Maximization of Approximately Submodular Functions
We study the problem of maximizing a function that is approximately submodular under a cardinality constraint. Approximate submodularity implicitly appears in a wide range of applications as in many cases errors in evaluation of a submodular function break submodularity. Say that F is ε-approximately submodular if there exists a submodular function f such that (1−ε)f (S) ≤ F (S) ≤ (1+ε)f (S) for all subsets S. We are interested in characterizing the query-complexity of maximizing F subject to a cardinality constraint k as a function of the error level ε > 0. We provide both lower and upper bounds: for ε > n −1/2 we show an exponential query-complexity lower bound. In contrast, when ε < 1/k or under a stronger bounded curvature assumption, we give constant approximation algorithms.
This paper considers the problem of maximizing a set function, f, that is an eps approximation to a monotone submodular function, that is there exists a monotone submodular function F such that (1 – eps) f(S) \leq F(S) \leq (1 + eps) f(S) for all sets S, under a cardinality constraint given access to an oracle for evaluating f. While it has been known that for any constant eps and exponential number of calls to the oracle are needed to obtain a constant approximation when F is submodular, this paper furthers the study of this question by providing several new and interesting facts.  • An exponential number of queries are needed to obtain better than O(1/n^beta) when eps \geq 1/(n^(1/2 – beta).  • An exponential number of queries are needed to obtain better than O(1/n^beta) when eps \geq 1/(n^(1/3 – beta) in the special case where F is a coverage function.  • Greedy achieves 1 – 1/eps – O(delta) approx. when eps = delta/ k and there is a cardinality constraint of k. Moreover, they show this is essentially tight characterization of greedy.  • A (1-c)(1-eps)^2/(1+eps)^2 algorithm when F has c curvature. 
This paper furthers the study of a natural question in optimization and machine learning, namely that of a maximizing a function that is approximately monotone submodular under a cardinality constraint. They strengthen previous lower bounds providing dimension dependent bounds on the amount of approximation until which an exponential number of queries. Moreover, they provide improved upper bounds on the quality of approximation achieved as dependent upon k and the curvature of F.  While the question studied by this paper is natural and possibly important and the writing is clear, it is less clear to what degree the results are novel and required new insight over previous work.  There is little discussion of how hard it what to achieve the lower bound or extend previous lower bound work to achieve their bounds. In other words, the paper would benefit from more clearly illustrating whether they provide a novel analysis or lower bound technique, versus simply applying previous machinery. Moreover, the upper bound they achieve for greedy feels somewhat natural and ultimately it is unclear if the paper actually proposes a new algorithm for performing approximate submodular maximization. Finally, it appears that there is a gap between the lower bounds and the upper bounds they provide again, making it difficult to ascertain whether or not the paper simply follows the natural analysis of previous work or provides fundamental new insights.  That said, the question of closing this gap is tantalizing, ultimately resulting in this borderline review.  Detailed Comments:  - Lines 8-10: are the inequalities on epsilon in the wrong direction? - Line 60  - 63: although it was said that F is assumed to be monotone, here I would be clear about whether there result applied to monotone submodular functions. - Lines 90-96: a little more discussion of best known running times under the assumption would be beneficial for comparison. - Line 145: say a little more about how this lower bound compares to previous lower bounds and the techniques they use. - Line 150: In the line after “an general” should be “a general” - Lines 273 – 281: say more about what the algorithm actually is and how compares to other algorithms. - General: discuss the gaps in upper bounds and lower bounds left open by this work.