The Power of Adaptivity in Experimental Design

Given n experiment subjects with potentially heterogeneous covariates and two possible treatments, namely active treatment and control, this paper addresses the fundamental question of determining the optimal accuracy in estimating the treatment effect. Furthermore, we propose an experimental design that approaches this optimal accuracy, giving a (non-asymptotic) answer to this fundamental yet still open question. The methodological contribution is listed as following. First, we establish an idealized optimal estimator with minimal variance as benchmark, and then demonstrate that adaptive experiment is necessary to achieve near-optimal estimation accuracy. Secondly, by incorporating the concept of doubly robust method into sequential experimental design, we frame the optimal estimation problem as an online bandit learning problem, bridging the two fields of statistical estimation and bandit learning. Using tools and ideas from both bandit algorithm design and adaptive statistical estimation, we propose a general low switching adaptive experiment framework, which could be a generic research paradigm for a wide range of adaptive experimental design. Through information-theoretic lower bound combined with Bayes risk analysis, we demonstrate the optimality of our proposed experiment. Numerical result indicates that the estimation accuracy approaches optimal with as few as two or three policy updates.

Experimenting on Markov Decision Processes with Local Treatments

As service systems grow increasingly complex and dynamic, many interventions become localized, available and taking effect only in specific states. This paper investigates experiments with local treatments on a widely-used class of dynamic models, Markov Decision Processes (MDPs). Particularly, we focus on utilizing the local structure to improve the inference efficiency of the average treatment effect. We begin by demonstrating the efficiency of classical inference methods, including model-based estimation and temporal difference learning under a fixed policy, as well as classical A/B testing with general treatments. We then introduce a variance reduction technique that exploits the local treatment structure by sharing information for states unaffected by the treatment policy. Our new estimator effectively overcomes the variance lower bound for general treatments while matching the more stringent lower bound incorporating the local treatment structure. Furthermore, our estimator can optimally achieve a linear reduction with the number of test arms for a major part of the variance. Finally, we explore scenarios with perfect knowledge of the control arm and design estimators that further improve inference efficiency.

Offline Oracle-Efficient Learning for Contextual MDPs via Layerwise Exploration-Exploitation Tradeoff

Motivated by the recent discovery of a statistical and computational reduction from contextual bandits to offline regression (Simchi-Levi and Xu, 2021), we address the general (stochastic) Contextual Markov Decision Process (CMDP) problem with horizon H (as known as CMDP with H layers). In this paper, we introduce a reduction from CMDPs to offline density estimation under the realizability assumption, i.e., a model class M containing the true underlying CMDP is provided in advance. We develop an efficient, statistically near-optimal algorithm requiring only O(HlogT) calls to an offline density estimation algorithm (or oracle) across all T rounds of interaction. This number can be further reduced to O(HloglogT) if T is known in advance. Our results mark the first efficient and near-optimal reduction from CMDPs to offline density estimation without imposing any structural assumptions on the model class. A notable feature of our algorithm is the design of a layerwise exploration-exploitation tradeoff tailored to address the layerwise structure of CMDPs. Additionally, our algorithm is versatile and applicable to pure exploration tasks in reward-free reinforcement learning.

On the Optimal Regret of Locally Private Linear Contextual Bandit

Contextual bandit with linear reward functions is among one of the most extensively studied models in bandit and online learning research. Recently, there has been increasing interest in designing \emph{locally private} linear contextual bandit algorithms, where sensitive information contained in contexts and rewards is protected against leakage to the general public. While the classical linear contextual bandit algorithm admits cumulative regret upper bounds of $\tilde O(\sqrt{T})$ via multiple alternative methods, it has remained open whether such regret bounds are attainable in the presence of local privacy constraints, with the state-of-the-art result being $\tilde O(T^{3/4})$. In this paper, we show that it is indeed possible to achieve an $\tilde O(\sqrt{T})$ regret upper bound for locally private linear contextual bandit. Our solution relies on several new algorithmic and analytical ideas, such as the analysis of mean absolute deviation errors and layered principal component regression in order to achieve small mean absolute deviation errors.

Online Local False Discovery Rate Control: A Resource Allocation Approach

We consider the problem of online local false discovery rate (FDR) control where multiple tests are conducted sequentially, with the goal of maximizing the total expected number of discoveries. We formulate the problem as an online resource allocation problem with accept/reject decisions, which from a high level can be viewed as an online knapsack problem, with the additional uncertainty of random budget replenishment. We start with general arrival distributions and propose a simple policy that achieves a $O(\sqrt{T})$ regret. We complement the result by showing that such regret rate is in general not improvable. We then shift our focus to discrete arrival distributions. We find that many existing re-solving heuristics in the online resource allocation literature, albeit achieve bounded loss in canonical settings, may incur a $\Omega(\sqrt{T})$ or even a $\Omega(T)$ regret. With the observation that canonical policies tend to be too optimistic and over accept arrivals, we propose a novel policy that incorporates budget buffers. We show that small additional logarithmic buffers suffice to reduce the regret from $\Omega(\sqrt{T})$ or even $\Omega(T)$ to $O(\ln^2 T)$. Numerical experiments are conducted to validate our theoretical findings. Our formulation may have wider applications beyond the problem considered in this paper, and our results emphasize how effective policies should be designed to reach a balance between circumventing wrong accept and reducing wrong reject in online resource allocation problems with uncertain budgets.

Privacy Preserving Adaptive Experiment Design

Adaptive experiment is widely adopted to estimate conditional average treatment effect (CATE) in clinical trials and many other scenarios. While the primary goal in experiment is to maximize estimation accuracy, due to the imperative of social welfare, it's also crucial to provide treatment with superior outcomes to patients, which is measured by regret in contextual bandit framework. These two objectives often lead to contrast optimal allocation mechanism. Furthermore, privacy concerns arise in clinical scenarios containing sensitive data like patients health records. Therefore, it's essential for the treatment allocation mechanism to incorporate robust privacy protection measures. In this paper, we investigate the tradeoff between loss of social welfare and statistical power in contextual bandit experiment. We propose a matched upper and lower bound for the multi-objective optimization problem, and then adopt the concept of Pareto optimality to mathematically characterize the optimality condition. Furthermore, we propose differentially private algorithms which still matches the lower bound, showing that privacy is "almost free". Additionally, we derive the asymptotic normality of the estimator, which is essential in statistical inference and hypothesis testing.

Utility Fairness in Contextual Dynamic Pricing with Demand Learning

This paper introduces a novel contextual bandit algorithm for personalized pricing under utility fairness constraints in scenarios with uncertain demand, achieving an optimal regret upper bound. Our approach, which incorporates dynamic pricing and demand learning, addresses the critical challenge of fairness in pricing strategies. We first delve into the static full-information setting to formulate an optimal pricing policy as a constrained optimization problem. Here, we propose an approximation algorithm for efficiently and approximately computing the ideal policy. We also use mathematical analysis and computational studies to characterize the structures of optimal contextual pricing policies subject to fairness constraints, deriving simplified policies which lays the foundations of more in-depth research and extensions. Further, we extend our study to dynamic pricing problems with demand learning, establishing a non-standard regret lower bound that highlights the complexity added by fairness constraints. Our research offers a comprehensive analysis of the cost of fairness and its impact on the balance between utility and revenue maximization. This work represents a step towards integrating ethical considerations into algorithmic efficiency in data-driven dynamic pricing.

Regret Distribution in Stochastic Bandits: Optimal Trade-off between Expectation and Tail Risk

We study the trade-off between expectation and tail risk for regret distribution in the stochastic multi-armed bandit problem. We fully characterize the interplay among three desired properties for policy design: worst-case optimality, instance-dependent consistency, and light-tailed risk. We show how the order of expected regret exactly affects the decaying rate of the regret tail probability for both the worst-case and instance-dependent scenario. A novel policy is proposed to characterize the optimal regret tail probability for any regret threshold. Concretely, for any given $\alpha\in[1/2, 1)$ and $\beta\in[0, \alpha]$, our policy achieves a worst-case expected regret of $\tilde O(T^\alpha)$ (we call it $\alpha$-optimal) and an instance-dependent expected regret of $\tilde O(T^\beta)$ (we call it $\beta$-consistent), while enjoys a probability of incurring an $\tilde O(T^\delta)$ regret ($\delta\geq\alpha$ in the worst-case scenario and $\delta\geq\beta$ in the instance-dependent scenario) that decays exponentially with a polynomial $T$ term. Such decaying rate is proved to be best achievable. Moreover, we discover an intrinsic gap of the optimal tail rate under the instance-dependent scenario between whether the time horizon $T$ is known a priori or not. Interestingly, when it comes to the worst-case scenario, this gap disappears. Finally, we extend our proposed policy design to (1) a stochastic multi-armed bandit setting with non-stationary baseline rewards, and (2) a stochastic linear bandit setting. Our results reveal insights on the trade-off between regret expectation and regret tail risk for both worst-case and instance-dependent scenarios, indicating that more sub-optimality and inconsistency leave space for more light-tailed risk of incurring a large regret, and that knowing the planning horizon in advance can make a difference on alleviating tail risks.

A Simple and Optimal Policy Design for Online Learning with Safety against Heavy-tailed Risk

We design simple and optimal policies that ensure safety against heavy-tailed risk in the classical multi-armed bandit problem. We start by showing that some widely used policies such as the standard Upper Confidence Bound policy and the Thompson Sampling policy incur heavy-tailed risk; that is, the worst-case probability of incurring a linear regret slowly decays at a polynomial rate of $1/T$, where $T$ is the time horizon. We further show that this heavy-tailed risk exists for all "instance-dependent consistent" policies. To ensure safety against such heavy-tailed risk, for the two-armed bandit setting, we provide a simple policy design that (i) has the worst-case optimality for the expected regret at order $\tilde O(\sqrt{T})$ and (ii) has the worst-case tail probability of incurring a linear regret decay at an exponential rate $\exp(-\Omega(\sqrt{T}))$. We further prove that this exponential decaying rate of the tail probability is optimal across all policies that have worst-case optimality for the expected regret. Finally, we improve the policy design and analysis to the general $K$-armed bandit setting. We provide detailed characterization of the tail probability bound for any regret threshold under our policy design. Namely, the worst-case probability of incurring a regret larger than $x$ is upper bounded by $\exp(-\Omega(x/\sqrt{KT}))$. Numerical experiments are conducted to illustrate the theoretical findings. Our results reveal insights on the incompatibility between consistency and light-tailed risk, whereas indicate that worst-case optimality on expected regret and light-tailed risk are compatible.

PAC-Bayesian Based Adaptation for Regularized Learning

In this paper, we propose a PAC-Bayesian \textit{a posteriori} parameter selection scheme for adaptive regularized regression in Hilbert scales under general, unknown source conditions. We demonstrate that our approach is adaptive to misspecification, and achieves the optimal learning rate under subgaussian noise. Unlike existing parameter selection schemes, the computational complexity of our approach is independent of sample size. We derive minimax adaptive rates for a new, broad class of Tikhonov-regularized learning problems under general, misspecified source conditions, that notably do not require any conventional a priori assumptions on kernel eigendecay. Using the theory of interpolation, we demonstrate that the spectrum of the Mercer operator can be inferred in the presence of "tight" $L^{\infty}$ embeddings of suitable Hilbert scales. Finally, we prove, that under a $\Delta_2$ condition on the smoothness index functions, our PAC-Bayesian scheme can indeed achieve minimax rates. We discuss applications of our approach to statistical inverse problems and oracle-efficient contextual bandit algorithms.