  We consider a natural model of online preference aggregation, where sets of
preferred items $R_1, R_2, \ldots, R_t$ along with a demand for $k_t$ items in
each $R_t$, appear online. Without prior knowledge of $(R_t, k_t)$, the learner
maintains a ranking $\pi_t$ aiming that at least $k_t$ items from $R_t$ appear
high in $\pi_t$. This is a fundamental problem in preference aggregation with
applications to, e.g., ordering product or news items in web pages based on
user scrolling and click patterns. The widely studied Generalized
Min-Sum-Set-Cover (GMSSC) problem serves as a formal model for the setting
above. GMSSC is NP-hard and the standard application of no-regret online
learning algorithms is computationally inefficient, because they operate in the
space of rankings. In this work, we show how to achieve low regret for GMSSC in
polynomial-time. We employ dimensionality reduction from rankings to the space
of doubly stochastic matrices, where we apply Online Gradient Descent. A key
step is to show how subgradients can be computed efficiently, by solving the
dual of a configuration LP. Using oblivious deterministic and randomized
rounding schemes, we map doubly stochastic matrices back to rankings with a
small loss in the GMSSC objective.
