arXiv subjects

David Simchi-Levi

Publications and source records attributed to David Simchi-Levi.

At least 73 records · Page 4Linked to original sources

arXiv2024

Bayesian Mechanism Design for Blockchain Transaction Fee Allocation

In blockchain systems, the design of transaction fee mechanisms is essential for stability and satisfaction for both miners and users. A recent work has proven the impossibility of collusion-proof mechanisms that achieve both non-zero miner revenue and Dominating-Strategy-Incentive-Compatible (DSIC) for users. However, a positive miner revenue is important in practice to motivate miners. To address this challenge, we consider a Bayesian game setting and relax the DSIC requirement for users to Bayesian-Nash-Incentive-Compatibility (BNIC). In particular, we propose an auxiliary mechanism method that makes connections between BNIC and DSIC mechanisms. With the auxiliary mechanism method, we design a transaction fee mechanism (TFM) based on the multinomial logit (MNL) choice model, and prove that the TFM has both BNIC and collusion-proof properties with an asymptotic constant-factor approximation of optimal miner revenue for i.i.d. bounded valuations. Our result breaks the zero-revenue barrier while preserving truthfulness and collusion-proof properties.

cs.GT↗

arXiv2024

Prediction-Guided Active Experiments

In this work, we introduce a new framework for active experimentation, the Prediction-Guided Active Experiment (PGAE), which leverages predictions from an existing machine learning model to guide sampling and experimentation. Specifically, at each time step, an experimental unit is sampled according to a designated sampling distribution, and the actual outcome is observed based on an experimental probability. Otherwise, only a prediction for the outcome is available. We begin by analyzing the non-adaptive case, where full information on the joint distribution of the predictor and the actual outcome is assumed. For this scenario, we derive an optimal experimentation strategy by minimizing the semi-parametric efficiency bound for the class of regular estimators. We then introduce an estimator that meets this efficiency bound, achieving asymptotic optimality. Next, we move to the adaptive case, where the predictor is continuously updated with newly sampled data. We show that the adaptive version of the estimator remains efficient and attains the same semi-parametric bound under certain regularity assumptions. Finally, we validate PGAE's performance through simulations and a semi-synthetic experiment using data from the US Census Bureau. The results underscore the PGAE framework's effectiveness and superiority compared to other existing methods.

stat.ML↗

arXiv2024

Optimal Adaptive Experimental Design for Estimating Treatment Effect

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 novel lower bound techniques for non-i.i.d. data, 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.

stat.ML↗

arXiv2024

A Simple and Optimal Policy Design with Safety against Heavy-Tailed Risk for Stochastic Bandits

We study the stochastic multi-armed bandit problem and design new policies that enjoy both worst-case optimality for expected regret and light-tailed risk for regret distribution. Specifically, our policy design (i) enjoys the worst-case optimality for the expected regret at order $O(\sqrt{KT\ln T})$ and (ii) has the worst-case tail probability of incurring a regret larger than any $x>0$ being upper bounded by $\exp(-Ω(x/\sqrt{KT}))$, a rate that we prove to be best achievable with respect to $T$ for all worst-case optimal policies. Our proposed policy achieves a delicate balance between doing more exploration at the beginning of the time horizon and doing more exploitation when approaching the end, compared to standard confidence-bound-based policies. We also enhance the policy design to accommodate the "any-time" setting where $T$ is unknown a priori, and prove equivalently desired policy performances as compared to the "fixed-time" setting with known $T$. Numerical experiments are conducted to illustrate the theoretical findings. We find that from a managerial perspective, our new policy design yields better tail distributions and is preferable than celebrated policies especially when (i) there is a risk of under-estimating the volatility profile, or (ii) there is a challenge of tuning policy hyper-parameters. We conclude by extending our proposed policy design to the stochastic linear bandit setting that leads to both worst-case optimality in terms of expected regret and light-tailed risk on the regret distribution.

stat.ML↗

arXiv2024

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.

cs.LG↗

arXiv2024

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.

stat.ML↗

arXiv2024

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.

stat.ME↗

arXiv2023

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.

stat.ML↗

arXiv2023

Optimal Learning Rates for Regularized Least-Squares with a Fourier Capacity Condition

We derive minimax adaptive rates for a new, broad class of Tikhonov-regularized learning problems in Hilbert scales under general source conditions. Our analysis does not require the regression function to be contained in the hypothesis class, and most notably does not employ the conventional \textit{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'' $L^{\infty}(\mathcal{X})$ embeddings of suitable Hilbert scales. Our analysis utilizes a new Fourier isocapacitary condition, which captures the interplay of the kernel Dirichlet capacities and small ball probabilities via the optimal Hilbert scale function.

math.ST↗

arXiv2022

Offline Reinforcement Learning: Fundamental Barriers for Value Function Approximation

We consider the offline reinforcement learning problem, where the aim is to learn a decision making policy from logged data. Offline RL -- particularly when coupled with (value) function approximation to allow for generalization in large or continuous state spaces -- is becoming increasingly relevant in practice, because it avoids costly and time-consuming online data collection and is well suited to safety-critical domains. Existing sample complexity guarantees for offline value function approximation methods typically require both (1) distributional assumptions (i.e., good coverage) and (2) representational assumptions (i.e., ability to represent some or all $Q$-value functions) stronger than what is required for supervised learning. However, the necessity of these conditions and the fundamental limits of offline RL are not well understood in spite of decades of research. This led Chen and Jiang (2019) to conjecture that concentrability (the most standard notion of coverage) and realizability (the weakest representation condition) alone are not sufficient for sample-efficient offline RL. We resolve this conjecture in the positive by proving that in general, even if both concentrability and realizability are satisfied, any algorithm requires sample complexity polynomial in the size of the state space to learn a non-trivial policy. Our results show that sample-efficient offline reinforcement learning requires either restrictive coverage conditions or representation conditions that go beyond supervised learning, and highlight a phenomenon called over-coverage which serves as a fundamental barrier for offline value function approximation methods. A consequence of our results for reinforcement learning with linear function approximation is that the separation between online and offline RL can be arbitrarily large, even in constant dimension.

cs.LG↗

arXiv2022

Model-Free Non-Stationary RL: Near-Optimal Regret and Applications in Multi-Agent RL and Inventory Control

We consider model-free reinforcement learning (RL) in non-stationary Markov decision processes. Both the reward functions and the state transition functions are allowed to vary arbitrarily over time as long as their cumulative variations do not exceed certain variation budgets. We propose Restarted Q-Learning with Upper Confidence Bounds (RestartQ-UCB), the first model-free algorithm for non-stationary RL, and show that it outperforms existing solutions in terms of dynamic regret. Specifically, RestartQ-UCB with Freedman-type bonus terms achieves a dynamic regret bound of $\widetilde{O}(S^{\frac{1}{3}} A^{\frac{1}{3}} Δ^{\frac{1}{3}} H T^{\frac{2}{3}})$, where $S$ and $A$ are the numbers of states and actions, respectively, $Δ>0$ is the variation budget, $H$ is the number of time steps per episode, and $T$ is the total number of time steps. We further present a parameter-free algorithm named Double-Restart Q-UCB that does not require prior knowledge of the variation budget. We show that our algorithms are \emph{nearly optimal} by establishing an information-theoretical lower bound of $Ω(S^{\frac{1}{3}} A^{\frac{1}{3}} Δ^{\frac{1}{3}} H^{\frac{2}{3}} T^{\frac{2}{3}})$, the first lower bound in non-stationary RL. Numerical experiments validate the advantages of RestartQ-UCB in terms of both cumulative rewards and computational efficiency. We demonstrate the power of our results in examples of multi-agent RL and inventory control across related products.

cs.LG↗

arXiv2022

Sobolev Norm Learning Rates for Conditional Mean Embeddings

We develop novel learning rates for conditional mean embeddings by applying the theory of interpolation for reproducing kernel Hilbert spaces (RKHS). We derive explicit, adaptive convergence rates for the sample estimator under the misspecifed setting, where the target operator is not Hilbert-Schmidt or bounded with respect to the input/output RKHSs. We demonstrate that in certain parameter regimes, we can achieve uniform convergence rates in the output RKHS. We hope our analyses will allow the much broader application of conditional mean embeddings to more complex ML/RL settings involving infinite dimensional RKHSs and continuous state spaces.

stat.ML↗

arXiv2021

Offline Planning and Online Learning under Recovering Rewards

Motivated by emerging applications such as live-streaming e-commerce, promotions and recommendations, we introduce and solve a general class of non-stationary multi-armed bandit problems that have the following two features: (i) the decision maker can pull and collect rewards from up to $K\,(\ge 1)$ out of $N$ different arms in each time period; (ii) the expected reward of an arm immediately drops after it is pulled, and then non-parametrically recovers as the arm's idle time increases. With the objective of maximizing the expected cumulative reward over $T$ time periods, we design a class of ``Purely Periodic Policies'' that jointly set a period to pull each arm. For the proposed policies, we prove performance guarantees for both the offline problem and the online problems. For the offline problem when all model parameters are known, the proposed periodic policy obtains an approximation ratio that is at the order of $1-\mathcal O(1/\sqrt{K})$, which is asymptotically optimal when $K$ grows to infinity. For the online problem when the model parameters are unknown and need to be dynamically learned, we integrate the offline periodic policy with the upper confidence bound procedure to construct on online policy. The proposed online policy is proved to approximately have $\widetilde{\mathcal O}(N\sqrt{T})$ regret against the offline benchmark. Our framework and policy design may shed light on broader offline planning and online learning applications with non-stationary and recovering rewards.

stat.ML↗

arXiv2021

Dynamic Pricing and Demand Learning on a Large Network of Products: A PAC-Bayesian Approach

We consider a seller offering a large network of $N$ products over a time horizon of $T$ periods. The seller does not know the parameters of the products' linear demand model, and can dynamically adjust product prices to learn the demand model based on sales observations. The seller aims to minimize its pseudo-regret, i.e., the expected revenue loss relative to a clairvoyant who knows the underlying demand model. We consider a sparse set of demand relationships between products to characterize various connectivity properties of the product network. In particular, we study three different sparsity frameworks: (1) $L_0$ sparsity, which constrains the number of connections in the network, and (2) off-diagonal sparsity, which constrains the magnitude of cross-product price sensitivities, and (3) a new notion of spectral sparsity, which constrains the asymptotic decay of a similarity metric on network nodes. We propose a dynamic pricing-and-learning policy that combines the optimism-in-the-face-of-uncertainty and PAC-Bayesian approaches, and show that this policy achieves asymptotically optimal performance in terms of $N$ and $T$. We also show that in the case of spectral and off-diagonal sparsity, the seller can have a pseudo-regret linear in $N$, even when the network is dense.

stat.ML↗

arXiv2021

Online Pricing with Offline Data: Phase Transition and Inverse Square Law

This paper investigates the impact of pre-existing offline data on online learning, in the context of dynamic pricing. We study a single-product dynamic pricing problem over a selling horizon of $T$ periods. The demand in each period is determined by the price of the product according to a linear demand model with unknown parameters. We assume that before the start of the selling horizon, the seller already has some pre-existing offline data. The offline data set contains $n$ samples, each of which is an input-output pair consisting of a historical price and an associated demand observation. The seller wants to utilize both the pre-existing offline data and the sequential online data to minimize the regret of the online learning process. We characterize the joint effect of the size, location and dispersion of the offline data on the optimal regret of the online learning process. Specifically, the size, location and dispersion of the offline data are measured by the number of historical samples $n$, the distance between the average historical price and the optimal price $δ$, and the standard deviation of the historical prices $σ$, respectively. We show that the optimal regret is $\widetilde Θ\left(\sqrt{T}\wedge \frac{T}{(n\wedge T)δ^2+nσ^2}\right)$, and design a learning algorithm based on the "optimism in the face of uncertainty" principle, whose regret is optimal up to a logarithmic factor. Our results reveal surprising transformations of the optimal regret rate with respect to the size of the offline data, which we refer to as phase transitions. In addition, our results demonstrate that the location and dispersion of the offline data also have an intrinsic effect on the optimal regret, and we quantify this effect via the inverse-square law.

cs.LG↗

arXiv2021

Inventory Balancing with Online Learning

We study a general problem of allocating limited resources to heterogeneous customers over time under model uncertainty. Each type of customer can be serviced using different actions, each of which stochastically consumes some combination of resources, and returns different rewards for the resources consumed. We consider a general model where the resource consumption distribution associated with each (customer type, action)-combination is not known, but is consistent and can be learned over time. In addition, the sequence of customer types to arrive over time is arbitrary and completely unknown. We overcome both the challenges of model uncertainty and customer heterogeneity by judiciously synthesizing two algorithmic frameworks from the literature: inventory balancing, which "reserves" a portion of each resource for high-reward customer types which could later arrive, and online learning, which shows how to "explore" the resource consumption distributions of each customer type under different actions. We define an auxiliary problem, which allows for existing competitive ratio and regret bounds to be seamlessly integrated. Furthermore, we show that the performance guarantee generated by our framework is tight, that is, we provide an information-theoretic lower bound which shows that both the loss from competitive ratio and the loss for regret are relevant in the combined problem. Finally, we demonstrate the efficacy of our algorithms on a publicly available hotel data set. Our framework is highly practical in that it requires no historical data (no fitted customer choice models, nor forecasting of customer arrival patterns) and can be used to initialize allocation strategies in fast-changing environments.

cs.AI↗

arXiv2021

Privacy-Preserving Dynamic Personalized Pricing with Demand Learning

The prevalence of e-commerce has made detailed customers' personal information readily accessible to retailers, and this information has been widely used in pricing decisions. When involving personalized information, how to protect the privacy of such information becomes a critical issue in practice. In this paper, we consider a dynamic pricing problem over $T$ time periods with an \emph{unknown} demand function of posted price and personalized information. At each time $t$, the retailer observes an arriving customer's personal information and offers a price. The customer then makes the purchase decision, which will be utilized by the retailer to learn the underlying demand function. There is potentially a serious privacy concern during this process: a third party agent might infer the personalized information and purchase decisions from price changes from the pricing system. Using the fundamental framework of differential privacy from computer science, we develop a privacy-preserving dynamic pricing policy, which tries to maximize the retailer revenue while avoiding information leakage of individual customer's information and purchasing decisions. To this end, we first introduce a notion of \emph{anticipating} $(\varepsilon, δ)$-differential privacy that is tailored to dynamic pricing problem. Our policy achieves both the privacy guarantee and the performance guarantee in terms of regret. Roughly speaking, for $d$-dimensional personalized information, our algorithm achieves the expected regret at the order of $\tilde{O}(\varepsilon^{-1} \sqrt{d^3 T})$, when the customers' information is adversarially chosen. For stochastic personalized information, the regret bound can be further improved to $\tilde{O}(\sqrt{d^2T} + \varepsilon^{-2} d^2)$

cs.CR↗

arXiv2021

Learning to Optimize under Non-Stationarity

We introduce algorithms that achieve state-of-the-art \emph{dynamic regret} bounds for non-stationary linear stochastic bandit setting. It captures natural applications such as dynamic pricing and ads allocation in a changing environment. We show how the difficulty posed by the non-stationarity can be overcome by a novel marriage between stochastic and adversarial bandits learning algorithms. Defining $d,B_T,$ and $T$ as the problem dimension, the \emph{variation budget}, and the total time horizon, respectively, our main contributions are the tuned Sliding Window UCB (\texttt{SW-UCB}) algorithm with optimal $\widetilde{O}(d^{2/3}(B_T+1)^{1/3}T^{2/3})$ dynamic regret, and the tuning free bandit-over-bandit (\texttt{BOB}) framework built on top of the \texttt{SW-UCB} algorithm with best $\widetilde{O}(d^{2/3}(B_T+1)^{1/4}T^{3/4})$ dynamic regret.

cs.LG↗