Geometric rescaling algorithms for submodular function minimization

We present a new class of polynomial-time algorithms for submodular function minimization (SFM) as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige–Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. First, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound O((n4 · EO + n5)log(nL)). Second, we exhibit a general combinatorial black box approach to turn εL-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige–Wolfe algorithm. Combined with the geometric rescaling technique, the black box approach provides an O((n5 · EO + n6)log2n) algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee et al., yielding a simplified variant of their O(n3log2n · EO + n4logO(1)n) algorithm

Online Subset Selection using α\alpha-Core with no Augmented Regret

We consider the problem of sequential sparse subset selections in an online learning setup. Assume that the set $[N]$ consists of $N$ distinct elements. On the $t^{\text{th}}$ round, a monotone reward function $f_t: 2^{[N]} \to \mathbb{R}_+,$ which assigns a non-negative reward to each subset of $[N],$ is revealed to a learner. The learner selects (perhaps randomly) a subset $S_t \subseteq [N]$ of $k$ elements before the reward function $f_t$ for that round is revealed $(k \leq N)$. As a consequence of its choice, the learner receives a reward of $f_t(S_t)$ on the $t^{\text{th}}$ round. The learner's goal is to design an online subset selection policy to maximize its expected cumulative reward accrued over a given time horizon. In this connection, we propose an online learning policy called SCore (Subset Selection with Core) that solves the problem for a large class of reward functions. The proposed SCore policy is based on a new concept of $\alpha$-Core, which is a generalization of the notion of Core from the cooperative game theory literature. We establish a learning guarantee for the SCore policy in terms of a new performance metric called $\alpha$-augmented regret. In this new metric, the power of the offline benchmark is suitably augmented compared to the online policy. We give several illustrative examples to show that a broad class of reward functions, including submodular, can be efficiently learned with the SCore policy. We also outline how the SCore policy can be used under a semi-bandit feedback model and conclude the paper with a number of open problems