Users derive value from a recommender system (RS) only to the extent that it is able to surface content (or items) that meet their needs/preferences. While RSs often have a comprehensive view of user preferences across the entire user base, content providers, by contrast, generally have only a local view of the preferences of users that have interacted with their content. This limits a provider's ability to offer new content to best serve the broader population. In this work, we tackle this information asymmetry with content prompting policies. A content prompt is a hint or suggestion to a provider to make available novel content for which the RS predicts unmet user demand. A prompting policy is a sequence of such prompts that is responsive to the dynamics of a provider's beliefs, skills and incentives. We aim to determine a joint prompting policy that induces a set of providers to make content available that optimizes user social welfare in equilibrium, while respecting the incentives of the providers themselves. Our contributions include: (i) an abstract model of the RS ecosystem, including content provider behaviors, that supports such prompting; (ii) the design and theoretical analysis of sequential prompting policies for individual providers; (iii) a mixed integer programming formulation for optimal joint prompting using path planning in content space; and (iv) simple, proof-of-concept experiments illustrating how such policies improve ecosystem health and user welfare.

Branch-and-cut is the most widely used algorithm for solving integer programs, employed by commercial solvers like CPLEX and Gurobi. Branch-and-cut has a wide variety of tunable parameters that have a huge impact on the size of the search tree that it builds, but are challenging to tune by hand. An increasingly popular approach is to use machine learning to tune these parameters: using a training set of integer programs from the application domain at hand, the goal is to find a configuration with strong predicted performance on future, unseen integer programs from the same domain. If the training set is too small, a configuration may have good performance over the training set but poor performance on future integer programs. In this paper, we prove sample complexity guarantees for this procedure, which bound how large the training set should be to ensure that for any configuration, its average performance over the training set is close to its expected future performance. Our guarantees apply to parameters that control the most important aspects of branch-and-cut: node selection, branching constraint selection, and cutting plane selection, and are sharper and more general than those found in prior research.

Cutting-plane methods have enabled remarkable successes in integer programming over the last few decades. State-of-the-art solvers integrate a myriad of cutting-plane techniques to speed up the underlying tree-search algorithm used to find optimal solutions. In this paper we prove the first guarantees for learning high-performing cut-selection policies tailored to the instance distribution at hand using samples. We first bound the sample complexity of learning cutting planes from the canonical family of Chv\'atal-Gomory cuts. Our bounds handle any number of waves of any number of cuts and are fine tuned to the magnitudes of the constraint coefficients. Next, we prove sample complexity bounds for more sophisticated cut selection policies that use a combination of scoring rules to choose from a family of cuts. Finally, beyond the realm of cutting planes for integer programming, we develop a general abstraction of tree search that captures key components such as node selection and variable selection. For this abstraction, we bound the sample complexity of learning a good policy for building the search tree.