Whole-Page Optimization and Submodular Welfare Maximization with Online Bidders

In the context of online ad serving, display ads may appear on different types of webpages, where each page includes several ad slots and therefore multiple ads can be shown on each page. The set of ads that can be assigned to ad slots of the same page needs to satisfy various prespecified constraints including exclusion constraints, diversity constraints, and the like. Upon arrival of a user, the ad serving system needs to allocate a set of ads to the current webpage respecting these per-page allocation constraints. Previous slot-based settings ignore the important concept of a page and may lead to highly suboptimal results in general. In this article, motivated by these applications in display advertising and inspired by the submodular welfare maximization problem with online bidders, we study a general class of page-based ad allocation problems, present the first (tight) constant-factor approximation algorithms for these problems, and confirm the performance of our algorithms experimentally on real-world datasets. A key technical ingredient of our results is a novel primal-dual analysis for handling free disposal, which updates dual variables using a “level function” instead of a single level and unifies with previous analyses of related problems. This new analysis method allows us to handle arbitrarily complicated allocation constraints for each page. Our main result is an algorithm that achieves a 1 &minus frac 1 e &minus o(1)-competitive ratio. Moreover, our experiments on real-world datasets show significant improvements of our page-based algorithms compared to the slot-based algorithms. Finally, we observe that our problem is closely related to the submodular welfare maximization (SWM) problem. In particular, we introduce a variant of the SWM problem with online bidders and show how to solve this problem using our algorithm for whole-page optimization.postprin

Whole-page Optimization and Submodular Welfare Maximization with Online Bidders

In the context of online ad serving, display ads may appear on different types of web-pages, where each page includes several ad slots and therefore multiple ads can be shown on each page. The set of ads that can be assigned to ad slots of the same page needs to satisfy various pre-specified constraints including exclusion constraints, diversity constraints, and the like. Upon arrival of a user, the ad serving system needs to allocate a set of ads to the current web-page respecting these per-page allocation constraints. Previous slot-based settings ignore the important concept of a page, and may lead to highly suboptimal results in general. In this paper, motivated by these applications in display advertising and inspired by the submodular welfare maximization problem with online bidders, we study a general class of page-based ad allocation problems, present the first (tight) constant-factor approximation algorithms for these problems, and confirm the performance of our algorithms experimentally on real-world data sets. A key technical ingredient of our results is a novel primal-dual analysis for handling free-disposal, which updates dual variables using a "level function" instead of a single level, and unifies with previous analyses of related problems. This new analysis method allows us to handle arbitrarily complicated allocation constraints for each page. Our main result is an algorithm that achieves a 1 − 1 e − o(1) competitive ratio. Moreover, our experiments on real-world data sets show significant improvements of our page-based algorithms compared to the slot-based algorithms. Finally, we observe that our problem is closely related to the submodular welfare maximization (SWM) problem. In particular, we introduce a variant of the SWM problem with online bidders, and show how to solve this problem using our algorithm for whole page optimization

Online Submodular Maximization Problem with Vector Packing Constraint

We consider the online vector packing problem in which we have a d dimensional knapsack and items u with weight vectors w_u in R_+^d arrive online in an arbitrary order. Upon the arrival of an item, the algorithm must decide immediately whether to discard or accept the item into the knapsack. When item u is accepted, w_u(i) units of capacity on dimension i will be taken up, for each i in [d]. To satisfy the knapsack constraint, an accepted item can be later disposed of with no cost, but discarded or disposed of items cannot be recovered. The objective is to maximize the utility of the accepted items S at the end of the algorithm, which is given by f(S) for some non-negative monotone submodular function f. For any small constant epsilon > 0, we consider the special case that the weight of an item on every dimension is at most a (1- epsilon) fraction of the total capacity, and give a polynomial-time deterministic O(k / epsilon^2)-competitive algorithm for the problem, where k is the (column) sparsity of the weight vectors. We also show several (almost) tight hardness results even when the algorithm is computationally unbounded. We first show that under the epsilon-slack assumption, no deterministic algorithm can obtain any o(k) competitive ratio, and no randomized algorithm can obtain any o(k / log k) competitive ratio. We then show that for the general case (when epsilon = 0), no randomized algorithm can obtain any o(k) competitive ratio. In contrast to the (1+delta) competitive ratio achieved in Kesselheim et al. [STOC 2014] for the problem with random arrival order of items and under large capacity assumption, we show that in the arbitrary arrival order case, even when |w_u|_infinity is arbitrarily small for all items u, it is impossible to achieve any o(log k / log log k) competitive ratio

Ranking for Relevance and Display Preferences in Complex Presentation Layouts

Learning to Rank has traditionally considered settings where given the relevance information of objects, the desired order in which to rank the objects is clear. However, with today's large variety of users and layouts this is not always the case. In this paper, we consider so-called complex ranking settings where it is not clear what should be displayed, that is, what the relevant items are, and how they should be displayed, that is, where the most relevant items should be placed. These ranking settings are complex as they involve both traditional ranking and inferring the best display order. Existing learning to rank methods cannot handle such complex ranking settings as they assume that the display order is known beforehand. To address this gap we introduce a novel Deep Reinforcement Learning method that is capable of learning complex rankings, both the layout and the best ranking given the layout, from weak reward signals. Our proposed method does so by selecting documents and positions sequentially, hence it ranks both the documents and positions, which is why we call it the Double-Rank Model (DRM). Our experiments show that DRM outperforms all existing methods in complex ranking settings, thus it leads to substantial ranking improvements in cases where the display order is not known a priori