Nian Si

Shengbo Wang, Nian Si, Jose Blanchet et al. · stanford

We consider a reinforcement learning setting in which the deployment environment is different from the training environment. Applying a robust Markov decision processes formulation, we extend the distributionally robust $Q$-learning framework studied in Liu et al. [2022]. Further, we improve the design and analysis of their multi-level Monte Carlo estimator. Assuming access to a simulator, we prove that the worst-case expected sample complexity of our algorithm to learn the optimal robust $Q$-function within an $ε$ error in the sup norm is upper bounded by $\tilde O(|S||A|(1-γ)^{-5}ε^{-2}p_{\wedge}^{-6}δ^{-4})$, where $γ$ is the discount rate, $p_{\wedge}$ is the non-zero minimal support probability of the transition kernels and $δ$ is the uncertainty size. This is the first sample complexity result for the model-free robust RL problem. Simulation studies further validate our theoretical results.

Shengbo Wang, Nian Si, Jose Blanchet et al. · stanford

Motivated by the need for a robust policy in the face of environment shifts between training and deployment, we contribute to the theoretical foundation of distributionally robust reinforcement learning (DRRL). This is accomplished through a comprehensive modeling framework centered around robust Markov decision processes (RMDPs). This framework obliges the decision maker to choose an optimal policy under the worst-case distributional shift orchestrated by an adversary. By unifying and extending existing formulations, we rigorously construct RMDPs that embrace various modeling attributes for both the decision maker and the adversary. These attributes include the structure of information availability-covering history-dependent, Markov, and Markov time-homogeneous dynamics-as well as constraints on the shifts induced by the adversary, with a focus on SA- and S-rectangularity. Within this RMDP framework, we investigate conditions for the existence or absence of the dynamic programming principle (DPP). From an algorithmic standpoint, the existence of DPP holds significant implications, as the vast majority of existing data and computationally efficient DRRL algorithms are reliant on the DPP. To investigate its existence, we systematically analyze various combinations of controller and adversary attributes, presenting streamlined proofs based on a unified methodology. We then construct counterexamples for settings where a fully general DPP fails to hold and establish asymptotically optimal history-dependent policies for key scenarios where the DPP is absent.

Baris Ata, J. Michael Harrison, Nian Si

Motivated by applications in queueing theory, we consider a stochastic control problem whose state space is the $d$-dimensional positive orthant. The controlled process $Z$ evolves as a reflected Brownian motion whose covariance matrix is exogenously specified, as are its directions of reflection from the orthant's boundary surfaces. A system manager chooses a drift vector $θ(t)$ at each time $t$ based on the history of $Z$, and the cost rate at time $t$ depends on both $Z(t)$ and $θ(t)$. In our initial problem formulation, the objective is to minimize expected discounted cost over an infinite planning horizon, after which we treat the corresponding ergodic control problem. Extending earlier work by Han et al. (Proceedings of the National Academy of Sciences, 2018, 8505-8510), we develop and illustrate a simulation-based computational method that relies heavily on deep neural network technology. For test problems studied thus far, our method is accurate to within a fraction of one percent, and is computationally feasible in dimensions up to at least $d=30$.

Yewen Fan, Nian Si, Kun Zhang

Calibration is defined as the ratio of the average predicted click rate to the true click rate. The optimization of calibration is essential to many online advertising recommendation systems because it directly affects the downstream bids in ads auctions and the amount of money charged to advertisers. Despite its importance, calibration optimization often suffers from a problem called "maximization bias". Maximization bias refers to the phenomenon that the maximum of predicted values overestimates the true maximum. The problem is introduced because the calibration is computed on the set selected by the prediction model itself. It persists even if unbiased predictions can be achieved on every datapoint and worsens when covariate shifts exist between the training and test sets. To mitigate this problem, we theorize the quantification of maximization bias and propose a variance-adjusting debiasing (VAD) meta-algorithm in this paper. The algorithm is efficient, robust, and practical as it is able to mitigate maximization bias problems under covariate shifts, neither incurring additional online serving costs nor compromising the ranking performance. We demonstrate the effectiveness of the proposed algorithm using a state-of-the-art recommendation neural network model on a large-scale real-world dataset.

Nanshan Jia, Ramesh Johari, Nian Si et al.

In data centers, tasks are dispatched to various servers to evenly distribute the workload. When a data center considers implementing a new scheduling algorithm, it typically conducts an A/B test prior to deployment to assess the real-world impact of this new method. However, a straightforward A/B test might be interfered with so-called ``Markovian'' interference. We utilized the Differences-in-Q estimator, as developed by Farias et al. (2022), and introduced mixed Differences-in-Q estimators grounded in Little's Law. We show that our A/B testing methods significantly reduce bias and variance when testing various scheduling policies. Extensive simulations were conducted under scenarios like non-stationary arrival rates, heterogeneous service rates, and communication delays. These simulations highlight the robustness and efficacy of our A/B testing approach.

Nian Si

In modern recommendation systems, the standard pipeline involves training machine learning models on historical data to predict user behaviors and improve recommendations continuously. However, these data training loops can introduce interference in A/B tests, where data generated by control and treatment algorithms, potentially with different distributions, are combined. To address these challenges, we introduce a novel approach called weighted training. This approach entails training a model to predict the probability of each data point appearing in either the treatment or control data and subsequently applying weighted losses during model training. We demonstrate that this approach achieves the least variance among all estimators that do not cause shifts in the training distributions. Through simulation studies, we demonstrate the lower bias and variance of our approach compared to other methods.

Zijun Chen, Shengbo Wang, Nian Si · stanford

Motivated by practical applications where stable long-term performance is critical-such as robotics, operations research, and healthcare-we study the problem of distributionally robust (DR) average-reward reinforcement learning. We propose two algorithms that achieve near-optimal sample complexity. The first reduces the problem to a DR discounted Markov decision process (MDP), while the second, Anchored DR Average-Reward MDP, introduces an anchoring state to stabilize the controlled transition kernels within the uncertainty set. Assuming the nominal MDP is uniformly ergodic, we prove that both algorithms attain a sample complexity of $\widetilde{O}\left(|\mathbf{S}||\mathbf{A}| t_{\mathrm{mix}}^2\varepsilon^{-2}\right)$ for estimating the optimal policy as well as the robust average reward under KL and $f_k$-divergence-based uncertainty sets, provided the uncertainty radius is sufficiently small. Here, $\varepsilon$ is the target accuracy, $|\mathbf{S}|$ and $|\mathbf{A}|$ denote the sizes of the state and action spaces, and $t_{\mathrm{mix}}$ is the mixing time of the nominal MDP. This represents the first finite-sample convergence guarantee for DR average-reward reinforcement learning. We further validate the convergence rates of our algorithms through numerical experiments.

Zitao Wang, Ziyuan Wang, Molei Liu et al.

Transfer learning is a popular strategy to leverage external knowledge and improve statistical efficiency, particularly with a limited target sample. We propose a novel knowledge-guided Wasserstein Distributionally Robust Optimization (KG-WDRO) framework that adaptively incorporates multiple sources of external knowledge to overcome the conservativeness of vanilla WDRO, which often results in overly pessimistic shrinkage toward zero. Our method constructs smaller Wasserstein ambiguity sets by controlling the transportation along directions informed by the source knowledge. This strategy can alleviate perturbations on the predictive projection of the covariates and protect against information loss. Theoretically, we establish the equivalence between our WDRO formulation and the knowledge-guided shrinkage estimation based on collinear similarity, ensuring tractability and geometrizing the feasible set. This also reveals a novel and general interpretation for recent shrinkage-based transfer learning approaches from the perspective of distributional robustness. In addition, our framework can adjust for scaling differences in the regression models between the source and target and accommodates general types of regularization such as lasso and ridge. Extensive simulations demonstrate the superior performance and adaptivity of KG-WDRO in enhancing small-sample transfer learning.

Shengbo Wang, Nian Si · stanford

Learning and optimal control under robust Markov decision processes (MDPs) have received increasing attention, yet most existing theory, algorithms, and applications focus on finite-horizon or discounted models. Long-run average-reward formulations, while natural in many operations research and management contexts, remain underexplored. This is primarily because the dynamic programming foundations are technically challenging and only partially understood, with several fundamental questions remaining open. This paper steps toward a general framework for average-reward robust MDPs by analyzing the constant-gain setting. We study the average-reward robust control problem with possible information asymmetries between the controller and an S-rectangular adversary. Our analysis centers on the constant-gain robust Bellman equation, examining both the existence of solutions and their relationship to the optimal average reward. Specifically, we identify when solutions to the robust Bellman equation characterize the optimal average reward and stationary policies, and we provide one-sided weak communication conditions ensuring solutions' existence. These findings expand the dynamic programming theory for average-reward robust MDPs and lay a foundation for robust dynamic decision making under long-run average criteria in operational environments.

Zhenghao Li, Shengbo Wang, Nian Si · stanford

Distributionally robust reinforcement learning (DR-RL) has recently gained significant attention as a principled approach that addresses discrepancies between training and testing environments. To balance robustness, conservatism, and computational traceability, the literature has introduced DR-RL models with SA-rectangular and S-rectangular adversaries. While most existing statistical analyses focus on SA-rectangular models, owing to their algorithmic simplicity and the optimality of deterministic policies, S-rectangular models more accurately capture distributional discrepancies in many real-world applications and often yield more effective robust randomized policies. In this paper, we study the empirical value iteration algorithm for divergence-based S-rectangular DR-RL and establish near-optimal sample complexity bounds of $\widetilde{O}(|\mathcal{S}||\mathcal{A}|(1-γ)^{-4}\varepsilon^{-2})$, where $\varepsilon$ is the target accuracy, $|\mathcal{S}|$ and $|\mathcal{A}|$ denote the cardinalities of the state and action spaces, and $γ$ is the discount factor. To the best of our knowledge, these are the first sample complexity results for divergence-based S-rectangular models that achieve optimal dependence on $|\mathcal{S}|$, $|\mathcal{A}|$, and $\varepsilon$ simultaneously. We further validate this theoretical dependence through numerical experiments on a robust inventory control problem and a theoretical worst-case example, demonstrating the fast learning performance of our proposed algorithm.

Hao Liu, Junze Tony Ye, Jose Blanchet et al.

We introduce ScoreFusion, a theoretically grounded method for fusing multiple pre-trained diffusion models that are assumed to generate from auxiliary populations. ScoreFusion is particularly useful for enhancing the generative modeling of a target population with limited observed data. Our starting point considers the family of KL barycenters of the auxiliary populations, which is proven to be an optimal parametric class in the KL sense, but difficult to learn. Nevertheless, by recasting the learning problem as score matching in denoising diffusion, we obtain a tractable way of computing the optimal KL barycenter weights. We prove a dimension-free sample complexity bound in total variation distance, provided that the auxiliary models are well-fitted for their own task and the auxiliary tasks combined capture the target well. The sample efficiency of ScoreFusion is demonstrated by learning handwritten digits. We also provide a simple adaptation of a Stable Diffusion denoising pipeline that enables sampling from the KL barycenter of two auxiliary checkpoints; on a portrait generation task, our method produces faces that enhance population heterogeneity relative to the auxiliary distributions.

Shengbo Wang, Jason Meng, Nian Si et al.

We study data-driven learning of robust stochastic control for infinite-horizon systems with potentially continuous state and action spaces. In many managerial settings--supply chains, finance, manufacturing, services, and dynamic games--the state-transition mechanism is determined by system design, while available data capture the distributional properties of the stochastic inputs from the environment. For modeling and computational tractability, a decision maker often adopts a Markov control model with i.i.d. environment inputs, which can render learned policies fragile to internal dependence or external perturbations. We introduce a distributionally robust stochastic control paradigm that promotes policy reliability by introducing adaptive adversarial perturbations to the environment input, while preserving the modeling, statistical, and computational tractability of the Markovian formulation. From a modeling perspective, we examine two adversary models--current-action-aware and current-action-unaware--leading to distinct dynamic behaviors and robust optimal policies. From a statistical learning perspective, we characterize optimal finite-sample minimax rates for uniform learning of the robust value function across a continuum of states under ambiguity sets defined by the $f_k$-divergence and Wasserstein distance. To efficiently compute the optimal robust policies, we further propose algorithms inspired by deep reinforcement learning methodologies. Finally, we demonstrate the applicability of the framework to real managerial problems.

Yewen Fan, Nian Si, Xiangchen Song et al.

The adoption of deep learning across various fields has been extensive, yet there is a lack of focus on evaluating the performance of deep learning pipelines. Typically, with the increased use of large datasets and complex models, the training process is run only once and the result is compared to previous benchmarks. This practice can lead to imprecise comparisons due to the variance in neural network evaluation metrics, which stems from the inherent randomness in the training process. Traditional solutions, such as running the training process multiple times, are often infeasible due to computational constraints. In this paper, we introduce a novel metric framework, the Calibrated Loss Metric, designed to address this issue by reducing the variance present in its conventional counterpart. Consequently, this new metric enhances the accuracy in detecting effective modeling improvements. Our approach is substantiated by theoretical justifications and extensive experimental validations within the context of Deep Click-Through Rate Prediction Models.

Shengbo Wang, Nian Si, Jose Blanchet et al.

Dynamic decision-making under distributional shifts is of fundamental interest in theory and applications of reinforcement learning: The distribution of the environment in which the data is collected can differ from that of the environment in which the model is deployed. This paper presents two novel model-free algorithms, namely the distributionally robust Q-learning and its variance-reduced counterpart, that can effectively learn a robust policy despite distributional shifts. These algorithms are designed to efficiently approximate the $q$-function of an infinite-horizon $γ$-discounted robust Markov decision process with Kullback-Leibler ambiguity set to an entry-wise $ε$-degree of precision. Further, the variance-reduced distributionally robust Q-learning combines the synchronous Q-learning with variance-reduction techniques to enhance its performance. Consequently, we establish that it attains a minimax sample complexity upper bound of $\tilde O(|\mathbf{S}||\mathbf{A}|(1-γ)^{-4}ε^{-2})$, where $\mathbf{S}$ and $\mathbf{A}$ denote the state and action spaces. This is the first complexity result that is independent of the ambiguity size $δ$, thereby providing new complexity theoretic insights. Additionally, a series of numerical experiments confirm the theoretical findings and the efficiency of the algorithms in handling distributional shifts.

Nian Si, Yifu Tang, Zeyu Zheng

We formulate selecting the best optimizing system (SBOS) problems and provide solutions for those problems. In an SBOS problem, a finite number of systems are contenders. Inside each system, a continuous decision variable affects the system's expected performance. An SBOS problem compares different systems based on their expected performances under their own optimally chosen decision to select the best, without advance knowledge of expected performances of the systems nor the optimizing decision inside each system. We design easy-to-implement algorithms that adaptively chooses a system and a choice of decision to evaluate the noisy system performance, sequentially eliminates inferior systems, and eventually recommends a system as the best after spending a user-specified budget. The proposed algorithms integrate the stochastic gradient descent method and the sequential elimination method to simultaneously exploit the structure inside each system and make comparisons across systems. For the proposed algorithms, we prove exponential rates of convergence to zero for the probability of false selection, as the budget grows to infinity. We conduct three numerical examples that represent three practical cases of SBOS problems. Our proposed algorithms demonstrate consistent and stronger performances in terms of the probability of false selection over benchmark algorithms under a range of problem settings and sampling budgets.

Nian Si, Karthyek Murthy, Jose Blanchet et al.

We present a statistical testing framework to detect if a given machine learning classifier fails to satisfy a wide range of group fairness notions. The proposed test is a flexible, interpretable, and statistically rigorous tool for auditing whether exhibited biases are intrinsic to the algorithm or due to the randomness in the data. The statistical challenges, which may arise from multiple impact criteria that define group fairness and which are discontinuous on model parameters, are conveniently tackled by projecting the empirical measure onto the set of group-fair probability models using optimal transport. This statistic is efficiently computed using linear programming and its asymptotic distribution is explicitly obtained. The proposed framework can also be used to test for testing composite fairness hypotheses and fairness with multiple sensitive attributes. The optimal transport testing formulation improves interpretability by characterizing the minimal covariate perturbations that eliminate the bias observed in the audit.

Viet Anh Nguyen, Nian Si, Jose Blanchet

We build a Bayesian contextual classification model using an optimistic score ratio for robust binary classification when there is limited information on the class-conditional, or contextual, distribution. The optimistic score searches for the distribution that is most plausible to explain the observed outcomes in the testing sample among all distributions belonging to the contextual ambiguity set which is prescribed using a limited structural constraint on the mean vector and the covariance matrix of the underlying contextual distribution. We show that the Bayesian classifier using the optimistic score ratio is conceptually attractive, delivers solid statistical guarantees and is computationally tractable. We showcase the power of the proposed optimistic score ratio classifier on both synthetic and empirical data.

Nian Si, Fan Zhang, Zhengyuan Zhou et al.

Policy learning using historical observational data is an important problem that has found widespread applications. Examples include selecting offers, prices, advertisements to send to customers, as well as selecting which medication to prescribe to a patient. However, existing literature rests on the crucial assumption that the future environment where the learned policy will be deployed is the same as the past environment that has generated the data -- an assumption that is often false or too coarse an approximation. In this paper, we lift this assumption and aim to learn a distributionally robust policy with incomplete observational data. We first present a policy evaluation procedure that allows us to assess how well the policy does under the worst-case environment shift. We then establish a central limit theorem type guarantee for this proposed policy evaluation scheme. Leveraging this evaluation scheme, we further propose a novel learning algorithm that is able to learn a policy that is robust to adversarial perturbations and unknown covariate shifts with a performance guarantee based on the theory of uniform convergence. Finally, we empirically test the effectiveness of our proposed algorithm in synthetic datasets and demonstrate that it provides the robustness that is missing using standard policy learning algorithms. We conclude the paper by providing a comprehensive application of our methods in the context of a real-world voting dataset.

Jose Blanchet, Karthyek Murthy, Nian Si

Wasserstein distributionally robust optimization estimators are obtained as solutions of min-max problems in which the statistician selects a parameter minimizing the worst-case loss among all probability models within a certain distance (in a Wasserstein sense) from the underlying empirical measure. While motivated by the need to identify optimal model parameters or decision choices that are robust to model misspecification, these distributionally robust estimators recover a wide range of regularized estimators, including square-root lasso and support vector machines, among others, as particular cases. This paper studies the asymptotic normality of these distributionally robust estimators as well as the properties of an optimal (in a suitable sense) confidence region induced by the Wasserstein distributionally robust optimization formulation. In addition, key properties of min-max distributionally robust optimization problems are also studied, for example, we show that distributionally robust estimators regularize the loss based on its derivative and we also derive general sufficient conditions which show the equivalence between the min-max distributionally robust optimization problem and the corresponding max-min formulation.