Optimal No-regret Learning in Repeated First-price Auctions

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 bidder's 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

Fighting Bandits with a New Kind of Smoothness

We define a novel family of algorithms for the adversarial multi-armed bandit problem, and provide a simple analysis technique based on convex smoothing. We prove two main results. First, we show that regularization via the \emph{Tsallis entropy}, which includes EXP3 as a special case, achieves the $\Theta(\sqrt{TN})$ minimax regret. Second, we show that a wide class of perturbation methods achieve a near-optimal regret as low as $O(\sqrt{TN \log N})$ if the perturbation distribution has a bounded hazard rate. For example, the Gumbel, Weibull, Frechet, Pareto, and Gamma distributions all satisfy this key property.Comment: In Proceedings of NIPS, 201

Online Influence Maximization under Independent Cascade Model with Semi-Bandit Feedback

We study the online influence maximization problem in social networks under the independent cascade model. Specifically, we aim to learn the set of "best influencers" in a social network online while repeatedly interacting with it. We address the challenges of (i) combinatorial action space, since the number of feasible influencer sets grows exponentially with the maximum number of influencers, and (ii) limited feedback, since only the influenced portion of the network is observed. Under a stochastic semi-bandit feedback, we propose and analyze IMLinUCB, a computationally efficient UCB-based algorithm. Our bounds on the cumulative regret are polynomial in all quantities of interest, achieve near-optimal dependence on the number of interactions and reflect the topology of the network and the activation probabilities of its edges, thereby giving insights on the problem complexity. To the best of our knowledge, these are the first such results. Our experiments show that in several representative graph topologies, the regret of IMLinUCB scales as suggested by our upper bounds. IMLinUCB permits linear generalization and thus is both statistically and computationally suitable for large-scale problems. Our experiments also show that IMLinUCB with linear generalization can lead to low regret in real-world online influence maximization.Comment: Compared with the previous version, this version has fixed a mistake. This version is also consistent with the NIPS camera-ready versio

Adaptive Bound Optimization for Online Convex Optimization

We introduce a new online convex optimization algorithm that adaptively chooses its regularization function based on the loss functions observed so far. This is in contrast to previous algorithms that use a fixed regularization function such as L2-squared, and modify it only via a single time-dependent parameter. Our algorithm's regret bounds are worst-case optimal, and for certain realistic classes of loss functions they are much better than existing bounds. These bounds are problem-dependent, which means they can exploit the structure of the actual problem instance. Critically, however, our algorithm does not need to know this structure in advance. Rather, we prove competitive guarantees that show the algorithm provides a bound within a constant factor of the best possible bound (of a certain functional form) in hindsight.Comment: Updates to match final COLT versio