arXiv subjects

Enlu Zhou

Publications and source records attributed to Enlu Zhou.

At least 19 recordsLinked to original sources

arXiv2026

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.

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arXiv2026

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.

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arXiv2026

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.

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arXiv2026

Quantum Grover Adaptive Search for Discrete Simulation Optimization

Quantum computing has advanced rapidly in recent years and has shown advantages in a variety of domains. In this paper, we investigate its potential for discrete simulation optimization in the fixed-confidence setting, a fundamental problem in the simulation literature. We first introduce a quantum simulation oracle that prepares a coherent superposition over all candidate solutions and provides the foundation for quantum algorithm design. Based on this oracle, we develop the first Grover-search-based quantum algorithm for discrete simulation optimization, called SOGAS. In particular, SOGAS uses a binary-search framework to progressively eliminate suboptimal solutions while carefully controlling the error probability, and eventually identifies a set of near-optimal solutions. We prove that SOGAS returns a near-optimal solution with probability at least the prescribed confidence level and achieves a quadratic speedup in the dependence of query complexity on the number of candidate solutions. Numerical experiments further show that SOGAS substantially outperforms classical benchmarks and provide empirical evidence for quantum advantage in discrete simulation optimization.

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arXiv2026

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.

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arXiv2026

Adaptive Distributionally Robust Optimal Control with Bayesian Ambiguity Sets

In stochastic optimal control (SOC), uncertainty may arise from incomplete knowledge of the true probability distribution of the underlying environment, which is known as Knightian or epistemic uncertainty. Distributionally robust optimal control (DROC) models are subsequently proposed to tackle this source of uncertainty. While such models are effective in some practical applications, most existing DROC models are offline and can be overly conservative when data are scarce. Moreover, they cannot be applied to the case when samples are generated episodically. Motivated by the Bayesian SOC framework recently proposed by Shapiro et al.~\cite{shapiro2025episodic}, we propose an adaptive DROC model in which the ambiguity set is updated via Bayesian learning from new data. Under some moderate conditions, we derive a tractable risk-averse reformulation, establish consistency of the optimal value function and optimal policy for an infinite-horizon SOC and establish a finite-sample posterior credibility guarantee for the policy value induced by the proposed episodic Bayesian DROC model. We also study the stability and statistical robustness of the proposed model with respect to sample perturbations that often arise in data-driven environments. To solve the episodic Bayesian DROC model, we propose a Bellman-operator cutting-plane (BOCP) algorithm that is computationally efficient and provably convergent. Numerical results on an inventory control problem demonstrate the effectiveness, adaptivity, and robust performance of the proposed model and algorithm.

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arXiv2026

Stochastic Optimal Control with Side Information and Bayesian Learning

We study infinite-horizon stochastic optimal control problems with observable side information: a Markov chain that modulates an unknown context-conditional randomness distribution. Since this distribution is unknown, we propose a Bayesian reformulation based on a parametric density model and posterior predictive dynamics, which yields a Bayesian Bellman equation. We prove posterior consistency under Markov samples and, under correct specification and identifiability, uniform convergence of the Bayesian value function. Finally, we establish Bernstein--von Mises-type asymptotic normality for the data-driven contextual optimal value.

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arXiv2026

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.

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arXiv2026

Pragmatic Curiosity: A Unified Framework for Hybrid Learning and Optimization via Active Inference

Many engineering and scientific workflows rely on expensive black-box evaluations, requiring sequential decisions that must both improve task performance and reduce uncertainty. Bayesian optimization (BO) and Bayesian experimental design (BED) provide powerful but largely separate treatments of goal-directed optimization and information-seeking experimentation, leaving limited guidance for hybrid settings in which learning and optimization are intrinsically coupled. We propose Pragmatic Curiosity (PraC), a unified framework for hybrid learning and optimization via active inference. PraC evaluates candidate queries by trading information gain about a task-relevant latent symbol against an expected regret-based potential over outcomes. This formulation exposes three operational design choices: which latent quantity should be clarified, how task value is encoded as regret, and how strongly information gain should be exchanged against pragmatic value. We instantiate PraC across three regimes of increasing complexity: decision-oriented plume monitoring with fixed global symbols and known downstream losses, targeted active search with induced local symbols and evolving coverage goals, and composite Bayesian optimization with hierarchical regret learning under unknown preferences. Across these regimes, PraC reduces downstream decision risk, improves coverage of critical outcome regions, and jointly learns predictive and preference structures without relying on task-specific staging rules.

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arXiv2025

Bayesian Risk-averse Model Predictive Control with Consistency and Stability Guarantees

Model Predictive Control (MPC) is a powerful framework for constrained control, but its performance and safety can be severely degraded when the prediction model is learned online and thus remains uncertain. In this work, we develop a Bayesian risk-averse MPC framework for stochastic, discrete-time, nonlinear systems that provides theoretical guarantees on the consistency of Bayesian learning and closed-loop stability. First, we study Bayesian learning under the conditionally independent state transitions induced by feedback control and establish explicit conditions for Bayesian consistency on an infinitely countable parameter space. Second, we introduce a general notion of risk-averse asymptotic stability (RAAS), defined via comparison function classes and independent of any specific coherent risk measure or convergence rate, and we derive a risk-averse Lyapunov stability theorem together with MPC-specific stability conditions. Third, building on these foundations, we design a practical Bayesian risk-averse MPC scheme that separates epistemic (parametric) and aleatoric (disturbance) uncertainty: additive disturbances are treated in a risk-neutral fashion, while parametric uncertainty is managed via dynamically shrinking ambiguity sets constructed from Bayesian credible intervals, approximated online using particle filtering. To enable real-time implementation, we propose both an optimal and a sub-optimal receding-horizon control policy, the latter obtained by warm-starting from the previous solution, and prove that asymptotic RAAS is recovered as the Bayesian estimator becomes consistent.

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arXiv2025

Bayesian Risk-Sensitive Policy Optimization For MDPs With General Loss Functions

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 $\mathcal{O}(T^{-1/2}+r^{-1/2})$, 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 $\mathcal{O}(\epsilon)$ in each episode.

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arXiv2025

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.

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arXiv2025

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.

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arXiv2025

Fixed Confidence and Fixed Tolerance Bi-level Optimization for Selecting the Best Optimized System

In this paper, we study a fixed-confidence, fixed-tolerance formulation of a class of stochastic bi-level optimization problems, where the upper-level problem selects from a finite set of systems based on a performance metric, and the lower-level problem optimizes continuous decision variables for each system. Notably, the objective functions for the upper and lower levels can differ. This class of problems has a wide range of applications, including model selection, ranking and selection under input uncertainty, and optimal design. To address this, we propose a multi-stage Pruning-Optimization framework that alternates between comparing the performance of different systems (Pruning) and optimizing systems (Optimization). % In the Pruning stage, we design a sequential algorithm that identifies and eliminates inferior systems through systematic performance evaluations. In the Optimization stage, the goal is to solve for a near-optimal solution that meets specified confidence and tolerance requirements. This multi-stage framework is designed to enhance computational efficiency by pruning inferior systems with high tolerance early on, thereby avoiding unnecessary computational efforts. We demonstrate the effectiveness of the proposed algorithm through both theoretical analysis of statistical validity and sample complexity and numerical experiments.

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arXiv2024

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.

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arXiv2023

Episodic Bayesian Optimal Control with Unknown Randomness Distributions

Stochastic optimal control with unknown randomness distributions has been studied for a long time, encompassing robust control, distributionally robust control, and adaptive control. We propose a new episodic Bayesian approach that incorporates Bayesian learning with optimal control. In each episode, the approach learns the randomness distribution with a Bayesian posterior and subsequently solves the corresponding Bayesian average estimate of the true problem. The resulting policy is exercised during the episode, while additional data/observations of the randomness are collected to update the Bayesian posterior for the next episode. We show that the resulting episodic value functions and policies converge almost surely to their optimal counterparts of the true problem if the parametrized model of the randomness distribution is correctly specified. We further show that the asymptotic convergence rate of the episodic value functions is of the order $O(N^{-1/2})$, where $N$ is the number of episodes given that only one data point is collected in each episode. We develop an efficient computational method based on stochastic dual dynamic programming (SDDP) for a class of problems that have convex cost functions and linear state dynamics. Our numerical results on a classical inventory control problem verify the theoretical convergence results, and numerical comparison with two other methods demonstrate the effectiveness of the proposed Bayesian approach.

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arXiv2023

Bayesian Risk-Averse Q-Learning with Streaming Observations

We consider a robust reinforcement learning problem, where a learning agent learns from a simulated training environment. To account for the model mis-specification between this training environment and the real environment due to lack of data, we adopt a formulation of Bayesian risk MDP (BRMDP) with infinite horizon, which uses Bayesian posterior to estimate the transition model and impose a risk functional to account for the model uncertainty. Observations from the real environment that is out of the agent's control arrive periodically and are utilized by the agent to update the Bayesian posterior to reduce model uncertainty. We theoretically demonstrate that BRMDP balances the trade-off between robustness and conservativeness, and we further develop a multi-stage Bayesian risk-averse Q-learning algorithm to solve BRMDP with streaming observations from real environment. The proposed algorithm learns a risk-averse yet optimal policy that depends on the availability of real-world observations. We provide a theoretical guarantee of strong convergence for the proposed algorithm.

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arXiv2023

Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes

We consider infinite-horizon Markov Decision Processes where parameters, such as transition probabilities, are unknown and estimated from data. The popular distributionally robust approach to addressing the parameter uncertainty can sometimes be overly conservative. In this paper, we utilize the recently proposed formulation, Bayesian risk Markov Decision Process (BR-MDP), to address parameter (or epistemic) uncertainty in MDPs. To solve the infinite-horizon BR-MDP with a class of convex risk measures, we propose a computationally efficient approach called approximate bilevel difference convex programming (ABDCP). The optimization is performed offline and produces the optimal policy that is represented as a finite state controller with desirable performance guarantees. We also demonstrate the empirical performance of the BR-MDP formulation and the proposed algorithm.

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