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Register a new userDistributed Submodular Maximization
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Added by
Baharan Mirzasoleiman
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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Many large-scale machine learning problems--clustering, non-parametric learning, kernel machines, etc.--require selecting a small yet representative subset from a large dataset. Such problems can often be reduced to maximizing a submodular set function subject to various constraints. Classical approaches to submodular optimization require centralized access to the full dataset, which is impractical for truly large-scale problems. In this paper, we consider the problem of submodular function maximization in a distributed fashion. We develop a simple, two-stage protocol GreeDi, that is easily implemented using MapReduce style computations. We theoretically analyze our approach, and show that under certain natural conditions, performance close to the centralized approach can be achieved. We begin with monotone submodular maximization subject to a cardinality constraint, and then extend this approach to obtain approximation guarantees for (not necessarily monotone) submodular maximization subject to more general constraints including matroid or knapsack constraints. In our extensive experiments, we demonstrate the effectiveness of our approach on several applications, including sparse Gaussian process inference and exemplar based clustering on tens of millions of examples using Hadoop.
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Continuous submodular functions are a category of generally non-convex/non-concave functions with a wide spectrum of applications. The celebrated property of this class of functions - continuous submodularity - enables both exact minimization and approximate maximization in poly. time. Continuous submodularity is obtained by generalizing the notion of submodularity from discrete domains to continuous domains. It intuitively captures a repulsive effect amongst different dimensions of the defined multivariate function. In this paper, we systematically study continuous submodularity and a class of non-convex optimization problems: continuous submodular function maximization. We start by a thorough characterization of the class of continuous submodular functions, and show that continuous submodularity is equivalent to a weak version of the diminishing returns (DR) property. Thus we also derive a subclass of continuous submodular functions, termed continuous DR-submodular functions, which enjoys the full DR property. Then we present operations that preserve continuous (DR-)submodularity, thus yielding general rules for composing new submodular functions. We establish intriguing properties for the problem of constrained DR-submodular maximization, such as the local-global relation. We identify several applications of continuous submodular optimization, ranging from influence maximization, MAP inference for DPPs to provable mean field inference. For these applications, continuous submodularity formalizes valuable domain knowledge relevant for optimizing this class of objectives. We present inapproximability results and provable algorithms for two problem settings: constrained monotone DR-submodular maximization and constrained non-monotone DR-submodular maximization. Finally, we extensively evaluate the effectiveness of the proposed algorithms.
This paper presents a novel Jacobi-style iteration algorithm for solving the problem of distributed submodular maximization, in which each agent determines its own strategy from a finite set so that the global submodular objective function is jointly maximized. Building on the multi-linear extension of the global submodular function, we expect to achieve the solution from a probabilistic, rather than deterministic, perspective, and thus transfer the considered problem from a discrete domain into a continuous domain. Since it is observed that an unbiased estimation of the gradient of multi-linear extension function~can be obtained by sampling the agents local decisions, a projected stochastic gradient algorithm is proposed to solve the problem. Our algorithm enables the distributed updates among all individual agents and is proved to asymptotically converge to a desirable equilibrium solution. Such an equilibrium solution is guaranteed to achieve at least 1/2-suboptimal bound, which is comparable to the state-of-art in the literature. Moreover, we further enhance the proposed algorithm by handling the scenario in which agents communication delays are present. The enhanced algorithmic framework admits a more realistic distributed implementation of our approach. Finally, a movie recommendation task is conducted on a real-world movie rating data set, to validate the numerical performance of the proposed algorithms.
In this paper, we introduce a novel technique for constrained submodular maximization, inspired by barrier functions in continuous optimization. This connection not only improves the running time for constrained submodular maximization but also provides the state of the art guarantee. More precisely, for maximizing a monotone submodular function subject to the combination of a $k$-matchoid and $ell$-knapsack constraint (for $ellleq k$), we propose a potential function that can be approximately minimized. Once we minimize the potential function up to an $epsilon$ error it is guaranteed that we have found a feasible set with a $2(k+1+epsilon)$-approximation factor which can indeed be further improved to $(k+1+epsilon)$ by an enumeration technique. We extensively evaluate the performance of our proposed algorithm over several real-world applications, including a movie recommendation system, summarization tasks for YouTube videos, Twitter feeds and Yelp business locations, and a set cover problem.
In this work we consider the problem of online submodular maximization under a cardinality constraint with differential privacy (DP). A stream of $T$ submodular functions over a common finite ground set $U$ arrives online, and at each time-step the decision maker must choose at most $k$ elements of $U$ before observing the function. The decision maker obtains a payoff equal to the function evaluated on the chosen set, and aims to learn a sequence of sets that achieves low expected regret. In the full-information setting, we develop an $(varepsilon,delta)$-DP algorithm with expected $(1-1/e)$-regret bound of $mathcal{O}left( frac{k^2log |U|sqrt{T log k/delta}}{varepsilon} right)$. This algorithm contains $k$ ordered experts that learn the best marginal increments for each item over the whole time horizon while maintaining privacy of the functions. In the bandit setting, we provide an $(varepsilon,delta+ O(e^{-T^{1/3}}))$-DP algorithm with expected $(1-1/e)$-regret bound of $mathcal{O}left( frac{sqrt{log k/delta}}{varepsilon} (k (|U| log |U|)^{1/3})^2 T^{2/3} right)$. Our algorithms contains $k$ ordered experts that learn the best marginal item to select given the items chosen her predecessors, while maintaining privacy of the functions. One challenge for privacy in this setting is that the payoff and feedback of expert $i$ depends on the actions taken by her $i-1$ predecessors. This particular type of information leakage is not covered by post-processing, and new analysis is required. Our techniques for maintaining privacy with feedforward may be of independent interest.
Submodular continuous functions are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation guarantees. Specifically, i) We introduce the weak DR property that gives a unified characterization of submodularity for all set, integer-lattice and continuous functions; ii) for maximizing monotone DR-submodular continuous functions under general down-closed convex constraints, we propose a Frank-Wolfe variant with $(1-1/e)$ approximation guarantee, and sub-linear convergence rate; iii) for maximizing general non-monotone submodular continuous functions subject to box constraints, we propose a DoubleGreedy algorithm with $1/3$ approximation guarantee. Submodular continuous functions naturally find applications in various real-world settings, including influence and revenue maximization with continuous assignments, sensor energy management, multi-resolution data summarization, facility location, etc. Experimental results show that the proposed algorithms efficiently generate superior solutions compared to baseline algorithms.
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