Vikram Krishnamurthy

Adit Jain, Vikram Krishnamurthy

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.

Luke Snow, Vikram Krishnamurthy

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.

Vikram Krishnamurthy, Luke Snow

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.

George Yin, Vikram Krishnamurthy

We analyze the finite sample regret of a decreasing step size stochastic gradient algorithm. We assume correlated noise and use a perturbed Lyapunov function as a systematic approach for the analysis. Finally we analyze the escape time of the iterates using large deviations theory.