Stochastic Bandits for Multi-platform Budget Optimization in Online Advertising

We study the problem of an online advertising system that wants to optimally spend an advertisers given budget for a campaign across multiple platforms, without knowing the value for showing an ad to the users on those platforms. We model this challenging practical application as a Stochastic Bandits with Knapsacks problem over $T$ rounds of bidding with the set of arms given by the set of distinct bidding $m$-tuples, where $m$ is the number of platforms. We modify the algorithm proposed in Badanidiyuru emph{et al.,} to extend it to the case of multiple platforms to obtain an algorithm for both the discrete and continuous bid-spaces. Namely, for discrete bid spaces we give an algorithm with regret $Oleft(OPT sqrt {frac{mn}{B} }+ sqrt{mn OPT}right)$, where $OPT$ is the performance of the optimal algorithm that knows the distributions. For continuous bid spaces the regret of our algorithm is $tilde{O}left(m^{1/3} cdot minleft{ B^{2/3}, (m T)^{2/3} right} right)$. When restricted to this special-case, this bound improves over Sankararaman and Slivkins in the regime $OPT ll T$, as is the case in the particular application at hand. Second, we show an $ Omegaleft (sqrt {m OPT} right)$ lower bound for the discrete case and an $Omegaleft( m^{1/3} B^{2/3}right)$ lower bound for the continuous setting, almost matching the upper bounds. Finally, we use a real-world data set from a large internet online advertising company with multiple ad platforms and show that our algorithms outperform common benchmarks and satisfy the required properties warranted in the real-world application.