arXiv subjects

Vikram Krishnamurthy

Publications and source records attributed to Vikram Krishnamurthy.

At least 19 recordsLinked to original sources

arXiv2026

Mitigating LLM sycophancy with RL-based fine-tuning: Bayesian Truth Serum approach

Large language models (LLMs) frequently exhibit \emph{sycophancy}: they adapt their answers to a user's stated beliefs or preferences instead of reporting what they hold to be true, which lowers factual accuracy and can amplify misinformation. This paper proposes a methodology for mitigating sycophancy that employs the Bayesian Truth Serum (BTS), a peer-prediction mechanism, as the reward in Group Relative Policy Optimization (GRPO) to fine-tune an LLM. BTS pays an answer for being \emph{surprisingly common}, that is, more frequent among respondents than those respondents themselves predicted. We treat a group of responses from a model for one question as those respondents, so the reward is a function of the model's own outputs and fine-tuning needs neither labels nor preference annotations. We prove that in the large-group limit a sycophantic response earns strictly lower expected reward than an honest one. We also prove that if the entire group agrees in advance on a symmetric answering rule, it cannot earn a higher information score than under truthful reporting. On our true/false benchmark the reference model's answer-flip rate under user pressure decreases from 23% to 4%, and its accuracy under that pressure increases from 80% to 93%. Our reward outperforms SMART and is comparable to synthetic-data fine-tuning and to pinpoint tuning, all three of which train on labels. It spends considerably more compute in exchange, which makes it suitable when labeled data is scarce. Peer Truth Serum, which also pays a premium for a rare answer but elicits no prediction report, reproduces the effect. A peer-prediction reward computed inside a single GRPO group therefore reduces sycophancy without labels, and comparing mechanisms suggests that the premium paid for a rarer answer drives the effect.

cs.LG↗

arXiv2026

Emergence of Preferential Attachment and Glass-Ceiling Effects in Autonomous Networks of LLMs

We investigate the emergence of structural disparities in networks of collaborating large language model (LLM) agents. When LLM agents autonomously choose collaborators, the resulting communication network exhibits preferential-attachment dynamics: agents that are already prominent become increasingly likely to attract additional connections. In some cases, weaker LLM agents (agents with smaller base model or older version) can disproportionately occupy central and influential network positions relative to stronger LLM agents. We interpret this as a type-dependent glass-ceiling effect (GCE). We model the network of LLM agents as a time-evolving sequence of directed weighted graphs, where the vector-valued edge weights represent cumulative tokens exchanged, number of interaction rounds, and reasoning effort. Using a contraction mapping argument on the mean-field dynamics, we prove that the importance (centrality) of each agent type converges to a unique stable equilibrium. To ground the model in LLM decision mechanisms, we introduce a cross-attention-inspired utility for collaborator selection. This utility specifies the local connection dynamics and, together with the mean-field model, yields a predictive characterization of the limiting network structure and its type-dependent centrality gaps. To validate the theory, we develop an experimental testbed with 100 LLM agents. Our experiments show that autonomous network formation can generate persistent centrality disparities, with their magnitude and direction depending on model family, model size, system-prompt design, and task context. They further show that the effect of preferential attachment depends on its alignment with model capability: reinforcing it improves collective performance when stronger agents become central, whereas weakening it improves performance when network dynamics instead favor weaker agents.

cs.SI↗

arXiv2026

Malliavin Calculus for Counterfactual Gradient Estimation in Adaptive Inverse Reinforcement Learning

Inverse reinforcement learning (IRL) recovers the loss function of a forward learner from its observed responses. Adaptive IRL aims to reconstruct the loss function of a forward learner by passively observing its gradients as it performs reinforcement learning (RL). This paper proposes a novel passive Langevin-based algorithm that achieves adaptive IRL. The key difficulty in adaptive IRL is that the required gradients in the passive algorithm are counterfactual, that is, they are conditioned on events of probability zero under the forward learner's trajectory. Therefore, naive Monte Carlo estimators are prohibitively inefficient, and kernel smoothing, though common, suffers from slow convergence. We overcome this by employing Malliavin calculus to efficiently estimate the required counterfactual gradients. We reformulate the counterfactual conditioning as a ratio of unconditioned expectations involving Malliavin quantities, thus recovering standard estimation rates. We derive the necessary Malliavin derivatives and their adjoint Skorohod integral formulations for a general Langevin structure, and provide a concrete algorithmic approach which exploits these for counterfactual gradient estimation.

cs.LG↗

arXiv2026

Efficient Counterfactual Estimation of Conditional Greeks via Malliavin-based Weak Derivatives

We study counterfactual gradient estimation of conditional loss functionals of diffusion processes. In quantitative finance, these gradients are known as conditional Greeks: the sensitivity of expected market values, conditioned on some event of interest. The difficulty is that when the conditioning event has vanishing or zero probability, naive Monte Carlo estimators are prohibitively inefficient; kernel smoothing, though common, suffers from slow convergence. We propose a two-stage kernel-free methodology. First, we show using Malliavin calculus that the conditional loss functional of a diffusion process admits an exact representation as a Skorohod integral, yielding classical Monte-Carlo estimator variance and convergence rates. Second, we establish that a weak derivative estimate of the conditional loss functional with respect to model parameters can be evaluated algorithmically with constant variance, in contrast to the widely used score function method whose variance grows linearly in the sample path length. Together, these results yield an efficient framework for counterfactual conditional stochastic gradient algorithms and financial Greek computations in rare-event regimes.

cs.CE↗

arXiv2026

LLMs as High-Dimensional Nonlinear Autoregressive Models with Attention: Training, Alignment and Inference

Large language models (LLMs) based on transformer architectures are typically described through collections of architectural components and training procedures, obscuring their underlying computational structure. This review article provides a concise mathematical reference for researchers seeking an explicit, equation-level description of LLM training, alignment, and generation. We formulate LLMs as high-dimensional nonlinear autoregressive models with attention-based dependencies. The framework encompasses pretraining via next-token prediction, alignment methods such as reinforcement learning from human feedback (RLHF), direct preference optimization (DPO), rejection sampling fine-tuning (RSFT), and reinforcement learning from verifiable rewards (RLVR), as well as autoregressive generation during inference. Self-attention emerges naturally as a repeated bilinear--softmax--linear composition, yielding highly expressive sequence models. This formulation enables principled analysis of alignment-induced behaviors (including sycophancy), inference-time phenomena (such as hallucination, in-context learning, chain-of-thought prompting, and retrieval-augmented generation), and extensions like continual learning, while serving as a concise reference for interpretation and further theoretical development.

cs.LG↗

arXiv2026

Emergence of Structural Disparities in the Web of Scientific Citations

Scientific attention is unevenly distributed, creating inequities in recognition and distorting access to opportunities. Using citations as a proxy, we quantify disparities in attention by gender and institutional prestige. We find that women receive systematically fewer citations than men, and that attention is increasingly concentrated among authors from elite institutions -- patterns not fully explained by underrepresentation alone. To explain these dynamics, we introduce a model of citation network growth that incorporates homophily (tendency to cite similar authors), preferential attachment (favoring highly cited authors) and group size (underrepresentation). The model shows that disparities arise not only from group size imbalances but also from cumulative advantage amplifying biased citation preferences. Importantly, increasing representation alone is often insufficient to reduce disparities. Effective strategies should also include reducing homophily, amplifying the visibility of underrepresented groups, and supporting equitable integration of newcomers. Our findings highlight the challenges of mitigating inequities in asymmetric networks like citations, where recognition flows in one direction. By making visible the mechanisms through which attention is distributed, we contribute to efforts toward a more responsible web of science that is fairer, more transparent, and more inclusive, and that better sustains innovation and knowledge production.

physics.soc-ph↗

arXiv2025

Why Most Optimism Bandit Algorithms Have the Same Regret Analysis: A Simple Unifying Theorem

Several optimism-based stochastic bandit algorithms -- including UCB, UCB-V, linear UCB, and finite-arm GP-UCB -- achieve logarithmic regret using proofs that, despite superficial differences, follow essentially the same structure. This note isolates the minimal ingredients behind these analyses: a single high-probability concentration condition on the estimators, after which logarithmic regret follows from two short deterministic lemmas describing radius collapse and optimism-forced deviations. The framework yields unified, near-minimal proofs for these classical algorithms and extends naturally to many contemporary bandit variants.

cs.LG↗

arXiv2025

Collaborative QA using Interacting LLMs. Impact of Network Structure, Node Capability and Distributed Data

In this paper, we model and analyze how a network of interacting LLMs performs collaborative question-answering (CQA) in order to estimate a ground truth given a distributed set of documents. This problem is interesting because LLMs often hallucinate when direct evidence to answer a question is lacking, and these effects become more pronounced in a network of interacting LLMs. The hallucination spreads, causing previously accurate LLMs to hallucinate. We study interacting LLMs and their hallucination by combining novel ideas of mean-field dynamics (MFD) from network science and the randomized utility model from economics to construct a useful generative model. We model the LLM with a latent state that indicates if it is truthful or not with respect to the ground truth, and extend a tractable analytical model considering an MFD to model the diffusion of information in a directed network of LLMs. To specify the probabilities that govern the dynamics of the MFD, we propose a randomized utility model. For a network of LLMs, where each LLM has two possible latent states, we posit sufficient conditions for the existence and uniqueness of a fixed point and analyze the behavior of the fixed point in terms of the incentive (e.g., test-time compute) given to individual LLMs. We experimentally study and analyze the behavior of a network of $100$ open-source LLMs with respect to data heterogeneity, node capability, network structure, and sensitivity to framing on multiple semi-synthetic datasets.

cs.AI↗

arXiv2025

Inferring Group Intent as a Cooperative Game. An NLP-based Framework for Trajectory Analysis

This paper studies group target trajectory intent as the outcome of a cooperative game where the complex-spatio trajectories are modeled using an NLP-based generative model. In our framework, the group intent is specified by the characteristic function of a cooperative game, and allocations for players in the cooperative game are specified by either the core, the Shapley value, or the nucleolus. The resulting allocations induce probability distributions that govern the coordinated spatio-temporal trajectories of the targets that reflect the group's underlying intent. We address two key questions: (1) How can the intent of a group trajectory be optimally formalized as the characteristic function of a cooperative game? (2) How can such intent be inferred from noisy observations of the targets? To answer the first question, we introduce a Fisher-information-based characteristic function of the cooperative game, which yields probability distributions that generate coordinated spatio-temporal patterns. As a generative model for these patterns, we develop an NLP-based generative model built on formal grammar, enabling the creation of realistic multi-target trajectory data. To answer the second question, we train a Graph Transformer Neural Network (GTNN) to infer group trajectory intent-expressed as the characteristic function of the cooperative game-from observational data with high accuracy. The self-attention function of the GTNN depends on the track estimates. Thus, the formulation and algorithms provide a multi-layer approach that spans target tracking (Bayesian signal processing) and the GTNN (for group intent inference).

eess.SP↗

arXiv2025

Malliavin Calculus with Weak Derivatives for Counterfactual Stochastic Optimization

We study counterfactual stochastic optimization of conditional loss functionals under misspecified and noisy gradient information. The difficulty is that when the conditioning event has vanishing or zero probability, naive Monte Carlo estimators are prohibitively inefficient; kernel smoothing, though common, suffers from slow convergence. We propose a two-stage kernel-free methodology. First, we show using Malliavin calculus that the conditional loss functional of a diffusion process admits an exact representation as a Skorohod integral, yielding variance comparable to classical Monte-Carlo variance. Second, we establish that a weak derivative estimate of the conditional loss functional with respect to model parameters can be evaluated with constant variance, in contrast to the widely used score function method whose variance grows linearly in the sample path length. Together, these results yield an efficient framework for counterfactual conditional stochastic gradient algorithms in rare-event regimes.

math.OC↗

arXiv2025

Multi-Agent Inverse Reinforcement Learning for Identifying Pareto-Efficient Coordination -- A Distributionally Robust Approach

Multi-agent inverse reinforcement learning (IRL) aims to identify Pareto-efficient behavior in a multi-agent system, and reconstruct utility functions of the individual agents. Motivated by the problem of detecting UAV coordination, how can we construct a statistical detector for Pareto-efficient behavior given noisy measurements of the decisions of a multi-agent system? This paper approaches this IRL problem by deriving necessary and sufficient conditions for a dataset of multi-agent system dynamics to be consistent with Pareto-efficient coordination, and providing algorithms for recovering utility functions which are consistent with the system dynamics. We derive an optimal statistical detector for determining Pareto-efficient coordination from noisy system measurements, which minimizes Type-I statistical detection error. Then, we provide a utility estimation algorithm which minimizes the worst-case estimation error over a statistical ambiguity set centered at empirical observations; this min-max solution achieves distributionally robust IRL, which is crucial in adversarial strategic interactions. We illustrate these results in a detailed example for detecting Pareto-efficient coordination among multiple UAVs given noisy measurement recorded at a radar. We then reconstruct the utility functions of the UAVs in a distributionally robust sense.

eess.SY↗

arXiv2025

Asymmetric Network Games: $\alpha$-Potential Function and Learning

In a network game, players interact over a network and the utility of each player depends on his own action and on an aggregate of his neighbours' actions. Many real world networks of interest are asymmetric and involve a large number of heterogeneous players. This paper analyzes static network games using the framework of $\alpha$-potential games. Under mild assumptions on the action sets (compact intervals) and the utility functions (twice continuously differentiable) of the players, we derive an expression for an inexact potential function of the game, called the $\alpha$-potential function. Using such a function, we show that modified versions of the sequential best-response algorithm and the simultaneous gradient play algorithm achieve convergence of players' actions to a $2\alpha$-Nash equilibrium. For linear-quadratic network games, we show that $\alpha$ depends on the maximum asymmetry in the network and is well-behaved for a wide range of networks of practical interest. Further, we derive bounds on the social welfare of the $\alpha$-Nash equilibrium corresponding to the maximum of the $\alpha$-potential function, under suitable assumptions. We numerically illustrate the convergence of the proposed algorithms and properties of the learned $2\alpha$-Nash equilibria.

cs.GT↗

arXiv2025

Multi-Agent Inverse Learning for Sensor Networks: Identifying Coordination in UAV Networks

Suppose there is an adversarial UAV network being tracked by a radar. How can the radar determine whether the UAVs are coordinating, in some well-defined sense? How can the radar infer the objectives of the individual UAVs and the network as a whole? We present an abstract interpretation of such a strategic interaction, allowing us to conceptualize coordination as a linearly constrained multi-objective optimization problem. Then, we present some tools from microeconomic theory that allow us to detect coordination and reconstruct individual UAV objective functions, from radar tracking signals. This corresponds to performing inverse multi-objective optimization. We present details for how the abstract microeconomic interpretation corresponds to, and naturally arises from, physical-layer radar waveform modulation and multi-target filtering. This article serves as a tutorial, bringing together concepts from several established research contributions in an expository style.

eess.SY↗

arXiv2025

Inverse Reinforcement Learning using Revealed Preferences and Passive Stochastic Optimization

This monograph, spanning three chapters, explores Inverse Reinforcement Learning (IRL). The first two chapters view inverse reinforcement learning (IRL) through the lens of revealed preferences from microeconomics while the third chapter studies adaptive IRL via Langevin dynamics stochastic gradient algorithms. Chapter uses classical revealed preference theory (Afriat's theorem and extensions) to identify constrained utility maximizers based on observed agent actions. This allows for the reconstruction of set-valued estimates of an agent's utility. We illustrate this procedure by identifying the presence of a cognitive radar and reconstructing its utility function. The chapter also addresses the construction of a statistical detector for utility maximization behavior when agent actions are corrupted by noise. Chapter 2 studies Bayesian IRL. It investigates how an analyst can determine if an observed agent is a rationally inattentive Bayesian utility maximizer (i.e., simultaneously optimizing its utility and observation likelihood). The chapter discusses inverse stopping-time problems, focusing on reconstructing the continuation and stopping costs of a Bayesian agent operating over a random horizon. We then apply this IRL methodology to identify the presence of a Bayes-optimal sequential detector. Additionally, Chapter 2 provides a concise overview of discrete choice models, inverse Bayesian filtering, and inverse stochastic gradient algorithms for adaptive IRL. Finally, Chapter 3 introduces an adaptive IRL approach utilizing passive Langevin dynamics. This method aims to track time-varying utility functions given noisy and misspecified gradients. In essence, the adaptive IRL algorithms presented in Chapter 3 can be conceptualized as inverse stochastic gradient algorithms, as they learn the utility function in real-time while a stochastic gradient algorithm is in operation.

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

On Word-of-Mouth and Private-Prior Sequential Social Learning

Social learning constitutes a fundamental framework for studying interactions among rational agents who observe each other's actions but lack direct access to individual beliefs. This paper investigates a specific social learning paradigm known as Word-of-Mouth (WoM), where a series of agents seeks to estimate the state of a dynamical system. The first agent receives noisy measurements of the state, while each subsequent agent relies solely on a degraded version of her predecessor's estimate. A defining feature of WoM is that the final agent's belief is publicly broadcast and subsequently adopted by all agents, in place of their own. We analyze this setting theoretically and through numerical simulations, noting that some agents benefit from using the belief of the last agent, while others experience performance deterioration.

cs.MA↗

arXiv2025

Efficient Neural SDE Training using Wiener-Space Cubature

A neural stochastic differential equation (SDE) is an SDE with drift and diffusion terms parametrized by neural networks. The training procedure for neural SDEs consists of optimizing the SDE vector field (neural network) parameters to minimize the expected value of an objective functional on infinite-dimensional path-space. Existing training techniques focus on methods to efficiently compute path-wise gradients of the objective functional with respect to these parameters, then pair this with Monte-Carlo simulation to estimate the gradient expectation. In this work we introduce a novel training technique which bypasses and improves upon this Monte-Carlo simulation; we extend results in the theory of Wiener space cubature to approximate the expected objective functional value by a weighted sum of functional evaluations of deterministic ODE solutions. Our main mathematical contribution enabling this approximation is an extension of cubature bounds to the setting of Lipschitz-nonlinear functionals acting on path-space. Our resulting constructive algorithm allows for more computationally efficient training along several lines. First, it circumvents Brownian motion simulation and enables the use of efficient parallel ODE solvers, thus decreasing the complexity of path-functional evaluation. Furthermore, and more surprisingly, we show that the number of paths required to achieve a given (expected loss functional oracle value) approximation can be reduced in this deterministic cubature regime. Specifically, we show that under reasonable regularity assumptions we can observe a O(1/n) convergence rate, where n is the number of path evaluations; in contrast with the standard O(1/sqrt(n)) rate of naive Monte-Carlo or the O(log(n)^d /n) rate of quasi-Monte-Carlo.

cs.LG↗