NeurIPS 2020

Continuous Submodular Maximization: Beyond DR-Submodularity

Review 1

Summary and Contributions: The paper studies the problem of maximizing a monotone continuous submodular function subject to an l1 constraint. This is a generalization of the problem of maximizing a monotone discrete submodular function subject to a cardinality constraint. The paper shows a near optimal 1-1/e-epsilon approximation. The key technique is to relate this problem to the usual discrete submodular problem with a knapsack constraint.

Strengths: - This is a rare work that generalizes to the continuous submodular function without DR-submodularity. Before this work, there was only a result for box-constrained maximization of non-monotone functions.

Weaknesses: The result seems purely theoretical at the moment without useful models where this generality is required.

Correctness: The claims and proofs seem correct.

Clarity: The paper is reasonably well-written.

Relation to Prior Work: The previous works are clearly discussed. They are missing a reference to Soma and Yoshida but hopefully it will be added to the revision.

Reproducibility: Yes

Additional Feedback: =====After rebuttal====== Please add a discussion on the comparison with Soma-Yoshida. In particular, I think the two settings are very related. The setting in this paper assumes smoothness, which is a very strong assumption compared with bounded gradient, a sufficient condition for approximating the continuous function with a grid. Consider a feasible point x. We will show there is a grid point that is not much worse in objective. Notice that if we subtract epsilon<=B/n^2 from the largest coordinate, wlog assume that is the first coordinate, and add epsilon/n to the rest, it is still feasible. Thus, consider a new point y that is the same as x everywhere except y_1 >= x_1 - epsilon and note that x_1 >= B/n >= n*epsilon. We just need to show that f(y) is not much smaller than f(x) (our grid point even dominates x in coordinates 2,3,.. so its value is even larger than f(y)). Let h be the one dimensional function where h(a) = f(a, x_2, x_3, ..). Let g=h'(y_1). By smoothness and monotonicity, h(y_1)-h(0) >= h(y_1) - h(y_1 - min(y_1, g/L)) >= min(g^2/(2L), y_1*(g-L*y_1/2)) h(x_1)-h(y_1) <= epsilon*g + L(epsilon)^2/2 Case 1. g < y_1*L. We have h(x_1)-h(y_1) <= epsilon*y_1*L + L(epsilon)^2/2 <= L*epsilon*x_1. Case 2. g >= y_1*L. We have h(x_1)-h(y_1) <= 2*epsilon/y_1 * (h(y_1)-h(0)) + L*epsilon^2/2 <= 2*(h(y_1)-h(0))/(n-1) + L*epsilon*x_1/n.

Review 2

Summary and Contributions: This paper studies an optimization problem in which the goal is to maximize a continuous submodular (not necessarily DR-submodular) function under a simple linear constraint. The authors have proposed two algorithms, namely COORDINATE-ASCENT+ and COORDINATE-ASCENT++, to solve this problem and in particular, the latter obtains the tight (1-1/e-\epsilon) approximation ratio. -------------- Update: I read the authors' feedback, thanks for your careful and detailed responses. In particular, I am convinced about the applications of (non-DR) continuous submodular functions and hence the importance of the algorithms provided in this paper. I've increased my score for the paper.

Strengths: This work is significant in several respects. First of all, the continuous submodular maximization problem has not been studied before in the literature. To be precise, continuous DR-submodular maximization, where the objective function is coordinate-wise concave along with continuous submodular, has been extensively studied before, however, the study of continuous submodular (not necessarily DR-submodular) maximization in this paper is novel and could be of high interest to the NeurIPS community. Moreover, although the computational complexity of the proposed algorithm is rather high, the algorithm is parallelizable which makes it useful in practice. The work contains a significant amount of novel proof ideas (mostly provided in the appendix) and overall, the theoretical contributions of the paper are significant.

Weaknesses: Despite all the theoretical strengths of this work, the paper has the following weaknesses: - The main text of the paper is very technical and it is hard for someone with limited exposure to the previous literature on this topic to follow the paper. For instance, it would have been useful to provide the diminishing returns type definition of continuous submodularity and compare it to that of DR-submodular functions. Moreover, a brief overview of techniques for continuous DR-submodular maximization and the reasons why they fail to apply to continuous submodular maximization would have been very beneficial. - The authors have failed to provide set of examples of continuous submodular functions that are not DR-submodular and are used in practice to motivate the study of maximizing such functions. I realize that the provided example $\phi_{i,j}(x_i-x_j)$, where $\phi_{i,j}$ is convex, is indeed continuous (non-DR)-submodular, but in my view, this point needed to be emphasized in the paper. - This paper considers a very simple linear constraint and the authors have failed to mention why their proposed algorithms could not be used for more general convex constraints and what the challenges are to design algorithms for such constraints. - This paper does not contain any numerical experiments to verify the efficiency of their proposed algorithms. In particular, the COORDINATE-ASCENT++ algorithm has a rather high computational complexity and it would have been useful to implement the algorithm and show the claim "the algorithm is easily parallelizable" in practice.

Correctness: All the theoretical claims and results of this paper have been well justified through mathematically sound arguments and proofs. I took a quick look at the proofs in the appendix as well and they looked fine to me. The paper does not contain any numerical experiments.

Clarity: As mentioned earlier, the main text of the paper is a bit too technical and it's hard to follow for the general audience with limited familiarity with the topic. I'm well familiar with the literature and nonetheless, occasionally, I had difficulty verifying the arguments and proof steps. To remedy this issue, I think Algorithm 5 and Algorithm 6, provided in the appendix, need to move to the main text of the paper and be discussed in more detail to help the reader better understand the COORDINATE-ASCENT+ and COORDINATE-ASCENT++ algorithms.

Relation to Prior Work: The authors have done a great job reviewing the previous literature on continuous DR-submodular maximization and have differentiated their contributions to the more general problem of continuous (non-DR)submodular maximization.

Reproducibility: Yes

Additional Feedback: - Discuss the challenges of solving continuous submodular maximization problem subject to general convex constraints. - Provide a list of continuous (non-DR)-submodular functions that are used in practice in different domains to further emphasize the significance of the framework of the paper. - Provide more details and explanations for the steps of the proofs to make it easier to follow.

Review 3

Summary and Contributions: This paper studies maximization of a monotone continuous submodular function under the cardinality constraint. The technical contributions are summarized as - ((e-1)/(2e-1)-ε)-approximation algorithm which performs O(n/ε) iterations with O(n√B/ε + n log n) cost per iteration - (1-1/e-ε)-approximation algorithm which performs O(n/ε) iterations with O(n^3√B/ε^2.5 + n^3 log n/ε^2) cost per iteration. This algorithm can be parallelized to reduce the cost to O(n√(B/ε) + n log n) per iteration, where ε is a user constant and n is the dimension. Previous work focused on a subclass of continuous submodular functions, i.e., DR-submodular functions, which assumes the natural diminishing return property. In contrast, continuous submodular functions satisfy only a weaker condition.

Strengths: - First approximation algorithms for continuous submodular maximization under the cardinality constraint - Approximation ratio is tight

Weaknesses: - A similar work exists in the literature whose time complexity is better (Update: the previous work has a flaw)

Correctness: I found no mathematical error.

Clarity: Clearly written.

Relation to Prior Work: They missed a relevant reference of lattice submodular maximization under cardinality constraint.

Reproducibility: Yes

Additional Feedback: The results presented in this paper look reasonable. However, I found a very similar result in the literature (Soma and Yoshida 2018, Mathematical Programming). They studied maximization of monotone lattice submodular function (hence it corresponds to continuous submodular maximization in this paper) subject to a cardinality constraint. Taking a fine enough discretization of each axis of the domain, one can reduce the continuous setting to the lattice setting. Soma-Yoshida's algorithm outputs a (1-1/e-ε)-approximate solution in O(n/ε^2 log ||u||_∞ log B/ε log τ) time, where τ is the minimum positive increase of the objective function. Therefore, it is faster than the proposed algorithms above. Indeed, Algorithm 1 in this NeurIPS manuscript looks very similar to Soma-Yoshida's Algorithm 3: Both algorithms are greedy with approximate step size search for each axis direction. Furthermore, their subroutine (Proposition 3.1) indeed does discrete step size search for each axis, so these two algorithms are similar even conceptually. Of course, the continuous submodular setting has several benefits over the integer lattice setting such as wider applications in ML and more natural integration with continuous optimization (Frank-Wolfe, SGD, step size, etc). But this paper focuses on the purely algorithmic side and the algorithms itself are based on discretization, so the advantage over the integer lattice is quite unclear. Unfortunately, the authors seem to have missed the Soma-Yoshida paper and did not give a detailed comparison. I would like to see differences (if there is any) in the rebuttal. ### update after rebuttal ### I have read the authors' feedback and agree that the previous result in the reference I pointed out were not completely true. The algorithm needs to be modified with partial enumeration, which is also used in this submission. This makes the complexity of the previous lattice setting algorithm to something at least n^3, which addresses my first comment. Also, it was nice the authors addressed my second comment by explaining the difference between their setting and the integer lattice setting. I hope the authors include them in the revised version. I raised my score.

Review 4

Summary and Contributions: This paper proposes a variant of the coordinate ascent algorithm for monotone continuous submodular maximization with a linear constraint. The proposed algorithm achieves (1-1/e-epsilon)-approximation without DR-submodularity.

Strengths: This paper deals with the problem of maximizing F(x) subject to ||x||_1 <= B, where F is a monotone L-smooth continuous submodular (not necessarily DR-submodular) function. Since the coordinate-wise concavity does not hold for continuous submodular functions, it is hard to apply existing methods such as continuous greedy to this problem. The authors propose two variants of the coordinate ascent, which achieve ((e-1)/(2e-1)-eps)-approximation in O(n/epsilon) time and (1-1/e-eps)-approximation in O(n^3/epsilon^2.5 + n^3 log(n)/epsilon^2) time, respectively. To my knowledge, the idea of applying the coordinate ascent to continuous submodular maximization is novel.

Weaknesses: In terms of applications, continuous submodular maximization is not highly motivated. The problem this paper tackles is continuous submodular maximization, which is a generalization of DR-submodular maximization. There are already efficient and effective algorithms for DR-submodular maximization, and continuous submodular functions that appear in most applications satisfy DR-submodularity. Therefore it is hard to say the proposed algorithm is ready to be applied to real world problems immediately. Also, this paper contains small typos, though they are not critical.

Correctness: I skimmed all the proofs and found no error.

Clarity: The overall writing quality is good.

Relation to Prior Work: This paper sufficiently surveys existing work on continuous submodular maximization. I think it would be better to mention the following work by Soma and Yoshida. Maximizing monotone submodular functions over the integer lattice. Mathematical Programming (2018). In this Soma--Yoshida paper, they provide a (1 - 1/e)-approximation algorithm for monotone submodular (not DR-submodular) maximization on the integer lattice with a cardinality constraint. This problem can be regarded as a discrete analogue of the problem tackled in this paper.

Reproducibility: Yes

Additional Feedback: - p.4, l.141--143: It would be helpful if the authors mention the case when x_i > y_i. In this case, x_i can never be equal to y_i, which deviate from the explanation of ``good'', but all analyses in the proof hold. - p.5, l.162: ] could be removed. - p.6, l.197: ||_1 could be removed. - p.6, l.198 and l.205: |_1 could be added. - p.6, l.201--206: \ell might be replaced with \ell'. - p.7, l.247: OPT could be replaced with opt. - p.17, l.531--532: B could be replaced with B^0.5 in three places. # Update after the author feedback My main concern is about the applications of (non-DR) continuous submodular functions. In the feedback, the authors mentioned the problem of finding the mode of MTP2 distributions. I think this is a reasonable application and agree with the acceptance of this paper.