Ningyuan Chen

Ningyuan Chen, Ming Hu, Wenhao Li

Commercial AI solutions provide analysts and managers with data-driven business intelligence for a wide range of decisions, such as demand forecasting and pricing. However, human analysts may have their own insights and experiences about the decision-making that is at odds with the algorithmic recommendation. In view of such a conflict, we provide a general analytical framework to study the augmentation of algorithmic decisions with human knowledge: the analyst uses the knowledge to set a guardrail by which the algorithmic decision is clipped if the algorithmic output is out of bound, and seems unreasonable. We study the conditions under which the augmentation is beneficial relative to the raw algorithmic decision. We show that when the algorithmic decision is asymptotically optimal with large data, the non-data-driven human guardrail usually provides no benefit. However, we point out three common pitfalls of the algorithmic decision: (1) lack of domain knowledge, such as the market competition, (2) model misspecification, and (3) data contamination. In these cases, even with sufficient data, the augmentation from human knowledge can still improve the performance of the algorithmic decision.

Ningyuan Chen, Setareh Farajollahzadeh, Guan Wang

Product bundling is a common selling mechanism used in online retailing. To set profitable bundle prices, the seller needs to learn consumer preferences from the transaction data. When customers purchase bundles or multiple products, classical methods such as discrete choice models cannot be used to estimate customers' valuations. In this paper, we propose an approach to learn the distribution of consumers' valuations toward the products using bundle sales data. The approach reduces it to an estimation problem where the samples are censored by polyhedral regions. Using the EM algorithm and Monte Carlo simulation, our approach can recover the distribution of consumers' valuations. The framework allows for unobserved no-purchases and clustered market segments. We provide theoretical results on the identifiability of the probability model and the convergence of the EM algorithm. The performance of the approach is also demonstrated numerically.

Ningyuan Chen, Wenhao Li

We consider a decision maker allocating one unit of renewable and divisible resource in each period on a number of arms. The arms have unknown and random rewards whose means are proportional to the allocated resource and whose variances are proportional to an order $b$ of the allocated resource. In particular, if the decision maker allocates resource $A_i$ to arm $i$ in a period, then the reward $Y_i$ is$Y_i(A_i)=A_i μ_i+A_i^b ξ_{i}$, where $μ_i$ is the unknown mean and the noise $ξ_{i}$ is independent and sub-Gaussian. When the order $b$ ranges from 0 to 1, the framework smoothly bridges the standard stochastic multi-armed bandit and online learning with full feedback. We design two algorithms that attain the optimal gap-dependent and gap-independent regret bounds for $b\in [0,1]$, and demonstrate a phase transition at $b=1/2$. The theoretical results hinge on a novel concentration inequality we have developed that bounds a linear combination of sub-Gaussian random variables whose weights are fractional, adapted to the filtration, and monotonic.

Huiling Meng, Ningyuan Chen, Xuefeng Gao

Online platforms routinely compare multi-armed bandit algorithms, such as UCB and Thompson Sampling, to select the best-performing policy. Unlike standard A/B tests for static treatments, each run of a bandit algorithm over $T$ users produces only one dependent trajectory, because the algorithm's decisions depend on all past interactions. Reliable inference therefore demands many independent restarts of the algorithm, making experimentation costly and delaying deployment decisions. We propose Artificial Replay (AR) as a new experimental design for this problem. AR first runs one policy and records its trajectory. When the second policy is executed, it reuses a recorded reward whenever it selects an action the first policy already took, and queries the real environment only otherwise. We develop a new analytical framework for this design and prove three key properties of the resulting estimator: it is unbiased; it requires only $T + o(T)$ user interactions instead of $2T$ for a run of the treatment and control policies, nearly halving the experimental cost when both policies have sub-linear regret; and its variance grows sub-linearly in $T$, whereas the estimator from a naïve design has a linearly-growing variance. Numerical experiments with UCB, Thompson Sampling, and $ε$-greedy policies confirm these theoretical gains.

Ziyuan Zhang, Darcy Wang, Ningyuan Chen et al.

Large language models (LLMs) are increasingly used to simulate or automate human behavior in complex sequential decision-making settings. A natural question is then whether LLMs exhibit similar decision-making behavior to humans, and can achieve comparable (or superior) performance. In this work, we focus on the exploration-exploitation (E&E) tradeoff, a fundamental aspect of dynamic decision-making under uncertainty. We employ canonical multi-armed bandit (MAB) experiments introduced in the cognitive science and psychiatry literature to conduct a comparative study of the E&E strategies of LLMs, humans, and MAB algorithms. We use interpretable choice models to capture the E&E strategies of the agents and investigate how enabling thinking traces, through both prompting strategies and thinking models, shapes LLM decision-making. We find that enabling thinking in LLMs shifts their behavior toward more human-like behavior, characterized by a mix of random and directed exploration. In a simple stationary setting, thinking-enabled LLMs exhibit similar levels of random and directed exploration compared to humans. However, in more complex, non-stationary environments, LLMs struggle to match human adaptability, particularly in effective directed exploration, despite achieving similar regret in certain scenarios. Our findings highlight both the promise and limits of LLMs as simulators of human behavior and tools for automated decision-making and point to potential areas for improvement.

Yi Xiong, Ningyuan Chen, Xuefeng Gao

When two players are engaged in a repeated game with unknown payoff matrices, they may use single-agent multi-armed bandit algorithms to choose the actions independent of each other. We show that when the players use Thompson sampling, the game dynamics converges to the Nash equilibrium under a mild assumption on the payoff matrices. Therefore, algorithmic collusion doesn't arise in this case despite the fact that the players do not intentionally deploy competitive strategies. To prove the convergence result, we find that the framework developed in stochastic approximation doesn't apply, because of the sporadic and infrequent updates of the inferior actions and the lack of Lipschitz continuity. We develop a novel sample-path-wise approach to show the convergence. However, when the payoff matrices do not satisfy the assumption, the game may converge to collusive outcomes.

Huiling Meng, Ningyuan Chen, Xuefeng Gao

Intensity control is a type of continuous-time dynamic optimization problems with many important applications in Operations Research including queueing and revenue management. In this study, we adapt the reinforcement learning framework to intensity control using choice-based network revenue management as a case study, which is a classical problem in revenue management that features a large state space, a large action space and a continuous time horizon. We show that by utilizing the inherent discretization of the sample paths created by the jump points, a unique and defining feature of intensity control, one does not need to discretize the time horizon in advance, which was believed to be necessary because most reinforcement learning algorithms are designed for discrete-time problems. As a result, the computation can be facilitated and the discretization error is significantly reduced. We lay the theoretical foundation for the Monte Carlo and temporal difference learning algorithms for policy evaluation and develop policy-gradient-based actor-critic algorithms for intensity control. Via a comprehensive numerical study, we demonstrate the benefit of our approach versus other state-of-the-art benchmarks.

Tianyu Wang, Ningyuan Chen, Chun Wang

In contextual optimization, a decision-maker leverages contextual information, often referred to as covariates, to better resolve uncertainty and make informed decisions. In this paper, we examine the challenges of contextual decision-making under covariate shift, a phenomenon where the distribution of covariates differs between the training and test environments. Such shifts can lead to inaccurate upstream estimations for test covariates that lie far from the training data, ultimately resulting in suboptimal downstream decisions. To tackle these challenges, we propose a novel approach called Intersection Wasserstein-balls DRO (IW-DRO), which integrates multiple estimation methods into the distributionally robust optimization (DRO) framework. At the core of our approach is an innovative ambiguity set defined as the intersection of two Wasserstein balls, with their centers constructed using appropriate nonparametric and parametric estimators. On the computational side, we reformulate the IW-DRO problem as a tractable convex program and develop an approximate algorithm tailored for large-scale problems to enhance computational efficiency. From a theoretical perspective, we demonstrate that IW-DRO achieves superior performance compared to single Wasserstein-ball DRO models. We further establish performance guarantees by analyzing the coverage of the intersection ambiguity set and the measure concentration of both estimators under the Wasserstein distance. Notably, we derive a finite-sample concentration result for the Nadaraya-Watson kernel estimator under covariate shift. The proposed IW-DRO framework offers practical value for decision-makers operating in uncertain environments affected by covariate shifts.

Ningyuan Chen, Shuoguang Yang, Hailun Zhang

In the multi-armed bandit framework, there are two formulations that are commonly employed to handle time-varying reward distributions: adversarial bandit and nonstationary bandit. Although their oracles, algorithms, and regret analysis differ significantly, we provide a unified formulation in this paper that smoothly bridges the two as special cases. The formulation uses an oracle that takes the best-fixed arm within time windows. Depending on the window size, it turns into the oracle in hindsight in the adversarial bandit and dynamic oracle in the nonstationary bandit. We provide algorithms that attain the optimal regret with the matching lower bound.

Ningyuan Chen, Xuefeng Gao, Yi Xiong

It has been recently shown in the literature that the sample averages from online learning experiments are biased when used to estimate the mean reward. To correct the bias, off-policy evaluation methods, including importance sampling and doubly robust estimators, typically calculate the conditional propensity score, which is ill-defined for non-randomized policies such as UCB. This paper provides a procedure to debias the samples using bootstrap, which doesn't require the knowledge of the reward distribution and can be applied to any adaptive policies. Numerical experiments demonstrate the effective bias reduction for samples generated by popular multi-armed bandit algorithms such as Explore-Then-Commit (ETC), UCB, Thompson sampling (TS) and $ε$-greedy (EG). We analyze and provide theoretical justifications for the procedure under the ETC algorithm, including the asymptotic convergence of the bias decay rate in the real and bootstrap worlds.

Yi Xiong, Ningyuan Chen, Xuefeng Gao et al.

We study the model-based undiscounted reinforcement learning for partially observable Markov decision processes (POMDPs). The oracle we consider is the optimal policy of the POMDP with a known environment in terms of the average reward over an infinite horizon. We propose a learning algorithm for this problem, building on spectral method-of-moments estimations for hidden Markov models, the belief error control in POMDPs and upper-confidence-bound methods for online learning. We establish a regret bound of $O(T^{2/3}\sqrt{\log T})$ for the proposed learning algorithm where $T$ is the learning horizon. This is, to the best of our knowledge, the first algorithm achieving sublinear regret with respect to our oracle for learning general POMDPs.

Tianyu Wang, Ningyuan Chen, Chun Wang

In prescriptive analytics, the decision-maker observes historical samples of $(X, Y)$, where $Y$ is the uncertain problem parameter and $X$ is the concurrent covariate, without knowing the joint distribution. Given an additional covariate observation $x$, the goal is to choose a decision $z$ conditional on this observation to minimize the cost $\mathbb{E}[c(z,Y)|X=x]$. This paper proposes a new distributionally robust approach under Wasserstein ambiguity sets, in which the nominal distribution of $Y|X=x$ is constructed based on the Nadaraya-Watson kernel estimator concerning the historical data. We show that the nominal distribution converges to the actual conditional distribution under the Wasserstein distance. We establish the out-of-sample guarantees and the computational tractability of the framework. Through synthetic and empirical experiments about the newsvendor problem and portfolio optimization, we demonstrate the strong performance and practical value of the proposed framework.

Ningyuan Chen

In many online learning or multi-armed bandit problems, the taken actions or pulled arms are ordinal and required to be monotone over time. Examples include dynamic pricing, in which the firms use markup pricing policies to please early adopters and deter strategic waiting, and clinical trials, in which the dose allocation usually follows the dose escalation principle to prevent dose limiting toxicities. We consider the continuum-armed bandit problem when the arm sequence is required to be monotone. We show that when the unknown objective function is Lipschitz continuous, the regret is $O(T)$. When in addition the objective function is unimodal or quasiconcave, the regret is $\tilde O(T^{3/4})$ under the proposed algorithm, which is also shown to be the optimal rate. This deviates from the optimal rate $\tilde O(T^{2/3})$ in the continuous-armed bandit literature and demonstrates the cost to the learning efficiency brought by the monotonicity requirement.

Ningyuan Chen, Anran Li, Shuoguang Yang

We consider the revenue maximization problem for an online retailer who plans to display in order a set of products differing in their prices and qualities. Consumers have attention spans, i.e., the maximum number of products they are willing to view, and inspect the products sequentially before purchasing a product or leaving the platform empty-handed when the attention span gets exhausted. Our framework extends the well-known cascade model in two directions: the consumers have random attention spans instead of fixed ones, and the firm maximizes revenues instead of clicking probabilities. We show a nested structure of the optimal product ranking as a function of the attention span when the attention span is fixed. \sg{Using this fact, we develop an approximation algorithm when only the distribution of the attention spans is given. Under mild conditions, it achieves $1/e$ of the revenue of the clairvoyant case when the realized attention span is known. We also show that no algorithms can achieve more than 0.5 of the revenue of the same benchmark. The model and the algorithm can be generalized to the ranking problem when consumers make multiple purchases.} When the conditional purchase probabilities are not known and may depend on consumer and product features, we devise an online learning algorithm that achieves $\tilde{\mathcal{O}}(\sqrt{T})$ regret relative to the approximation algorithm, despite the censoring of information: the attention span of a customer who purchases an item is not observable. Numerical experiments demonstrate the outstanding performance of the approximation and online learning algorithms.

Wenhao Li, Ningyuan Chen, L. Jeff Hong

We consider a contextual online learning (multi-armed bandit) problem with high-dimensional covariate $\mathbf{x}$ and decision $\mathbf{y}$. The reward function to learn, $f(\mathbf{x},\mathbf{y})$, does not have a particular parametric form. The literature has shown that the optimal regret is $\tilde{O}(T^{(d_x+d_y+1)/(d_x+d_y+2)})$, where $d_x$ and $d_y$ are the dimensions of $\mathbf x$ and $\mathbf y$, and thus it suffers from the curse of dimensionality. In many applications, only a small subset of variables in the covariate affect the value of $f$, which is referred to as \textit{sparsity} in statistics. To take advantage of the sparsity structure of the covariate, we propose a variable selection algorithm called \textit{BV-LASSO}, which incorporates novel ideas such as binning and voting to apply LASSO to nonparametric settings. Our algorithm achieves the regret $\tilde{O}(T^{(d_x^*+d_y+1)/(d_x^*+d_y+2)})$, where $d_x^*$ is the effective covariate dimension. The regret matches the optimal regret when the covariate is $d^*_x$-dimensional and thus cannot be improved. Our algorithm may serve as a general recipe to achieve dimension reduction via variable selection in nonparametric settings.

Ningyuan Chen, Chun Wang, Longlin Wang

A standard assumption adopted in the multi-armed bandit (MAB) framework is that the mean rewards are constant over time. This assumption can be restrictive in the business world as decision-makers often face an evolving environment where the mean rewards are time-varying. In this paper, we consider a non-stationary MAB model with $K$ arms whose mean rewards vary over time in a periodic manner. The unknown periods can be different across arms and scale with the length of the horizon $T$ polynomially. We propose a two-stage policy that combines the Fourier analysis with a confidence-bound-based learning procedure to learn the periods and minimize the regret. In stage one, the policy correctly estimates the periods of all arms with high probability. In stage two, the policy explores the periodic mean rewards of arms using the periods estimated in stage one and exploits the optimal arm in the long run. We show that our learning policy incurs a regret upper bound $\tilde{O}(\sqrt{T\sum_{k=1}^K T_k})$ where $T_k$ is the period of arm $k$. Moreover, we establish a general lower bound $Ω(\sqrt{T\max_{k}\{ T_k\}})$ for any policy. Therefore, our policy is near-optimal up to a factor of $\sqrt{K}$.

Xiang Zhou, Yi Xiong, Ningyuan Chen et al.

We study a multi-armed bandit problem where the rewards exhibit regime switching. Specifically, the distributions of the random rewards generated from all arms are modulated by a common underlying state modeled as a finite-state Markov chain. The agent does not observe the underlying state and has to learn the transition matrix and the reward distributions. We propose a learning algorithm for this problem, building on spectral method-of-moments estimations for hidden Markov models, belief error control in partially observable Markov decision processes and upper-confidence-bound methods for online learning. We also establish an upper bound $O(T^{2/3}\sqrt{\log T})$ for the proposed learning algorithm where $T$ is the learning horizon. Finally, we conduct proof-of-concept experiments to illustrate the performance of the learning algorithm.

Ningyuan Chen, Guillermo Gallego, Zhuodong Tang

Problem definition. In retailing, discrete choice models (DCMs) are commonly used to capture the choice behavior of customers when offered an assortment of products. When estimating DCMs using transaction data, flexible models (such as machine learning models or nonparametric models) are typically not interpretable and hard to estimate, while tractable models (such as the multinomial logit model) tend to misspecify the complex behavior represeted in the data. Methodology/results. In this study, we use a forest of binary decision trees to represent DCMs. This approach is based on random forests, a popular machine learning algorithm. The resulting model is interpretable: the decision trees can explain the decision-making process of customers during the purchase. We show that our approach can predict the choice probability of any DCM consistently and thus never suffers from misspecification. Moreover, our algorithm predicts assortments unseen in the training data. The mechanism and errors can be theoretically analyzed. We also prove that the random forest can recover preference rankings of customers thanks to the splitting criterion such as the Gini index and information gain ratio. Managerial implications. The framework has unique practical advantages. It can capture customers' behavioral patterns such as irrationality or sequential searches when purchasing a product. It handles nonstandard formats of training data that result from aggregation. It can measure product importance based on how frequently a random customer would make decisions depending on the presence of the product. It can also incorporate price information and customer features. Our numerical experiments using synthetic and real data show that using random forests to estimate customer choices can outperform existing methods.

Wenhao Li, Ningyuan Chen, L. Jeff Hong

In contextual continuum-armed bandits, the contexts $x$ and the arms $y$ are both continuous and drawn from high-dimensional spaces. The payoff function to learn $f(x,y)$ does not have a particular parametric form. The literature has shown that for Lipschitz-continuous functions, the optimal regret is $\tilde{O}(T^{\frac{d_x+d_y+1}{d_x+d_y+2}})$, where $d_x$ and $d_y$ are the dimensions of contexts and arms, and thus suffers from the curse of dimensionality. We develop an algorithm that achieves regret $\tilde{O}(T^{\frac{d_x+1}{d_x+2}})$ when $f$ is globally concave in $y$. The global concavity is a common assumption in many applications. The algorithm is based on stochastic approximation and estimates the gradient information in an online fashion. Our results generate a valuable insight that the curse of dimensionality of the arms can be overcome with some mild structures of the payoff function.

Ningyuan Chen, Guillermo Gallego

We consider the problem of a firm seeking to use personalized pricing to sell an exogenously given stock of a product over a finite selling horizon to different consumer types. We assume that the type of an arriving consumer can be observed but the demand function associated with each type is initially unknown. The firm sets personalized prices dynamically for each type and attempts to maximize the revenue over the season. We provide a learning algorithm that is near-optimal when the demand and capacity scale in proportion. The algorithm utilizes the primal-dual formulation of the problem and learns the dual optimal solution explicitly. It allows the algorithm to overcome the curse of dimensionality (the rate of regret is independent of the number of types) and sheds light on novel algorithmic designs for learning problems with resource constraints.

Ningyuan Chen, Guillermo Gallego

Personalized pricing analytics is becoming an essential tool in retailing. Upon observing the personalized information of each arriving customer, the firm needs to set a price accordingly based on the covariates such as income, education background, past purchasing history to extract more revenue. For new entrants of the business, the lack of historical data may severely limit the power and profitability of personalized pricing. We propose a nonparametric pricing policy to simultaneously learn the preference of customers based on the covariates and maximize the expected revenue over a finite horizon. The policy does not depend on any prior assumptions on how the personalized information affects consumers' preferences (such as linear models). It is adaptively splits the covariate space into smaller bins (hyper-rectangles) and clusters customers based on their covariates and preferences, offering similar prices for customers who belong to the same cluster trading off granularity and accuracy. We show that the algorithm achieves a regret of order $O(\log(T)^2 T^{(2+d)/(4+d)})$, where $T$ is the length of the horizon and $d$ is the dimension of the covariate. It improves the current regret in the literature \citep{slivkins2014contextual}, under mild technical conditions in the pricing context (smoothness and local concavity). We also prove that no policy can achieve a regret less than $O(T^{(2+d)/(4+d)})$ for a particular instance and thus demonstrate the near optimality of the proposed policy.

Donald K. K. Lee, Ningyuan Chen, Hemant Ishwaran

Given functional data from a survival process with time-dependent covariates, we derive a smooth convex representation for its nonparametric log-likelihood functional and obtain its functional gradient. From this, we devise a generic gradient boosting procedure for estimating the hazard function nonparametrically. An illustrative implementation of the procedure using regression trees is described to show how to recover the unknown hazard. The generic estimator is consistent if the model is correctly specified; alternatively, an oracle inequality can be demonstrated for tree-based models. To avoid overfitting, boosting employs several regularization devices. One of them is step-size restriction, but the rationale for this is somewhat mysterious from the viewpoint of consistency. Our work brings some clarity to this issue by revealing that step-size restriction is a mechanism for preventing the curvature of the risk from derailing convergence.

Ningyuan Chen, Donald K. K. Lee, Sahand Negahban

Exploiting the fact that most arrival processes exhibit cyclic behaviour, we propose a simple procedure for estimating the intensity of a nonhomogeneous Poisson process. The estimator is the super-resolution analogue to Shao 2010 and Shao & Lii 2011, which is a sum of $p$ sinusoids where $p$ and the frequency, amplitude, and phase of each wave are not known and need to be estimated. This results in an interpretable yet flexible specification that is suitable for use in modelling as well as in high resolution simulations. Our estimation procedure sits in between classic periodogram methods and atomic/total variation norm thresholding. Through a novel use of window functions in the point process domain, our approach attains super-resolution without semidefinite programming. Under suitable conditions, finite sample guarantees can be derived for our procedure. These resolve some open questions and expand existing results in spectral estimation literature.