arXiv subjects

Moran Feldman

Publications and source records attributed to Moran Feldman.

At least 37 records · Page 2Linked to original sources

arXiv2022

Streaming Submodular Maximization under Matroid Constraints

Recent progress in (semi-)streaming algorithms for monotone submodular function maximization has led to tight results for a simple cardinality constraint. However, current techniques fail to give a similar understanding for natural generalizations, including matroid constraints. This paper aims at closing this gap. For a single matroid of rank $k$ (i.e., any solution has cardinality at most $k$), our main results are: 1) a single-pass streaming algorithm that uses $\widetilde{O}(k)$ memory and achieves an approximation guarantee of $0.3178$, and 2) a multi-pass streaming algorithm that uses $\widetilde{O}(k)$ memory and achieves an approximation guarantee of $(1-1/e - \varepsilon)$ by taking a constant (depending on $\varepsilon$) number of passes over the stream. This improves on the previously best approximation guarantees of $1/4$ and $1/2$ for single-pass and multi-pass streaming algorithms, respectively. In fact, our multi-pass streaming algorithm is tight in that any algorithm with a better guarantee than $1/2$ must make several passes through the stream and any algorithm that beats our guarantee of $1-1/e$ must make linearly many passes (as well as an exponential number of value oracle queries). Moreover, we show how the approach we use for multi-pass streaming can be further strengthened if the elements of the stream arrive in uniformly random order, implying an improved result for $p$-matchoid constraints.

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arXiv2021

Maximum Matching sans Maximal Matching: A New Approach for Finding Maximum Matchings in the Data Stream Model

The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1/2-approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned maximal matching. One contribution of this paper continuous this line of work by presenting new three-pass semi-streaming algorithms that work along these lines and obtain improved approximation ratios of 0.6111 and 0.5694 for triangle-free and general graphs, respectively. Unfortunately, a recent work (Konrad and Naidu, 2021) shows that the strategy of constructing a maximal matching in the first pass and then improving it in further passes has limitations. Additionally, this technique is unlikely to get us closer to single-pass semi-streaming algorithms obtaining a better than 1/2-approximation. Therefore, it is interesting to come up with algorithms that do something else with their first pass (we term such algorithms non-maximal-matching-first algorithms). No such algorithms are currently known (to the best of our knowledge), and the main contribution of this paper is describing such algorithms that obtain approximation ratios of 0.5384 and 0.5555 in two and three passes, respectively, for general graphs (the result for three passes improves over the previous state-of-the-art, but is worse than the result of this paper mentioned in the previous paragraph for general graphs).

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arXiv2021

How Do You Want Your Greedy: Simultaneous or Repeated?

We present SimultaneousGreedys, a deterministic algorithm for constrained submodular maximization. At a high level, the algorithm maintains $\ell$ solutions and greedily updates them in a simultaneous fashion. SimultaneousGreedys achieves the tightest known approximation guarantees for both $k$-extendible systems and the more general $k$-systems, which are $(k+1)^2/k = k + \mathcal{O}(1)$ and $(1 + \sqrt{k+2})^2 = k + \mathcal{O}(\sqrt{k})$, respectively. This is in contrast to previous algorithms, which are designed to provide tight approximation guarantees in one setting, but not both. We also improve the analysis of RepeatedGreedy, showing that it achieves an approximation ratio of $k + \mathcal{O}(\sqrt{k})$ for $k$-systems when allowed to run for $\mathcal{O}(\sqrt{k})$ iterations, an improvement in both the runtime and approximation over previous analyses. We demonstrate that both algorithms may be modified to run in nearly linear time with an arbitrarily small loss in the approximation. Both SimultaneousGreedys and RepeatedGreedy are flexible enough to incorporate the intersection of $m$ additional knapsack constraints, while retaining similar approximation guarantees: both algorithms yield an approximation guarantee of roughly $k + 2m + \mathcal{O}(\sqrt{k+m})$ for $k$-systems and SimultaneousGreedys enjoys an improved approximation guarantee of $k+2m + \mathcal{O}(\sqrt{m})$ for $k$-extendible systems. To complement our algorithmic contributions, we provide a hardness result which states that no algorithm making polynomially many oracle queries can achieve an approximation better than $k + 1/2 + \varepsilon$. We also present SubmodularGreedy.jl, a Julia package which implements these algorithms and may be downloaded at https://github.com/crharshaw/SubmodularGreedy.jl . Finally, we test the effectiveness of these algorithms on real datasets.

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arXiv2021

Submodular + Concave

It has been well established that first order optimization methods can converge to the maximal objective value of concave functions and provide constant factor approximation guarantees for (non-convex/non-concave) continuous submodular functions. In this work, we initiate the study of the maximization of functions of the form $F(x) = G(x) +C(x)$ over a solvable convex body $P$, where $G$ is a smooth DR-submodular function and $C$ is a smooth concave function. This class of functions is a strict extension of both concave and continuous DR-submodular functions for which no theoretical guarantee is known. We provide a suite of Frank-Wolfe style algorithms, which, depending on the nature of the objective function (i.e., if $G$ and $C$ are monotone or not, and non-negative or not) and on the nature of the set $P$ (i.e., whether it is downward closed or not), provide $1-1/e$, $1/e$, or $1/2$ approximation guarantees. We then use our algorithms to get a framework to smoothly interpolate between choosing a diverse set of elements from a given ground set (corresponding to the mode of a determinantal point process) and choosing a clustered set of elements (corresponding to the maxima of a suitable concave function). Additionally, we apply our algorithms to various functions in the above class (DR-submodular + concave) in both constrained and unconstrained settings, and show that our algorithms consistently outperform natural baselines.

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arXiv2021

The Power of Subsampling in Submodular Maximization

We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set, and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual offline setting, we present SampleGreedy, which obtains a $(p + 2 + o(1))$-approximation for maximizing a submodular function subject to a $p$-extendible system using $O(n + nk/p)$ evaluation and feasibility queries, where $k$ is the size of the largest feasible set. The approximation ratio improves to $p+1$ and $p$ for monotone submodular and linear objectives, respectively. In the streaming setting, we present SampleStreaming, which obtains a $(4p +2 - o(1))$-approximation for maximizing a submodular function subject to a $p$-matchoid using $O(k)$ memory and $O(km/p)$ evaluation and feasibility queries per element, where $m$ is the number of matroids defining the $p$-matchoid. The approximation ratio improves to $4p$ for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks.

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arXiv2021

Practical Budgeted Submodular Maximization

We consider the problem of maximizing a non-negative monotone submodular function subject to a knapsack constraint, which is also known as the Budgeted Submodular Maximization (BSM) problem. Sviridenko (2004) showed that by guessing 3 appropriate elements of an optimal solution, and then executing a greedy algorithm, one can obtain the optimal approximation ratio of $α=1-1/e\approx 0.632$ for BSM. However, the need to guess (by enumeration) 3 elements makes the algorithm of Sviridenko impractical as it leads to a time complexity of $O(n^5)$ (which can be slightly improved using the thresholding technique of Badanidiyuru & Vondrak (2014), but only to roughly $O(n^4)$). Our main results in this paper show that fewer guesses suffice. Specifically, by making only 2 guesses, we get the same optimal approximation ratio of $α$ with an improved time complexity of roughly $O(n^3)$. Furthermore, by making only a single guess, we get an almost as good approximation ratio of $0.6174>0.9767α$ in roughly $O(n^2)$ time. Prior to our work, the only algorithms that were known to obtain an approximation ratio close to $α$ for BSM were the algorithm of Sviridenko and an algorithm of Ene & Nguyen (2019) that achieves $(α-ε)$-approximation. However, the algorithm of Ene & Nguyen requires ${(1/ε)}^{O(1/ε^4)}n\log^2 n$ time, and hence, is of theoretical interest only as ${(1/ε)}^{O(1/ε^4)}$ is huge even for moderate values of $ε$. In contrast, all the algorithms we analyze are simple and parallelizable, which makes them good candidates for practical use. Recently, Tang et al. (2020) studied a simple greedy algorithm that already has a long research history, and proved that its approximation ratio is at least 0.405. We improve over this result, and show that the approximation ratio of this algorithm is within the range [0.427, 0.462].

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arXiv2020

Optimal Streaming Algorithms for Submodular Maximization with Cardinality Constraints

We study the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model. Our main contribution is a single-pass (semi-)streaming algorithm that uses roughly $O(k / \varepsilon^2)$ memory, where $k$ is the size constraint. At the end of the stream, our algorithm post-processes its data structure using any offline algorithm for submodular maximization, and obtains a solution whose approximation guarantee is $\fracα{1+α}-\varepsilon$, where $α$ is the approximation of the offline algorithm. If we use an exact (exponential time) post-processing algorithm, this leads to $\frac{1}{2}-\varepsilon$ approximation (which is nearly optimal). If we post-process with the algorithm of Buchbinder and Feldman (Math of OR 2019), that achieves the state-of-the-art offline approximation guarantee of $α=0.385$, we obtain $0.2779$-approximation in polynomial time, improving over the previously best polynomial-time approximation of $0.1715$ due to Feldman et al. (NeurIPS 2018). It is also worth mentioning that our algorithm is combinatorial and deterministic, which is rare for an algorithm for non-monotone submodular maximization, and enjoys a fast update time of $O(\frac{\log k + \log (1/α)}{\varepsilon^2})$ per element.

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arXiv2020

Continuous Submodular Maximization: Beyond DR-Submodularity

In this paper, we propose the first continuous optimization algorithms that achieve a constant factor approximation guarantee for the problem of monotone continuous submodular maximization subject to a linear constraint. We first prove that a simple variant of the vanilla coordinate ascent, called Coordinate-Ascent+, achieves a $(\frac{e-1}{2e-1}-\varepsilon)$-approximation guarantee while performing $O(n/\varepsilon)$ iterations, where the computational complexity of each iteration is roughly $O(n/\sqrt{\varepsilon}+n\log n)$ (here, $n$ denotes the dimension of the optimization problem). We then propose Coordinate-Ascent++, that achieves the tight $(1-1/e-\varepsilon)$-approximation guarantee while performing the same number of iterations, but at a higher computational complexity of roughly $O(n^3/\varepsilon^{2.5} + n^3 \log n / \varepsilon^2)$ per iteration. However, the computation of each round of Coordinate-Ascent++ can be easily parallelized so that the computational cost per machine scales as $O(n/\sqrt{\varepsilon}+n\log n)$.

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arXiv2020

The One-way Communication Complexity of Submodular Maximization with Applications to Streaming and Robustness

We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multi-player model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the above-mentioned connections, have multiple implications in the data stream and robust setting. Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above $1/2$ can be achieved in our model, if only queries to feasible sets are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight $2/3$-approximation taking exponential time, and an efficient $0.514$-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the above-mentioned link to the robust setting, both of these algorithms improve on the current state-of-the-art for robust submodular maximization, showing that approximation factors beyond $1/2$ are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight $1/2+\varepsilon$ hardness result, based on the construction of a new family of coverage functions. This improves on a prior $1-1/e+\varepsilon$ hardness and matches, up to an arbitrarily small margin, the best known approximation algorithm.

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arXiv2020

Regularized Submodular Maximization at Scale

In this paper, we propose scalable methods for maximizing a regularized submodular function $f = g - \ell$ expressed as the difference between a monotone submodular function $g$ and a modular function $\ell$. Indeed, submodularity is inherently related to the notions of diversity, coverage, and representativeness. In particular, finding the mode of many popular probabilistic models of diversity, such as determinantal point processes, submodular probabilistic models, and strongly log-concave distributions, involves maximization of (regularized) submodular functions. Since a regularized function $f$ can potentially take on negative values, the classic theory of submodular maximization, which heavily relies on the non-negativity assumption of submodular functions, may not be applicable. To circumvent this challenge, we develop the first one-pass streaming algorithm for maximizing a regularized submodular function subject to a $k$-cardinality constraint. It returns a solution $S$ with the guarantee that $f(S)\geq(ϕ^{-2}-ε) \cdot g(OPT)-\ell (OPT)$, where $ϕ$ is the golden ratio. Furthermore, we develop the first distributed algorithm that returns a solution $S$ with the guarantee that $\mathbb{E}[f(S)] \geq (1-ε) [(1-e^{-1}) \cdot g(OPT)-\ell(OPT)]$ in $O(1/ ε)$ rounds of MapReduce computation, without keeping multiple copies of the entire dataset in each round (as it is usually done). We should highlight that our result, even for the unregularized case where the modular term $\ell$ is zero, improves the memory and communication complexity of the existing work by a factor of $O(1/ ε)$ while arguably provides a simpler distributed algorithm and a unifying analysis. We also empirically study the performance of our scalable methods on a set of real-life applications, including finding the mode of distributions, data summarization, and product recommendation.

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arXiv2020

Streaming Submodular Maximization under a $k$-Set System Constraint

In this paper, we propose a novel framework that converts streaming algorithms for monotone submodular maximization into streaming algorithms for non-monotone submodular maximization. This reduction readily leads to the currently tightest deterministic approximation ratio for submodular maximization subject to a $k$-matchoid constraint. Moreover, we propose the first streaming algorithm for monotone submodular maximization subject to $k$-extendible and $k$-set system constraints. Together with our proposed reduction, we obtain $O(k\log k)$ and $O(k^2\log k)$ approximation ratio for submodular maximization subject to the above constraints, respectively. We extensively evaluate the empirical performance of our algorithm against the existing work in a series of experiments including finding the maximum independent set in randomly generated graphs, maximizing linear functions over social networks, movie recommendation, Yelp location summarization, and Twitter data summarization.

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arXiv2019

Making a Sieve Random: Improved Semi-Streaming Algorithm for Submodular Maximization under a Cardinality Constraint

In this paper we consider the problem of maximizing a non-negative submodular function subject to a cardinality constraint in the data stream model. Previously, the best known algorithm for this problem was a $5.828$-approximation semi-streaming algorithm based on a local search technique (Feldman et al., 2018). For the special case of this problem in which the objective function is also monotone, the state-of-the-art semi-streaming algorithm is an algorithm known as Sieve-Streaming, which is based on a different technique (Badanidiyuru, 2014). Adapting the technique of Sieve-Streaming to non-monotone objective functions has turned out to be a challenging task, which has so far prevented an improvement over the local search based $5.828$-approximation. In this work, we overcome the above challenge, and manage to adapt Sieve-Streaming to non-monotone objective functions by introducing a "just right" amount of randomness into it. Consequently, we get a semi-streaming polynomial time $4.282$-approximation algorithm for non-monotone objectives. Moreover, if one allows our algorithm to run in super-polynomial time, then its approximation ratio can be further improved to $3 + \varepsilon$.

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arXiv2019

Adaptive Sequence Submodularity

In many machine learning applications, one needs to interactively select a sequence of items (e.g., recommending movies based on a user's feedback) or make sequential decisions in a certain order (e.g., guiding an agent through a series of states). Not only do sequences already pose a dauntingly large search space, but we must also take into account past observations, as well as the uncertainty of future outcomes. Without further structure, finding an optimal sequence is notoriously challenging, if not completely intractable. In this paper, we view the problem of adaptive and sequential decision making through the lens of submodularity and propose an adaptive greedy policy with strong theoretical guarantees. Additionally, to demonstrate the practical utility of our results, we run experiments on Amazon product recommendation and Wikipedia link prediction tasks.

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arXiv2019

Almost Optimal Semi-streaming Maximization for k-Extendible Systems

In this paper we consider the problem of finding a maximum weight set subject to a $k$-extendible constraint in the data stream model. The only non-trivial algorithm known for this problem to date---to the best of our knowledge---is a semi-streaming $k^2(1 + \varepsilon)$-approximation algorithm (Crouch and Stubbs, 2014), but semi-streaming $O(k)$-approximation algorithms are known for many restricted cases of this general problem. In this paper, we close most of this gap by presenting a semi-streaming $O(k \log k)$-approximation algorithm for the general problem, which is almost the best possible even in the offline setting (Feldman et al., 2017).

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arXiv2019

Submodular Maximization Beyond Non-negativity: Guarantees, Fast Algorithms, and Applications

It is generally believed that submodular functions -- and the more general class of $γ$-weakly submodular functions -- may only be optimized under the non-negativity assumption $f(S) \geq 0$. In this paper, we show that once the function is expressed as the difference $f = g - c$, where $g$ is monotone, non-negative, and $γ$-weakly submodular and $c$ is non-negative modular, then strong approximation guarantees may be obtained. We present an algorithm for maximizing $g - c$ under a $k$-cardinality constraint which produces a random feasible set $S$ such that $\mathbb{E} \left[ g(S) - c(S) \right] \geq (1 - e^{-γ} - ε) g(OPT) - c(OPT)$, whose running time is $O (\frac{n}ε \log^2 \frac{1}ε)$, i.e., independent of $k$. We extend these results to the unconstrained setting by describing an algorithm with the same approximation guarantees and faster $O(\frac{n}ε \log\frac{1}ε)$ runtime. The main techniques underlying our algorithms are two-fold: the use of a surrogate objective which varies the relative importance between $g$ and $c$ throughout the algorithm, and a geometric sweep over possible $γ$ values. Our algorithmic guarantees are complemented by a hardness result showing that no polynomial-time algorithm which accesses $g$ through a value oracle can do better. We empirically demonstrate the success of our algorithms by applying them to experimental design on the Boston Housing dataset and directed vertex cover on the Email EU dataset.

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arXiv2018

Online Submodular Maximization: Beating 1/2 Made Simple

The Submodular Welfare Maximization problem (SWM) captures an important subclass of combinatorial auctions and has been studied extensively from both computational and economic perspectives. In particular, it has been studied in a natural online setting in which items arrive one-by-one and should be allocated irrevocably upon arrival. In this setting, it is well known that the greedy algorithm achieves a competitive ratio of 1/2, and recently Kapralov et al. (2013) showed that this ratio is optimal for the problem. Surprisingly, despite this impossibility result, Korula et al. (2015) were able to show that the same algorithm is 0.5052-competitive when the items arrive in a uniformly random order, but unfortunately, their proof is very long and involved. In this work, we present an (arguably) much simpler analysis that provides a slightly better guarantee of 0.5096-competitiveness for the greedy algorithm in the random-arrival model. Moreover, this analysis applies also to a generalization of online SWM in which the sets defining a (simple) partition matroid arrive online in a uniformly random order, and we would like to maximize a monotone submodular function subject to this matroid. Furthermore, for this more general problem, we prove an upper bound of 0.576 on the competitive ratio of the greedy algorithm, ruling out the possibility that the competitiveness of this natural algorithm matches the optimal offline approximation ratio of 1-1/e.

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arXiv2018

Unconstrained Submodular Maximization with Constant Adaptive Complexity

In this paper, we consider the unconstrained submodular maximization problem. We propose the first algorithm for this problem that achieves a tight $(1/2-\varepsilon)$-approximation guarantee using $\tilde{O}(\varepsilon^{-1})$ adaptive rounds and a linear number of function evaluations. No previously known algorithm for this problem achieves an approximation ratio better than $1/3$ using less than $Ω(n)$ rounds of adaptivity, where $n$ is the size of the ground set. Moreover, our algorithm easily extends to the maximization of a non-negative continuous DR-submodular function subject to a box constraint and achieves a tight $(1/2-\varepsilon)$-approximation guarantee for this problem while keeping the same adaptive and query complexities.

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arXiv2018

Guess Free Maximization of Submodular and Linear Sums

We consider the problem of maximizing the sum of a monotone submodular function and a linear function subject to a general solvable polytope constraint. Recently, Sviridenko et al. (2017) described an algorithm for this problem whose approximation guarantee is optimal in some intuitive and formal senses. Unfortunately, this algorithm involves a guessing step which makes it less clean and significantly affects its time complexity. In this work we describe a clean alternative algorithm that uses a novel weighting technique in order to avoid the problematic guessing step while keeping the same approximation guarantee as the algorithm of Sviridenko et al.

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