Learning on sets is increasingly gaining attention in the machine learning community, due to its widespread applicability. Typically, representations over sets are computed by using fixed aggregation functions such as sum or maximum. However, recent results showed that universal function representation by sum- (or max-) decomposition requires either highly discontinuous (and thus poorly learnable) mappings, or a latent dimension equal to the maximum number of elements in the set. To mitigate this problem, we introduce LAF (Learning Aggregation Function), a learnable aggregator for sets of arbitrary cardinality. LAF can approximate several extensively used aggregators (such as average, sum, maximum) as well as more complex functions (e.g. variance and skewness). We report experiments on semi-synthetic and real data showing that LAF outperforms state-of-the-art sum- (max-) decomposition architectures such as DeepSets and library-based architectures like Principal Neighborhood Aggregation, and can be effectively combined with attention-based architectures.

We propose a new recommendation system for service and product bundling in the domain of telecommunication and multimedia. Using this system, users can easily generate a combined service plan that best suits their needs within a vast range of candidates. The system exploits the recent constructive preference elicitation framework, which allows us to flexibly model the exponentially large domain of bundle offers as an implicitly defined set of variables and constraints. The user preferences are modeled by a utility function estimated via coactive learning interaction, while iteratively generating high-utility recommendations through constraint optimization. In this paper, we detail the structure of our system, as well as the methodology and results of an empirical validation study which involved more than 130 participants. The system turned out to be highly usable with respect to both time and number of interactions, and its outputs were found much more satisfactory than those obtained with standard techniques used in the market.