Accurate estimation of treatment effects is essential for decision-making across various scientific fields. This task, however, becomes challenging in areas like social sciences and online marketplaces, where treating one experimental unit can influence outcomes for others through direct or indirect interactions. Such interference can lead to biased treatment effect estimates, particularly when the structure of these interactions is unknown. We address this challenge by introducing a new class of estimators based on causal message-passing, specifically designed for settings with pervasive, unknown interference. Our estimator draws on information from the sample mean and variance of unit outcomes and treatments over time, enabling efficient use of observed data to estimate the evolution of the system state. Concretely, we construct non-linear features from the moments of unit outcomes and treatments and then learn a function that maps these features to future mean and variance of unit outcomes. This allows for the estimation of the treatment effect over time. Extensive simulations across multiple domains, using synthetic and real network data, demonstrate the efficacy of our approach in estimating total treatment effect dynamics, even in cases where interference exhibits non-monotonic behavior in the probability of treatment.

Linear dynamical systems that obey stochastic differential equations are canonical models. While optimal control of known systems has a rich literature, the problem is technically hard under model uncertainty and there are hardly any such result. We initiate study of this problem and aim to learn (and simultaneously deploy) optimal actions for minimizing a quadratic cost function. Indeed, this work is the first that comprehensively addresses the crucial challenge of balancing exploration versus exploitation in continuous-time systems. We present online policies that learn optimal actions fast by carefully randomizing the parameter estimates, and establish their performance guarantees: a regret bound that grows with square-root of time multiplied by the number of parameters. Implementation of the policy for a flight-control task demonstrates its efficacy. Further, we prove sharp stability results for inexact system dynamics and tightly specify the infinitesimal regret caused by sub-optimal actions. To obtain the results, we conduct a novel eigenvalue-sensitivity analysis for matrix perturbation, establish upper-bounds for comparative ratios of stochastic integrals, and introduce the new method of policy differentiation. Our analysis sheds light on fundamental challenges in continuous-time reinforcement learning and suggests a useful cornerstone for similar problems.

Diffusion processes that evolve according to linear stochastic differential equations are an important family of continuous-time dynamic decision-making models. Optimal policies are well-studied for them, under full certainty about the drift matrices. However, little is known about data-driven control of diffusion processes with uncertain drift matrices as conventional discrete-time analysis techniques are not applicable. In addition, while the task can be viewed as a reinforcement learning problem involving exploration and exploitation trade-off, ensuring system stability is a fundamental component of designing optimal policies. We establish that the popular Thompson sampling algorithm learns optimal actions fast, incurring only a square-root of time regret, and also stabilizes the system in a short time period. To the best of our knowledge, this is the first such result for Thompson sampling in a diffusion process control problem. We validate our theoretical results through empirical simulations with real matrices. Moreover, we observe that Thompson sampling significantly improves (worst-case) regret, compared to the state-of-the-art algorithms, suggesting Thompson sampling explores in a more guarded fashion. Our theoretical analysis involves characterization of a certain \emph{optimality manifold} that ties the local geometry of the drift parameters to the optimal control of the diffusion process. We expect this technique to be of broader interest.