Cardinality constrained submodular maximization for random streams

We consider the problem of maximizing submodular functions in single-pass streaming and secretaries-with-shortlists models , both with random arrival order . For cardinality constrained monotone functions , Agrawal , Shadravan , and Stein [ ASS19 ] gave a single-pass ( 1 − 1/e − ε ) -approximation algorithm using only linear memory , but their exponential dependence on ε makes it impractical even for ε = 0.1 . We simplify both the algorithm and the analysis , obtaining an exponential improvement in the ε-dependence ( in particular , O ( k/ε ) memory ) . Extending these techniques , we also give a simple ( 1/e − ε ) -approximation for non-monotone functions in O ( k/ε ) memory . For the monotone case , we also give a corresponding unconditional hardness barrier of 1 − 1/e + ε for single-pass algorithms in randomly ordered streams , even assuming unlimited computation . Finally , we show that the algorithms are simple to implement and work well on real world datasets . 1 Introduction . Over the past few decades , submodularity has become recognized as a useful property occurring in a wide variety of discrete optimization problems . Submodular functions model the property of diminishing returns , whereby the gain in a utility function decreases as the set of items considered increases . This property occurs naturally in machine learning , information retrieval , and influence maximization , to name a few ( see [ IKBA20 ] and the references within ) . In many settings , the data is not available in a random-access model ; either for external reasons ( customers arriving online ) or because of the massive amount of data . When the data becomes too big to store in memory , we look for streaming algorithms that pass ( once ) through the data , and efficiently decide for each element whether to discard it or keep it in a small buffer in memory . In this work , we consider algorithms that process elements arriving in a random order . Note that the classical greedy algorithm which iteratively adds the best element [ NWF78 ] can not be used here , and hence we must look for new algorithmic techniques . The motivation for considering random element arrival comes from the prevalence of submodularity in big-data applications , in which data is often logged in batches that can be modelled as random samples from an underlying 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . distribution.1 The problem of streaming submodular maximization has recently received significant attention both for random arrival order and the more pessimistic worst-case order arrival [ ASS19 , AEF+20 , AF19 , BMKK14 , BV14 , CGQ15 , FKK18 , FNSZ20 , HKMY20 , IV19 , KMZ+19 , MV19 , NTM+18 , HTW20 , Sha20 ] . 1.1 Submodular functions and the streaming model . Let f : 2E → R+ be a non-negative set function satisfying f ( S∪ { e } ) −f ( S ) ≥ f ( T ∪ { e } ) −f ( T ) for all S ⊆ T ⊆ E\ { e } . Such a function is called submodular . For simplicity , we assume f ( ∅ ) = 0.2 We use the shorthand f ( e|S ) : = f ( S ∪ { e } ) − f ( S ) to denote the marginal of e on top of S. When f ( S ) ≤ f ( T ) for all S ⊆ T , f is called monotone . We consider the following optimization problem : OPT : = max |S|≤k f ( S ) where k is a cardinality constraint for the solution size . Our focus on submodular maximization in the streaming setting . In this setting , an algorithm is given a single pass over a dataset in a streaming fashion , where the stream is a some permutation of the input dataset and each element is seen once . The stream is in random order when the permutation is uniformly random . When there are no constraints on the stream order , we call the stream adversarial . At each step of the stream , our algorithm is allowed to maintain a buffer of input elements . When a new element is streamed in , the algorithm can choose to add the element to its buffer . To be as general as possible , we assume an oracle model — that is , we assume there is an oracle that returns the value of f ( S ) for any set S. The decision of the algorithm to add an element is based only on queries to the oracle on subsets of the buffer elements . The algorithm may also choose to throw away buffered elements at any given time . The goal in the streaming model is to minimize memory use , and the complexity of the algorithm is the maximum number of input elements the algorithm stores in the buffer at any given time . For the oracle model , an important distinction is between weak oracle access or and strong oracle access . In the weak oracle setting , the algorithm is only allowed to query sets of feasible elements ( sets that have cardinality less than k ) . In the strong oracle setting however , the algorithm is allowed to query any set of elements . All our results apply to both the weak and strong oracle models . Our aim will be to develop algorithms that only make one pass over the data stream , using Õε ( k ) memory , where k is the maximum size of a solution set in I . We assume that k is small relative to n : = |E| , the size of the ground set . 1.2 Our contributions . On the algorithmic side , we give ( 1−1/e−ε ) and ( 1/e−ε ) approximation for cardinality constrained monotone and non-monotone submodular maximization respectively , both using O ( k/ε ) memory . The monotone result has an exponential improvement in memory requirements compared to Agrawal et al . [ ASS19 ] ( in terms of the dependence on ε ) , while the non-monotone result is the first to appear in the random-order streaming model , and improves upon the best known polynomialtime approximations under adversarial orders [ AEF+20 ] . The algorithms are extremely simple to implement , and perform well on real world data ( see Section 5 ) , even compared to offline greedy algorithms . On the hardness side , we prove that a ( 1 − 1/e + ε ) -approximation for monotone submodular maximization would require Ω ( n ) memory ( even with unlimited queries and computational power ) . This improves the hardness bound of 7/8 from [ ASS19 ] . 1.3 Related work . Prior work on this problem has focused on both the adversarial and random-order streaming setting . Algorithmic and hardness results further depend on whether the function f is monotone or non- 1Note that the random order assumption generalizes the typical assumption of i.i.d . sampling stream elements from a known distribution , as the random order assumption applies equally well to unknown distributions . 2Submodular functions f with f ( ∅ ) > 0 only improves the approximation ratio of our algorithms . monotone , and whether f has explicit structure ( e.g . such as by presenting a set system for f in the coverage case ) , or accessible only via oracle queries . Table 1 describes all the relevant results . Algorithmic results . Submodular maximization in the streaming setting was first considered by Badanidiyuru et al . [ BMKK14 ] who gave a ( 1/2− ) -approximation in O ( k log k/ ) memory for monotone submodular functions under a cardinality constraint , using a thresholding idea with parallel enumeration of possible thresholds . This work led to a number of subsequent developments with the current best being a ( 1/2− ) -approximation in O ( k/ ) memory [ KMZ+19 ] . It turns out that the factor of 1/2 is the best possible in the adversarial setting ( with a weak oracle ) , but an improvement is possible in the random order model ( the input is ordered by a uniformly random permutation ) . This was first shown by Norouzi-Fard et al . [ NTM+18 ] , who proved that the 1/2-hardness barrier for cardinality constraints can be broken , exhibiting a ( 1/2 + c ) -approximation in O ( k/ε log k ) memory where c < 10−13 . In a breakthrough work , Agrawal , Shadravan , and Stein [ ASS19 ] gave a ( 1 − 1/e − ε ) -approximation using k 2poly ( 1/ε ) memory and running time . We note that this is arbitrarily close to the optimal factor of 1 − 1/e , but the algorithm is not practical , due to its dependence on ε ( even for = 0.1 , the resulting constants are astronomical ) . Lower bounds . A few lower bounds are known for monotone functions in the adversarial order model : with a weak oracle , any ( 1/2 + ) -approximation would require Ω ( n/k ) memory [ NTM+18 ] . Under a strong oracle , a lower bound of Ω ( n/k3 ) memory for any ( 1/2+ ) -approximation algorithm was shown in a recent paper by Feldman et al . [ FNSZ20 ] . Another recent lower bound was proved by McGregor and Vu [ MV19 ] : a ( 1− 1/e+ ε ) -approximation for coverage functions requires Ω ( n/k2 ) memory ( this lower bound holds for explicitly given inputs , via communication complexity ; we note that this is incomparable to the computational ( 1− 1/e+ ε ) -hardness of maximum coverage [ Fei98 ] ) . For non-monotone functions , Alaluf et al . [ AEF+20 ] proved an Ω ( n ) memory lower bound for the adversarial order model with unbounded computation . In the random-order model , Agrawal et al . [ ASS19 ] show that beating 7/8 ( for monotone submodular functions ) requires Ω ( n ) memory . In contrast , we show that same construction as McGregor and Vu [ MV19 ] also applies to randomly ordered streams : ( 1− 1/e+ ε ) for coverage functions requires Ω ( n/k2 ) memory even in the random-order model . Submodular maximization in related models . A closely related model is the secretary with shortlists model [ ASS19 ] , where an algorithm is allowed to store a shortlist of more than k items ( where k is the cardinality constraint ) . Unlike the streaming model however , once an element goes into the shortlist , it can not be removed . Then , after seeing the entire stream , the algorithm chooses a subset of size k from the shortlist and returns that to the user . We note that the algorithms developed in this paper apply almost without modification to the shortlists model . 1.4 Overview of our techniques . Main algorithmic techniques . The primary impetus for our algorithmic work was an effort to avoid the extensive enumeration involved in the algorithm of Agrawal et al . [ ASS19 ] which leads to memory requirements exponential in poly ( 1/ε ) . To make things concrete , let us consider the input divided into disjoint windows of consecutive elements . The windows containing actual optimal elements play a special role — let ’ s call them active windows — these are the windows where we make quantifiable progress . When the stream is randomly ordered , we would ideally like to have each new element sampled independently and uniformly from the input . This leads to the intuition that the optimal elements are evenly spread out through all the windows . This can not be literally true , since conditioned on the history of the stream , some elements have already appeared and can not appear again . However , a key idea of Agrawal et al . [ ASS19 ] allows us to circumvent this by reinserting the elements that we have already seen and that played a role in the selection process . What needs to be proved is that elements that were not selected can still appear in the future , conditioned on the history of the algorithm ; that turns out to be true , provided that our algorithm operates in a certain greedy-like manner . To ensure progress was made regardless of the positioning of the optimal elements , previous work made use of exponentially large enumerations to essentially guess which windows the optimal elements arrive in . Where we depart from previous work is the way we build our solution . The idea is to use an evolving family of solutions which are updated in parallel , so that we obtain a quantifiable gain regardless of where the optimal elements arrived . Specifically , we grow k solutions in parallel , where solution Li has cardinality i . In each window , we attempt to extend a collection of solutions Li ( for varying i ) by a new element e ; if e is beneficial on average to every Li in the collection , we replace each Li+1 with the new solution Li ∪ { e } . Regardless of which windows happen to be active , we will show that the average gain over our evolving collection of solutions is analogous to the greedy algorithm . This is the basis of the analysis that leads to a factor of 1− 1/e . In addition to our candidate solutions Li , we maintain a pool of elements H that our algorithm has ever included in some candidate solution . We then use H to reintroduce elements artificially back into the input ; this makes it possible to assume that every input element still appears in a future window with the same probability , which is key to the probabilistic analysis leading to 1− 1/e . Non-monotone functions . Our algorithm for non-monotone submodular functions is similar , with the caveat that here we also have to be careful about not including any element in the solution with large probability . This is an important aspect of the randomized greedy algorithm for ( offline ) non-monotone submodular maximization [ BFNS14 ] which randomly includes in each step one of the top k elements in terms of marginal values . We achieve a similar property by choosing the top element from the current window and a random subset of the pool of prior elements H . Hardness results . Our hardness instances have the following general structure : there is a special subset of k good elements , and the remaining n − k elements are bad . The good elements are indistinguishable from each other , and ditto for the bad elements . In the monotone case , any k bad elements are a ( 1 − 1/e ) -factor worse than the optimal solution ( k good elements ) . Suppose furthermore that for parameter r > 0 , as long as we never query the function on a subset with ≥ r good elements , the good elements are indistinguishable from bad elements . The only way to collect ≥ r good elements in the memory buffer is by chance – until we ’ ve collected the required number of good elements , they are indistinguishable from bad elements , so the subset in the memory buffer is random . The classic work of Nemhauser and Wolsey [ NW78 ] constructs a pathological monotone submodular function with r = Ω ( k ) , which we use to prove that without memory Ω ( n ) the algorithm can not beat 1− 1/e . McGregor and Vu [ MV19 ] construct a simple example of a coverage function with r = 2 , which we use for our Ω ( n/k2 ) bound . For an exponential-size ground set , we extend their construction to r = 3 which translates to the improved lower bound of Ω ( n/k3/2 ) .

Summary: This paper studies the problem of maximizing a submodular function. For monotone submodular functions subject to a cardinality constraint, the paper gives a $(1-1/e-\epsilon)$-approximation. The space requirement is $O(k/\epsilon)$ where $k$ is the cardinality constraint. This is in contrast to a previous algorithm of Agrawal et al. which gave an algorithm with the same approximation ratio but with an exponential dependence on $\epsilon$. Hence, the new algorithm obtains an exponential improvement on the dependence on $\epsilon$. The new algorithm is also straightforward to implement. For the case of non-monotone submodular maximization subject to a cardinality constraint of size $k$, the authors gives an algorithm that achieves a $(1/e - \epsilon)$-approximation with space complexity $O(k / \epsilon)$.  The authors also prove a $(1-1/e+\epsilon)$ hardness result for monotone submodular maximization. This improves upon the previous best hardness result of $0.7$.  Finally, the authors show some experimental results and it is apparent that their algorithm performs quite well when compared to offline greedy.