Many social phenomena are triggered by public opinion that is formed in the process of opinion exchange among individuals. To date, from the engineering point of view, a large body of work has been devoted to studying how to manipulate individual opi
nions so as to guide public opinion towards the desired state. Recently, Abebe et al. (KDD 2018) have initiated the study of the impact of interventions at the level of susceptibility rather than the interventions that directly modify individual opinions themselves. For the model, Chan et al. (The Web Conference 2019) designed a local search algorithm to find an optimal solution in polynomial time. However, it can be seen that the solution obtained by solving the above model might not be implemented in real-world scenarios. In fact, as we do not consider the amount of changes of the susceptibility, it would be too costly to change the susceptibility values for agents based on the solution. In this paper, we study an opinion optimization model that is able to limit the amount of changes of the susceptibility in various forms. First we introduce a novel opinion optimization model, where the initial susceptibility values are given as additional input and the feasible region is defined using the $ell_p$-ball centered at the initial susceptibility vector. For the proposed model, we design a projected gradient method that is applicable to the case where there are millions of agents. Finally we conduct thorough experiments using a variety of real-world social networks and demonstrate that the proposed algorithm outperforms baseline methods.
The importance of aggregated count data, which is calculated from the data of multiple individuals, continues to increase. Collective Graphical Model (CGM) is a probabilistic approach to the analysis of aggregated data. One of the most important oper
ations in CGM is maximum a posteriori (MAP) inference of unobserved variables under given observations. Because the MAP inference problem for general CGMs has been shown to be NP-hard, an approach that solves an approximate problem has been proposed. However, this approach has two major drawbacks. First, the quality of the solution deteriorates when the values in the count tables are small, because the approximation becomes inaccurate. Second, since continuous relaxation is applied, the integrality constraints of the output are violated. To resolve these problems, this paper proposes a new method for MAP inference for CGMs on path graphs. First we show that the MAP inference problem can be formulated as a (non-linear) minimum cost flow problem. Then, we apply Difference of Convex Algorithm (DCA), which is a general methodology to minimize a function represented as the sum of a convex function and a concave function. In our algorithm, important subroutines in DCA can be efficiently calculated by minimum convex cost flow algorithms. Experiments show that the proposed method outputs higher quality solutions than the conventional approach.
Non-convex sparse minimization (NSM), or $ell_0$-constrained minimization of convex loss functions, is an important optimization problem that has many machine learning applications. NSM is generally NP-hard, and so to exactly solve NSM is almost impo
ssible in polynomial time. As regards the case of quadratic objective functions, exact algorithms based on quadratic mixed-integer programming (MIP) have been studied, but no existing exact methods can handle more general objective functions including Huber and logistic losses; this is unfortunate since those functions are prevalent in practice. In this paper, we consider NSM with $ell_2$-regularized convex objective functions and develop an algorithm by leveraging the efficiency of best-first search (BFS). Our BFS can compute solutions with objective errors at most $Deltage0$, where $Delta$ is a controllable hyper-parameter that balances the trade-off between the guarantee of objective errors and computation cost. Experiments demonstrate that our BFS is useful for solving moderate-size NSM instances with non-quadratic objectives and that BFS is also faster than the MIP-based method when applied to quadratic objectives.
We derive properties of powers of a function satisfying a second-order linear differential equation. In particular we prove that the n-th power of the function satisfies an (n+1)-th order differential equation and give a simple method for obtaining t
he differential equation. Also we determine the exponents of the differential equation and derive a bound for the degree of the polynomials, which are coefficients in the differential equation. The bound corresponds to the order of differential equation satisfied by the n-fold convolution of the Fourier transform of the function. These results are applied to some probability density functions used in statistics.
