Improving Bayesian Optimization via Training-Aware Conditional Diffusion Models

Bayesian optimization (BO) is a widely used approach for black-box optimization that uses a Gaussian process (GP) as a surrogate and guides sequential evaluations via an acquisition function, with the ultimate goal of locating the global optimum $\mathbf{x}^{\star}$. To align with this goal, information-based acquisition functions such as Predictive Entropy Search (PES) model $\mathbf{x}^{\star}$ as a random variable and reduce the entropy of its distribution, but approximating this distribution via traditional GP posterior sampling is computationally expensive. To address this limitation, we leverage Conditional Diffusion Models (CDMs) to efficiently approximate the distribution of $\mathbf{x}^{\star}$ and develop BO-inherent training strategies for CDMs. Motivated by the structural properties of the CDM-learned distribution, we further develop an acquisition strategy termed Diffusion-based Mode Seeking (DMS) to guide the sequential evaluation. We establish a sub-optimality guarantee for the CDM-learned distribution and demonstrate through extensive experiments that DMS outperforms standard BO baselines.

Optimal Data Acquisition for Reinforcement Learning: A Large Deviations Perspective

Data acquisition efficiency is a central challenge in deploying reinforcement learning in business and healthcare operations, where interactions are costly, slow, and often involve humans in the loop. This paper develops a unified large deviations framework for data acquisition in infinite-horizon reinforcement learning. We introduce the exponential decay rate of the policy-selection error probability as a principled efficiency metric and derive a variational characterization of this rate via large deviations theory for Markov chains, yielding a nested optimization problem. Based on this characterization, we formalize two complementary notions of optimality in terms of the optimal solution of the nested problem. Because the resulting program is implicit and generally intractable, we propose a tractable convex relaxation with explicit constraints. We then develop a lazy one-step projected subgradient method to solve the relaxed problem and use its iterates to construct an adaptive data acquisition policy. We prove that the resulting reinforcement learning algorithm is near-robustly optimal under our optimality criterion, up to a constant factor. Finally, we extend the framework to linear function approximation to improve scalability, and numerical experiments support the effectiveness of the proposed approach.

Evolving Robustness--Exploration Trade-off in Online Reinforcement Learning via Quantile Bayesian Risk MDPs

In online reinforcement learning, data scarcity creates epistemic uncertainty that makes robustness important early in learning, whereas sufficient exploration is needed to learn the true-environment optimal policy. We study this time-varying robustness--exploration trade-off through a quantile Bayesian risk-aware Markov decision process (BR-MDP), in which the quantile level controls how posterior uncertainty enters the Bellman backup. We characterize this control through an asymptotic normality result for the difference between the quantile BR-MDP value and the value in the true environment. The result implies that upper/lower-tail quantiles induce optimism/pessimism towards epistemic uncertainty, and the magnitude of the optimism/pessimism decreases as data accumulate. Building on this characterization, we propose an online Bayesian risk-aware algorithm with an adaptive quantile schedule that emphasizes robustness early and gradually encourages exploration of less-visited state--action pairs. We establish sublinear Bayesian regret bounds with respect to both the true optimal value and the optimal BR-MDP robust value. Numerical experiments demonstrate strong performance in both exploration-demanding and exploration-costly environments.

Adaptive Simulation Experiment for LLM Policy Optimization

Large language models (LLMs) have significant potential to improve operational efficiency in operations management. Deploying these models requires specifying a policy that governs response quality, shapes user experience, and influences operational value. In this research, we treat LLMs as stochastic simulators and propose a pairwise comparison-based adaptive simulation experiment framework for identifying the optimal policy from a finite set of candidates. We consider two policy spaces: an unstructured space with no parametric assumption, and a structured space in which the data are generated from a preference model. For both settings, we characterize the fundamental data requirements for identifying the optimal policy with high probability. In the unstructured case, we derive a closed-form expression for the optimal sampling proportions, together with a clear operational interpretation. In the structured case, we formulate a regularized convex program to compute the optimal proportions. We then develop an adaptive experimental procedure, termed LLM-PO, for both policy spaces, and prove that it identifies the optimal policy with the desired statistical guarantee while asymptotically attaining the fundamental data requirements. Numerical experiments demonstrate that LLM-PO consistently outperforms benchmark methods and improves LLM performance.

Curiosity is Knowledge: Self-Consistent Learning and No-Regret Optimization with Active Inference

Active inference (AIF) unifies exploration and exploitation by minimizing the Expected Free Energy (EFE), balancing epistemic value (information gain) and pragmatic value (task performance) through a curiosity coefficient. Yet it has been unclear when this balance yields both coherent learning and efficient decision-making: insufficient curiosity can drive myopic exploitation and prevent uncertainty resolution, while excessive curiosity can induce unnecessary exploration and regret. We establish the first theoretical guarantee for EFE-minimizing agents, showing that a single requirement--sufficient curiosity--simultaneously ensures self-consistent learning (Bayesian posterior consistency) and no-regret optimization (bounded cumulative regret). Our analysis characterizes how this mechanism depends on initial uncertainty, identifiability, and objective alignment, thereby connecting AIF to classical Bayesian experimental design and Bayesian optimization within one theoretical framework. We further translate these theories into practical design guidelines for tuning the epistemic-pragmatic trade-off in hybrid learning-optimization problems, validated through real-world experiments.

Pragmatic Curiosity: A Hybrid Learning-Optimization Paradigm via Active Inference

Many engineering and scientific workflows depend on expensive black-box evaluations, requiring decision-making that simultaneously improves performance and reduces uncertainty. Bayesian optimization (BO) and Bayesian experimental design (BED) offer powerful yet largely separate treatments of goal-seeking and information-seeking, providing limited guidance for hybrid settings where learning and optimization are intrinsically coupled. We propose "pragmatic curiosity," a hybrid learning-optimization paradigm derived from active inference, in which actions are selected by minimizing the expected free energy--a single objective that couples pragmatic utility with epistemic information gain. We demonstrate the practical effectiveness and flexibility of pragmatic curiosity on various real-world hybrid tasks, including constrained system identification, targeted active search, and composite optimization with unknown preferences. Across these benchmarks, pragmatic curiosity consistently outperforms strong BO-type and BED-type baselines, delivering higher estimation accuracy, better critical-region coverage, and improved final solution quality.

Policy Gradient Optimzation for Bayesian-Risk MDPs with General Convex Losses

Motivated by many application problems, we consider Markov decision processes (MDPs) with a general loss function and unknown parameters. To mitigate the epistemic uncertainty associated with unknown parameters, we take a Bayesian approach to estimate the parameters from data and impose a coherent risk functional (with respect to the Bayesian posterior distribution) on the loss. Since this formulation usually does not satisfy the interchangeability principle, it does not admit Bellman equations and cannot be solved by approaches based on dynamic programming. Therefore, We propose a policy gradient optimization method, leveraging the dual representation of coherent risk measures and extending the envelope theorem to continuous cases. We then show the stationary analysis of the algorithm with a convergence rate of $O(T^{-1/2}+r^{-1/2})$, where $T$ is the number of policy gradient iterations and $r$ is the sample size of the gradient estimator. We further extend our algorithm to an episodic setting, and establish the global convergence of the extended algorithm and provide bounds on the number of iterations needed to achieve an error bound $O(\epsilon)$ in each episode.

Online Bayesian Risk-Averse Reinforcement Learning

In this paper, we study the Bayesian risk-averse formulation in reinforcement learning (RL). To address the epistemic uncertainty due to a lack of data, we adopt the Bayesian Risk Markov Decision Process (BRMDP) to account for the parameter uncertainty of the unknown underlying model. We derive the asymptotic normality that characterizes the difference between the Bayesian risk value function and the original value function under the true unknown distribution. The results indicate that the Bayesian risk-averse approach tends to pessimistically underestimate the original value function. This discrepancy increases with stronger risk aversion and decreases as more data become available. We then utilize this adaptive property in the setting of online RL as well as online contextual multi-arm bandits (CMAB), a special case of online RL. We provide two procedures using posterior sampling for both the general RL problem and the CMAB problem. We establish a sub-linear regret bound, with the regret defined as the conventional regret for both the RL and CMAB settings. Additionally, we establish a sub-linear regret bound for the CMAB setting with the regret defined as the Bayesian risk regret. Finally, we conduct numerical experiments to demonstrate the effectiveness of the proposed algorithm in addressing epistemic uncertainty and verifying the theoretical properties.

Ranking and Selection with Simultaneous Input Data Collection

In this paper, we propose a general and novel formulation of ranking and selection with the existence of streaming input data. The collection of multiple streams of such data may consume different types of resources, and hence can be conducted simultaneously. To utilize the streaming input data, we aggregate simulation outputs generated under heterogeneous input distributions over time to form a performance estimator. By characterizing the asymptotic behavior of the performance estimators, we formulate two optimization problems to optimally allocate budgets for collecting input data and running simulations. We then develop a multi-stage simultaneous budget allocation procedure and provide its statistical guarantees such as consistency and asymptotic normality. We conduct several numerical studies to demonstrate the competitive performance of the proposed procedure.

Reusing Historical Trajectories in Natural Policy Gradient via Importance Sampling: Convergence and Convergence Rate

Reinforcement learning provides a mathematical framework for learning-based control, whose success largely depends on the amount of data it can utilize. The efficient utilization of historical trajectories obtained from previous policies is essential for expediting policy optimization. Empirical evidence has shown that policy gradient methods based on importance sampling work well. However, existing literature often neglect the interdependence between trajectories from different iterations, and the good empirical performance lacks a rigorous theoretical justification. In this paper, we study a variant of the natural policy gradient method with reusing historical trajectories via importance sampling. We show that the bias of the proposed estimator of the gradient is asymptotically negligible, the resultant algorithm is convergent, and reusing past trajectories helps improve the convergence rate. We further apply the proposed estimator to popular policy optimization algorithms such as trust region policy optimization. Our theoretical results are verified on classical benchmarks.