Two-level monotonic multistage recommender systems

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A recommender system learns to predict the user-specific preference or intention over many items simultaneously for all users, making personalized recommendations based on a relatively small number of observations. One central issue is how to leverage three-way interactions, referred to as user-item-stage dependencies on a monotonic chain of events, to enhance the prediction accuracy. A monotonic chain of events occurs, for instance, in an article sharing dataset, where a “follow” action implies a “like” action, which in turn implies a “view” action. In this article, we develop a multistage recommender system utilizing a two-level monotonic property characterizing a monotonic chain of events for personalized prediction. Particularly, we derive a large-margin classifier based on a nonnegative additive latent factor model in the presence of a high percentage of missing observations, particularly between stages, reducing the number of model parameters for personalized prediction while guaranteeing prediction consistency. On this ground, we derive a regularized cost function to learn user-specific behaviors at different stages, linking decision functions to numerical and categorical covariates to model user-item-stage interactions. Computationally, we derive an algorithm based on blockwise coordinate descent. Theoretically, we show that the two-level monotonic property enhances the accuracy of learning as compared to a standard method treating each stage individually and an ordinal method utilizing only one-level monotonicity. Finally, the proposed method compares favorably with existing methods in simulations and an article sharing dataset.

Original languageEnglish (US)
Pages (from-to)5545-5569
Number of pages25
JournalElectronic Journal of Statistics
Issue number2
StatePublished - 2021

Bibliographical note

Funding Information:
∗Research supported in part by NSF grants DMS-1712564, DMS-1721216, DMS-1952539, and NIH grants 1R01GM126002, R01HL105397, R01AG069895.

Publisher Copyright:
© 2021, Institute of Mathematical Statistics. All rights reserved.


  • Monotonicity
  • implicit feedback
  • nonconvex minimization
  • nonnegative latent factor model


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