Original: Efficient Submodular Optimization under Noise: Local Search is Robust
Revised: Efficient Submodular Optimization under Noise: Local Search is Robust

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AffiliationAddressemail
Revised: AffiliationAddress

Original: Abstract
Revised: Abstract

Original: The problem of monotone submodular maximization has been studied extensivelydue to its wide range of applications.
Revised: The problem of monotone submodular maximization has been studied extensivelydue to its wide range of applications.

Original: However, there are cases where one canonly access the objective function in a distorted or noisy form because of theuncertain nature or the errors involved in the evaluation.
Revised: However, there are cases where one canonly access the objective function in a distorted or noisy form because of theuncertain nature or the errors involved in the evaluation.

Original: This paper considers theproblem of constrained monotone submodular maximization with noisy oraclesintroduced by [9].
Revised: This paper considers theproblem of constrained monotone submodular maximization with noisy oraclesintroduced by [9].

Original: For a cardinality constraint, we propose an algorithm achievinga near-optimal (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation guarantee with only a polynomialnumber of queries to the noisy value oracle, which improves the exponential querycomplexity of [18].
Revised: For a cardinality constraint, we propose an algorithm achievinga near-optimal (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation guarantee (for arbitrary ε > 0 ) withonly a polynomial number of queries to the noisy value oracle, which improvesthe exponential query complexity of [18].

Original: For general matroid constraints, we show the first constantapproximation algorithm in the presence of noise.
Revised: For general matroid constraints, weshow the first constant approximation algorithm in the presence of noise.

Original: Our main approaches are todesign a novel local search framework that can handle the effect of noise and toconstruct certain smoothing surrogate functions for noise reduction.
Revised: Ourmain approaches are to design a novel local search framework that can handle theeffect of noise and to construct certain smoothing surrogate functions for noisereduction.

Original:  Introduction
Revised:  Introduction

Original: Consider the following problems in machine learning and operations research: (1) selecting a setof locations to open up facilities with the goal of maximizing their overall user coverage [13]; (2)reducing the number of features in a machine learning model while retaining as much informationas possible [21]; and (3) identifying a small set of seed nodes that can achieve the largest overallinfluence in a social network [12].
Revised: Consider the following problems in machine learning and operations research: (1) selecting a setof locations to open up facilities with the goal of maximizing their overall user coverage [13]; (2)reducing the number of features in a machine learning model while retaining as much informationas possible [21]; and (3) identifying a small set of seed nodes that can achieve the largest overallinfluence in a social network [12].

Original: Solving these problems all involves maximizing a monotonesubmodular set function f : 2 N (cid:55)→ R subject to certain constraints.
Revised: Solving these problems all involves maximizing a monotonesubmodular set function f : 2 N (cid:55)→ R subject to certain constraints.

Original: Intuitively, submodularitycaptures the property of diminishing returns.
Revised: Intuitively, submodularitycaptures the property of diminishing returns.

Original: For example, a newly opened facility will contributeless to the overall user coverage if we have already opened many facilities and more if we haveonly opened a few.
Revised: For example, a newly opened facility will contributeless to the overall user coverage if we have already opened many facilities and more if we haveonly opened a few.

Original: Although the general problem of monotone submodular maximization subjectto a cardinality or general matroid constraint is NP-hard [4], the greedy algorithm, which selects anelement with the largest margin at each step, can approximately solve this problem under a cardinalityconstraint by a factor of 1 − 1 /e , and this approximation ratio is tight [16].
Revised: Although the general problem of monotone submodular maximization subjectto a cardinality or general matroid constraint is NP-hard [4], the greedy algorithm, which selects anelement with the largest margin at each step, can approximately solve this problem under a cardinalityconstraint by a factor of 1 − 1 /e , and this approximation ratio is tight [16].

Original: Moreover, a non-obliviouslocal search algorithm, which iteratively alters one element to improve an auxiliary objective function,is guaranteed to achieve an approximate ratio of 1 − 1 /e for general matroid constraints [5].
Revised: Moreover, a non-obliviouslocal search algorithm, which iteratively alters one element to improve an auxiliary objective function,is guaranteed to achieve an approximation ratio of 1 − 1 /e for general matroid constraints [5].

Original: In the literature, the submodular optimization problem usually assumes a value oracle to the objectivefunction f , which means one is allowed to query the exact value of f ( S ) for any S ⊆ N .
Revised: In the literature, the submodular optimization problem usually assumes a value oracle to the objectivefunction f , which means one is allowed to query the exact value of f ( S ) for any S ⊆ N .

Original: However, inmany applications, due to the uncertain nature of the objective or the errors involved in the evaluation,one can only access the function value in a distorted or noisy form.
Revised: However, inmany applications, due to the uncertain nature of the objective or the errors involved in the evaluation,one can only access the function value in a distorted or noisy form.

Original: For example, Globerson and
Revised: For example, Globerson and

Original: Roweis [7] pointed out that selecting features to construct a robust learning model is particularlyimportant in domains with non-stationary feature distributions or input sensor failures.
Revised: Roweis [7] pointed out that selecting features to construct a robust learning model is particularlyimportant in domains with non-stationary feature distributions or input sensor failures.

Original: It is knownthat without any assumption on the noise, direct adaptions of the greedy and local search methodsmentioned above may yield arbitrarily poor performance [10].
Revised: It is known that without any assumption on the noise, direct adaptions of the greedy and local search methods mentioned above may yield arbitrarily poor performance [10].

Original: To address this issue, Hassidim and
Revised: To address this issue, Hassidim and

Original: Singer [9] introduced and studied the following problem of monotone submodular maximization undernoise : For a monotone submodular function f , given a noisy value oracle ˜ f satisfying ˜ f
Revised: Singer [9] introduced and studied the following problem of monotone submodular maximization under noise : For a monotone submodular function f , given a noisy value oracle ˜ f satisfying ˜ f

Original: ( S ) =
Revised: ( S ) =

Original: ξ S f ( S ) for each set S ⊆ N , where the noise multiplier ξ S is independently drawn from a certain distribution,the goal is to find a set S maximizing f ( S ) under certain constraints.
Revised: ξ S f ( S )for each set S ⊆ N , where the noise multiplier ξ S is independently drawn from a certain distribution,the goal is to find a set S maximizing f ( S ) under certain constraints.

Original: Hassidim and Singer [9] showedthat under a sufficiently large cardinality constraint, a variant of the greedy algorithm achieves anear-optimal approximation ratio of 1 − 1 e − O ( ε ) for arbitrary ε > 0 .
Revised: Hassidim and Singer [9] showedthat under a sufficiently large cardinality constraint, a variant of the greedy algorithm achieves anear-optimal approximation ratio of 1 − 1 e − O ( ε ) for arbitrary ε > 0 .

Original: The problem becomes morechallenging when the cardinality constraint is relatively small since there is less room for mistakes.
Revised: The problem becomes morechallenging when the cardinality constraint is relatively small since there is less room for mistakes.

Original: In a subsequent work, Singer and Hassidim [18] developed another greedy-based algorithm for smallcardinality constraints and showed a near-tight approximation guarantee.
Revised: In a subsequent work, Singer and Hassidim [18] developed another greedy-based algorithm for smallcardinality constraints and showed a near-tight approximation guarantee.

Original: Despite these encouraging results, we want to point out two directions along this line of monotonesubmodular maximization under noise that still have room for improvements:
Revised: Despite these encouraging results, we want to point out two directions along this line of monotonesubmodular maximization under noise that still have room for improvements:

Original: • In the algorithm from [18], the query complexity to the noisy value oracle is exponential inε − 1 , which is costly when a near-optimal solution is needed, i.e., when the parameter ε isclose to 0 .
Revised: • In the algorithm from [18], the query complexity to the noisy value oracle is exponential inε − 1 , which is costly when a near-optimal solution is needed, i

Revised: e., when the parameter ε isclose to 0 .

Original: Is it possible to obtain near-optimal approximations for monotone submodularmaximization under cardinality constraints with the number of queries polynomial in ϵ − 1 ?
Revised: Is it possible to obtain near-optimal approximations for monotone submodularmaximization under cardinality constraints with the number of queries polynomial in ε − 1 ?

Original: • All previous works in submodular maximization under noise consider only the cardinalityconstraint, and no approximation guarantee is known under other constraints.
Revised: • All previous works in submodular maximization under noise consider only the cardinalityconstraint, and no approximation guarantee is known under other constraints.

Original: 1 Is there anyalgorithm that can achieve a constant approximation for submodular maximization undernoise for more general constraints, such as commonly studied matroid constraints [14, 24]?
Revised: 1 Is there anyalgorithm that can achieve a constant approximation for submodular maximization undernoise for more general constraints, such as commonly studied matroid constraints [14, 24]?

Original: In this paper, we provide answers to both questions above.
Revised: In this paper, we provide answers to both questions above.

Original: 1.1 Our contributions
Revised: 1.1 Our contributions

Original: We study the problem of constrained monotone submodular maximization under noise (Problem 2.6).
Revised: We study the problem of constrained monotone submodular maximization under noise (Problem 2.6).

Original: Following prior works [9, 18], we assume generalized exponential tail noises (Definition 2.5) andconsider the solutions subject to cardinality constraints (Definition 2.2) and matroid constraints(Definition 2.3).
Revised: Following prior works [9, 18], we assume generalized exponential tail noises (Definition 2.5) andconsider the solutions subject to cardinality constraints (Definition 2.2) and matroid constraints(Definition 2.3).

Original: The main contribution of this work is to show that for optimizing a monotonesubmodular function under a cardinality constraint, (cid:0) 1 − 1 e − O
Revised: The main contribution of this work is to show that for optimizing a monotonesubmodular function under a cardinality constraint, (cid:0) 1 − 1 e − O

Original: ( ε ) (cid:1) -approximations can be obtainedwith high probability by querying the noisy oracle only Poly (cid:0) n, 1 ε (cid:1) times.
Revised: ( ε ) (cid:1) -approximations can be obtainedwith high probability by querying the noisy oracle only Poly (cid:0) n, 1 ε (cid:1) times.

Original: Theorem 1.1 ( Informal, see Theorems 4.1 and 4.6 ) .
Revised: Theorem 1.1 ( Informal, see Theorems 4.1 and 4.6 ) .

Original: Let ε > 0 and assume n is sufficiently large.
Revised: Let ε > 0 and assume n is sufficiently large.

Original: For any r ∈ Ω (cid:0) 1 ε (cid:1) , there exists an algorithm that returns a (cid:0) 1 − 1 e − O
Revised: For any r ∈ Ω (cid:0) 1 ε (cid:1) , there exists an algorithm that returns a (cid:0) 1 − 1 e − O

Original: ( ε ) (cid:1) -approximation
Revised: ( ε ) (cid:1) -approximation

Original: forthe monotone submodular maximization problem under a r -cardinality constraint, with probability1 − o (1) and query complexity Poly (cid:0) n, 1 ε (cid:1) to ˜ f .
Revised: forthe monotone submodular maximization problem under a r -cardinality constraint, with probability1 − o (1) and query complexity Poly (cid:0) n, 1 ε (cid:1) to ˜ f .

Original: For a cardinality constraint, this paper and prior works [9, 18] all achieve near-optimal approximations.
Revised: For a cardinality constraint, this paper and prior works [9, 18] all achieve near-optimal approximations.

Original: However, our result is applicable for a larger range Ω (cid:0) 1 ε (cid:1) of cardinalities than Ω (cid:0) log log n · ε − 2 (cid:1)in [9].
Revised: However, our result is applicable for a larger range Ω (cid:0) 1 ε (cid:1) of cardinalities than Ω (cid:0) log log n · ε − 2 (cid:1)in [9].

Original: Moreover, we only require Poly (cid:0) n, 1 ε (cid:1) queries to the noisy value oracle, which improves thequery complexity Ω( n 1 ε ) of [18].
Revised: Moreover, we only require Poly (cid:0) n, 1 ε (cid:1) queries to the noisy value oracle, which improves thequery complexity Ω( n 1 ε ) of [18].

Original: Our main idea to address this problem is to employ a local search procedure, whereas prior methodsare all variants of the greedy algorithm.
Revised: Our main idea to address this problem is to employ a local search procedure, whereas prior methodsare all variants of the greedy algorithm.

Original: Intuitively, local search is more robust than the greedy sincethe marginal functions are more sensitive to noise than the value functions.
Revised: Intuitively, local search is more robust than the greedy sincethe marginal functions are more sensitive to noise than the value functions.

Original: To achieve a sufficientgain in each iteration, the greedy algorithm must identify the element with maximum margin.
Revised: To achieve a sufficientgain in each iteration, the greedy algorithm must identify the element with maximum margin.

Original: Incontrast, local search only needs to estimate the sets’ values.
Revised: Incontrast, local search only needs to estimate the sets’ values.

Original: This is why we can improve the querycomplexity to Poly (cid:0) n, 1 ε (cid:1) for small cardinality constraints.
Revised: This is why we can improve the querycomplexity to Poly (cid:0) n, 1 ε (cid:1) for small cardinality constraints.

Original: We present a unified framework (Algorithm 1) for enhancing the local search to cope with noise.
Revised: We present a unified framework (Algorithm 1) for enhancing the local search to cope with noise.

Original: One of the main differences between our framework and the non-oblivious local search proposedby [5] is that we use an approximation of the auxiliary function (Definition 3.2) rather than the exactone due to the presence of noise.
Revised: One of the main differences between our framework and the non-oblivious local search proposedby [5] is that we use an approximation of the auxiliary function (Definition 3.2) rather than the exactone due to the presence of noise.

Original: We analyze the impact of the inaccuracies on the approximationperformance and query complexity of the local search (Theorem 3.3).
Revised: We analyze the impact of the inaccuracies on the approximationperformance and query complexity of the local search (Theorem 3.3).

Original: Another difference is the construction of the auxiliary functions used to guide the local search.
Revised: Another difference is the construction of the auxiliary functions used to guide the local search.

Original: Weconstruct the auxiliary function not based on the objective function f , but on smoothing surrogatefunctions that are "close enough" to f and robust to noise.
Revised: Weconstruct the auxiliary function not based on the objective function f , but on smoothing surrogatefunctions that are "close enough" to f and robust to noise.

Original: For a set with size r ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:0) n 1 / 3 (cid:1) ,we define the smoothing surrogate function h as the expectation of f ’s value when a random element in N is added to the set (Definition 4.2).
Revised: For a set with size r ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:0) n 1 / 3 (cid:1) , we define the smoothing surrogate function h as the expectation of f ’s value when a random elementin N is added to the set (Definition 4.2).

Original: This surrogate function is robust for a relatively smallcardinality as it is based on a rather large averaging set with size nearly n , but too concentrated fora large cardinality close to n .
Revised: This surrogate function is robust for a relatively smallcardinality as it is based on a rather large averaging set with size nearly n , but too concentrated fora large cardinality close to n .

Original: Thus, we consider another smoothing surrogate function h H for sizer ∈ Ω (cid:0) n 1 / 3 (cid:1) , defined as the average value combined with all subsets of a certain small-size set H(Definition 4.7).
Revised: Thus, we consider another smoothing surrogate function h H for sizer ∈ Ω (cid:0) n 1 / 3 (cid:1) , defined as the average value combined with all subsets of a certain small-size set H(Definition 4.7).

Original: The auxiliary functions constructed on both smoothing surrogates are shown to havealmost accurate approximations (Lemma 4.4 and 4.9).
Revised: The auxiliary functions constructed on both smoothing surrogates are shown to havealmost accurate approximations (Lemma 4.4 and 4.9).

Original: Consequently, we can apply our unified localsearch framework (Algorithm 1) in both cases (Algorithm 2 and 3), and guarantee to achieve nearlytight approximate solutions (Theorem 4.1 and 4.6).
Revised: Consequently, we can apply our unified localsearch framework (Algorithm 1) in both cases (Algorithm 2 and 3), and guarantee to achieve nearlytight approximate solutions (Theorem 4.1 and 4.6).

Original: Another contribution of this paper is a constant approximation result for maximizing monotonesubmodular functions with noisy oracles under general matroid constraints.
Revised: Another contribution of this paper is a constant approximation result for maximizing monotonesubmodular functions with noisy oracles under general matroid constraints.

Original: Theorem 1.2 ( Informal, see Theorems 5.1 and 5.2 ) .
Revised: Theorem 1.2 ( Informal, see Theorems 5.1 and 5.2 ) .

Original: Let ε > 0 and assume n is sufficiently large.
Revised: Let ε > 0 and assume n is sufficientlylarge.

Original: For any r ∈ Ω (cid:0)  1  ε  (cid:1) , there exists an algorithm that returns a (cid:0) 12 (cid:0) 1 − 1 e (cid:1) − O( ε ) (cid:1)
Revised: For any r ∈ Ω (cid:0) ε − 1 log( ε − 1 ) (cid:1) , there exists an algorithm that returns a (cid:0) 12 (cid:0) 1 − 1 e (cid:1) − O

Original: -approximationfor the monotone submodular maximization problem under a matroid constraint with rank r , withprobability 1 − o (1) and query complexity at most Poly (cid:0) n, 1 ε (cid:1) to ˜ f .
Revised: ( ε ) (cid:1) -approximation for the monotone submodular maximization problem under a matroid constraint withrank r , with probability 1 − o (1) and query complexity at most Poly (cid:0) n, 1 ε (cid:1) to ˜ f .

Original: To the best of our knowledge, this is the first result showing that constant approximation guaranteesare obtainable under general matroid constraints in the presence of noise.
Revised: To the best of our knowledge, this is the first result showing that constant approximation guaranteesare obtainable under general matroid constraints in the presence of noise.

Original: To cope with noise, onecommon approach for cardinality constraints is to incorporate some extra elements to gain robustnessand include these elements in the final solutions.
Revised: To cope with noise, onecommon approach for cardinality constraints is to incorporate some extra elements to gain robustnessand include these elements in the final solutions.

Original: However, for a matroid, additional elements mayundermine the independence of a set.
Revised: However, for a matroid, additional elements mayundermine the independence of a set.

Original: To address this difficulty, we develop a technique for comparingthe values of independent sets in the presence of noise, which allows us to select either the local searchsolutions or the additional elements for robustness and leads to an approximate ratio of 12 (cid:0) 1 − 1 e (cid:1) .
Revised: To address this difficulty, we develop a technique for comparingthe values of independent sets in the presence of noise, which allows us to select either the localsearch solutions or the additional elements for robustness and leads to an approximation ratio of12 (cid:0) 1 − 1 e (cid:1) .

Original: 1.2 Related work
Revised: 1.2 Related work

Original: Research has been conducted on monotone submodular maximization in the presence of noise.
Revised: Research has been conducted on monotone submodular maximization in the presence of noise.

Original: We say a noisy oracle is inconsistent if it returns different answers when repeatedly queried.
Revised: We say a noisy oracle is inconsistent if it returns different answers when repeatedly queried.

Original: Forinconsistent oracles, noise often does not present a barrier to optimization, since concentrationassumptions can eliminate the noise after a sufficient number of queries [19, 11].
Revised: Forinconsistent oracles, noise often does not present a barrier to optimization, since concentrationassumptions can eliminate the noise after a sufficient number of queries [19, 11].

Original: When identicalqueries always obtain the same answer, the problem becomes more challenging.
Revised: When identicalqueries always obtain the same answer, the problem becomes more challenging.

Original: Aside from the i.i.dnoise adopted in [9, 18] and this paper, Horel and Singer [10] study submodular optimization undernoise adversarially generated from [1 − ε/r, 1 + ε/r ] , where the greedy algorithm achieves a ratio of1 − 1 /e − O ( ε ) .
Revised: Aside from the i.i.dnoise adopted in [9, 18] and this paper, Horel and Singer [10] study submodular optimization undernoise adversarially generated from [1 − ε/r, 1 + ε/r ] , where the greedy algorithm achieves a ratio of1 − 1 /e − O ( ε ) .

Original: No algorithm can obtain a constant approximation if noise is not bounded in thisrange.
Revised: No algorithm can obtain a constant approximation if noise is not bounded in thisrange.

Original: Besides cardinality and single matroid constraints, more complex constraints have also been considered in the context of submodular optimization before.

Original: Fisher et al.

Original: [6] present an algorithm achievinga 1 / ( k + 1) -approximation for monotone submodular maximization under k matroid constraints.

Original: Leeet al.

Original: [15] subsequently improve the approximation guarantee to 1 / ( k + ε ) for k > 2 .

Original: Sviridenko[20] provides a (cid:0) 1 − 1 e (cid:1) -approximation under knapsack constraints.

Original: With the multilinear relaxationtechnique, Chekuri et al.

Original: [3] obtain a 0 .

Original: 38 /k -approximation for maximizing a monotone submodularfunction subject to k matroids and a constant number of knapsack constraints.

Original:  The model
Revised:  The model

Original: This section formally defines our model (Problem 2.6) of maximizing a monotone submodularfunction (Definition 2.1) under a cardinality constraint (Definition 2.2) or a matroid constraint(Definition 2.3), given access to a noisy value oracle (Definition 2.4).
Revised: This section formally defines our model (Problem 2.6) of maximizing a monotone submodularfunction (Definition 2.1) under a cardinality constraint (Definition 2.2) or a matroid constraint(Definition 2.3), given access to a noisy value oracle (Definition 2.4).

Original: Let N be the ground set withsize | N | = n , and we use the shorthands S + x = S ∪ { x } and S − x = S \{ x } throughout thispaper.
Revised: Let N be the ground set withsize | N | = n , and we use the shorthands S + x = S ∪ { x } and S − x = S \{ x } throughout thispaper.

Original: We first review the definition of monotone submodular functions.
Revised: We first review the definition of monotone submodular functions.

Original: Definition 2.1 ( Monontone submodular functions).
Revised: Definition 2.1 ( Monontone submodular functions).

Original: A function f : 2 N → R ≥ 0 is monotonesubmodular if 1) (monotonicity)
Revised: A function f : 2 N → R ≥ 0 is monotonesubmodular if 1) (monotonicity)

Original: f ( A ) ≤ f ( B ) for any A ⊆ B ⊆ N ; 2) (submodularity) for anysubset A, B ⊆ N and x ∈ N : f
Revised: f ( A ) ≤ f ( B ) for any A ⊆ B ⊆ N ; 2) (submodularity) for anysubset A, B ⊆ N and x ∈ N : f

Original: ( A + x ) − f ( A ) ≥ f ( B + x ) − f ( B ) .
Revised: ( A + x ) − f ( A ) ≥ f ( B + x ) − f ( B ) .

Original: There are various examples of monotone submodular functions in optimization, such as budgetadditive functions, coverage functions, cut functions and rank functions [1].
Revised: There are various examples of monotone submodular functions in optimization, such as budgetadditive functions, coverage functions, cut functions and rank functions [1].

Original: Since the description ofa submodular function may be exponential in the size N , we usually assume access of a value oraclethat answers f ( S ) for each S ⊆ N .
Revised: Since the description ofa submodular function may be exponential in the size N , we usually assume access of a value oraclethat answers f ( S ) for each S ⊆ N .

Original: Let I denote a collection of feasible subsets S ⊆ N .
Revised: Let I denote a collection of feasible subsets S ⊆ N .

Original: The goal of a constrained monotone submodular maximization is to find a subset S ⊆
Revised: The goal ofa constrained monotone submodular maximization is to find a subset S ⊆

Original: I to maximize f ( S ) .
Revised: I to maximize f ( S ) .

Original: Theobjective f is further assumed to be normalized, i.e., f ( ∅ ) = 0 .
Revised: Theobjective f is further assumed to be normalized, i.e., f ( ∅ ) = 0 .

Original: We consider two types of constraintsin our models: cardinality constraints and matroid constraints.
Revised: We consider two types of constraintsin our models: cardinality constraints and matroid constraints.

Original: Definition 2.2 ( Carinality constraints).
Revised: Definition 2.2 ( Carinality constraints).

Original: Given a ground set N and an integer r ≥ 1 , a cardinalityconstraint is of the form I ( r ) =
Revised: Given a ground set N and an integer r ≥ 1 , a cardinalityconstraint is of the form I ( r ) =

Original: { S ⊆ N : | S | ≤ r } .
Revised: { S ⊆ N : | S | ≤ r } .

Original: Definition 2.3 ( Matroids and Matroid constraints).
Revised: Definition 2.3 ( Matroids and Matroid constraints).

Original: Given a ground set N , a matroid M isrepresented by an ordered pair ( N, I ( M )) satisfying that 1) ∅ ∈
Revised: Given a ground set N , a matroid M isrepresented by an ordered pair ( N, I ( M )) satisfying that 1) ∅ ∈

Original: I ( M ) ; 2) If I ∈ I ( M ) and I ′ ⊆
Revised: I ( M ) ; 2) If I ∈ I ( M ) and I ′ ⊆

Original: I ,then I ′ ∈
Revised: I ,then I ′ ∈

Original: I ( M ) ; 3) If I 1 , I 2 ⊆
Revised: I ( M ) ; 3) If I 1 , I 2 ⊆

Original: I ( M ) and | I 1 | < | I 2 | , then there must exist an element e ∈
Revised: I ( M ) and | I 1 | < | I 2 | , then there must exist an element e ∈

Original: I 2 \
Revised: I 2 \

Original: I 1such that I 1 ∪ { e } ∈ I ( M ) .
Revised: I 1such that I 1 ∪ { e } ∈ I ( M ) .

Original: Each I ∈ I ( M ) is called an independent set.
Revised: Each I ∈ I ( M ) is called an independent set.

Original: The maximum size of anindependent set is called the rank of M .
Revised: The maximum size of anindependent set is called the rank of M .

Original: We call the collection
Revised: We call the collection

Original: I ( M ) a matroid constraint.
Revised: I ( M ) a matroid constraint.

Original: Assume we are given a membership oracle of I ( M ) that for any set S ⊆
Revised: Assume we are given a membership oracle of I ( M ) that for any set S ⊆

Original: N answers whether S ∈
Revised: N answers whether S ∈

Original: I ( M ) .
Revised: I ( M ) .

Original: As a widely used combinatorial structure, there is an extensive study on matroids [26, 17, 25].
Revised: As a widely used combinatorial structure, there is an extensive study on matroids [26, 17, 25].

Original: Common matroids include uniform matroids, partition matroids, regular matroids, etc.
Revised: Common matroids include uniform matroids, partition matroids, regular matroids, etc.

Original: See [17] formore discussions.
Revised: See [17] formore discussions.

Original: Specifically, a uniform matroid constraint is equivalent to a cardinality constraint,implying that cardinality constraints are a special case of matroid constraints.
Revised: Specifically, a uniform matroid constraint is equivalent to a cardinality constraint,implying that cardinality constraints are a special case of matroid constraints.

Original: Noisy value oracle.
Revised: Noisy value oracle.

Original: It is well known that given an exact value oracle to f , for any ε > 0 , there existsa randomized (1 − 1 /e − ε ) -approximate algorithm for the submodular optimization problem undera matroid constraint, i.e., max S ⊆I f
Revised: It is well known that given an exact value oracle to f , for any ε > 0 , there existsa randomized (1 − 1 /e − ε ) -approximate algorithm for the submodular optimization problem undera matroid constraint, i.e., max S ⊆I f

Original: ( S ) [2, 5].
Revised: ( S ) [2, 5].

Original: However, as discussed earlier, the value oracle off may be imperfect, and we may only have a noisy value oracle ˜ f
Revised: However, as discussed earlier, the value oracle off may be imperfect, and we may only have a noisy value oracle ˜ f

Original: instead of f .
Revised: instead of f .

Original: We consider thefollowing noisy value oracle that has also been investigated in [10, 9].
Revised: We consider thefollowing noisy value oracle that has also been investigated in [10, 9].

Original: Definition 2.4 ( (Multiplicative) noisy value oracle [9]).
Revised: Definition 2.4 ( (Multiplicative) noisy value oracle [9]).

Original: We call ˜ f : 2 N → R ≥ 0 a (multiplicative)noisy value oracle of f , if there exists some distribution D s.t. for any S ⊆ N , ˜ f ( S ) =
Revised: We call ˜ f : 2 N → R ≥ 0 a (multiplicative)noisy value oracle of f , if there exists some distribution D s.t. for any S ⊆ N , ˜ f ( S ) =

Original: ξ S f ( S ) whereξ S is i.i.d.
Revised: ξ S f ( S ) whereξ S is i.i.d.

Original: drawn from D .
Revised: drawn from D .

Original: Throughout this paper, we consider a general class of noise distributions, called generalized exponential tail distributions, defined in [18].
Revised: Throughout this paper, we consider a general class of noise distributions, called generalized exponential tail distributions, defined in [18].

Original: Definition 2.5 ( Generalized exponential tail distributions [18]).
Revised: Definition 2.5 ( Generalized exponential tail distributions [18]).

Original: A noise distribution D has ageneralized exponential tail if there exists some x 0 such that for every x > x 0 the probability densityfunction ρ ( x ) =
Revised: A noise distribution D has ageneralized exponential tail if there exists some x 0 such that for every x > x 0 the probability densityfunction ρ ( x ) =

Original: e − g ( x ) , where g ( x ) =
Revised: e − g ( x ) , where g ( x ) =

Original: (cid:80) i c i x γ i for some (not necessarily integers) γ 0 ≥ γ 1 ≥. . .
Revised: (cid:80) i c i x γ i for some (not necessarily integers) γ 0 ≥ γ 1 ≥. . .

Original: , s.t.γ 0 ≥ 1 and c 0 > 0 .
Revised: , s.t.γ 0 ≥ 1 and c 0 > 0 .

Original: If D has bounded support we only require that either it has an atom at itssupremum or that ρ is continuous and non-zero at the supremum.
Revised: If D has bounded support we only require that either it has an atom at itssupremum or that ρ is continuous and non-zero at the supremum.

Original: W.l.o.g.
Revised: W.l.o.g.

Original: , we assume E
Revised: , we assume E

Original: [ D ] = 1 . 

Revised: [ D ] = 1 .

Original: As mentioned in [18], the class of generalized exponential taildistributions contains Gaussian distributions, exponential distributions, and all distributions withbounded support that is independent of n .
Revised: As mentioned in [18], the class of generalized exponential taildistributions contains Gaussian distributions, exponential distributions, and all distributions withbounded support that is independent of n .

Original: The model.
Revised: The model.

Original: We are ready to propose the main problem, which has already been considered in [9, 18].
Revised: We are ready to propose the main problem, which has already been considered in [9, 18].

Original: Problem 2.6 ( Constrained submodular optimization under noise).
Revised: Problem 2.6 ( Constrained submodular optimization under noise).

Original: Given a noisy value oracle˜ f : 2 N → R ≥ 0 (with a certain generalized exponential tail distribution D ) to an underlying monotonesubmodular function f , and a cardinality constraint I =
Revised: Given a noisy value oracle˜ f : 2 N → R ≥ 0 (with a certain generalized exponential tail distribution D ) to an underlying monotonesubmodular function f , and a cardinality constraint I =

Original: I ( r ) or a matroid constraint I
Revised: I ( r ) or a matroid constraint I

Original: = I ( M ) ,the goal is to find S ⊆
Revised: = I ( M ) ,the goal is to find S ⊆

Original: I to maximize f ( S ) .
Revised: I to maximize f ( S ) .

Original:  Local search with approximate evaluation oracles to auxiliary functions
Revised:  Local search with approximate evaluation oracles to auxiliary functions

Original: This section proposes a non-oblivious local search framework with a noisy value oracle (Algorithm 1),and gives an analysis (Theorem 3.3) of its performance.
Revised: This section proposes a non-oblivious local search framework with a noisy value oracle (Algorithm 1),and gives an analysis (Theorem 3.3) of its performance.

Original: This algorithm is a generalization of thatin [5] and will be used as a meta-algorithm for solving Problem 2.6 in Sections 4 and 5.
Revised: This algorithm is a generalization of thatin [5] and will be used as a meta-algorithm for solving Problem 2.6 in Sections 4 and 5.

Original: Roughlyspeaking, a local search framework first constructs a "good" initial solution via a standard greedyalgorithm and then iteratively improves the solution w.r.t.
Revised: Roughlyspeaking, a local search framework first constructs a "good" initial solution via a standard greedyalgorithm and then iteratively improves the solution w.r.t.

Original: to an auxiliary function (Definition 3.1)by swapping an element at each step.
Revised: to an auxiliary function (Definition 3.1)by swapping an element at each step.

Original: The following auxiliary function proposed by [5] is a linearcombination of the original function over all subsets.
Revised: The following auxiliary function proposed by [5] is a linearcombination of the original function over all subsets.

Original: Definition 3.1 ( Auxiliary function [5]).
Revised: Definition 3.1 ( Auxiliary function [5]).

Original: Given a monotone submodular function h : 2 N → R ≥ 0 ,its auxiliary function φ h : 2 N → R ≥ 0 is defined as φ h ( S ) =
Revised: Given a monotone submodular function h : 2 N → R ≥ 0 ,its auxiliary function φ h : 2 N → R ≥ 0 is defined as φ h ( S ) =

Original: (cid:80) T ⊆
Revised: (cid:80) T ⊆

Original: S m | S |− 1 , | T |− 1 · h ( T ) , where{+wherem s,t = (cid:82) 1 
+}
Revised: S m | S |− 1 , | T |− 1 · h ( T ) , {+wherem s,t = (cid:82) 1 
+}

Original: [Equation]
Revised: e p ( e − 1) − 1 p t (1 − p )

Original:  for all s ≥ t [Equation]
Revised: s − t dp for all s ≥ t ≥ 0 .

Original: We use h instead of f because we may do local search on certain smoothing surrogate function h off (see examples in Definitions 4.2 and 4.7).
Revised: We use h instead of f because we may do local search on certain smoothing surrogate function h off (see examples in Definitions 4.2 and 4.7).

Original: Note that φ h is also a monotone submodular function.
Revised: Note that φ h is also a monotone submodular function.

Original: Given an exact value oracle of φ h and a matroid constraint I ( M ) , Filmus and Ward [5] show thatthere exists a local search algorithm that outputs a (1 − 1 /e )
Revised: Given an exact value oracle of φ h and a matroid constraint I ( M ) , Filmus and Ward [5] show thatthere exists a local search algorithm that outputs a (1 − 1 /e )

Original: -approximate solution for the optimizationproblem max S ⊆I ( M ) h ( S ) .
Revised: -approximate solution for the optimization problem max S ⊆I ( M ) h ( S ) .

Original: However, as mentioned before, we do not have an exact value oracle ofh .
Revised: However, as mentioned before, we do not have an exact value oracle ofh .

Original: Thus, it may not be possible to construct an exact value oracle of φ h .
Revised: Thus, it may not be possible to construct an exact value oracle of φ h .

Original: To understand how theseinaccuracies can affect the performance of the local search algorithm, we introduce the followingapproximation for the auxiliary function φ h .
Revised: To understand how theseinaccuracies can affect the performance of the local search algorithm, we introduce the followingapproximation for the auxiliary function φ h .

Original: Definition 3.2 ( ( α, δ, I ( M ))
Revised: Definition 3.2 ( ( α, δ, I ( M ))

Original: -approximation of φ h ).
Revised: -approximation of φ h ).

Original: Given an auxiliary function φ h , a matroidconstraint I ( M ) and constants α, δ > 0 , we say a randomized function (cid:99) φ h is an ( α, δ, I ( M )) -approximation of φ h if for any independent set A ∈ I ( M ) ,
Revised: Given an auxiliary function φ h , a matroidconstraint I ( M ) and constants α, δ > 0 , we say a randomized function (cid:99) φ h is an ( α, δ, I ( M )) -approximation of φ h if for any independent set A ∈ I ( M ) ,

Original: [Equation]
Revised: [Equation]

Original: Intuitively, (cid:99) φ h is a randomized version of φ h with an additive concentration guarantee, where therandomness may come from noise of h or sampling error for estimating φ h .
Revised: Intuitively, (cid:99) φ h is a randomized version of φ h with an additive concentration guarantee, where therandomness may come from noise of h or sampling error for estimating φ h .

Original: As α, δ tend to 0,(cid:99)
Revised: As α, δ tend to 0,(cid:99)

Original: φ h is a better approximation of φ h .
Revised: φ h is a better approximation of φ h .

Original: Specifically, when α = δ = 0 and I = 2 N , (cid:99)
Revised: Specifically, when α = δ = 0 and I = 2 N , (cid:99)

Original: φ h is exactlyequivalent to φ h .
Revised: φ h is exactlyequivalent to φ h .

Original: Now we are ready to provide our local search framework (Algorithm 1), which is ageneralization of [5, Algorithm 2].
Revised: Now we are ready to provide our local search framework (Algorithm 1), which is ageneralization of [5, Algorithm 2].

Original: The main difference is that we use an approximation (cid:99) φ
Revised: The main difference is that we use an approximation (cid:99) φ

Original: h insteadof φ h for evaluation at each greedy step (Line 4) and local search step (Line 12).
Revised: h insteadof φ h for evaluation at each greedy step (Line 4) and local search step (Line 12).

Original: The performance of
Revised: The performance of

Original: Algorithm 1 is summarized in Theorem 3.3.
Revised: Algorithm 1 is summarized in Theorem 3.3.

Original:  α  r +
Revised: 1 Set I ← (cid:108) log 1+∆ (cid:16) 2(1+ α ) 1 − 2( r +1)

Original: α {+(cid:17)(cid:109) 
+}
Revised: α {+(cid:17)(cid:109) 
+}

Original: 2 Initialize: U 0 ← ∅ 
Revised: Initialize: U 0 ← ∅ , i =3 while i ≤ r do u

Original: ir do u i ← arg max e : U i − 1 + e ∈I ( M )  φ 
Revised: i ← arg max e : U i − 1 + e ∈I ( M ) (cid:99) φ h ( U i − 1 + e ) U i ←

Original: U i −  
 
Revised: U i − 1 + u i i ← i +7 S 0 ← U r ▷ Initial solution by greedy 8 for i = 
 to I − 1 do ▷ Local search for I iterations for each element x ∈ S i and y ∈ N \ S i do S ′

Original: + i 
Revised: i ← S

Original: Local search for I iterations
Revised: i − x + y if S ′ i ∈ I ( M ) then if (cid:99) φ h ( S ′ i ) ≥ (1 + ∆) · (cid:99) φ h

Original:  An improved solution S i  found
Revised: ( S i ) then ▷ An improved solution S i was found

Revised: S i +1 ← S

Revised: i break and continue to the next iteration of i 15 return S

Revised: I

Original: Theorem 3.3 ( Performance of noisy local search ) .
Revised: Theorem 3.3 ( Performance of noisy local search ) .

Original: Let I = log 1+∆ (cid:16) 2(1+ α ) 1 − 2( r +1)
Revised: Let I = log 1+∆ (cid:16) 2(1+ α ) 1 − 2( r +1)

Original: α (cid:17) .
Revised: α (cid:17) .

Original: With probabilityat least 1 − ( I + 1) rnδ , the output S I of Algorithm 1 is a (cid:0) 1 − 1 e (cid:1) (1 − r ln( er ) ((2 + ∆)
Revised: With probabilityat least 1 − ( I + 1) rnδ , the output S I of Algorithm 1 is a (cid:0) 1 − 1 e (cid:1) (1 − r ln( er ) ((2 + ∆)

Original: α + ∆)) -approximate solution for problem max S ⊆I ( M ) h ( S ) , with at most ( I + 1) rn calls to (cid:99) φ h .
Revised: α + ∆)) -approximate solution for problem max S ⊆I ( M ) h ( S ) , with at most ( I + 1) rn calls to (cid:99) φ h .

Original: Note that if α, ∆ ∈ O ( ε r log r ) , Algorithm 1 achieves an approximate ratio of 1 − 1 e − O ( ε ) , and thenumber of calls to (cid:99) φ h can be upper bounded by ˜ O ( r 2 n · ε − 1 ) . The success probability is close to 1for a sufficiently small δ , e.g., δ = O (1 /n ) .
Revised: We defer the proof to Section C.

Original: We defer the proof to Section B. Roughly, given a nearly accurate approximation (cid:99) φ h , we can showthat the initial solution S 0 is likely to be a constant approximation for max S ∈I ( M )
Revised: Roughly, given a nearly accurate approximation (cid:99) φ h , we can showthat the initial solution S 0 is likely to be a constant approximation for max S ∈I ( M )

Original: φ h ( S ) (Lemma
Revised: φ h ( S ) (Lemma

Original: B.1).
Revised: C.1).

Original: This guarantees that we will reach a local optimal solution w.r.t.
Revised: This guarantees that we will reach a local optimal solution w.r.t.

Original: φ h in I iterations with highprobability (Claim B.3).
Revised: φ h in I iterations with highprobability (Claim C.3).

Original: Then combining with basic properties of φ h shown in [5] and the fact that(cid:99) φ h is "close" to φ h , we obtain Theorem 3.3. 
Revised: Then combining with basic properties of φ h shown in [5] and the fact that(cid:99) φ h is "close" to φ h , we obtain Theorem 3.3. Corollary 3.4.

Revised: Assume n ∈ Ω (cid:16) exp (cid:16)(cid:0) 1 ε (cid:1) O (1) (cid:17)(cid:17) , if α, ∆ = ε 4 r log r , we have I ≤ 5 r ln r ε , and withprobability at least 1 − ( I + 1) rnδ , the output S I of Algorithm 1 is a (cid:0) 1 − 1 e − ε (cid:1) -approximatesolution for problem max S

Revised: ⊆I ( M ) h ( S ) , with at most ( I + 1) rn calls to (cid:99) φ h .

Original:  Our algorithms and main theorems for cardinality constraints
Revised:  Our algorithms and main theorems for cardinality constraints

Original: For different ranges of cardinalities, this section presents algorithms (Algorithm 2 and 3) that return a 216 (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation for Problem 2.6 with only Poly (cid:0) n, 1 ε (cid:1) queries to the noisy oracle ˜ f .
Revised: For different ranges of cardinalities, this section presents algorithms (Algorithm 2 and 3) that return a 212 (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation for Problem 2.6 with only Poly (cid:0) n, 1 ε (cid:1) queries to the noisy oracle ˜ f .

Original: The analyses (Theorem 4.1 and 4.6) of the algorithms constitute a proof of Theorem 1.1. 4.1 Algorithmic results for 
Revised: The analyses (Theorem 4.1 and 4.6) of the algorithms constitute a proof of Theorem 1.1.4.1 Algorithmic results for cardinality constraints when r ∈ Ω (cid:0) 1 ε (cid:1)

Original: cardinality constraints when r ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:16) n 13 (cid:17)
Revised: ∩ O (cid:16) n 13 (cid:17)

Original: We first present an algorithm (Algorithm 2) and its analysis (Theorem 4.1) that are applicable for allr ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:16) n 13 (cid:17) when n is sufficiently large.
Revised: We first present an algorithm (Algorithm 2) and its analysis (Theorem 4.1) that are applicable for allr ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:16) n 13 (cid:17) when n is sufficiently large.

Original: All missing proofs can be found in Section C.
Revised: All missing proofs can be found in Section D.

Original: Theorem 4.1 ( Algorithmic results for cardinality constraints when r ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:16) n 13 (cid:17) ) .
Revised: Theorem 4.1 ( Algorithmic results for cardinality constraints when r ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:16) n 13 (cid:17) ) .

Original: Letε > 0 and assume n ∈ Ω (cid:16) exp (cid:16)(cid:0) 

Revised: Letε > 0 and assume n ∈ Ω (cid:16) exp (cid:16)(cid:0)

Original: ε (cid:1) O (1) (cid:17)(cid:17) is sufficiently large.
Revised: ε (cid:1) O (1) (cid:17)(cid:17) is sufficiently large.

Original: For any r ∈ Ω( 1 ε ) ∩ O (cid:16) n 13 (cid:17) ,there exists an algorithm that returns a (cid:0) 1 − 1 e − O
Revised: For any r ∈ Ω( 1 ε ) ∩ O (cid:16) n 13 (cid:17) ,there exists an algorithm that returns a (cid:0) 1 − 1 e − O

Original: ( ε ) (cid:1) -approximation for Problem 2.6 under ar -cardinality constraint, with probability at least 1 − O (cid:16) 1 log n (cid:17) and query complexity at most
Revised: ( ε ) (cid:1) -approximation for Problem 2.6 under ar -cardinality constraint, with probability at least 1 − O (cid:16) 1 log n (cid:17) and query complexity at most

Original: O (cid:16) r 2 log 2 r · n 32 ε − 1 max { r, log n } (cid:17) to ˜ f .
Revised: O (cid:16) r 2 log 2 r · n 32 ε − 1 max { r, log n } (cid:17) to ˜ f .

Original: The assumption that n is sufficiently large is necessary and has also been adopted in prior works onnoisy submodular optimization [9, 18].
Revised: The assumption that n is sufficiently large is necessary and has also been adopted in prior works onnoisy submodular optimization [9, 18].

Original: We achieve a tight approximate ratio of 1 − 1 /e − O ( ε ) .
Revised: We achieve a tight approximation ratio of 1 − 1 /e − O ( ε ) .

Original: Furthermore, we only require Poly ( n, 1 /ε ) queries to ˜ f , in contrast to Ω (cid:0) n 1 /ε (cid:1) for the prior greedyalgorithm [18].
Revised: Furthermore, we only require Poly ( n, 1 /ε ) queries to ˜ f , in contrast to Ω (cid:0) n 1 /ε (cid:1) for the prior greedyalgorithm [18].

Original: 4.1.1 Useful notations and useful facts for Theorem 4.1
Revised: 4.1.1 Useful notations and useful facts for Theorem 4.1

Original: Our algorithm and analysis are based on the following smoothing surrogate function h .
Revised: Our algorithm and analysis are based on the following smoothing surrogate function h .

Original: Definition 4.2 ( Smoothing surrogate function I).
Revised: Definition 4.2 ( Smoothing surrogate function I).

Original: For any set S ⊆ N , we define the smoothingsurrogate function h ( S ) as the expectation of f ( S + e ) over a random element e ∈ N , i.e., h ( S ) =
Revised: For any set S ⊆ N , we define the smoothingsurrogate function h ( S ) as the expectation of f ( S + e ) over a random element e ∈ N , i.e., h ( S ) =

Original: 1 n (cid:80) e ∈ N f
Revised: 1 n (cid:80) e ∈ N f

Original: ( S + e ) .
Revised: ( S + e ) .

Original: The surrogate function h ( S ) is robust to noise when | S | is relatively small since at this time h ( S ) isbased on a rather large averaging set with size nearly n .
Revised: The surrogate function h ( S ) is robust to noise when | S | is relatively small since at this time h ( S ) isbased on a rather large averaging set with size nearly n .

Original: Note that h ( S ) becomes too concentrated onf ( S ) as | S | ≈ n , and hence, we consider this surrogate h for the range r ∈ O (cid:16) n 13 (cid:17) .
Revised: Note that h ( S ) becomes too concentrated onf ( S ) as | S | ≈ n , and hence, we consider this surrogate h for the range r ∈ O (cid:16) n 13 (cid:17) .

Original: We now showthat h shares some basic properties with f .
Revised: We now showthat h shares some basic properties with f .

Original: Lemma 4.3 ( Properties of h ) .
Revised: Lemma 4.3 ( Properties of h ) .

Original: The smoothing surrogate function h is monotone and submodular.
Revised: The smoothing surrogate function h is monotone and submodular.

Original: Note that h is implicitly constructed since we only have a value oracle to ˜ f instead of f .
Revised: Note that h is implicitly constructed since we only have a value oracle to ˜ f instead of f .

Original: Hence, weconstruct an approximation of its auxiliary function φ h ; summarized by the following lemma.
Revised: Hence, weconstruct an approximation of its auxiliary function φ h ; summarized by the following lemma.

Original: Lemma 4.4 ( Approximation of φ h ) .
Revised: Lemma 4.4 ( Approximation of φ h ) .

Original: Let α, δ ∈ (0 , 1 / 2) and assume n ∈ Ω( α − 2 log( δ − 1 )) .
Revised: Let α, δ ∈ (0 , 1 / 2) and assume n ∈ Ω( α − 2 log( δ − 1 )) .

Original: Thereexists a value oracle O to an ( α, δ, I ( r − 1))
Revised: Thereexists a value oracle O to an ( α, δ, I ( r − 1))

Original: -approximation (cid:99) φ h of φ h , which answering O
Revised: -approximation (cid:99) φ h of φ h , which answering O

Original: ( A )queries at most M = Θ (cid:16) log r · n 12 max { r, log n } (cid:17) times to ˜ f for each set
Revised: ( A )queries at most M = Θ (cid:16) log r · n 12 max { r, log n } (cid:17) times to ˜ f for each set

Original: A ∈ I ( r − 1) .
Revised: A ∈ I ( r − 1) .

Original: The above lemma indicates that for a sufficiently large n , we can achieve an arbitrary accurateapproximation (cid:99) φ h of φ h , which queries ˜ f only ˜ O ( rn 12 ) times for each (cid:99) φ h ( A ) .
Revised: The above lemma indicates that for a sufficiently large n , we can achieve an arbitrary accurateapproximation (cid:99) φ h of φ h , which queries ˜ f only ˜ O ( rn 12 ) times for each (cid:99) φ h ( A ) .

Original: 4.1.2 Algorithm for Theorem 4.1
Revised: 4.1.2 Algorithm for Theorem 4.1

Original: We present Algorithm 2 that contains two phases: a local search phase (Line 2) and a ˜ f -maximizationphase (Line 3).
Revised: We present Algorithm 2 that contains two phases: a local search phase (Line 2) and a ˜ f -maximizationphase (Line 3).

Original: We first run a non-oblivious local search procedure (Algorithm 1) under the ( r − 1) -cardinality constraint (Line 2).
Revised: We first run a non-oblivious local search procedure (Algorithm 1) under the ( r − 1) -cardinality constraint (Line 2).

Original: We setα = ∆ = ε 4 r ln r and δ =1 ( I +1)
Revised: We set δ = ( I +1)

Original: ( r − 1) n 2 by Theorem 3.3 whenapplying Algorithm 1 to obtain a (cid:0) 1 − 1 e −O ( ε) (cid:1) -approximation solution S L for h -maximization.
Revised: ( r − 1) n 2 by Corollary 3.4 when applying Algorithm 1to obtain a (cid:0) 1 − 1 e − ε (cid:1) -approximation solution S L for h -maximization.

Original: At the ˜ f -maximization phase (Line 3), we obtain S M by selecting an additional element e ∈ N \ S Lthat maximizes ˜ f
Revised: At the ˜ f -maximization phase (Line 3), we obtain S M by selecting an additional element e ∈ N \ S Lthat maximizes ˜ f

Original: ( S L + e ) .
Revised: ( S L + e ) .

Original: This guarantees that f ( S M ) ≥ (1 − ε ) h ( S L ) with high probability(Lemma 4.5), which results in the approximate ratio 1 − 1 e − O ( ε ) in Theorem 4.1.
Revised: This guarantees that f ( S M ) ≥ (1 − ε ) h ( S L ) with high probability(Lemma 4.5), which results in the approximation ratio 1 − 1 e − O ( ε ) in Theorem 4.1.

Original: Input : a value oracle to ˜ f , budget r ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:16) n 13 (cid:17) and ε ∈ (0 , 1 / 2) 1 Let (cid:99) φ h be a (cid:16) α =
Revised: Input : a value oracle to ˜ f , budget r ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:16) n 13 (cid:17) and ε ∈ (0 , 1 / 2) 1 Let (cid:99) φ h be a (cid:16) α =

Original: ε 4 r ln r , δ = 

Revised: ε 4 r ln r , δ =

Original: ( I +1)
Revised: ( I +1)

Original: ( r − 1) n 2 , I ( r − 1) (cid:17) -approximation of φ h as in Lemma 4.
Revised: ( r − 1) n 2 , I ( r − 1) (cid:17) -approximation of φ h as in Lemma 4.

Original: 2 S L ← NLS (cid:0)(cid:99)
Revised: 2 S L ← [Equation] (cid:0)(cid:99)

Original: [Equation]
Revised: [Equation]

Original: h , I (
Revised: , I (

Original: [Equation]
Revised: [Equation]

Original: ▷ Local search phase
Revised: ▷ Local search phase

Original: [Equation]
Revised: [Equation]

Original: e ) ▷ ˜ f -maximization phase 4 return S M
Revised: e [Equation] ▷ ˜ f -maximization phase 4 return S M

Original: 4.1.3 Proof sketchof Theorem 4.
Revised: 4.1.3 Proof sketch

Revised: of Theorem

Original: We first analyze the query complexity and then prove the approximation performance.
Revised: We first analyze the query complexity and then prove the approximation performance.

Original: Query complexity of Algorithm 2.
Revised: Query complexity of Algorithm 2.

Original: In Line 3, we make ( n − r +1) calls to the noisy oracle ˜ f in total.
Revised: In Line 3, we make ( n − r + 1) calls to the noisy oracle ˜ fin total.

Original: By Theorem 3.3, the total number of queries to (cid:99) φ h in Line 2 is at most ( I + 1)( r − 1) n .
Revised: By Corollary 3.4, the total number of queries to (cid:99) φ h in Line 2 is at most ( I + 1)( r − 1) nand I ≤ 5 r ln r ε .

Original: Combiningwith Lemma 4.4, the query complexity to ˜ f is upper bounded by ( n − r + 1)
Revised: Combining with Lemma 4.4, the query complexity to ˜ f is upper bounded by( n − r +1)+ M

Original: + M ( I + 1)( r − 1) n = O (cid:16) r 2 log 2 r · n 32 ε − 1 max { r, log n } (cid:17) .
Revised: ( I +1) ( r − 1) n = O (cid:16) r 2 log 2 r · n 32 ε − 1 max { r, log n } (cid:17) .

Original: This matches Theorem 4.1. Approximate ratio and success probability of Algorithm 2.
Revised: This matches Theorem 4.1.approximation ratio and success probability of Algorithm 2.

Original: Let O h ∈ arg max S ∈I ( r − 1) h ( S ) represent the ( r − 1) -set whose value of h is the largest.
Revised: Let O h ∈ arg max S ∈I ( r − 1) h ( S )represent the ( r − 1) -set whose value of h is the largest.

Original: Following from Theorem 3.3, we have h ( S L ) ≥263 (cid:0) [-
-] −  e − ε (cid:1) h
Revised: Following from Theorem 3.3, wehave h ( S L ) ≥ (cid:0) [-
-]1 − 1 e − ε (cid:1) h

Original: ( O h ) with probability at least 1 − 1 n .
Revised: ( O h ) with probability at least 1 − 1 n .

Original: We denote by O f ∈ arg max S ∈I ( r ) f ( S ) theoptimal solution to f .
Revised: We denote by O f ∈arg max S ∈I ( r ) f ( S ) the optimal solution to f .

Original: By the submodularity of f , O f has a subset ˜ O f with r − 1 elements such thatf ( ˜ O f ) ≥ (cid:0) 1 − 1 r (cid:1) f
Revised: By the submodularity of f , O f has a subset ˜ O fwith r − 1 elements such that f ( ˜ O f ) ≥ (cid:0) 1 − 1 r (cid:1) f

Original: ( O f ) .
Revised: ( O f ) .

Original: Then we have [Equation] S 
Revised: Then we haveh ( S L ) ≥

Original: [Equation]
Revised: [Equation]

Original: (1) where the first inequality follows from h ( S L ) ≥ (cid:0) 1 − 1 e − ε (cid:1)
Revised: (1) where the first inequality follows from h ( S L ) ≥ (cid:0) 1 − 1 e − ε (cid:1)

Original: h
Revised: h

Original: ( O h ) and the second from monotonicity of f .
Revised: ( O h ) and the second from monotonicity of f .

Original: Since r is assumed to be Ω (cid:0) 

Revised: Since r is assumed to be Ω (cid:0)

Original: ε (cid:1) , h ( S L ) is a (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation of f ( O f ) .
Revised: ε (cid:1) , h ( S L ) is a (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation of f ( O f ) .

Original: Recall that h ( S L ) is the expectation of f ( S L + e ) over a random element e ∈ N .
Revised: Recall that h ( S L ) is the expectation of f ( S L + e ) over a random element e ∈ N .

Original: Ineq.
Revised: Ineq.

Original: (1) alreadyproves a claim that uniformly randomly adding an element e ∈ N to S L achieves an approximateratio of 1 − 1 e − O ( ε ) in expectation for maximizing f , i.e., E [ f ( S L + e )] ≥ (cid:0) 1 − 1 e − O ( ε ) (cid:1) f
Revised: (1) alreadyproves a claim that uniformly randomly adding an element e ∈ N to S L achieves an approximationratio of 1 − 1 e − O ( ε ) in expectation for maximizing f , i.e., E [ f ( S L + e )] ≥ (cid:0) 1 − 1 e − O ( ε ) (cid:1) f

Original: ( O f ) .
Revised: ( O f ) .

Original: Moreover, we can convert this claim to a with-high-probability claim by the following lemma.
Revised: Moreover, we can convert this claim to a with-high-probability claim by the following lemma.

Original: Lemma 4.5 ( Approximation at the ˜ f -maximization phase ) .
Revised: Lemma 4.5 ( Approximation at the ˜ f -maximization phase ) .

Original: With probability 1 − O (cid:16) 1 log n (cid:17) , wehave f ( S M ) ≥ (1 − ε ) h ( S L ) .
Revised: With probability 1 − O (cid:16) 1 log n (cid:17) , wehave f ( S M ) ≥ (1 − ε ) h ( S L ) .

Original: This lemma indicates that a bad element e ∈ N \ S L with f ( S L + e ) <
Revised: This lemma indicates that a bad element e ∈ N \ S L with f ( S L + e ) <

Original: (1 − ε ) h ( S L ) is unlikelyto be chosen at the ˜ f -maximization phase.
Revised: (1 − ε ) h ( S L ) is unlikelyto be chosen at the ˜ f -maximization phase.

Original: This is because the local search procedure guaranteesthat the number of good elements with f ( S L + e ) ≥ (1 − ε ) h ( S L ) is almost of the same order asthe bad ones.
Revised: This is because the local search procedure guaranteesthat the number of good elements with f ( S L + e ) ≥ (1 − ε ) h ( S L ) is almost of the same order asthe bad ones.

Original: Consequently, we can show that the selected element is likely to be good due to thegeneralized exponential tail noise, using similar proof idea as that of [18, Lemma 3.4].
Revised: Consequently, we can show that the selected element is likely to be good due to thegeneralized exponential tail noise, using similar proof idea as that of [18, Lemma 3.4].

Original: By Ineq.
Revised: By Ineq.

Original: (1)and Lemma 4.5, we complete the proof.
Revised: (1)and Lemma 4.5, we complete the proof.

Original: 4.2 Algorithmic results for cardinality constraints when r ∈ Ω (cid:16) n 13 (cid:17)
Revised: 4.2 Algorithmic results for cardinality constraints when r ∈ Ω (cid:16) n 13 (cid:17)

Original: We present in this subsection an algorithm (Algorithm 3) and its analysis (Theorem 4.6) that can beapplied to r ∈ Ω (cid:16) n 13 (cid:17) when n is sufficiently large.
Revised: We present in this subsection an algorithm (Algorithm 3) and its analysis (Theorem 4.6) that can beapplied to r ∈ Ω (cid:16) n 13 (cid:17) when n is sufficiently large.

Original: All missing proofs can be found in Section D.
Revised: All missing proofs can be found in Section E.

Original: Theorem 4.6 ( Algorithmic results for cardinality constraints when r ∈ Ω (cid:16) n 1 3 (cid:17) ) .
Revised: Theorem 4.6 ( Algorithmic results for cardinality constraints when r ∈ Ω (cid:16) n 1 3 (cid:17) ) .

Original: Let ε > 0 andassume n ∈ Ω (cid:0) 1 ε 4 (cid:1) is sufficiently large.
Revised: Let ε > 0 andassume n ∈ Ω (cid:0) 1 ε 4 (cid:1) is sufficiently large.

Original: For any r ∈ Ω (cid:16) n 13 (cid:17) , there exists an algorithm that returnsa (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation for Problem 2.6 under a r -cardinality constraint, with probability1 − O (cid:0) 1 n 2 (cid:1) and query complexity at most O ( n 6 ε − 1 ) to ˜ f .
Revised: For any r ∈ Ω (cid:16) n 13 (cid:17) , there exists an algorithm that returnsa (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation for Problem 2.6 under a r -cardinality constraint, with probability1 − O (cid:0) 1 n 2 (cid:1) and query complexity at most O ( n 6 ε − 1 ) to ˜ f .

Original: The above theorem addresses the remaining range of r in Theorem 4.1, and also achieves a tightapproximate ratio of 1 − 1 e − O ( ε ) with Poly ( n, 1 /ε ) queries to ˜ f .
Revised: The above theorem addresses the remaining range of r in Theorem 4.1, and also achieves a tightapproximation ratio of 1 − 1 e − O ( ε ) with Poly ( n, 1 /ε ) queries to ˜ f .

Original: 4.2.1 Useful notations and useful facts for  
Revised: 4.2.1 Useful notations and useful facts for Theorem 4.

Original: Theorem

Original: Our algorithm and analysis in this subsection are based on another smoothing surrogate function h .
Revised: Our algorithm and analysis in this subsection are based on another smoothing surrogate function h .

Original: Definition 4.7 ( Smoothing surrogate function II).
Revised: Definition 4.7 ( Smoothing surrogate function II).

Original: Given a subset H ⊆ N , for any set S ⊆ N \ H ,we define the smoothing surrogate function h H ( S ) as h H ( S ) =
Revised: Given a subset H ⊆ N , for any set S ⊆ N \ H ,we define the smoothing surrogate function h H ( S ) as h H ( S ) =

Original: (cid:80)
Revised: (cid:80)

Original: H j ⊆ H f
Revised: H j ⊆ H f

Original: ( S ∪ H j ) / 2 | H | .
Revised: ( S ∪ H j ) / 2 | H | .

Original: Intuitively, if | H | is sufficiently large, the surrogate function h is robust to noise as it is based on alarge averaging set with size 2 | H | .
Revised: Intuitively, if | H | is sufficiently large, the surrogate function h is robust to noise as it is based on alarge averaging set with size 2 | H | .

Original: Throughout the rest of this subsection, we consider the case that| H | ∈ Ω(log n ) .
Revised: Throughout the rest of this subsection, we consider the case that| H | ∈ Ω(log n ) .

Original: Similar to Section 4.1.1, we give some basic properties of h H .
Revised: Similar to Section 4.1.1, we give some basic properties of h H .

Original: Lemma 4.8 ( Properties of h H ) .
Revised: Lemma 4.8 ( Properties of h H ) .

Original: For any H ⊆ N , the smoothing surrogate function h H is monotoneand submodular, and for all S ⊆ N \ H , h H ( S ) ≥ 12 f ( S ∪ H )
Revised: For any H ⊆ N , the smoothing surrogate function h H is monotoneand submodular, and for all S ⊆ N \ H , h H ( S ) ≥ 12 f ( S ∪ H )

Original: + 12 f ( S ) .
Revised: + 12 f ( S ) .

Original: Given a cardinality r and a set H ⊆ N , we define a ( r − | H | ) -cardinality constraint confined on
Revised: Given a cardinality r and a set H ⊆ N , we define a ( r − | H | ) -cardinality constraint confined on

Original: N \ H as I H ( r ) =
Revised: N \ H as I H ( r ) =

Original: { S ⊆ N \ H : | S | = r − | H |} .
Revised: { S ⊆ N \ H : | S | = r − | H |} .

Original: Similar to Section 4.1.1, we provide the followinglemma indicating that the auxiliary function φ h H can be well approximated.
Revised: Similar to Section 4.1.1, we provide the followinglemma indicating that the auxiliary function φ h H can be well approximated.

Original: Lemma 4.9 ( Approximation of φ h H ).
Revised: Lemma 4.9 ( Approximation of φ h H ).

Original: Let ε > 0 and assume n ∈ Ω (cid:0) ε − 4 (cid:1) .
Revised: Let ε > 0 and assume n ∈ Ω (cid:0) ε − 4 (cid:1) .

Original: For a set H with | H | ≥3 ln n , there exists a value oracle O to a (cid:0) α
Revised: For a set H with | H | ≥3 ln n , there exists a value oracle O to a (cid:0) α

Original: = ε 4 r ln r , δ = 3 n 6 , I H ( r ) (cid:1) -approximation (cid:100)
Revised: = ε 4 r ln r , δ = 3 n 6 , I H ( r ) (cid:1) -approximation (cid:100)

Original: φ h H of φ h H ,which answering O
Revised: φ h H of φ h H ,which answering O

Original: ( A ) queries ˜ f ’s oracle O (cid:16)
Revised: ( A ) queries ˜ f ’s oracle O (cid:16)

Original: rε − 1 log 52 n log 2 r (cid:17) times for each set A ∈ I H ( r ) .
Revised: rε − 1 log 52 n log 2 r (cid:17) times for each set A ∈ I H ( r ) .

Original: The above lemma indicates that for sufficiently large n and | H | , we can achieve an almost accurateapproximation (cid:100)
Revised: The above lemma indicates that for sufficiently large n and | H | , we can achieve an almost accurateapproximation (cid:100)

Original: φ h H of φ h H , which queries ˜ f only ˜ O ( nε − 1 ) times for each (cid:100) φ h H ( A ) .
Revised: φ h H of φ h H , which queries ˜ f only ˜ O ( nε − 1 ) times for each (cid:100) φ h H ( A ) .

Original: By Theorem 3.3,we setα = ∆ = ε 4 r ln r and δ = 3 n 6 when applying Algorithm 1.
Revised: By Corollay 3.4,we set δ = 3 n 6 when applying Algorithm 1.

Original: 4.2.2 Algorithm forTheorem 4.6
Revised: 4.2.2 Algorithm for

Revised: Theorem

Original: We present Algorithm 3 that mainly contains a local search phase (Line 3).
Revised: We present Algorithm 3 that mainly contains a local search phase (Line 3).

Original: The main differencefrom Algorithm 2 is that Algorithm 3 arbitrarily select a redundant set H (Line 1), and use this set toconstruct a smoothing surrogate function h H for local search (Line 3).
Revised: The main differencefrom Algorithm 2 is that Algorithm 3 arbitrarily select a redundant set H (Line 1), and use this set toconstruct a smoothing surrogate function h H for local search (Line 3).

Original: Then Algorithm 3 returns
Revised: Then Algorithm 3 returns

Original: S L ∪ H directly (Line 4) without a ˜ f -maximization phase.
Revised: S L ∪ H directly (Line 4) without a ˜ f -maximization phase.

Original: Algorithm 3:

Original: Input : a value oracle to ˜ f , budget r ∈ Ω (cid:16) n 13 (cid:17) , and ε ∈ (0 , 1 / 2) 

Revised: Input : a value oracle to ˜ f , budget r ∈ Ω (cid:16) n 13 (cid:17) , and ε ∈ (0 , 1 / 2)

Original: Arbitrarily select a subset H ⊆ N with | H | = ⌈ 3 ln n ⌉ 2 Let (cid:100) φ h H be a (cid:0) α
Revised: Arbitrarily select a subset H ⊆ N with | H | = ⌈ 3 ln n ⌉ 2 Let (cid:100) φ h H be a (cid:0) α

Original: = ε 4 r ln r , δ = 3 n 6 , I H ( r ) (cid:1) -approximation of φ h H as in Lemma 4.
Revised: = ε 4 r ln r , δ = 3 n 6 , I H ( r ) (cid:1) -approximation of φ h H as in Lemma 4.

Original: 3 S L ← NLS (cid:0) (cid:100) φ h H , I H ( r ) , ∆ = ε 4 r ln r (cid:1)
Revised: 3 S L ← NLS (cid:0) (cid:100) φ h H , I H ( r ) , ∆ = ε 4 r ln r (cid:1)

Original: ▷ Local search phase 4 return S L ∪ H
Revised: ▷ Local search phase 4 return S L ∪ H

Original: 4.2.3 Proofsketch of Theorem 4.
Revised: 2.3 Proof

Original: {+sketch of Theorem 4.
+}
Revised: {+sketch of Theorem 4.
+}

Original: Again, we first analyze the query complexity and then prove the approximation performance.
Revised: Again, we first analyze the query complexity and then prove the approximation performance.

Original: Query complexity of Algorithm 3.
Revised: Query complexity of Algorithm 3.

Original: Since∆ = α = ε 4 r ln r and n ∈ Ω (cid:0) 1 ε 4 (cid:1) , we have I ∈ O ( n 2 )in Theorem 3.3.
Revised: Since n ∈ Ω (cid:0) 1 ε 4 (cid:1) , we have I ∈ O ( n 2 ) in Corollary 3.4. Hence

Original: Hence Line 4 calls the oracle of (cid:100) φ h H at most O ( rn 3 ) times by Theorem 3.3, andeach call to (cid:100) φ h H queries ˜ f at most O (cid:16) rε − 1 log 52 n log 2 r (cid:17) times by Lemma 4.9.
Revised: Line 4 calls the oracle of (cid:100) φ h H at most O ( rn 3 ) times by Corollary 3.4, and each call to (cid:100) φ h H queries˜ f at most O (cid:16) rε − 1 log 52 n log 2 r (cid:17) times by Lemma 4.9.

Original: Thus we query ˜ fat most O ( n 6 ε − 1 ) times. Approximate ratio and success probability of Algorithm 3.
Revised: Thus we query ˜ f at most O ( n 6 ε − 1 ) times.approximation ratio and success probability of Algorithm 3.

Original: By Theorem 3.3, we have h H ( S L ) ≥ 319 (cid:0) 1 − 1 e − ε (cid:1) · max S ⊆I H ( r ) h H ( S ) with probability 1 − O (cid:0) 1 n 2 (cid:1) .
Revised: By Corollary 3.4, we have h H ( S L ) ≥ 314 (cid:0) 1 − 1 e − ε (cid:1) · max S ⊆I H ( r ) h H ( S ) with probability 1 − O (cid:0) 1 n 2 (cid:1) .

Original: Furthermore, f ( S L ∪ H ) ≥ h H ( S L )follows from the monotonicity of f .
Revised: Furthermore, f ( S L ∪ H ) ≥ h H ( S L )follows from the monotonicity of f .

Original: Then we can demonstrate the approximation performance of
Revised: Then we can demonstrate the approximation performance of

Original: Algorithm 3 by the following lemma.
Revised: Algorithm 3 by the following lemma.

Original: Lemma 4.10.
Revised: Lemma 4.10.

Original: Let O ⋆ = arg max S ∈I ( r ) f
Revised: Let O ⋆ = arg max S ∈I ( r ) f

Original: ( S ) .
Revised: ( S ) .

Original: We have max S ∈I H ( r ) h H ( S ) ≥ (cid:16) 1 − | H | r (cid:17)
Revised: We have max S ∈I H ( r ) h H ( S ) ≥ (cid:16) 1 − | H | r (cid:17)

Original: f ( O ⋆ ) .
Revised: f ( O ⋆ ) .

Original: Roughly speaking, there exists a subset A ⊆ O ⋆ of size | H | such that f ( O ⋆ \ A ) ≥ (cid:16) 1 − | H | r (cid:17)
Revised: Roughly speaking, there exists a subset A ⊆ O ⋆ of size | H | such that f ( O ⋆ \ A ) ≥ (cid:16) 1 − | H | r (cid:17)

Original: f
Revised: f

Original: ( O ⋆ )by submodularity of f .
Revised: ( O ⋆ )by submodularity of f .

Original: The claim follows from max S L ∈I H ( r ) h H ( S ) ≥ h
Revised: The claim follows from max S L ∈I H ( r ) h H ( S ) ≥ h

Original: ( O ⋆ \ A ) ≥ f
Revised: ( O ⋆ \ A ) ≥ f

Original: ( O ⋆ \ A ) .
Revised: ( O ⋆ \ A ) .

Original: Combining the fact that f ( S L ∪ H ) ≥ (cid:0) 1 − 1 e − ε (cid:1) max S ⊆I H ( r ) h H ( S ) with Lemma 4.10, wecomplete the proof of Theorem 4.6.
Revised: Combining the fact that f ( S L ∪ H ) ≥ (cid:0) 1 − 1 e − ε (cid:1) max S ⊆I H ( r ) h H ( S ) with Lemma 4.10, wecomplete the proof of Theorem 4.6.

Original:  Our algorithms and main theorems for general matroid constraints
Revised:  Our algorithms and main theorems for general matroid constraints

Original: In this section, we consider Problem 2.6 under matroid constraints.
Revised: In this section, we consider Problem 2.6 under matroid constraints.

Original: Similarly, for different ranges ofmatroid ranks, we prove there are (1 − 1 e − O ( ε )) / 2 -approximate algorithms (Theorems 5.1 and 5.2).
Revised: Similarly, for different ranges ofmatroid ranks, we prove there are (1 − 1 e − O ( ε )) / 2 -approximate algorithms (Theorems 5.1 and 5.2).

Original: W.l.o.g.
Revised: W.l.o.g.

Original: , we assume that all single elements are feasible for the given matroid constraints.
Revised: , we assume that all single elements are feasible for the given matroid constraints.

Original: Theorem 5.1 ( Algorithmic results for matroid constraints with rank r ∈ Ω (cid:0) 1 ε (cid:1) ∩ O (cid:16) n 13 (cid:17) ) .
Revised: Theorem 5.1 ( Algorithmic results for matroid constraints with rank r ∈ Ω (cid:0) 1 ε log (cid:0) 1 ε (cid:1)(cid:1) ∩ O (cid:16) n 13 (cid:17) ) .

Original: Let ε > 0 is sufficiently small and assume n ∈ Ω (cid:16) exp (cid:16)(cid:0) 

Revised: Let ε > 0 is sufficiently small and assume n ∈ Ω (cid:16) exp (cid:16)(cid:0)

Original: ε (cid:1) O (1) (cid:17)(cid:17) is sufficiently large.
Revised: ε (cid:1) O (1) (cid:17)(cid:17) is sufficiently large.

Original: For anyr ∈ Ω(  1  ε ) ∩ O (cid:16) n 13 (cid:17) , there exists an algorithm that returns a (cid:0)(cid:0) 1 − 1 e (cid:1) / 2 − O ( ε ) (cid:1) -approximationfor
Revised: For anyr ∈ Ω( ε − 1 log( ε − 1 )) ∩ O (cid:16) n 13 (cid:17) , there exists an algorithm that returns a (cid:0)(cid:0) 1 − 1 e (cid:1) / 2 − O ( ε ) (cid:1) -approximation for Problem 2.6 under a matroid constraint

Original: Problem 2.6 under a matroid constraint with rank r , with probability at least 1 − O (cid:0) ε 4 (cid:1) andquery complexity at most O (cid:16) r 2 log 2 r · n 32 ε − 1 max { r, log n } (cid:17) to ˜ f .
Revised: I ( M ) with rank r , with probability atleast 1 − O (cid:0) ε 4 (cid:1) and query complexity at most O (cid:16) r 2 log 2 r · n 32 ε − 1 max { r, log n } (cid:17) to ˜ f .

Original: The proof can be found in Section E.
Revised: The proof can be found in Section F.

Original: The main difference from Theorem 4.1 is that the approximateratio is (cid:0) 1 − 1 e (cid:1) / 2 − O ( ε ) instead of 1 − 1 e − O ( ε ) .
Revised: The main difference from Theorem 4.1 is that the approximationratio is (cid:0) 1 − 1 e (cid:1) / 2 − O ( ε ) instead of 1 − 1 e − O ( ε ) .

Original: The reason is that to maintain the feasibilityof the output, we may not be able to add an element to S L as in Line 3 of Algorithm 2.
Revised: The reason is that to maintain the feasibilityof the output, we may not be able to add an element to S L as in Line 3 of Algorithm 2.

Original: To addressthis issue, we design a comparison procedure (Lines 4-8 of Algorithm 7) to evaluate the value off ( S L ) and f ( S M \ S L ) and output the larger one, which results in a loss on the approximate ratio.
Revised: To addressthis issue, we design a comparison procedure (Lines 4-8 of Algorithm 7) to evaluate the value off ( S L ) and f ( S M \ S L ) and output the larger one, which results in a loss on the approximation ratio.

Original: Moreover, the failure probability of the comparison procedure is upper bounded by O ( ε 4 ) .
Revised: Moreover, the failure probability of the comparison procedure is upper bounded by O ( ε 4 ) .

Original: Theorem 5.2 ( Algorithmic results for matroid constraints with rank r ∈ Ω (cid:16) n 13 (cid:17) ) .
Revised: Theorem 5.2 ( Algorithmic results for matroid constraints with rank r ∈ Ω (cid:16) n 13 (cid:17) ) .

Original: Let ε > 0and assume n ∈ Ω (cid:0) 1 ε 4 (cid:1) is sufficiently large.
Revised: Let ε > 0and assume n ∈ Ω (cid:0) 1 ε 4 (cid:1) is sufficiently large.

Original: For r ∈ Ω (cid:16) n 13 (cid:17) , there exists an algorithm returninga (cid:0)(cid:0) 1 − 1 e (cid:1) / 2 − O ( ε ) (cid:1) -approximation for Problem 2.6 under a matroid constraint  withrank r ,with probability 1 − O (cid:0) 1 n 2 (cid:1) and query complexity O ( n 6 ε − 1 ) to ˜ f .
Revised: For r ∈ Ω (cid:16) n 13 (cid:17) , there exists an algorithm returning a(cid:0)(cid:0) 1 − 1 e (cid:1) / 2 − O ( ε ) (cid:1) -approximation for Problem 2.6 under a matroid constraint I ( M ) with rankr , with probability 1 − O (cid:0) 1 n 2 (cid:1) and query complexity O ( n 6 ε − 1 ) to ˜ f .

Original: Specifically, for a stronglybase-orderable matroid (Definition G.1), the approximate ratio can be improved to 1 − 1 e − O ( ε ) .
Revised: Specifically, for a stronglybase-orderable matroid (Definition H.1), the approximation ratio can be improved to 1 − 1 e − O ( ε ) .

Original: The proof for general matroid constraints can be found in Section F.
Revised: The proof for general matroid constraints can be found in Section G.

Original: The main difference from
Revised: The main difference from

Original: Theorem 4.6 is again the approximate ratio.
Revised: Theorem 4.6 is again the approximation ratio.

Original: For maintaing the feasibility, we divide an arbitrarybase into two equal-size subsets, apply Algorithm 3 to both subsets, and output the larger one.
Revised: For maintaing the feasibility, we divide an arbitrarybase into two equal-size subsets, apply Algorithm 3 to both subsets, and output the larger one.

Original: Consequently, we achieve a (cid:0)(cid:0) 1 − 1 e (cid:1) / 2 − O ( ε ) (cid:1) -approximate ratio.
Revised: Consequently, we achieve a (cid:0)(cid:0) 1 − 1 e (cid:1) / 2 − O ( ε ) (cid:1) -approximation ratio.

Original: Specifically, for a stronglybase-orderable matroid, we can use its good exchangable property to maintain the feasibility withoutsacrificing the approximate ratio; see Section G for a complete proof.
Revised: Specifically, for a stronglybase-orderable matroid, we can use its good exchangable property to maintain the feasibility withoutsacrificing the approximation ratio; see Section H for a complete proof.

Original:  Conclusions
Revised:  Conclusions

Original: In this work, we study the problem of constrained monotone submodular maximization with noisyoracles.
Revised: In this work, we study the problem of constrained monotone submodular maximization with noisyoracles.

Original: We design a unified local search framework that allows for inaccuracy in the objectivefunction.
Revised: We design a unified local search framework that allows for inaccuracy in the objectivefunction.

Original: Under this framework, we construct several smoothing surrogate functions to average thenoise out.
Revised: Under this framework, we construct several smoothing surrogate functions to average thenoise out.

Original: For cardinality constraints, the local search framework results in algorithms that achieve a 358 (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation with Poly (cid:0) n, 1 ε (cid:1) query complexity.
Revised: For cardinality constraints, the local search framework results in algorithms that achieve a 353 (cid:0) 1 − 1 e − O ( ε ) (cid:1) -approximation with Poly (cid:0) n, 1 ε (cid:1) query complexity.

Original: Moreover, for general matroidconstraints, the framework obtains an approximate ratio arbitrarily close to (cid:0) 1 − 1 e (cid:1) / 2 , which is thefirst constant approximation result to our knowledge.
Revised: Moreover, for general matroidconstraints, the framework obtains an approximation ratio arbitrarily close to (cid:0) 1 − 1 e (cid:1) / 2 , which isthe first constant approximation result to our knowledge.

Original: There are many directions in which this work could be extended.
Revised: There are many directions in which this work could be extended.

Original: For submodular maximizationwith noisy oracles under general matroid constraints, there is a gap between the approximate ratioof (1 − 1 /e ) /
Revised: For submodular maximizationwith noisy oracles under general matroid constraints, there is a gap between the approximationratio of (1 − 1 /e ) /

Original: 2 − O ( ε ) provided in this paper and impossibility results [4, 18].
Revised: 2 − O ( ε ) provided in this paper and impossibility results [4, 18].

Original: The first openquestion is to close this gap.
Revised: The first openquestion is to close this gap.

Original: In addition, it would be interesting to consider more complex constraintsunder noise including knapsack constraints [3].
Revised: In addition, it would be interesting to consider more complex constraintsunder noise including knapsack constraints [3].

Original: Another limitation of this work is that we onlyconsider independent noise.
Revised: Another limitation of this work is that we onlyconsider independent noise.

Original: It is meaningful to investigate submodular maximization with correlatednoise.
Revised: It is meaningful to investigate submodular maximization with correlatednoise.

Original: Moreover, it is also worthwhile to investigate the robustness of other submodular optimizationapproaches, e.g., multi-linear extension [23].
Revised: Moreover, it is also worthwhile to investigate the robustness of other submodular optimizationapproaches, e.g., multi-linear extension [23].

Original: Checklist
Revised: Checklist

Original: For all authors...
Revised: For all authors...

Original: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
Revised: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s

Original: contributions and scope?
Revised: contributions and scope?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you describe the limitations of your work?
Revised: (b) Did you describe the limitations of your work?

Original: [Yes] See Section 6.
Revised: [Yes] See Section 6.

Original: (c) Did you discuss any potential negative societal impacts of your work?
Revised: (c) Did you discuss any potential negative societal impacts of your work?

Original: [N/A]
Revised: [N/A](d)

Original: (d) Have you read the ethics review guidelines and ensured that your paper conforms tothem?
Revised: Have you read the ethics review guidelines and ensured that your paper conforms tothem?

Original: [Yes]
Revised: [Yes]

Original: If you are including theoretical results...
Revised: If you are including theoretical results...

Original: (a) Did you state the full set of assumptions of all theoretical results?
Revised: (a) Did you state the full set of assumptions of all theoretical results?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you include complete proofs of all theoretical results?
Revised: (b) Did you include complete proofs of all theoretical results?

Original: [Yes] All missing proofs can be found in the appendix.
Revised: [Yes] All missing proofscan be found in the appendix.3.

Original: If you ran experiments...
Revised: If you ran experiments...

Original: (a) Did you include the code, data, and instructions needed to reproduce the main experi-
Revised: (a) Did you include the code, data, and instructions needed to reproduce the main experi-

Original: mental results (either in the supplemental material or as a URL)?
Revised: mental results (either in the supplemental material or as a URL)?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you specify all the training details (e
Revised: (b) Did you specify all the training details (e

Original: g., data splits, hyperparameters, how theywere chosen)?
Revised: g., data splits, hyperparameters, how they were chosen)?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you report error bars (e
Revised: (c) Did you report error bars (e

Original: g., with respect to the random seed after running experiments multiple times)?
Revised: g., with respect to the random seed after running experiments multiple times)?

Original: [N/A]
Revised: [N/A]

Original: (d) Did you include the total amount of compute and the type of resources used (e g., typeof GPUs, internal cluster, or cloud provider)?
Revised: (d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?

Original: [N/A]
Revised: [N/A]

Original: If you are using existing assets (e
Revised: If you are using existing assets (e

Original: g., code, data, models) or curating/releasing new assets...
Revised: g., code, data, models) or curating/releasing new assets...

Original: (a) If your work uses existing assets, did you cite the creators?
Revised: (a) If your work uses existing assets, did you cite the creators?

Original: [Yes]
Revised: [Yes]

Original: (b) Did you mention the license of the assets?
Revised: (b) Did you mention the license of the assets?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include any new assets either in the supplemental material or as a URL?
Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [No]
Revised: [No]

Original: (d) Did you discuss whether and how consent was obtained from people whose data you’re
Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Original: using/curating?
Revised: using/curating?

Original: [N/A]
Revised: [N/A]

Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable
Revised: (e) Did you discuss whether the data you are using/curating contains personally identifiable

Original: information or offensive content?
Revised: information or offensive content?

Original: [N/A]
Revised: [N/A]

Original: If you used crowdsourcing or conducted research with human subjects...
Revised: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of instructions given to participants and screenshots, if
Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Original: applicable?
Revised: applicable?

Original: [N/A]
Revised: [N/A]

Original: (b) Did you describe any potential participant risks, with links to Institutional Review
Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?
Revised: Board (IRB) approvals, if applicable?

Original: [N/A]
Revised: [N/A]

Original: (c) Did you include the estimated hourly wage paid to participants and the total amount
Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Original: spent on participant compensation?
Revised: spent on participant compensation?

Original: [N/A]
Revised: [N/A]