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Added by
Jamie Morgenstern
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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We present a general framework for proving polynomial sample complexity bounds for the problem of learning from samples the best auction in a class of simple auctions. Our framework captures all of the most prominent examples of simple auctions, including anonymous and non-anonymous item and bundle pricings, with either a single or multiple buyers. The technique we propose is to break the analysis of auctions into two natural pieces. First, one shows that the set of allocation rules have large amounts of structure; second, fixing an allocation on a sample, one shows that the set of auctions agreeing with this allocation on that sample have revenue functions with low dimensionality. Our results effectively imply that whenever its possible to compute a near-optimal simple auction with a known prior, it is also possible to compute such an auction with an unknown prior (given a polynomial number of samples).
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In this paper, we investigate the problem about how to bid in repeated contextual first price auctions. We consider a single bidder (learner) who repeatedly bids in the first price auctions: at each time $t$, the learner observes a context $x_tin mathbb{R}^d$ and decides the bid based on historical information and $x_t$. We assume a structured linear model of the maximum bid of all the others $m_t = alpha_0cdot x_t + z_t$, where $alpha_0in mathbb{R}^d$ is unknown to the learner and $z_t$ is randomly sampled from a noise distribution $mathcal{F}$ with log-concave density function $f$. We consider both emph{binary feedback} (the learner can only observe whether she wins or not) and emph{full information feedback} (the learner can observe $m_t$) at the end of each time $t$. For binary feedback, when the noise distribution $mathcal{F}$ is known, we propose a bidding algorithm, by using maximum likelihood estimation (MLE) method to achieve at most $widetilde{O}(sqrt{log(d) T})$ regret. Moreover, we generalize this algorithm to the setting with binary feedback and the noise distribution is unknown but belongs to a parametrized family of distributions. For the full information feedback with emph{unknown} noise distribution, we provide an algorithm that achieves regret at most $widetilde{O}(sqrt{dT})$. Our approach combines an estimator for log-concave density functions and then MLE method to learn the noise distribution $mathcal{F}$ and linear weight $alpha_0$ simultaneously. We also provide a lower bound result such that any bidding policy in a broad class must achieve regret at least $Omega(sqrt{T})$, even when the learner receives the full information feedback and $mathcal{F}$ is known.
We study online learning in repeated first-price auctions with censored feedback, where a bidder, only observing the winning bid at the end of each auction, learns to adaptively bid in order to maximize her cumulative payoff. To achieve this goal, the bidder faces a challenging dilemma: if she wins the bid--the only way to achieve positive payoffs--then she is not able to observe the highest bid of the other bidders, which we assume is iid drawn from an unknown distribution. This dilemma, despite being reminiscent of the exploration-exploitation trade-off in contextual bandits, cannot directly be addressed by the existing UCB or Thompson sampling algorithms in that literature, mainly because contrary to the standard bandits setting, when a positive reward is obtained here, nothing about the environment can be learned. In this paper, by exploiting the structural properties of first-price auctions, we develop the first learning algorithm that achieves $O(sqrt{T}log^2 T)$ regret bound when the bidders private values are stochastically generated. We do so by providing an algorithm on a general class of problems, which we call monotone group contextual bandits, where the same regret bound is established under stochastically generated contexts. Further, by a novel lower bound argument, we characterize an $Omega(T^{2/3})$ lower bound for the case where the contexts are adversarially generated, thus highlighting the impact of the contexts generation mechanism on the fundamental learning limit. Despite this, we further exploit the structure of first-price auctions and develop a learning algorithm that operates sample-efficiently (and computationally efficiently) in the presence of adversarially generated private values. We establish an $O(sqrt{T}log^3 T)$ regret bound for this algorithm, hence providing a complete characterization of optimal learning guarantees for this problem.
We study the design of multi-item mechanisms that maximize expected profit with respect to a distribution over buyers values. In practice, a full description of the distribution is typically unavailable. Therefore, we study the setting where the designer only has samples from the distribution and the goal is to find a high-profit mechanism within a class of mechanisms. If the class is complex, a mechanism may have high average profit over the samples but low expected profit. This raises the question: how many samples are sufficient to ensure that a mechanisms average profit is close to its expected profit? To answer this question, we uncover structure shared by many pricing, auction, and lottery mechanisms: for any set of buyers values, profit is piecewise linear in the mechanisms parameters. Using this structure, we prove new bounds for mechanism classes not yet studied in the sample-based mechanism design literature and match or improve over the best known guarantees for many classes. Finally, we provide tools for optimizing an important tradeoff: more complex mechanisms typically have higher average profit over the samples than simpler mechanisms, but more samples are required to ensure that average profit nearly matches expected profit.
The combinatorial auction (CA) is an efficient mechanism for resource allocation in different fields, including cloud computing. It can obtain high economic efficiency and user flexibility by allowing bidders to submit bids for combinations of different items instead of only for individual items. However, the problem of allocating items among the bidders to maximize the auctioneers revenue, i.e., the winner determination problem (WDP), is NP-complete to solve and inapproximable. Existing works for WDPs are generally based on mathematical optimization techniques and most of them focus on the single-unit WDP, where each item only has one unit. On the contrary, few works consider the multi-unit WDP in which each item may have multiple units. Given that the multi-unit WDP is more complicated but prevalent in cloud computing, we propose leveraging machine learning (ML) techniques to develop a novel low-complexity algorithm for solving this problem with negligible revenue loss. Specifically, we model the multi-unit WDP as an augmented bipartite bid-item graph and use a graph neural network (GNN) with half-convolution operations to learn the probability of each bid belonging to the optimal allocation. To improve the sample generation efficiency and decrease the number of needed labeled instances, we propose two different sample generation processes. We also develop two novel graph-based post-processing algorithms to transform the outputs of the GNN into feasible solutions. Through simulations on both synthetic instances and a specific virtual machine (VM) allocation problem in a cloud computing platform, we validate that our proposed method can approach optimal performance with low complexity and has good generalization ability in terms of problem size and user-type distribution.
We study a family of convex polytopes, called SIM-bodies, which were introduced by Giannakopoulos and Koutsoupias (2018) to analyze so-called Straight-Jacket Auctions. First, we show that the SIM-bodies belong to the class of generalized permutahedra. Second, we prove an optimality result for the Straight-Jacket Auctions among certain deterministic auctions. Third, we employ computer algebra methods and mathematical software to explicitly determine optimal prices and revenues.
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