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Abstract:We study a collaborative learning problem where $m$ agents estimate a vector $\mu\in\mathbb{R}^d$ by collecting samples from normal distributions, with each agent $i$ incurring a cost $c_{i,k} \in (0, \infty]$ to sample from the $k^{\text{th}}$ distribution $\mathcal{N}(\mu_k, \sigma^2)$. Instead of working on their own, agents can collect data that is cheap to them, and share it with others in exchange for data that is expensive or even inaccessible to them, thereby simultaneously reducing data collection costs and estimation error. However, when agents have different collection costs, we need to first decide how to fairly divide the work of data collection so as to benefit all agents. Moreover, in naive sharing protocols, strategic agents may under-collect and/or fabricate data, leading to socially undesirable outcomes. Our mechanism addresses these challenges by combining ideas from cooperative and non-cooperative game theory. We use ideas from axiomatic bargaining to divide the cost of data collection. Given such a solution, we develop a Nash incentive-compatible (NIC) mechanism to enforce truthful reporting. We achieve a $\mathcal{O}(\sqrt{m})$ approximation to the minimum social penalty (sum of agent estimation errors and data collection costs) in the worst case, and a $\mathcal{O}(1)$ approximation under favorable conditions. We complement this with a hardness result, showing that $\Omega(\sqrt{m})$ is unavoidable in any NIC mechanism.

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Abstract:We study a data pricing problem, where a seller has access to $N$ homogeneous data points (e.g. drawn i.i.d. from some distribution). There are $m$ types of buyers in the market, where buyers of the same type $i$ have the same valuation curve $v_i:[N]\rightarrow [0,1]$, where $v_i(n)$ is the value for having $n$ data points. \textit{A priori}, the seller is unaware of the distribution of buyers, but can repeat the market for $T$ rounds so as to learn the revenue-optimal pricing curve $p:[N] \rightarrow [0, 1]$. To solve this online learning problem, we first develop novel discretization schemes to approximate any pricing curve. When compared to prior work, the size of our discretization schemes scales gracefully with the approximation parameter, which translates to better regret in online learning. Under assumptions like smoothness and diminishing returns which are satisfied by data, the discretization size can be reduced further. We then turn to the online learning problem, both in the stochastic and adversarial settings. On each round, the seller chooses an \emph{anonymous} pricing curve $p_t$. A new buyer appears and may choose to purchase some amount of data. She then reveals her type \emph{only if} she makes a purchase. Our online algorithms build on classical algorithms such as UCB and FTPL, but require novel ideas to account for the asymmetric nature of this feedback and to deal with the vastness of the space of pricing curves. Using the improved discretization schemes previously developed, we are able to achieve $\tilde{O}\left(m\sqrt{T}\right)$ regret in the stochastic setting and $\tilde{O}\left(m^{\frac{3}{2}}\sqrt{T}\right)$ regret in the adversarial setting.

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Abstract:We study a multi-round mechanism design problem, where we interact with a set of agents over a sequence of rounds. We wish to design an incentive-compatible (IC) online learning scheme to maximize an application-specific objective within a given class of mechanisms, without prior knowledge of the agents' type distributions. Even if each mechanism in this class is IC in a single round, if an algorithm naively chooses from this class on each round, the entire learning process may not be IC against non-myopic buyers who appear over multiple rounds. On each round, our method randomly chooses between the recommendation of a weakly differentially private online learning algorithm (e.g., Hedge), and a commitment mechanism which penalizes non-truthful behavior. Our method is IC and achieves $O(T^{\frac{1+h}{2}})$ regret for the application-specific objective in an adversarial setting, where $h$ quantifies the long-sightedness of the agents. When compared to prior work, our approach is conceptually simpler,it applies to general mechanism design problems (beyond auctions), and its regret scales gracefully with the size of the mechanism class.

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Abstract:We study a sequential profit-maximization problem, optimizing for both price and ancillary variables like marketing expenditures. Specifically, we aim to maximize profit over an arbitrary sequence of multiple demand curves, each dependent on a distinct ancillary variable, but sharing the same price. A prototypical example is targeted marketing, where a firm (seller) wishes to sell a product over multiple markets. The firm may invest different marketing expenditures for different markets to optimize customer acquisition, but must maintain the same price across all markets. Moreover, markets may have heterogeneous demand curves, each responding to prices and marketing expenditures differently. The firm's objective is to maximize its gross profit, the total revenue minus marketing costs. Our results are near-optimal algorithms for this class of problems in an adversarial bandit setting, where demand curves are arbitrary non-adaptive sequences, and the firm observes only noisy evaluations of chosen points on the demand curves. We prove a regret upper bound of $\widetilde{\mathcal{O}}\big(nT^{3/4}\big)$ and a lower bound of $\Omega\big((nT)^{3/4}\big)$ for monotonic demand curves, and a regret bound of $\widetilde{\Theta}\big(nT^{2/3}\big)$ for demands curves that are monotonic in price and concave in the ancillary variables.

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Abstract:We study actively labeling streaming data, where an active learner is faced with a stream of data points and must carefully choose which of these points to label via an expensive experiment. Such problems frequently arise in applications such as healthcare and astronomy. We first study a setting when the data's inputs belong to one of $K$ discrete distributions and formalize this problem via a loss that captures the labeling cost and the prediction error. When the labeling cost is $B$, our algorithm, which chooses to label a point if the uncertainty is larger than a time and cost dependent threshold, achieves a worst-case upper bound of $O(B^{\frac{1}{3}} K^{\frac{1}{3}} T^{\frac{2}{3}})$ on the loss after $T$ rounds. We also provide a more nuanced upper bound which demonstrates that the algorithm can adapt to the arrival pattern, and achieves better performance when the arrival pattern is more favorable. We complement both upper bounds with matching lower bounds. We next study this problem when the inputs belong to a continuous domain and the output of the experiment is a smooth function with bounded RKHS norm. After $T$ rounds in $d$ dimensions, we show that the loss is bounded by $O(B^{\frac{1}{d+3}} T^{\frac{d+2}{d+3}})$ in an RKHS with a squared exponential kernel and by $O(B^{\frac{1}{2d+3}} T^{\frac{2d+2}{2d+3}})$ in an RKHS with a Mat\'ern kernel. Our empirical evaluation demonstrates that our method outperforms other baselines in several synthetic experiments and two real experiments in medicine and astronomy.

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Abstract:In online marketplaces, customers have access to hundreds of reviews for a single product. Buyers often use reviews from other customers that share their type -- such as height for clothing, skin type for skincare products, and location for outdoor furniture -- to estimate their values, which they may not know a priori. Customers with few relevant reviews may hesitate to make a purchase except at a low price, so for the seller, there is a tension between setting high prices and ensuring that there are enough reviews so that buyers can confidently estimate their values. Simultaneously, sellers may use reviews to gauge the demand for items they wish to sell. In this work, we study this pricing problem in an online setting where the seller interacts with a set of buyers of finitely-many types, one-by-one, over a series of $T$ rounds. At each round, the seller first sets a price. Then a buyer arrives and examines the reviews of the previous buyers with the same type, which reveal those buyers' ex-post values. Based on the reviews, the buyer decides to purchase if they have good reason to believe that their ex-ante utility is positive. Crucially, the seller does not know the buyer's type when setting the price, nor even the distribution over types. We provide a no-regret algorithm that the seller can use to obtain high revenue. When there are $d$ types, after $T$ rounds, our algorithm achieves a problem-independent $\tilde O(T^{2/3}d^{1/3})$ regret bound. However, when the smallest probability $q_{\text{min}}$ that any given type appears is large, specifically when $q_{\text{min}} \in \Omega(d^{-2/3}T^{-1/3})$, then the same algorithm achieves a $\tilde O(T^{1/2}q_{\text{min}}^{-1/2})$ regret bound. We complement these upper bounds with matching lower bounds in both regimes, showing that our algorithm is minimax optimal up to lower order terms.

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Abstract:The sharing of scarce resources among multiple rational agents is one of the classical problems in economics. In exchange economies, which are used to model such situations, agents begin with an initial endowment of resources and exchange them in a way that is mutually beneficial until they reach a competitive equilibrium (CE). CE allocations are Pareto efficient and fair. Consequently, they are used widely in designing mechanisms for fair division. However, computing CEs requires the knowledge of agent preferences which are unknown in several applications of interest. In this work, we explore a new online learning mechanism, which, on each round, allocates resources to the agents and collects stochastic feedback on their experience in using that allocation. Its goal is to learn the agent utilities via this feedback and imitate the allocations at a CE in the long run. We quantify CE behavior via two losses and propose a randomized algorithm which achieves $\bigOtilde(\sqrt{T})$ loss after $T$ rounds under both criteria. Empirically, we demonstrate the effectiveness of this mechanism through numerical simulations.

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Authors:Brijen Thananjeyan, Kirthevasan Kandasamy, Ion Stoica, Michael I. Jordan, Ken Goldberg, Joseph E. Gonzalez

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Abstract:We study $(\epsilon, \delta)$-PAC best arm identification, where a decision-maker must identify an $\epsilon$-optimal arm with probability at least $1 - \delta$, while minimizing the number of arm pulls (samples). Most of the work on this topic is in the sequential setting, where there is no constraint on the time taken to identify such an arm; this allows the decision-maker to pull one arm at a time. In this work, the decision-maker is given a deadline of $T$ rounds, where, on each round, it can adaptively choose which arms to pull and how many times to pull them; this distinguishes the number of decisions made (i.e., time or number of rounds) from the number of samples acquired (cost). Such situations occur in clinical trials, where one may need to identify a promising treatment under a deadline while minimizing the number of test subjects, or in simulation-based studies run on the cloud, where we can elastically scale up or down the number of virtual machines to conduct as many experiments as we wish, but need to pay for the resource-time used. As the decision-maker can only make $T$ decisions, she may need to pull some arms excessively relative to a sequential algorithm in order to perform well on all possible problems. We formalize this added difficulty with two hardness results that indicate that unlike sequential settings, the ability to adapt to the problem difficulty is constrained by the finite deadline. We propose Elastic Batch Racing (EBR), a novel algorithm for this setting and bound its sample complexity, showing that EBR is optimal with respect to both hardness results. We present simulations evaluating EBR in this setting, where it outperforms baselines by several orders of magnitude.

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Abstract:We describe mechanisms for the allocation of a scarce resource among multiple users in a way that is efficient, fair, and strategy-proof, but when users do not know their resource requirements. The mechanism is repeated for multiple rounds and a user's requirements can change on each round. At the end of each round, users provide feedback about the allocation they received, enabling the mechanism to learn user preferences over time. Such situations are common in the shared usage of a compute cluster among many users in an organisation, where all teams may not precisely know the amount of resources needed to execute their jobs. By understating their requirements, users will receive less than they need and consequently not achieve their goals. By overstating them, they may siphon away precious resources that could be useful to others in the organisation. We formalise this task of online learning in fair division via notions of efficiency, fairness, and strategy-proofness applicable to this setting, and study this problem under three types of feedback: when the users' observations are deterministic, when they are stochastic and follow a parametric model, and when they are stochastic and nonparametric. We derive mechanisms inspired by the classical max-min fairness procedure that achieve these requisites, and quantify the extent to which they are achieved via asymptotic rates. We corroborate these insights with an experimental evaluation on synthetic problems and a web-serving task.

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Authors:Brijen Thananjeyan, Kirthevasan Kandasamy, Ion Stoica, Michael I. Jordan, Ken Goldberg, Joseph E. Gonzalez

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Abstract:We study exploration in stochastic multi-armed bandits when we have access to a divisible resource, and can allocate varying amounts of this resource to arm pulls. By allocating more resources to a pull, we can compute the outcome faster to inform subsequent decisions about which arms to pull. However, since distributed environments do not scale linearly, executing several arm pulls in parallel, and hence less resources per pull, may result in better throughput. For example, in simulation-based scientific studies, an expensive simulation can be sped up by running it on multiple cores. This speed-up is, however, partly offset by the communication among cores and overheads, which results in lower throughput than if fewer cores were allocated to run more trials in parallel. We explore these trade-offs in the fixed confidence setting, where we need to find the best arm with a given success probability, while minimizing the time to do so. We propose an algorithm which trades off between information accumulation and throughout and show that the time taken can be upper bounded by the solution of a dynamic program whose inputs are the squared gaps between the suboptimal and optimal arms. We prove a matching hardness result which demonstrates that the above dynamic program is fundamental to this problem. Next, we propose and analyze an algorithm for the fixed deadline setting, where we are given a time deadline and need to maximize the success probability of finding the best arm. We corroborate these theoretical insights with an empirical evaluation.

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