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Over the last decade, digital media (web or app publishers) generalized the use of real time ad auctions to sell their ad spaces. Multiple auction platforms, also called Supply-Side Platforms (SSP), were created. Because of this multiplicity, publishers started to create competition between SSPs. In this setting, there are two successive auctions: a second price auction in each SSP and a secondary, first price auction, called header bidding auction, between SSPs.In this paper, we consider an SSP competing with other SSPs for ad spaces. The SSP acts as an intermediary between an advertiser wanting to buy ad spaces and a web publisher wanting to sell its ad spaces, and needs to define a bidding strategy to be able to deliver to the advertisers as many ads as possible while spending as little as possible. The revenue optimization of this SSP can be written as a contextual bandit problem, where the context consists of the information available about the ad opportunity, such as properties of the internet user or of the ad placement.Using classical multi-armed bandit strategies (such as the original versions of UCB and EXP3) is inefficient in this setting and yields a low convergence speed, as the arms are very correlated. In this paper we design and experiment a version of the Thompson Sampling algorithm that easily takes this correlation into account. We combine this bayesian algorithm with a particle filter, which permits to handle non-stationarity by sequentially estimating the distribution of the highest bid to beat in order to win an auction. We apply this methodology on two real auction datasets, and show that it significantly outperforms more classical approaches.The strategy defined in this paper is being developed to be deployed on thousands of publishers worldwide.

We extend Fano's inequality, which controls the average probability of events in terms of the average of some f -divergences, to work with arbitrary events (not necessarily forming a partition) and even with arbitrary [0, 1]-valued random variables, possibly in continuously infinite number. We provide two applications of these extensions, in which the consideration of random variables is particularly handy: we offer new and elegant proofs for existing lower bounds, on Bayesian posterior concentration (minimax or distribution-dependent) rates and on the regret in non-stochastic sequential learning.

We consider the problem of maximizing a non-concave Lipschitz multivariate function f over a compact domain. We provide regret guarantees (i.e., optimization error bounds) for a very natural algorithm originally designed by Piyavskii and Shubert in 1972. Our results hold in a general setting in which values of f can only be accessed approximately. In particular, they yield state-of-the-art regret bounds both when f is observed exactly and when evaluations are perturbed by an independent subgaussian noise.