field approximation and a singularly perturbed approximation of the two-time-scale system. To mitigate hallucination and improve the influence of truthful nodes, we propose a reputation-based preferential attachment mechanism that reconfigures the network based on LLMs' evaluations of their neighbors. Numerical experiments on an open-source LLM (LLaMA-3.1-8B) validate the efficacy of our preferential attachment mechanism and demonstrate the optimization of a cost function for the two-time-scale system.

cs.SI↗
arXiv2025

Detecting Structural Shifts in Multivariate Hawkes Processes with Fréchet Statistics

This paper proposes a new approach for change point detection in multivariate Hawkes processes using Fréchet statistic of a network. The method splits the point process into overlapping windows, estimates kernel matrices in each window, and reconstructs the signed Laplacians by treating the kernel matrices as the adjacency matrices of the causal network. We demonstrate the effectiveness of our method through experiments on both simulated and cryptocurrency datasets. Our results show that our method is capable of accurately detecting and characterizing changes in the causal structure of multivariate Hawkes processes, and may have potential applications in fields such as finance and neuroscience. The proposed method is an extension of previous work on Fréchet statistics in point process settings and represents an important contribution to the field of change point detection in multivariate point processes.

stat.ML↗
arXiv2025

Finite-Sample Bounds for Adaptive Inverse Reinforcement Learning using Passive Langevin Dynamics

This paper provides a finite-sample analysis of a passive stochastic gradient Langevin dynamics (PSGLD) algorithm. This algorithm is designed to achieve adaptive inverse reinforcement learning (IRL). Adaptive IRL aims to estimate the cost function of a forward learner performing a stochastic gradient algorithm (e.g., policy gradient reinforcement learning) by observing their estimates in real-time. The PSGLD algorithm is considered passive because it incorporates noisy gradients provided by an external stochastic gradient algorithm (forward learner), of which it has no control. The PSGLD algorithm acts as a randomized sampler to achieve adaptive IRL by reconstructing the forward learner's cost function nonparametrically from the stationary measure of a Langevin diffusion. This paper analyzes the non-asymptotic (finite-sample) performance; we provide explicit bounds on the 2-Wasserstein distance between PSGLD algorithm sample measure and the stationary measure encoding the cost function, and provide guarantees for a kernel density estimation scheme which reconstructs the cost function from empirical samples. Our analysis uses tools from the study of Markov diffusion operators. The derived bounds have both practical and theoretical significance. They provide finite-time guarantees for an adaptive IRL mechanism, and substantially generalize the analytical framework of a line of research in passive stochastic gradient algorithms.

cs.LG↗
arXiv2024

Slow Convergence of Interacting Kalman Filters in Word-of-Mouth Social Learning

We consider word-of-mouth social learning involving $m$ Kalman filter agents that operate sequentially. The first Kalman filter receives the raw observations, while each subsequent Kalman filter receives a noisy measurement of the conditional mean of the previous Kalman filter. The prior is updated by the $m$-th Kalman filter. When $m=2$, and the observations are noisy measurements of a Gaussian random variable, the covariance goes to zero as $k^{-1/3}$ for $k$ observations, instead of $O(k^{-1})$ in the standard Kalman filter. In this paper we prove that for $m$ agents, the covariance decreases to zero as $k^{-(2^m-1)}$, i.e, the learning slows down exponentially with the number of agents. We also show that by artificially weighing the prior at each time, the learning rate can be made optimal as $k^{-1}$. The implication is that in word-of-mouth social learning, artificially re-weighing the prior can yield the optimal learning rate.

cs.LG↗
arXiv2024

Distributionally Robust Inverse Reinforcement Learning for Identifying Multi-Agent Coordinated Sensing

We derive a minimax distributionally robust inverse reinforcement learning (IRL) algorithm to reconstruct the utility functions of a multi-agent sensing system. Specifically, we construct utility estimators which minimize the worst-case prediction error over a Wasserstein ambiguity set centered at noisy signal observations. We prove the equivalence between this robust estimation and a semi-infinite optimization reformulation, and we propose a consistent algorithm to compute solutions. We illustrate the efficacy of this robust IRL scheme in numerical studies to reconstruct the utility functions of a cognitive radar network from observed tracking signals.

cs.LG↗
arXiv2024

Large Deviations Analysis For Regret Minimizing Stochastic Approximation Algorithms

Motivated by learning of correlated equilibria in non-cooperative games, we perform a large deviations analysis of a regret minimizing stochastic approximation algorithm. The regret minimization algorithm we consider comprises multiple agents that communicate over a graph to coordinate their decisions. We derive an exponential decay rate towards the algorithm's stable point using large deviations theory. Our analysis leverages the variational representation of the Laplace functionals and weak convergence methods to characterize the exponential decay rate.

math.OC↗
arXiv2024

Eclipse Attack Detection on a Blockchain Network as a Non-Parametric Change Detection Problem

This paper introduces a novel non-parametric change detection algorithm to identify eclipse attacks on a blockchain network; the non-parametric algorithm relies only on the empirical mean and variance of the dataset, making it highly adaptable. An eclipse attack occurs when malicious actors isolate blockchain users, disrupting their ability to reach consensus with the broader network, thereby distorting their local copy of the ledger. To detect an eclipse attack, we monitor changes in the Fréchet mean and variance of the evolving blockchain communication network connecting blockchain users. First, we leverage the Johnson-Lindenstrauss lemma to project large-dimensional networks into a lower-dimensional space, preserving essential statistical properties. Subsequently, we employ a non-parametric change detection procedure, leading to a test statistic that converges weakly to a Brownian bridge process in the absence of an eclipse attack. This enables us to quantify the false alarm rate of the detector. Our detector can be implemented as a smart contract on the blockchain, offering a tamper-proof and reliable solution. Finally, we use numerical examples to compare the proposed eclipse attack detector with a detector based on the random forest model.

cs.CR↗
arXiv2024

Structured Reinforcement Learning for Incentivized Stochastic Covert Optimization

This paper studies how a stochastic gradient algorithm (SG) can be controlled to hide the estimate of the local stationary point from an eavesdropper. Such problems are of significant interest in distributed optimization settings like federated learning and inventory management. A learner queries a stochastic oracle and incentivizes the oracle to obtain noisy gradient measurements and perform SG. The oracle probabilistically returns either a noisy gradient of the function} or a non-informative measurement, depending on the oracle state and incentive. The learner's query and incentive are visible to an eavesdropper who wishes to estimate the stationary point. This paper formulates the problem of the learner performing covert optimization by dynamically incentivizing the stochastic oracle and obfuscating the eavesdropper as a finite-horizon Markov decision process (MDP). Using conditions for interval-dominance on the cost and transition probability structure, we show that the optimal policy for the MDP has a monotone threshold structure. We propose searching for the optimal stationary policy with the threshold structure using a stochastic approximation algorithm and a multi-armed bandit approach. The effectiveness of our methods is numerically demonstrated on a covert federated learning hate-speech classification task.

cs.LG↗
arXiv2024

Identifying Hate Speech Peddlers in Online Platforms. A Bayesian Social Learning Approach for Large Language Model Driven Decision-Makers

This paper studies the problem of autonomous agents performing Bayesian social learning for sequential detection when the observations of the state belong to a high-dimensional space and are expensive to analyze. Specifically, when the observations are textual, the Bayesian agent can use a large language model (LLM) as a map to get a low-dimensional private observation. The agent performs Bayesian learning and takes an action that minimizes the expected cost and is visible to subsequent agents. We prove that a sequence of such Bayesian agents herd in finite time to the public belief and take the same action disregarding the private observations. We propose a stopping time formulation for quickest time herding in social learning and optimally balance privacy and herding. Structural results are shown on the threshold nature of the optimal policy to the stopping time problem. We illustrate the application of our framework when autonomous Bayesian detectors aim to sequentially identify if a user is a hate speech peddler on an online platform by parsing text observations using an LLM. We numerically validate our results on real-world hate speech datasets. We show that autonomous Bayesian agents designed to flag hate speech peddlers in online platforms herd and misclassify the users when the public prior is strong. We also numerically show the effect of a threshold policy in delaying herding.

cs.SI↗
arXiv2024

Fisher Information Approach for Masking the Sensing Plan: Applications in Multifunction Radars

How to design a Markov Decision Process (MDP) based radar controller that makes small sacrifices in performance to mask its sensing plan from an adversary? The radar controller purposefully minimizes the Fisher information of its emissions so that an adversary cannot identify the controller's model parameters accurately. Unlike classical open loop statistical inference, where the Fisher information serves as a lower bound for the achievable covariance, this paper employs the Fisher information as a design constraint for a closed loop radar controller to mask its sensing plan. We analytically derive a closed-form expression for the determinant of the Fisher Information Matrix (FIM) pertaining to the parameters of the MDP-based controller. Subsequently, we constrain the MDP with respect to the determinant of the FIM. Numerical results show that the introduction of minor perturbations to the MDP's transition kernel and the total operation cost can reduce the Fisher Information of the emissions. Consequently, this reduction amplifies the variability in policy and transition kernel estimation errors, thwarting the adversary's accuracy in estimating the controller's sensing plan.

eess.SY↗
arXiv2023

Anomalous Edge Detection in Edge Exchangeable Social Network Models

This paper studies detecting anomalous edges in directed graphs that model social networks. We exploit edge exchangeability as a criterion for distinguishing anomalous edges from normal edges. Then we present an anomaly detector based on conformal prediction theory; this detector has a guaranteed upper bound for false positive rate. In numerical experiments, we show that the proposed algorithm achieves superior performance to baseline methods.

cs.SI↗
arXiv2023

Multiple-stopping time Sequential Detection for Energy Efficient Mining in Blockchain-Enabled IoT

What are the optimal times for an Internet of Things (IoT) device to act as a blockchain miner? The aim is to minimize the energy consumed by low-power IoT devices that log their data into a secure (tamper-proof) distributed ledger. We formulate a multiple stopping time Bayesian sequential detection problem to address energy-efficient blockchain mining for IoT devices. The objective is to identify $L$ optimal stops for mining, thereby maximizing the probability of successfully adding a block to the blockchain; we also present a model to optimize the number of stops (mining instants). The formulation is equivalent to a multiple stopping time POMDP. Since POMDPs are in general computationally intractable to solve, we show mathematically using submodularity arguments that the optimal mining policy has a useful structure: 1) it is monotone in belief space, and 2) it exhibits a threshold structure, which divides the belief space into two connected sets. Exploiting the structural results, we formulate a computationally-efficient linear mining policy for the blockchain-enabled IoT device. We present a policy gradient technique to optimize the parameters of the linear mining policy. Finally, we use synthetic and real Bitcoin datasets to study the performance of our proposed mining policy. We demonstrate the energy efficiency achieved by the optimal linear mining policy in contrast to other heuristic strategies.

cs.CR↗
arXiv2023

Controlling Federated Learning for Covertness

A learner aims to minimize a function $f$ by repeatedly querying a distributed oracle that provides noisy gradient evaluations. At the same time, the learner seeks to hide $\arg\min f$ from a malicious eavesdropper that observes the learner's queries. This paper considers the problem of \textit{covert} or \textit{learner-private} optimization, where the learner has to dynamically choose between learning and obfuscation by exploiting the stochasticity. The problem of controlling the stochastic gradient algorithm for covert optimization is modeled as a Markov decision process, and we show that the dynamic programming operator has a supermodular structure implying that the optimal policy has a monotone threshold structure. A computationally efficient policy gradient algorithm is proposed to search for the optimal querying policy without knowledge of the transition probabilities. As a practical application, our methods are demonstrated on a hate speech classification task in a federated setting where an eavesdropper can use the optimal weights to generate toxic content, which is more easily misclassified. Numerical results show that when the learner uses the optimal policy, an eavesdropper can only achieve a validation accuracy of $52\%$ with no information and $69\%$ when it has a public dataset with 10\% positive samples compared to $83\%$ when the learner employs a greedy policy.

cs.LG↗
arXiv2023

Unifying Revealed Preference and Revealed Rational Inattention

This paper unifies two key results from economic theory, namely, revealed rational inattention and classical revealed preference. Revealed rational inattention tests for rationality of information acquisition for Bayesian decision makers. On the other hand, classical revealed preference tests for utility maximization under known budget constraints. Our first result is an equivalence result - we unify revealed rational inattention and revealed preference through an equivalence map over decision parameters and partial order for payoff monotonicity over the decision space in both setups. Second, we exploit the unification result computationally to extend robustness measures for goodness-of-fit of revealed preference tests in the literature to revealed rational inattention. This extension facilitates quantifying how well a Bayesian decision maker's actions satisfy rational inattention. Finally, we illustrate the significance of the unification result on a real-world YouTube dataset comprising thumbnail, title and user engagement metadata from approximately 140,000 videos. We compute the Bayesian analog of robustness measures from revealed preference literature on YouTube metadata features extracted from a deep auto-encoder, i.e., a deep neural network that learns low-dimensional features of the metadata. The computed robustness values show that YouTube user engagement fits the rational inattention model remarkably well. All our numerical experiments are completely reproducible.

econ.TH↗
arXiv2023

Mitigating Misinformation Spread on Blockchain Enabled Social Media Networks

The paper develops a blockchain protocol for a social media network (BE-SMN) to mitigate the spread of misinformation. BE-SMN is derived based on the information transmission-time distribution by modeling the misinformation transmission as double-spend attacks on blockchain. The misinformation distribution is then incorporated into the SIR (Susceptible, Infectious, or Recovered) model, which substitutes the single rate parameter in the traditional SIR model. Then, on a multi-community network, we study the propagation of misinformation numerically and show that the proposed blockchain enabled social media network outperforms the baseline network in flattening the curve of the infected population.

cs.SI↗