NeurIPS 2019
Sun Dec 8th through Sat the 14th, 2019 at Vancouver Convention Center
Paper ID:2990
Title:Adaptive Influence Maximization with Myopic Feedback

Reviewer 1

This paper studies an important problem, adaptive influence maximization, in social network analysis. This problem was first formally considered by Golovin and Krause but very few theoretical results have been found since then. The results provided in this paper are solid and may have a significant impact on this area. First, the upper bound of the adaptive gap is the first result of this kind concerning the adaptive influence maximization problem. As a corollary, the approximation ratio of the simple adaptive greedy algorithm under the myopic model can be confirmed as at least 0.25*(1-1/e), which, as mentioned in the paper, solves the conjecture of Golovin and Krause. The proof is obtained based on a novel construction of an intermediate seeding strategy. Second, the author also provides the lower bound based on sophisticated construction, which, more importantly, opens a question of how to fill the gap between the upper and lower bound. Finally, the authors provide the upper bound of the approximation ratio of the greedy strategy under the myopic feedback model. The proof indicates that myopic feedback does not help adaptive greedy, or from another perspective, adaptivity does not fundamentally useful under myopic feedback. The results here immediately lead future research to investigate better feedback models. I have thoroughly checked all the proofs and to my best believe they are sound. Some minors: line 214 page 5, may need rephrasing. line 242 page 5, one pair of brackets missed line 260 page 6, phi^1 might be phi Eq 6 page 7, the notation for a realization uses a generalized version of that defined in line 270 page 6. The authors may define this explicitly. line 469 page 13, "lemmas" might be "lemma" Scored Review: (Excellent Very Good Good Fair) Originality: Excellent Quality: Excellent Clarity: Very Good Significance: Very Good Update after Author response: The clarifications are clear.

Reviewer 2

The paper considers the adaptive influence maximization problem with myopic feedback under the independent cascade model. In the adaptive influence maximization problem, the seed nodes are sequentially selected one by one. After each seed selection, some information (feedback) about the spread of influence is returned to the algorithm which could be used for the selection of the next seeds. In the myopic feedback, the set of activated immediate neighbors of the selected seed is returned as such feedback. Finally, in the independent cascade model, every edge of the graph has an independent probability of passing the influence. This problem is shown to not be adaptive submodular. Therefore, it was an open problem if a constant approximation can be achieved. The paper shows that the answer is yes and prove that the greedy algorithm gives a (1-1/e)/4-approximation. The main idea of the proof is to show that for each adaptive policy, we could construct a non-adaptive randomized policy, such that the adaptive influence spread of the adaptive policy is at most 4 times the non-adaptive influence spread of the randomized policy. This shows that the adaptivity gap of the problem is at most 4 and proves the approximation factor of the greedy algorithm. The paper then proceeds to show that an adaptive version of the greedy algorithm also achieves the same approximation factor. They also show that there exist cases where the adaptive greedy and non-adaptive greedy (classic greedy) cannot find an approximation factor better than (e^2+1)/(e+1)^2. I personally am not a fan of this last result because it only focuses on the greedy algorithm. It would have been better to give an actual hardness of approximation result. I haven’t checked the proofs presented in the appendix but I believe the results are very interesting and the (1-1/e)/4-approximation result guarantees the acceptance. However as I mentioned before, I would really like to see a hardness of approximation result since the problem is not adaptive submodular and the hardness might be different than 1-1/e+epsilon.

Reviewer 3

The problem dealt with in this paper is the adaptive influence maximization with myopic feedback model. This problem was proposed by Golovin--Krause (2011), but they did not provide any bound on the approximation ratio of the adaptive greedy algorithm. They conjectured that this algorithm achieves constant-factor approximation. In this paper, the authors affirmatively solve this conjecture by analyzing the adaptivity gap. The adaptivity gap is the largest ratio of the objective values achieved by an optimal adaptive policy and an optimal non-adaptive policy among all graphs. The authors provide lower bound 1/4 and upper bound (1-1/e) on the adaptivity gap. By using this bound, the authors show both the adaptive and non-adaptive greedy algorithms are (1-1/e)/4-approximation to the optimal adaptive policy. Also, it is shown that neither the adaptive greedy nor the non-adaptive greedy is better than (e^2+1)/(e+1)^2-approximation. This paper is generally very well-written. In the proofs, this paper utilizes several novel ideas such as the multiplied version \sigma^t of the influence function. The open problem solved by this paper is significant not only in theory but also in practice. The results would give an impact on future research on influence maximization. I would be for the acceptance of this paper. Small comments: - In the first formula of (38), G(t) could be replaced with G(w). - In the proof of Claim 5, though it is not theoretically essential, 2d/(e+1) might not be an integer. It would be better to add a flooring function. - The statement of Lemma 7 would be a little bit misleading. In the current form, it sounds to me that "for any graph, the approximation ratio achieved by the non-adaptive greedy is no worse than the approximation ratio achieved by the adaptive greedy *in the same graph*." However, it is not what the authors mean. In my understanding, the authors mean that "for the approximation ratio of the non-adaptive greedy in any graph, there is some graph for which the approximation ratio of the adaptive greedy is equal to this one." Update after the author feedback: I have checked the author feedback. The authors explain why they did not conduct experiments, and I am satisfied with their comment. My opinion about the theoretical results remains the same, so I don't change my score.