Online advertising is a vital revenue source for major internet platforms. Recently, joint advertising, which assigns a bundle of two advertisers in an ad slot instead of allocating a single advertiser, has emerged as an effective method for enhancing allocation efficiency and revenue. However, existing mechanisms for joint advertising fail to realize the optimality, as they tend to focus on individual advertisers and overlook bundle structures. This paper identifies an optimal mechanism for joint advertising in a single-slot setting. For multi-slot joint advertising, we propose \textbf{BundleNet}, a novel bundle-based neural network approach specifically designed for joint advertising. Our extensive experiments demonstrate that the mechanisms generated by \textbf{BundleNet} approximate the theoretical analysis results in the single-slot setting and achieve state-of-the-art performance in the multi-slot setting. This significantly increases platform revenue while ensuring approximate dominant strategy incentive compatibility and individual rationality.

Many-to-one maps are ubiquitous in machine learning, from the image recognition model that assigns a multitude of distinct images to the concept of "cat" to the time series forecasting model which assigns a range of distinct time-series to a single scalar regression value. While the primary use of such models is naturally to associate correct output to each input, in many problems it is also useful to be able to explore, understand, and sample from a model's fibers, which are the set of input values $x$ such that $f(x) = y$, for fixed $y$ in the output space. In this paper we show that popular generative architectures are ill-suited to such tasks. Motivated by this we introduce a novel generative architecture, a Bundle Network, based on the concept of a fiber bundle from (differential) topology. BundleNets exploit the idea of a local trivialization wherein a space can be locally decomposed into a product space that cleanly encodes the many-to-one nature of the map. By enforcing this decomposition in BundleNets and by utilizing state-of-the-art invertible components, investigating a network's fibers becomes natural.