Recent studies in reinforcement learning (RL) have made significant progress by leveraging function approximation to alleviate the sample complexity hurdle for better performance. Despite the success, existing provably efficient algorithms typically rely on the accessibility of immediate feedback upon taking actions. The failure to account for the impact of delay in observations can significantly degrade the performance of real-world systems due to the regret blow-up. In this work, we tackle the challenge of delayed feedback in RL with linear function approximation by employing posterior sampling, which has been shown to empirically outperform the popular UCB algorithms in a wide range of regimes. We first introduce Delayed-PSVI, an optimistic value-based algorithm that effectively explores the value function space via noise perturbation with posterior sampling.

While the setting of state-only demonstrations is not common, there are certain exceptions.

Machine learning models are increasingly trained or fine-tuned on synthetic data. Recursively training on such data has been observed to significantly degrade performance in a wide range of tasks, often characterized by a progressive drift away from the target distribution. In this work, we theoretically analyze this phenomenon in the setting of score-based diffusion models. For a realistic pipeline where each training round uses a combination of synthetic data and fresh samples from the target distribution, we obtain upper and lower bounds on the accumulated divergence between the generated and target distributions. This allows us to characterize different regimes of drift, depending on the score estimation error and the proportion of fresh data used in each generation. We also provide empirical results on synthetic data and images to illustrate the theory.

The problem of model collapse has presented new challenges in iterative training of generative models, where such training with synthetic data leads to an overall degradation of performance. This paper looks at the problem from a statistical viewpoint, illustrating that one can actually hope for improvement when models are trained on data contaminated with synthetic samples, as long as there is some amount of fresh information from the true target distribution. In particular, we consider iterative training on samples sourced from a mixture of the true target and synthetic distributions. We analyze the entire iterative evolution in a next-token prediction language model, capturing how the interplay between the mixture weights and the sample size controls the overall long-term performance. With non-trivial mixture weight of the true distribution, even if it decays over time, simply training the model in a contamination-agnostic manner with appropriate sample sizes can avoid collapse and even recover the true target distribution under certain conditions. Simulation studies support our findings and also show that such behavior is more general for other classes of models.

Stochastic simulation is widely used to study complex systems composed of various interconnected subprocesses, such as input processes, routing and control logic, optimization routines, and data-driven decision modules. In practice, these subprocesses may be inherently unknown or too computationally intensive to directly embed in the simulation model. Replacing these elements with estimated or learned approximations introduces a form of epistemic uncertainty that we refer to as submodel uncertainty. This paper investigates how submodel uncertainty affects the estimation of system performance metrics. We develop a framework for quantifying submodel uncertainty in stochastic simulation models and extend the framework to digital-twin settings, where simulation experiments are repeatedly conducted with the model initialized from observed system states. Building on approaches from input uncertainty analysis, we leverage bootstrapping and Bayesian model averaging to construct quantile-based confidence or credible intervals for key performance indicators. We propose a tree-based method that decomposes total output variability and attributes uncertainty to individual submodels in the form of importance scores. The proposed framework is model-agnostic and accommodates both parametric and nonparametric submodels under frequentist and Bayesian modeling paradigms. A synthetic numerical experiment and a more realistic digital-twin simulation of a contact center illustrate the importance of understanding how and how much individual submodels contribute to overall uncertainty.

Bayesian quadrature is a probabilistic, model-based approach to numerical integration, the estimation of intractable integrals, or expectations. Although Bayesian quadrature was popularised already in the 1980s, no systematic and comprehensive treatment has been published. The purpose of this survey is to fill this gap. We review the mathematical foundations of Bayesian quadrature from different points of view; present a systematic taxonomy for classifying different Bayesian quadrature methods along the three axes of modelling, inference, and sampling; collect general theoretical guarantees; and provide a controlled numerical study that explores and illustrates the effect of different choices along the axes of the taxonomy. We also provide a realistic assessment of practical challenges and limitations to application of Bayesian quadrature methods and include an up-to-date and nearly exhaustive bibliography that covers not only machine learning and statistics literature but all areas of mathematics and engineering in which Bayesian quadrature or equivalent methods have seen use.

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