Influence maximization (IM) has been extensively studied for better viral marketing. However, previous works put less emphasis on how balancedly the audience are affected across different communities and how diversely the seed nodes are selected. In this paper, we incorporate audience diversity and seed diversity into the IM task. From the model perspective, in order to characterize both influence spread and diversity in our objective function, we adopt three commonly used utilities in economics (i.e., Perfect Substitutes, Perfect Complements and Cobb-Douglas). We validate our choices of these three functions by showing their nice properties. From the algorithmic perspective, we present various approximation strategies to maximize the utilities. In audience diversification, we propose a solution-dependent approximation algorithm to circumvent the hardness results. In seed diversification, we prove a ( @math ) approximation ratio based on non-monotonic submodular maximization. Experimental results show that our framework outperforms other natural heuristics both in utility maximization and result diversification.
@cite_22 first formalize IM as a discrete optimization problem. Subsequent efforts following their framework can be divided into two directions. In one direction, researchers focus on how to accelerate the vanilla hill-climbing greedy algorithm @cite_10 @cite_36 @cite_37 @cite_9 @cite_17 @cite_25 @cite_18 @cite_33 . In the other direction, researchers propose new problem settings @cite_21 @cite_30 @cite_28 @cite_34 @cite_5 based on IM. Due to space limitation, we do not list all the variants here. One can refer to a recent tutorial @cite_40 for more details. Despite their success, the final cascade size is their primary criterion in selecting influential nodes. In contrast, our model incorporates diversity as a factor.
@cite_30 @cite_37 @cite_18 @cite_22 @cite_33 @cite_36 @cite_28 @cite_9 @cite_21 @cite_40 @cite_5 @cite_34 @cite_10 @cite_25 @cite_17
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We consider the budget allocation problem over bipartite influence model proposed by (, 2012). This problem can be viewed as the well-known influence maximization problem with budget constraints. We first show that this problem and its much more general form fall into a general setting; namely the monotone submodular function maximization over integer lattice subject to a knapsack constraint. Our framework includes 's model, even with a competitor and with cost. We then give a (1 - 1 e)-approximation algorithm for this more general problem. Furthermore, when influence probabilities are nonincreasing, we obtain a faster (1 - 1 e)-approximation algorithm, which runs essentially in linear time in the number of nodes. This allows us to implement our algorithm up to almost 10M edges (indeed, our experiments tell us that we can implement our algorithm up to 1 billion edges. It would approximately take us only 500 seconds.). We introduce a generic scheme for accelerating first-order optimization methods in the sense of Nesterov, which builds upon a new analysis of the accelerated proximal point algorithm. Our approach consists of minimizing a convex objective by approximately solving a sequence of well-chosen auxiliary problems, leading to faster convergence. This strategy applies to a large class of algorithms, including gradient descent, block coordinate descent, SAG, SAGA, SDCA, SVRG, Finito MISO, and their proximal variants. For all of these methods, we provide acceleration and explicit support for non-strongly convex objectives. In addition to theoretical speed-up, we also show that acceleration is useful in practice, especially for ill-conditioned problems where we measure significant improvements. In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. We present a novel, general, optimally fast, incremental way of searching for a universal algorithm that solves each task in a sequence of tasks. The Optimal Ordered Problem Solver (OOPS) continually organizes and exploits previously found solutions to earlier tasks, efficiently searching not only the space of domain-specific algorithms, but also the space of search algorithms. Essentially we extend the principles of optimal nonincremental universal search to build an incremental universal learner that is able to improve itself through experience. The initial bias is embodied by a task-dependent probability distribution on possible program prefixes. Prefixes are self-delimiting and executed in online fashion while being generated. They compute the probabilities of their own possible continuations. Let p^n denote a found prefix solving the first n tasks. It may exploit previously stored solutions p^i, i >n, by calling them as subprograms, or by copying them and editing the copies before applying them. We provide equal resources for two searches that run in parallel until p^ n+1 is discovered and stored. The first search is exhaustive; it systematically tests all possible prefixes on all tasks up to n+1. The second search is much more focused; it only searches for prefixes that start with p^n, and only tests them on task n+1, which is safe, because we already know that such prefixes solve all tasks up to n. Both searches are depth-first and bias-optimal: the branches of the search trees are program prefixes, and backtracking is triggered once the sum of the runtimes of the current prefix on all current tasks exceeds the prefix probability multiplied by the total search time so far. In illustrative experiments, our self-improver becomes the first general system that learns to solve all n disk Towers of Hanoi tasks (solution size 2^n-1) for n up to 30, profiting from previously solved, simpler tasks involving samples of a simple context free language.