Neurips_21_22_Limitation_OpenReview/neurips.1100.json

{
  "File Number": "1100",
  "Title": "Social-Inverse: Inverse Decision-making of Social Contagion Management with Task Migrations",
  "Limitation": "We close our paper with a discussion on the limitations and future directions of the presented work, with the related works being discussed in Appendix F. Future directions. Following the discussion on Figure 1, one immediate direction is to systemically investigate the necessary or sufficient condition in which the task migrations between DE and DC are manageable. In addition, settings in Problem 4 can be carried over to other contagion management tasks beyond DE and DC, such as effector detection [56] and diffusion-based community detection [57]. Finally, in its most general sense, the problem of inverse decision-making with task migrations can be conceptually defined over any two stochastic combinatorial optimization problems [40] sharing the same underlying model (e.g., graph distribution). For instance, considering the stochastic shortest path problem [58] and the minimum Steiner tree problem [59], with the information showing the shortest paths between some pairs of nodes, can we infer the minimum Steiner tree of a certain group of nodes with respect to the same graph distribution? Such problems are interesting and fundamental to many complex decision-making applications [60, 61]. Limitations. While our method offers promising performance in experiments, it is possible that deep architectures can be designed in a sophisticated manner so as to achieve improved results. In addition, while we believe that similar observations also hold for graphs that are not considered in our experiment, more experimental studies are required to support the universal superiority of Social-Inverse. In another issue, the assumption that the training set contains approximation solutions is the minimal one for the purpose of theoretical analysis, but in practice, such guarantees may never be known. Therefore, experimenting with samples of heuristic query-decision pairs is needed to further justify the practical utility of our method. Finally, we have not experimented with graphs of extreme scales (e.g., over 1M nodes) due to the limit in memory. We wish to explore the above issues in future work. Acknowledgments and Disclosure of Funding\nWe thank the reviewers for their time and insightful comments. This work is supported in part by a) National Science Foundation under Award IIS-2144285 and b) the University of Delaware.",
  "Reviewer Comment": "Reviewer_4: This paper addresses an important problem (task migration) in a very general setting that can be applied to many different instances (contagion management in social networks). Unfortunately I am unable to leave an educated review since this paper seems to be targeted at a highly technical research sub-community which I have had no prior exposure to.\nMy primary concern is that this paper might not be accessible to most of Neurips' audience and thus might be better received at more theoretical conferences like STOC or COLT. However, I do not feel as though this should be a disqualifying factor for acceptance, especially if there is a large enough community at Neurips that would find this paper useful.\nGoing through the theoretical results, the assumptions seem reasonable and I can't see any glaring red flags in their resulting theorems.\nI do not feel like I have enough context to critically interpret the significance of the improvements in the experimental results.\nOverall this paper seems sound to me, but I have concerns about the accessibility of the paper. Hence I recommend a borderline accept with very low confidence. I think the authors could significantly improve the paper if they spent some time providing more intuition in the first few sections and also added some concrete examples to aid the interpretation of their definitions.\nQuestions:\nHow closely do the assumptions of this work mirror real-world practices in social contagion management? For example, the introduction states that \"For example, a network manager may need to work on a rumor blocking task, but they only have historical data collected in solving viral marketing tasks.\" are there any existing cases where such a network manager would employ any methods that are remotely similar to social-inverse instead of just proceeding based on intuition/industry best practices for their interventions?\nLimitations:\nLimitations were well addressed by the author.\nEthics Flag: No\nSoundness: 3 good\nPresentation: 1 poor\nContribution: 3 good\n\nReviewer_5: Strengths:\nQuality: The paper is nicely presented with thorough empirical studies.\nOriginality: The paper solves an essentially \"graph-kernel\" problem using inspirations from random kitchen sink. This sounds like a reasonable idea.\nWeaknesses:\nTheory: The presentation is not entirely clear and I have some questions about some unexplained terms in the key algorithm and theoretical analysis.\nSignificance: We have to accept the assumption that the training data is presented as query-decision pairs, instead of the more commonly used decision-impact pairs, that is, the actual coverage caused by the seed nodes.\nEthics: The paper is missing ethics discussions on a potentially sensitive topic.\nQuestions:\nLine 172. Why do you introduce a Gaussian distribution after you learned the optimal combinations of realizations? Doesn't this introduce pure noise?\nLine 185. Related, the generalization error considers the mean of the sample you use for inference, yet in the algorithm description, you use only one point in the sample for inference. Does this generalization error cover the full risk?\nLine 191. I do not understand the use of beta. Also, why do you analyze a binary loss when your objective contains an optimization sub-problem?\nTable 1. Can you elaborate on Naive Bayes? It seems to be the second-best in the considered baselines, just below High Degree.\nLimitations:\nOverall, I can get some inspirations from the proposed method and empirical studies, but some key aspects in the algorithm were left unexplained, namely eta and the extra step of Gaussian sampling. The theory presentation does not meet the expectation of the NeurIPS community, perhaps because too many new concepts were introduced without good explanations.\nThough the paper is purely methodological, I flagged it for ethics reviews due to some word choices, such as information containment, which may lead to a limitation of people's access to opportunities. The authors should be also be advised on the creation of an ethics discussion section.\nEthics Flag: Yes\nEthics Review Area: Inappropriate Potential Applications & Impact  (e.g., human rights concerns)\nSoundness: 2 fair\nPresentation: 2 fair\nContribution: 3 good\n\nReviewer_6: Strengths: I find the idea of task migration on contagion management novel and important (although I have some concerns about the detailed setup (see weakness 1). The paper provides rigorous theoretical analysis and shows empirical effectiveness on four datasets.\nWeaknesses: (1) I think the setup that the platform only observes one type of task (either diffusion containment or enhancement) does not seem realistic. It is more likely that the platform will observe a combination of both. I think this is a crucial question: How realistic is the task migration problem proposed in this paper?\n(2) Given that many models have been proposed in the diffusion literature (also cited by the paper), designing benchmarks w.r.t. based on these models is important to show the effectiveness of the methods, rather than other ML methods that do not know the underlying diffusion process.\nQuestions:\n(1) In remark 1, the paper mentioned that the model generalizes popular diffusion models, including the linear threshold model, which requires a percentage or number of neighbors to adopt before the users adopt. However, in lines 79—80 on P2, the paper mentions that the node will only be activated by \"first in-attempting neighbors\" or activated by the cascade with the smallest index. These two scenarios both conflict with the linear threshold model. Why can't a node be activated by two or more cascades? In reality, a user only observes his/her neighbor's decisions, rather than which cascade resulted in the neighbors' decision.\n(2) L88–89, why is the subgraph weighted? From the description in Line 71—72, I think the subgraph is attributed instead of weighted?\n(3) There are two decision-making problems in the paper, one is the decision-making of users, and the other is the decision-making of the central planner (I believe referred to as agent by the paper). Given that both are important concepts yet are entirely different, I suggest distinguishing the two concepts to avoid confusing the readers, e.g., user decision-marking vs. planner decision-making).\n(4) \\beta is an important parameter that controls the trade-off between estimation and approximation errors. How can it be tuned? It seems that the experiment section directly uses beta = 1. If it cannot be systematically tuned, it might also help to show the robustness of the method w.r.t. different beta\n(5) Regarding the experiment, I think it will help add a benchmark, which assumes either a linear threshold model or an independent cascade. The parameters in these models can be learned based on different realizations. That is, since there is rich research on the diffusion model in sociology, it seems sensible to use them directly in such a migration task.\n(6) I wonder how realistic in practice do we assume the platform has information about all DC but no DE, or all DE but no DC? It seems more reasonable to observe a mix of both. Can you provide some insights on how the method performs (especially compared with the benchmarks) if this is the case?\n(7) Any insights on why the benchmarks perform this badly on ER?\n(8) Minor: There is a typo on line 84, \"For sample\" —> \"For example\"\nLimitations:\nNA\nEthics Flag: No\nSoundness: 3 good\nPresentation: 4 excellent\nContribution: 4 excellent",
  "Limitations_refined": "",
  "abstractText": "Considering two decision-making tasks A and B, each of which wishes to compute an effective decision Y for a given query X , can we solve task B by using querydecision pairs (X,Y ) of A without knowing the latent decision-making model? Such problems, called inverse decision-making with task migrations, are of interest in that the complex and stochastic nature of real-world applications often prevents the agent from completely knowing the underlying system. In this paper, we introduce such a new problem with formal formulations and present a generic framework for addressing decision-making tasks in social contagion management. On the theory side, we present a generalization analysis for justifying the learning performance of our framework. In empirical studies, we perform a sanity check and compare the presented method with other possible learning-based and graph-based methods. We have acquired promising experimental results, confirming for the first time that it is possible to solve one decision-making task by using the solutions associated with another one.",
  "1 Introduction": "Social contagion management. Social contagion, in its most general sense, describes the diffusion process of one or more information cascades spreading between a set of atomic entities through the underlying network [1, 2, 3, 4]. Prototypical applications of social contagion management include deploying advertising campaigns to maximize brand awareness [5, 6], broadcasting debunking information to minimize the negative impact of online misinformation [7, 8, 9], HIV prevention for homeless youth [10, 11], and the prevention of youth obesity [12, 13]. In these applications, a central problem is to launch new information cascades in response to certain input queries, with the goal of optimizing the agents’ objectives [14, 15]. In principle, most of these tasks fall into either diffusion enhancement, which seeks to maximize the influence of the to-be-generated cascade (e.g., marketing campaign [5, 16, 17] and public service announcement [12, 18, 19]), or diffusion containment, which aims to generate positive cascades to minimize the spread of negative cascades (e.g., misinformation [20, 21, 22] and violence-promoting messages [23, 24]).\nInverse decision-making with task migrations. Traditional research on social contagion management often adopts classic operational diffusion models with known parameters, and focuses on algorithmic development in overcoming the NP-hardness [25, 17, 26, 27, 28]. However, real-world contagions are often very complicated, and therefore, perfectly knowing the diffusion model is less realistic [29, 30, 31, 32, 33]. When presented with management tasks defined over unknown diffusion models, one can adopt the learn-and-optimize approach in which modeling methodologies and optimization schemes are designed separately; in such methods, the main issue is that the learning process is guided by model accuracy but not by the optimization effect [34, 35], suggesting that\n36th Conference on Neural Information Processing Systems (NeurIPS 2022).\nthe endeavors dedicated to model construction are neither necessary nor sufficient for successfully handling the downstream optimization problems. This motivates us to explore unified frameworks that can shape the learning pipeline towards effective approximations. Recently, it has been shown that for contagion management tasks like diffusion containment, it is possible to produce high-quality decisions for future queries by using query-decision pairs from the same management task without learning the diffusion model [36]. Such findings point out an interesting and fundamental question: with a fixed latent diffusion model, can we solve a target management task by using query-decision pairs from a different management task? This is of interest because the agents often simultaneously deal with several management tasks while it is less likely that they always have the proper empirical evidence concerning the target task. For example, a network manager may need to work on a rumor blocking task, but they only have historical data collected in solving viral marketing tasks. We call such a setting as inverse decision-making with task migrations.\nContribution. This paper presents a formal formulation of inverse decision making where the target task we wish to solve is different from the source task that generates samples, with a particular focus on social contagion management tasks. Our main contribution is a generic framework, called SocialInverse, for handling migrations between tasks of diffusion enhancement and diffusion containment. For Social-Inverse, we present theoretical analysis to obtain insights regarding how different contagion management tasks can be subtly correlated in order for samples from one task to help the optimization of another task. In empirical studies, we have observed encouraging results indicating that our method indeed works the way it is supposed to. Our main observations suggest that the task migrations are practically manageable to a satisfactory extent in many cases. In addition, we also explore the situations where the discrepancy between the target task and the source task is inherently essential, thereby making the samples from the source task less useful.\nRoadmap. In Sec. 2, we first provide preliminaries regarding social contagion models, and then discuss how to formalize the considered problem. The proposed method together with its theoretical analysis is presented in Sec. 3. In Sec. 4, we present our empirical studies. We close our paper with a discussion on limitations and future works (Sec. 5). The technical proofs, source code, pre-train models, data, and full experimental analysis can be found in the supplementary material. The data and source code is maintained online1.",
  "2.1 Stochastic diffusion model": "A social network is given by a directed graph G = (V,E), with V and E respectively denoting the user set and the edge set. In modeling the contagion process, let us assume that there are L ∈ Z information cascades {Ci}Li=1, each of which is associated with a seed set Si ⊆ V . Without loss of generality, we assume that Si ∩ Sj = ∅ for i 6= j. A diffusion modelM is governed by two sets of configurations: each node u ∈ V is associated with a distributionN u over 2N − u , where N−u is the set of the in-neighbors of u; each edge (u, v) ∈ E is associated with a distribution T (u,v) over (0,+∞) denoting the transmission time. During the diffusion process, a node can be inactive or Ci-active if activated by cascade Ci. Given the seed sets, the diffusion process unfolds as follows:\n• Initialization: Each node u samples a subset Au ⊆ N−u following N u, and each edge (u, v) samples a real number t(u,v) ≥ 0 following T (u,v). • Time 0: The nodes in Si become Ci-active at time 0, and other nodes are inactive. • Time t: When a node u becomes Ci-active at time t, for each inactive node v such that u is\nin Av, v will be activated by u and become Ci-active at time t+ t(u,v). Each node will be activated by the first in-neighbor attempting to activate them and never deactivated. When a node v is activated by two or more in-neighbors at the same time, v will be activated by the cascade with the smallest index.\nRemark 1. The considered model is in general highly expressive because N u and T (u,v) can be flexibly designed. For sample, it subsumes the classic independent cascade model [25] by makingN u sample each in-neighbor independently. When there is only one or two cascades, the above model generalizes a few popular diffusion models, including discrete-time independent cascade model [25],\n1https://github.com/cdslabamotong/social_inverse\ndiscrete-time linear threshold model [25], continuous-time independent cascade model [37], and independent multi-cascade model [38, 36].\nDefinition 1 (Realization). Notice that the initialization phase essentially samples a weighted subgraph, and the diffusion process becomes deterministic after the initialization phase. For an abstraction, we call each of such weighted subgraph a realization, and useRG to denote the space of weighted subgraphs of G. With the concept of realization, we may abstract a concrete stochastic diffusion model M as a collection of density functions, i.e., M = {N u : u ∈ V } ∪ {T (u,v) : (u, v) ∈ E}. Slightly abusing the notation, we also useM : RG → [0, 1] to denote the distribution over RG induced by the density functions specified byM. On top of a diffusion modelM, the distribution of the diffusion outcome depends on the seed sets of the cascades. An example for illustrating the diffusion process is given in Appendix A.",
  "2.2 Social contagion management tasks": "In this paper, we focus on the following two classes of social contagion management tasks.\nProblem 1 (Diffusion Enhancement (DE)). Given a diffusion modelM and a set of target users X ⊆ V , we consider the single-cascade case and let fDEM (X,Y ) be the expected number of users in X who are activated by a cascade from seed set Y ⊆ V . We would like to find a seed set with at most k ∈ Z nodes such that the total influence on X can be maximized, i.e.,\narg max Y⊆V,|Y |≤k\nfDEM (X,Y ). (1)\nProblem 2 (Diffusion Containment (DC)). Given a diffusion model M, we now consider the situation of competitive diffusion where there is a negative cascade C1 with a seed set X ⊆ V and a positive cascade C2 with a seed set Y ⊆ V . Let fDCM (X,Y ) be the expected number of users who are not activated by the negative cascade. Given the diffusion modelM and the seed set X of the negative cascade, we would like to find a seed set Y for the positive cascade with at most k ∈ Z nodes such that the impact of the negative cascade can be maximally limited, i.e.,\narg max Y⊆V,|Y |≤k\nfDCM (X,Y ). (2)\nContagion management tasks like Problems 1 and 2 might be viewed as decision-making problems aiming to infer an effective decision Y in response to a query X . In such a sense, we may abstract such problems in the following way:\nProblem 3 (Abstract Contagion Management Tasks). Given a diffusion modelM, an abstract management task T is specified by an objective function fTM(X,Y ) : 2\nV × 2V → R, a candidate space X T ⊆ 2V of the queries, and a candidate space YT ⊆ 2V of the decisions, where we wish to compute\nYM,T,X := arg max Y ∈YT\nfTM(X,Y ) (3)\nfor each input query X ∈ X T . We assume that X T and YT are matroids over V , subsuming common constraints such as the cardinality constraint or k-partition [39]. Since such optimization problems are often NP-hard, their approximate solutions are frequently used, and we denote by Y αM,T,X an α-approximation to Equation 3.\nDefinition 2 (Linearity over Kernel Functions). In addressing the above management tasks, it is worth noting that the objective function is calculated over the possible diffusion outcomes, which are determined in the initialization phase. Specifically, denoting by fTr (X,Y ) the objective value projected to a single realization r ∈ RG, the objective function can be expressed as\nfTM(X,Y ) = ∫ r∈RG M(r) · fTr (X,Y )dr. (4)\nThe function fTr (X,Y ) is called kernel function, in the sense that it transforms set structures into real numbers. For example, fDEr (X,Y ) denotes the number of users in X who are activated by a cascade generated from Y in a single realization r; fDCr (X,Y ) denotes the number of users in G who are not activated by the negative cascade (generated from X) in realization r when the positive cascade spreads from Y .",
  "2.3 Inverse decision-making of contagion management with task migrations": "Supposing that the diffusion modelM is given, DE and DC are purely combinatorial optimization problems, which have been extensively studied [25, 20]. In the case that the diffusion model is unknown, inverse decision-making is a potential solution, which seeks to solve contagion management tasks by directly learning from query-decision pairs [40]. In particular, with respect to a certain management task fTM associated with an unknown diffusion modelM, the agent receives a collection of pairs (Xi, Yi) where Yi is the optimal/suboptimal solution to maximizing fTM(Xi, Y ). Such empirical evidence can be mathematically characterized as\nSαM,T,m = { (Xi, Y α M,T,Xi) : f T M(Xi, Y\nα M,T,Xi) ≥ α · maxY ∈YT fTM(Xi, Y ) }m i=1\n(5)\nwhere α ∈ (0, 1] is introduced to measure the optimality of the sample decisions. For the purpose of theoretical analysis, the ratio α may be interpreted as the best approximation ratio that can be achieved by a polynomial-time algorithm under common complexity assumptions (e.g., NP 6= P ). For DE and DC, we have the best ratio as 1 − 1/e due to the well-known fact that their objective functions are submodular [25, 20]. Leveraging such empirical evidence, we wish to solve the same or a different management task for future queries: Problem 4 (Inverse Decision-making with Task Migrations). Suppose that there is an underlying diffusion modelMtrue. Consider two management tasks T s© and T t© defined in Problem 3, where T s© is the source task and T t© is the target task. With a collection S α s© Mtrue,T s©,m of samples concerning the source task T s© for some ratio α s© ∈ (0, 1], we aim to build a framework A : X T t© → YT t© that can make a prediction A(X) for each future query X of the target task T t©. Let l be a loss function l(X, Ŷ ) : X T t© ×YT t© → [0, 1] that measures the desirability of Ŷ with respect to X . We seek to minimize the generalization error L with respect to an unknown distribution D over X T t© :\nL(A,D, l) :=EX∼D [ l(X,A(X)) ] . (6)\nSinceMtrue and m are fixed, we denote S α s© Mtrue,T s©,m as S α s© T s©\nfor conciseness. We will focus on the case where the source task and the target task are selected from DE and DC. Remark 2. In general, the above problem appears to be challenging because the query-decision pairs of one optimization problem do not necessarily shed any clues on effectively solving another optimization problem. What makes our problem tractable is that the source task and the target task share the same underlying diffusion modelMtrue. With the hope that the query-decision pairs of the source task can identifyMtrue to a certain extent, we may solve the target task with statistical significance, as evidenced in experiments. In such a sense, our setting is called inverse as it implicitly infers the structure of the underlying model from solutions, in contrast to the forward decision-making pipeline that seeks solutions based on given models.",
  "3 Social-Inverse": "In this section, we present a learning framework called Social-Inverse for solving Problem 4. Our method is inspired by the classic structured prediction [41] coupled with randomized kernels [40], which may be ultimately credited to the idea of Random Kitchen Sink [42]. Social-Inverse starts by selecting an empirical distributionMem overRG and a hyperparameter K ∈ Z, and then proceeds with the following steps:\n• Hypothesis design. Sample K iid realizations RK = {r1, ..., rK} ⊆ RG followingMem, and obtain the hypothesis spaceFRK :={H t© RK ,w\n(X,Y ) : w = (w1, ..., wK) ∈ RK}where H\nt© RK ,w is the affine combination of fT t©r over the realizations in RK :\nH t© RK ,w (X,Y ) := K∑ i=1 wi · fT t©ri (X,Y ). (7)\n• Training. Compute a prior vector w̃ using the training set Sα s©T s© (Sec. 3.2), and sample the final parameter w from an isotropic Gaussian Q(γ · w̃, I) with a mean of w̃ scaled by γ. The selection of γ will be discussed in Sec. 3.1.\n• Inference. Given a future query X ∈ X T t© of the target task T t©, the prediction is made by solving the inference problem over the hypothesis in FRK associated with final weight w:\narg max Y ∈YT t©\nH t© RK ,w (X,Y ). (8)\nIt is often NP-hard to solve the above inference problem in optimal, and therefore, we assume that an α̃ t©-approximation to Equation 8 – denoted by Y α̃ t© RK ,w,X\n– is employed for some α̃ t© ∈ (0, 1]. Notice that the ratio α̃ t© herein represents the inference hardness, while the ratio α s© associated with the training set measures the hardness of the source task.\nIn completing the above procedure, it is left to determine a) the prior distribution Mem, b) the hyperparameter K, c) the scale factor γ, d) the training method for computing the prior vector w̃, and e) the inference algorithm for computing Y α̃ t©RK ,w,X . In what follows, we will first discuss how they may influence the generalization performance in theory, and then present methods for their selections. For the convenience of reading, the notations are summarized in Table 2 in Appendix B.",
  "3.1 Generalization analysis": "For Social-Inverse, given the fact the generation of w is randomized, the generalization error is further expressed as\nL(Social-Inverse,D, l) :=EX∼D,w∼Q [ l(X,Y α̃ t© RK ,w,X ) ] . (9)\nIn deriving an upper bound with respect to the prior vector w̃, let us notice that the empirical risk is given by l(Xi, Y α̃ t© RK ,w,Xi\n), which is randomized by w ∼ Q(γ · w̃, I). Thus, the prediction Y α̃ t©RK ,w,Xi associated with a training input Xi is most likely one of those centered around Y\nα̃ t© RK ,w̃,Xi\n, and we will measure such concentration by their difference in terms of a fraction of the empirical risk associated with w̃. More specifically, controlled by a hyperparameter β ∈ (0, α̃ t©), for an input query Xi, the potential predictions are those within the margin:\nI t© RK ,w̃,Xi,β := (10){ Y ∈ Y t© : α̃ t© ·H t©RK ,w̃(Xi, Y α̃ t© RK ,w̃,Xi )−H t©RK ,w̃(Xi, Y ) ≤ β ·H t© RK ,w̃ (Xi, Y α̃ t© RK ,w̃,Xi ) } .\nThe empirical risk is therefore given via the above margin:\nLem(w̃, RK , S α s© T s©\n, l) := 1\nm m∑ i=1 max Y ∈Y t© l(Xi, Y ) · 1I t© RK,w̃,Xi,β (Y ), (11)\nwhere 1S(x) ∈ {0, 1} is the indicator function: 1S(x) = 1 ⇐⇒ x ∈ S. With the above progressions, we have the following result concerning the generalization error. Theorem 1. For each w̃ = (w̃1, ..., w̃K), RK ⊆ RG, β ∈ (0, α̃ t©), and δ > 0, with probability at least 1− δ, we have\nL(Social-Inverse,D, l) ≤ Lem(w̃, RK , S α s© T s©\n, l) + ‖w̃‖2\nm +\n√ γ2 ‖w̃‖2 /2 + ln(m/δ)\n2(m− 1)\nprovided that\nγ = α̃2t© + 1\nminp | w̃p | · β · α̃ t©\n√ 2 ln 2mK\n‖w̃‖2 (12)\nThe proof follows from the standard analysis of the PAC-Bayesian framework [43] coupled with the approximate inference [44] based on a multiplicative margin [40]; the extra caution we need to handle is that our margin (Equation 10) is parameterized by β. Notice that when β decreases, the regularization term becomes larger, while the margin set I t©RK ,w̃,Xi,β becomes smaller – implying a low empirical risk (Equation 11). In this regard, Theorem 1 presents an intuitive trade-off between the estimation error and the approximation error controlled by β.\nHaving seen the result for a general loss, we now seek to understand the best possible generalization performance in terms of the approximation loss lapprox:\nlapprox(X, Ŷ ) := 1− f T t© Mtrue(X, Ŷ )\nmaxY ∈Y t© f T t© Mtrue(X,Y )\n∈ [0, 1]. (13)\nSuch questions essentially explore the realizability of the hypothesis space FRK , which is determined by the empirical distributionMem and the number of random realizations used to construct FRK . We will see shortly how these factors may impact the generalization performance. By Theorem 1, when infinite samples are available, the empirical risk approaches to\nEX∼D [\nmax Y ∈YT t© lapprox(X,Y ) · 1I t© RK,w̃,X,β\n(Y ) ] . (14)\nThe next result provides an analytical relationship between the complexity of the hypothesis space and the best possible generalization performance in terms of lapprox.\nTheorem 2. Let ∆ := supr Mtrue(r) Mem(r) ·\nmax f T t© r (X,Y ) min f T t© r (X,Y )\nmeasure the divergence betweenMtrue and Mem scaled by the range of the kernel function. For each > 0, δ1 > 0, δ2 > 0, andMem, when K is O( ∆ 2\n2·δ22 (ln | YT t© | + ln 1δ1 )), with probability at least 1 − δ1 over the selection of RK , there\nexists a desired weight w̃ such that\nPr X∼D\n[ max\nY ∈YT t© lapprox(X,Y ) · 1I t© RK,w̃,X,β (Y ) ≤ 1− α̃ t© · (α̃ t© − β) · (1− ) (1 + )\n] ≥ 1− δ2. (15)\nRemark 3. The above result has the implication that the best possible ratio in generalization is essentially bounded by O(α̃ t© · (α̃ t© − β)). On the other hand, one can easily see that the target task (Equation 3) and the inference problem (Equation 8) suffer the same approximation hardness, and therefore, one would not wish for a true approximation error that is better than α̃ t©; in this regard, the result in Theorem 2 is not very loose.\nThe results in this section demonstrate how the selections of K,Mem, w̃, and β may affect the generalization performance in theory. Since the true modelMtrue is unknown, the prior distributionMem can be selected to be uniform or Gaussian distribution. K and β can be taken as hyperparameters determining the model complexity. In addition, since the true loss lapprox is not accessible, one can take general loss functions. Given the fact that we are concerned with set structures rather than real numbers, we employ the zero-one loss, which is adopted also for the convenience of optimization. Therefore, it remains to figure out how to compute the prior vector w̃ from training samples as well as how to solve the inference problem (Equation 8), which will be discussed in the next part.",
  "3.2 Training method": "In computing the prior vector w̃, the main challenge caused by the task migration is that the target task on which we performance inference is different from the source task that generates training samples. Theorem 1 suggests that, ignoring the low-order terms, one may find the prior vector by minimizing the regularized empirical risk Lem(w̃, RK , S α s© T s© , l) + ‖w̃‖ 2\nm . Directly minimizing such a quantity would be notoriously hard because the optimization problem is bilevel: optimizing over w̃ involves the term Y α̃ t©RK ,w̃,Xi which is obtained by solving another optimization problem depending on w̃ (Equations 10 and 11). Notably, since H t©RK ,w̃(Xi, Y α̃ t© RK ,w̃,Xi ) is lower bounded by α̃ t© ·H t©RK ,w̃(Xi, YMtrue,T t©,Xi), replacing Y α̃ t© RK ,w̃,Xi\nwith YMtrue,T t©,Xi would allow for us to optimize an upper bound of the empirical risk. Seeking a large-margin formulation, this amounts to solving the following mathematical program under the zero-one loss [41, 45]:\nmin ‖w‖2 + C m∑ i=1 ξi/m s.t. α̃ t©(α̃ t© − β) ·H t©RK ,w(Xi, YMtrue,T t©,Xi)−H t© RK ,w\n(Xi, Y ) ≥ ξi, ∀i ∈ [m], ∀ Y ∈ Y t© w ≥ 0 (16)\nwhere ξi is the slack variable and C is a hyperparameter [46]. However, our dataset concerns only about the source task T s© without informing YMtrue,T t©,Xi or its approximation. In order to see where we could feed the training samples into the training process, let us notice that the constraints in Equation 16 have an intuitive meaning: with respect to the target the task T t©, a desired weight w̃ should lead to a score function H t©RK ,w̃ that can assign highest scores to the optimal solutions YMtrue,T t©,Xi . Similar arguments also apply to the source task T s©, as the weight w̃ implicitly estimates the true modelMtrue, which is independent of the management tasks. This enables us to reformulate the optimization problem with respect to the source task T s© by using the following constraints:\nα̃ t©(α̃ t© − β) ·H s©RK ,w(Xi, Y α s© Mtrue,T s©,Xi)−H s© RK ,w (Xi, Y ) ≥ ξi, ∀i ∈ [m], ∀Y ∈ Y t© (17)\nwhere H s©RK ,w(X,Y ) := ∑K i=1 wi · f T s© ri (X,Y ) is the score function corresponding to the source task T s©. As desired, pairs of (Xi, Y α s© Mtrue,T s©,Xi) are the exactly the information we have in the training data Sα s©T s© . One remaining issue is that the acquired program (Equation 17) has an exponential number of constrains [45], which can be reduced to linear (in sample size) if the following optimization problem can be solved for each w and Xi:\nmax Y ∈Y s©\nH s© RK ,w (Xi, Y ). (18)\nProvided that the above problem can be addressed, the entire program can be solved by several classic algorithms, such as the cutting plane algorithm [47] and the online subgradient algorithm [48]. Therefore, in completing the entire framework, it remains to solve Equations 8 and 18. For tasks of DE and DC, we delightedly have the following results. Theorem 3. When T s© and T t© are selected from DE and DC, Equations 8 and 18 are both NP-hard to solve in optimal, but both can be approximated within a ratio of 1− 1/e in polynomial time.\nA concrete example of using Social-Inverse to solve Problem 4 is provided in Appendix C.",
  "4 Empirical studies": "Although some theoretical properties of our framework can be justified (Sec. 3.1), it remains open whether or not the proposed method is practically effective, especially given the fact that no prior work has attempted to solve one optimization problem by using the solutions to another one. In this section, we present our empirical studies.",
  "4.1 Experimental settings": "We herein present the key logic of our experimental settings and provide details in Appendix E.1.\nThe latent model Mtrue and samples (Appendix E.1.1). To generate a latent diffusion model Mtrue, we first determine the graph structure and then fix the distributions N u and T (u,v) by generating random parameters. We adopt four graphs: a Kronecker graph [49], an Erdős-Rényi graph [50], a Higgs graph [51], and a Hep graph [52]). Given the underlying diffusion modelMtrue, for each of DE and DC, we generate a pool of query-decision pairs (Xi, Yi) for training and testing, where Xi is selected randomly from V and Yi is the approximate solution associated with Xi (Theorem 3). As for Problem 4, there are four possible target-source pairs: DE-DE, DC-DE, DE-DC, and DC-DC.\nSocial-Inverse (Appendix E.1.2). WithK and β being hyperparameters, to set up Social-Inverse, we need to specify the empirical distributionMem. We construct the empirical distribution by building three diffusion modelsMq with q ∈ {0.1, 0.5, 1}, where a smaller q implies thatMq is closer to Mtrue. In addition, we construct an empirical distributionM∞ which is totally random and not close toMtrue anywhere. For each empirical distribution, we generate a pool of realizations. Competitors (Appendix E.1.3). Given the fact that Problem 4 may be treated as a supervised learning problem with the ignorance of task migration, we have implemented Naive Bayes (NB) [53], graph neural networks (GNN) [54], and a deep set prediction network (DSPN) [55]. In addition, we consider the High-Degree (HD) method, which is a popular heuristic believing that selecting the high-degree users as the seed nodes can decently solve DE and DC. A random (Random) method is also used as the baseline.\nTraining, testing, and evaluation (Appendix E.1.4) The testing size is 540, and the training size m is selected from {90, 270, 1350}. Given the training size and the testing size, the samples are randomly selected from the pool we generate; similarly, given K, the realizations used in SocialInverse are also randomly selected from the pool we generate. For each method, the entire process is repeated five times, and we report the average performance ratio together with the standard deviation. The performance ratio is computed by comparing the predictions with the decisions in testing samples; larger is better.",
  "4.2 Main observations": "The main results on the Kronecker graph and the Erdős-Rényi graph are provided in Table 1. According to Table 1, it is clear that Social-Inverse performs better when K becomes larger or when the discrepancy betweenMem andMtrue becomes smaller (i.e., q is small), which suggests that Social-Inverse indeed works the way it is supposed to. In addition, while all the methods are non-trivially better than Random, one can also observe that Social-Inverse easily outperforms other methods by an evident margin as long as sufficient realizations are provided. We also see that learning-based methods do not perform well in many cases; this is not very surprising because the effectiveness of learning-based methods hinges on the assumption that different tasks share similar decisions for the same query, which however may not be the case especially on the Erdős-Rényi graph. Furthermore, Social-Inverse appears to be more robust than other methods in terms of standard deviation. Finally, the performance of standard learning methods (e.g., NB and GNN) are sensitive to graph structures; they are relatively good on the Kronecker graph but less satisfactory on the Erdős-Rényi graph, while Social-Inverse is consistently effective on all datasets.",
  "4.3 An in-depth investigation on task migration": "Notably, the effectiveness of Social-Inverse depends not only on the training samples Sα t©T t© (for tuning the weight w) but also on the expressiveness of the hypothesis space (determined by Mem and K). Therefore, with solely the results in Table 1, we are not ready to conclude that samples of DE (resp., DC) are really useful for solving DC (resp., DE). In fact, whenMem is identical toMtrue, no samples are needed because setting w = 1 can allow for us to perfectly recover the best decisions as long as K is sufficiently large. As a result, the usefulness of the samples should be assessed by examining how much they can help in delivering a high-quality w. To this end, for each testing query, we report the quality of two predictions made based, respectively, on the initial weight (before optimization) and on the final weight (after optimization).\nSuch results for DC-DE on the Kronecker graph are provided in Figure 1. As seen from Figure 1b, the efficacy of DC samples in solving DE is statistically significant underM0.1, which might be the first\npiece of experimental evidence confirming that it is indeed possible to solve one decision-making task by using the query-decision pairs of another one. In addition, according to Figures 1a, 1b, and 1c, the efficacy of the samples is limited when the sample size is too small, and it does not increase too much after sufficient samples are provided. On the other hand, with Figure 1e, we also see that the samples of DC are not very useful when the empirical distribution (e.g.,M∞) deviates too much from the true model, and in such a case, providing more samples can even cause performance degeneration (Figure 1f), which is an interesting observation calling for further investigations.\nThe full experimental study can be found in Appendix E.2, E.3 and E.4, including the results on Higgs and Hep, results with more realizations, results of DE-DE and DC-DC, results of GNN under different seeds, a discussion on the impact of β, and a discussion of training efficiency.",
  "5 Further discussion": "We close our paper with a discussion on the limitations and future directions of the presented work, with the related works being discussed in Appendix F.\nFuture directions. Following the discussion on Figure 1, one immediate direction is to systemically investigate the necessary or sufficient condition in which the task migrations between DE and DC are manageable. In addition, settings in Problem 4 can be carried over to other contagion management tasks beyond DE and DC, such as effector detection [56] and diffusion-based community detection [57]. Finally, in its most general sense, the problem of inverse decision-making with task migrations can be conceptually defined over any two stochastic combinatorial optimization problems [40] sharing the same underlying model (e.g., graph distribution). For instance, considering the stochastic shortest path problem [58] and the minimum Steiner tree problem [59], with the information showing the shortest paths between some pairs of nodes, can we infer the minimum Steiner tree of a certain group of nodes with respect to the same graph distribution? Such problems are interesting and fundamental to many complex decision-making applications [60, 61].\nLimitations. While our method offers promising performance in experiments, it is possible that deep architectures can be designed in a sophisticated manner so as to achieve improved results. In addition, while we believe that similar observations also hold for graphs that are not considered in our experiment, more experimental studies are required to support the universal superiority of Social-Inverse. In another issue, the assumption that the training set contains approximation solutions is the minimal one for the purpose of theoretical analysis, but in practice, such guarantees may never be known. Therefore, experimenting with samples of heuristic query-decision pairs is needed to further justify the practical utility of our method. Finally, we have not experimented with graphs of extreme scales (e.g., over 1M nodes) due to the limit in memory. We wish to explore the above issues in future work.\nAcknowledgments and Disclosure of Funding\nWe thank the reviewers for their time and insightful comments. This work is supported in part by a) National Science Foundation under Award IIS-2144285 and b) the University of Delaware.",
  "Reviewer Summary": "Reviewer_4: The authors consider the problem of task migration for decision making tasks in social networks for the purpose of managing contagion events. The authors begin by formally defining a social network (and associated social contagions) as a directed graph of users, where a set of seed users initiate a social contagion, with contagions spreading through edges at a rate defined by a distribution on each edge.\nThe authors then define four different contagion management problems, the last of which is the task migration problem. In this setting, we have two contagion management tasks (source and target) on the same social network and have samples from the source task. The goal is to minimize a loss with respect to the target task using those source task samples.\nThe authors propose an algorithm for this task migration problem which they dub social-inverse. They bound the generalization error of social inverse and compare their algorithm in simulated experiments to supervised learning algorithms and a heuristic that selects high-degree users as seed nodes.\n\nReviewer_5: The paper considers a semi-opaque-box influence network and the cumulative impacts from information dissemination at a few seed nodes. The network is semi-opaque in the sense that we know the general connectivity of the nodes, but not the actual realization of the influence, which is sampled from an unknown distribution with support by the observed connections. To make up for the observability gap, the paper suggests to use previous examples of information diffusion, collected in the form of which targets to reach (queries) and which seeds would maximize the coverage of the targets after diffusion (decisions). It then proposes to use random samples of possible realizations to solve a max-margin (SVM) optimization problem to find out the most likely linear combination of the realizations. The paper finally uses the learned weighted combination of realizations to decide for optimal solutions in new information diffusion tasks. Theoretical and empirical studies were included.\n\nReviewer_6: This paper studies an inverse decision-making problem for task migrations on contagion management. Specifically, it investigates how prior decision-making on diffusion containment can be migrated to diffusion enhancement; and vice versa. It provides a theoretical analysis of how different diffusion management tasks can be correlated such that samples from one task can be useful for another task. It performs empirical analysis on four graphs, comparing with several benchmarks (used for supervised learning) to evaluate the performance."
}