The Power of Second Chance: Personalized Submodular Maximization with Two Candidates
Jing Yuan, Shaojie Tang
TL;DR
The paper tackles personalized submodular maximization with two candidate solutions by defining the objective $\sum_i \max\{f_i(S_1), f_i(S_2)\}$ and proving that direct aggregation omits personalization. To cope with non-submodularity, it introduces a partition-based strategy: for constant $m$, an Enumeration-based Algorithm partitions $[m]$ into $A$ and $B$ and leverages $\alpha$-approximation subroutines on each side to obtain an $\alpha$-approximation to the target; for large $m$, a Sampling-based Algorithm with $T$ partitions yields an $\alpha/2 \cdot \gamma(T)$-approximation in expectation (with a refined bound $\max\{1/2, \gamma(T)(1/2+\epsilon/\sqrt{m})\}$ in the monotone submodular case). The results specialize to monotone submodular $f_i$, giving a $(1-1/e)$-factor baseline, and they discuss extending the framework to more than two candidates, enabling scalable personalization in two-stage contexts where per-user functions are drawn from a distribution.
Abstract
Most of existing studies on submodular maximization focus on selecting a subset of items that maximizes a \emph{single} submodular function. However, in many real-world scenarios, we might have multiple user-specific functions, each of which models the utility of a particular type of user. In these settings, our goal would be to choose a set of items that performs well across all the user-specific functions. One way to tackle this problem is to select a single subset that maximizes the sum of all of the user-specific functions. Although this aggregate approach is efficient in the sense that it avoids computation of sets for individual functions, it really misses the power of personalization - for it does not allow to choose different sets for different functions. In this paper, we introduce the problem of personalized submodular maximization with two candidate solutions. For any two candidate solutions, the utility of each user-specific function is defined as the better of these two candidates. Our objective is, therefore, to select the best set of two candidates that maximize the sum of utilities of all the user-specific functions. We have designed effective algorithms for this problem. We also discuss how our approach generalizes to multiple candidate solutions, increasing flexibility and personalization in our solution.