Search for: All records

In this paper, we consider the problem of online monotone DR-submodular maximization subject to long-term stochastic constraints. Specifically, at each round $$t\in [T]$$, after committing an action $$\bx_t$$, a random reward $$f_t(\bx_t)$$ and an unbiased gradient estimate of the point $$\widetilde{\nabla}f_t(\bx_t)$$ (semi-bandit feedback) are revealed. Meanwhile, a budget of $$g_t(\bx_t)$$, which is linear and stochastic, is consumed of its total allotted budget $$B_T$$. We propose a gradient ascent based algorithm that achieves $$\frac{1}{2}$$-regret of $$\mathcal{O}(\sqrt{T})$$ with $$\mathcal{O}(T^{3/4})$$ constraint violation with high probability. Moreover, when first-order full-information feedback is available, we propose an algorithm that achieves $(1-1/e)$-regret of $$\mathcal{O}(\sqrt{T})$$ with $$\mathcal{O}(T^{3/4})$$ constraint violation. These algorithms significantly improve over the state-of-the-art in terms of query complexity. 

We investigate the problems of maximizing k-submodular functions over total size constraints and over individual size constraints. k-submodularity is a generalization of submodularity beyond just picking items of a ground set, instead associating one of k types to chosen items. For sensor selection problems, for instance, this enables modeling of which type of sensor to put at a location, not simply whether to put a sensor or not. We propose and analyze threshold-greedy algorithms for both types of constraints. We prove that our proposed algorithms achieve the best known approximation ratios for both constraint types, up to a user-chosen parameter that balances computational complexity and the approximation ratio, while only using a number of function evaluations that depends linearly (up to poly-logarithmic terms) on the number of elements n, the number of types k, and the inverse of the user chosen parameter. Other algorithms that achieve the best-known deterministic approximation ratios require a number of function evaluations that depend linearly on the budget B, while our methods do not. We empirically demonstrate our algorithms' performance in applications of sensor placement with k types and influence maximization with k topics. 

We investigate the problem of combinatorial multi-armed bandits with stochastic submodular (in expectation) rewards and full-bandit feedback, where no extra information other than the reward of selected action at each time step $$t$$ is observed. We propose a simple algorithm, Explore-Then-Commit Greedy (ETCG) and prove that it achieves a $(1-1/e)$-regret upper bound of $$\mathcal{O}(n^\frac{1}{3}k^\frac{4}{3}T^\frac{2}{3}\log(T)^\frac{1}{2})$$ for a horizon $$T$$, number of base elements $$n$$, and cardinality constraint $$k$$. We also show in experiments with synthetic and real-world data that the ETCG empirically outperforms other full-bandit methods. 

Information and content can spread in social networks analogous to how diseases spread between organisms. Identifying the source of an outbreak is challenging when the infection times are unknown. We consider the problem of detecting the source of a rumor that spread randomly in a network according to a simple diffusion model, the susceptible-infected (SI) exponential time model. The infection times are unknown. Only the set of nodes that propagated the rumor before a certain time is known. Since evaluating the likelihood of spreads is computationally prohibitive, we propose a simple and efficient procedure to approximate the likelihood and select a candidate rumor source. We empirically demonstrate our method out-performs the Jordan center procedure in various random graphs and a real-world network.