Wenjing Chen

Wenjing Chen, Ruida Zhou, Chao Tian et al.

We analyze the performance of the Borda counting algorithm in a non-parametric model. The algorithm needs to utilize probabilistic rankings of the items within $m$-sized subsets to accurately determine which items are the overall top-$k$ items in a total of $n$ items. The Borda counting algorithm simply counts the cumulative scores for each item from these partial ranking observations. This generalizes a previous work of a similar nature by Shah et al. using probabilistic pairwise comparison data. The performance of the Borda counting algorithm critically depends on the associated score separation $Δ_k$ between the $k$-th item and the $(k+1)$-th item. Specifically, we show that if $Δ_k$ is greater than certain value, then the top-$k$ items selected by the algorithm is asymptotically accurate almost surely; if $Δ_k$ is below certain value, then the result will be inaccurate with a constant probability. In the special case of $m=2$, i.e., pairwise comparison, the resultant bound is tighter than that given by Shah et al., leading to a reduced gap between the error probability upper and lower bounds. These results are further extended to the approximate top-$k$ selection setting. Numerical experiments demonstrate the effectiveness and accuracy of the Borda counting algorithm, compared with the spectral MLE-based algorithm, particularly when the data does not necessarily follow an assumed parametric model.

Wenjing Chen, Shuo Xing, Samson Zhou et al.

Submodular optimization is a fundamental problem with many applications in machine learning, often involving decision-making over datasets with sensitive attributes such as gender or age. In such settings, it is often desirable to produce a diverse solution set that is fairly distributed with respect to these attributes. Motivated by this, we initiate the study of Fair Submodular Cover (FSC), where given a ground set $U$, a monotone submodular function $f:2^U\to\mathbb{R}_{\ge 0}$, a threshold $τ$, the goal is to find a balanced subset of $S$ with minimum cardinality such that $f(S)\geτ$. We first introduce discrete algorithms for FSC that achieve a bicriteria approximation ratio of $(\frac{1}ε, 1-O(ε))$. We then present a continuous algorithm that achieves a $(\ln\frac{1}ε, 1-O(ε))$-bicriteria approximation ratio, which matches the best approximation guarantee of submodular cover without a fairness constraint. Finally, we complement our theoretical results with a number of empirical evaluations that demonstrate the effectiveness of our algorithms on instances of maximum coverage.

Wenjing Chen, Victoria G. Crawford

Submodular optimization with bandit feedback has recently been studied in a variety of contexts. In a number of real-world applications such as diversified recommender systems and data summarization, the submodular function exhibits additional linear structure. We consider developing approximation algorithms for the maximization of a submodular objective function $f:2^U\to\mathbb{R}_{\geq 0}$, where $f=\sum_{i=1}^dw_iF_{i}$. It is assumed that we have value oracle access to the functions $F_i$, but the coefficients $w_i$ are unknown, and $f$ can only be accessed via noisy queries. We develop algorithms for this setting inspired by adaptive allocation algorithms in the best-arm identification for linear bandit, with approximation guarantees arbitrarily close to the setting where we have value oracle access to $f$. Finally, we empirically demonstrate that our algorithms make vast improvements in terms of sample efficiency compared to algorithms that do not exploit the linear structure of $f$ on instances of move recommendation.