We consider a dynamic pricing problem for repeated contextual second-price auctions with multiple strategic buyers who aim to maximize their long-term time discounted utility. The seller has limited information on buyers' overall demand curves which depends on a non-parametric market-noise distribution, and buyers may potentially submit corrupted bids (relative to true valuations) to manipulate the seller's pricing policy for more favorable reserve prices in the future. We focus on designing the seller's learning policy to set contextual reserve prices where the seller's goal is to minimize regret compared to the revenue of a benchmark clairvoyant policy that has full information of buyers' demand. We propose a policy with a phased-structure that incorporates randomized "isolation" periods, during which a buyer is randomly chosen to solely participate in the auction. We show that this design allows the seller to control the number of periods in which buyers significantly corrupt their bids. We then prove that our policy enjoys a $T$-period regret of $\widetilde{\mathcal{O}}(\sqrt{T})$ facing strategic buyers. Finally, we conduct numerical simulations to compare our proposed algorithm to standard pricing policies. Our numerical results show that our algorithm outperforms these policies under various buyer bidding behavior.
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Individualized treatment decisions can improve health outcomes, but using data to make these decisions in a reliable, precise, and generalizable way is challenging with a single dataset. Leveraging multiple randomized controlled trials allows for the combination of datasets with unconfounded treatment assignment to improve the power to estimate heterogeneous treatment effects. This paper discusses several non-parametric approaches for estimating heterogeneous treatment effects using data from multiple trials. We extend single-study methods to a scenario with multiple trials and explore their performance through a simulation study, with data generation scenarios that have differing levels of cross-trial heterogeneity. The simulations demonstrate that methods that directly allow for heterogeneity of the treatment effect across trials perform better than methods that do not, and that the choice of single-study method matters based on the functional form of the treatment effect. Finally, we discuss which methods perform well in each setting and then apply them to four randomized controlled trials to examine effect heterogeneity of treatments for major depressive disorder.
The study of market equilibria is central to economic theory, particularly in efficiently allocating scarce resources. However, the computation of equilibrium prices at which the supply of goods matches their demand typically relies on having access to complete information on private attributes of agents, e.g., suppliers' cost functions, which are often unavailable in practice. Motivated by this practical consideration, we consider the problem of setting equilibrium prices in the incomplete information setting wherein a market operator seeks to satisfy the customer demand for a commodity by purchasing the required amount from competing suppliers with privately known cost functions unknown to the market operator. In this incomplete information setting, we consider the online learning problem of learning equilibrium prices over time while jointly optimizing three performance metrics -- unmet demand, cost regret, and payment regret -- pertinent in the context of equilibrium pricing over a horizon of $T$ periods. We first consider the setting when suppliers' cost functions are fixed and develop algorithms that achieve a regret of $O(\log \log T)$ when the customer demand is constant over time, or $O(\sqrt{T} \log \log T)$ when the demand is variable over time. Next, we consider the setting when the suppliers' cost functions can vary over time and illustrate that no online algorithm can achieve sublinear regret on all three metrics when the market operator has no information about how the cost functions change over time. Thus, we consider an augmented setting wherein the operator has access to hints/contexts that, without revealing the complete specification of the cost functions, reflect the variation in the cost functions over time and propose an algorithm with sublinear regret in this augmented setting.
We consider the problem of identification of linear dynamical systems from a single trajectory. Recent results have predominantly focused on the setup where no structural assumption is made on the system matrix $A^* \in \mathbb{R}^{n \times n}$, and have consequently analyzed the ordinary least squares (OLS) estimator in detail. We assume prior structural information on $A^*$ is available, which can be captured in the form of a convex set $\mathcal{K}$ containing $A^*$. For the solution of the ensuing constrained least squares estimator, we derive non-asymptotic error bounds in the Frobenius norm which depend on the local size of the tangent cone of $\mathcal{K}$ at $A^*$. To illustrate the usefulness of this result, we instantiate it for the settings where, (i) $\mathcal{K}$ is a $d$ dimensional subspace of $\mathbb{R}^{n \times n}$, or (ii) $A^*$ is $k$-sparse and $\mathcal{K}$ is a suitably scaled $\ell_1$ ball. In the regimes where $d, k \ll n^2$, our bounds improve upon those obtained from the OLS estimator.
This work derives upper bounds on the convergence rate of the moment-sum-of-squares hierarchy with correlative sparsity for global minimization of polynomials on compact basic semialgebraic sets. The main conclusion is that both sparse hierarchies based on the Schm\"udgen and Putinar Positivstellens\"atze enjoy a polynomial rate of convergence that depends on the size of the largest clique in the sparsity graph but not on the ambient dimension. Interestingly, the sparse bounds outperform the best currently available bounds for the dense hierarchy when the maximum clique size is sufficiently small compared to the ambient dimension and the performance is measured by the running time of an interior point method required to obtain a bound on the global minimum of a given accuracy.
Platforms for online civic participation rely heavily on methods for condensing thousands of comments into a relevant handful, based on whether participants agree or disagree with them. These methods should guarantee fair representation of the participants, as their outcomes may affect the health of the conversation and inform impactful downstream decisions. To that end, we draw on the literature on approval-based committee elections. Our setting is novel in that the approval votes are incomplete since participants will typically not vote on all comments. We prove that this complication renders non-adaptive algorithms impractical in terms of the amount of information they must gather. Therefore, we develop an adaptive algorithm that uses information more efficiently by presenting incoming participants with statements that appear promising based on votes by previous participants. We prove that this method satisfies commonly used notions of fair representation, even when participants only vote on a small fraction of comments. Finally, an empirical evaluation using real data shows that the proposed algorithm provides representative outcomes in practice.
In the past decade, the technology industry has adopted online randomized controlled experiments (a.k.a. A/B testing) to guide product development and make business decisions. In practice, A/B tests are often implemented with increasing treatment allocation: the new treatment is gradually released to an increasing number of units through a sequence of randomized experiments. In scenarios such as experimenting in a social network setting or in a bipartite online marketplace, interference among units may exist, which can harm the validity of simple inference procedures. In this work, we introduce a widely applicable procedure to test for interference in A/B testing with increasing allocation. Our procedure can be implemented on top of an existing A/B testing platform with a separate flow and does not require a priori a specific interference mechanism. In particular, we introduce two permutation tests that are valid under different assumptions. Firstly, we introduce a general statistical test for interference requiring no additional assumption. Secondly, we introduce a testing procedure that is valid under a time fixed effect assumption. The testing procedure is of very low computational complexity, it is powerful, and it formalizes a heuristic algorithm implemented already in industry. We demonstrate the performance of the proposed testing procedure through simulations on synthetic data. Finally, we discuss one application at LinkedIn, where a screening step is implemented to detect potential interference in all their marketplace experiments with the proposed methods in the paper.
We propose an incentive mechanism for the sponsored content provider market in which the communication of users can be represented by a graph and the private information of the users is assumed to have a continuous distribution function. The content provider stipulates incentive rewards to encourage users to reveal their private information truthfully and increase their content demand, which leads to an increase in advertising revenue. We prove that all users gain a non-negative utility and disclose their private information truthfully. Moreover, we study the effectiveness and scalability of the proposed mechanism in a case study with different network structures.
We study identifying user clusters in contextual multi-armed bandits (MAB). Contextual MAB is an effective tool for many real applications, such as content recommendation and online advertisement. In practice, user dependency plays an essential role in the user's actions, and thus the rewards. Clustering similar users can improve the quality of reward estimation, which in turn leads to more effective content recommendation and targeted advertising. Different from traditional clustering settings, we cluster users based on the unknown bandit parameters, which will be estimated incrementally. In particular, we define the problem of cluster detection in contextual MAB, and propose a bandit algorithm, LOCB, embedded with local clustering procedure. And, we provide theoretical analysis about LOCB in terms of the correctness and efficiency of clustering and its regret bound. Finally, we evaluate the proposed algorithm from various aspects, which outperforms state-of-the-art baselines.
In this paper we develop inference for high dimensional linear models, with serially correlated errors. We examine Lasso under the assumption of strong mixing in the covariates and error process, allowing for fatter tails in their distribution. While the Lasso estimator performs poorly under such circumstances, we estimate via GLS Lasso the parameters of interest and extend the asymptotic properties of the Lasso under more general conditions. Our theoretical results indicate that the non-asymptotic bounds for stationary dependent processes are sharper, while the rate of Lasso under general conditions appears slower as $T,p\to \infty$. Further we employ the debiased Lasso to perform inference uniformly on the parameters of interest. Monte Carlo results support the proposed estimator, as it has significant efficiency gains over traditional methods.
Bid optimization for online advertising from single advertiser's perspective has been thoroughly investigated in both academic research and industrial practice. However, existing work typically assume competitors do not change their bids, i.e., the wining price is fixed, leading to poor performance of the derived solution. Although a few studies use multi-agent reinforcement learning to set up a cooperative game, they still suffer the following drawbacks: (1) They fail to avoid collusion solutions where all the advertisers involved in an auction collude to bid an extremely low price on purpose. (2) Previous works cannot well handle the underlying complex bidding environment, leading to poor model convergence. This problem could be amplified when handling multiple objectives of advertisers which are practical demands but not considered by previous work. In this paper, we propose a novel multi-objective cooperative bid optimization formulation called Multi-Agent Cooperative bidding Games (MACG). MACG sets up a carefully designed multi-objective optimization framework where different objectives of advertisers are incorporated. A global objective to maximize the overall profit of all advertisements is added in order to encourage better cooperation and also to protect self-bidding advertisers. To avoid collusion, we also introduce an extra platform revenue constraint. We analyze the optimal functional form of the bidding formula theoretically and design a policy network accordingly to generate auction-level bids. Then we design an efficient multi-agent evolutionary strategy for model optimization. Offline experiments and online A/B tests conducted on the Taobao platform indicate both single advertiser's objective and global profit have been significantly improved compared to state-of-art methods.
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