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We describe a model for polarization in multi-agent systems based on Esteban and Ray's standard family of polarization measures from economics. Agents evolve by updating their beliefs (opinions) based on an underlying influence graph, as in the standard DeGroot model for social learning, but under a confirmation bias; i.e., a discounting of opinions of agents with dissimilar views. We show that even under this bias polarization eventually vanishes (converges to zero) if the influence graph is strongly-connected. If the influence graph is a regular symmetric circulation, we determine the unique belief value to which all agents converge. Our more insightful result establishes that, under some natural assumptions, if polarization does not eventually vanish then either there is a disconnected subgroup of agents, or some agent influences others more than she is influenced. We also prove that polarization does not necessarily vanish in weakly-connected graphs under confirmation bias. Furthermore, we show how our model relates to the classic DeGroot model for social learning. We illustrate our model with several simulations of a running example about polarization over vaccines and of other case studies. The theoretical results and simulations will provide insight into the phenomenon of polarization.

We consider the problem of \textit{Influence Maximization} (IM), the task of selecting $k$ seed nodes in a social network such that the expected number of nodes influenced is maximized. We propose a community-aware divide-and-conquer framework that involves (i) learning the inherent community structure of the social network, (ii) generating candidate solutions by solving the influence maximization problem for each community, and (iii) selecting the final set of seed nodes using a novel progressive budgeting scheme. Our experiments on real-world social networks show that the proposed framework outperforms the standard methods in terms of run-time and the heuristic methods in terms of influence. We also study the effect of the community structure on the performance of the proposed framework. Our experiments show that the community structures with higher modularity lead the proposed framework to perform better in terms of run-time and influence.

We consider the classical fair division problem which studies how to allocate resources fairly and efficiently. We give a complete landscape on the computational complexity and approximability of maximizing the social welfare within (1) envy-free up to any item (EFX) and (2) envy-free up to one item (EF1) allocations of indivisible goods for both normalized and unnormalized valuations. We show that a partial EFX allocation may have a higher social welfare than a complete EFX allocation, while it is well-known that this is not true for EF1 allocations. Thus, our first group of results focuses on the problem of maximizing social welfare subject to (partial) EFX allocations. For $n=2$ agents, we provide a polynomial time approximation scheme (PTAS) and an NP-hardness result. For a general number of agents $n>2$, we present algorithms that achieve approximation ratios of $O(n)$ and $O(\sqrt{n})$ for unnormalized and normalized valuations, respectively. These results are complemented by the asymptotically tight inapproximability results. We also study the same constrained optimization problem for EF1. For $n=2$, we show a fully polynomial time approximation scheme (FPTAS) and complement this positive result with an NP-hardness result. For general $n$, we present polynomial inapproximability ratios for both normalized and unnormalized valuations. Our results also imply the price of EFX is $\Theta(\sqrt{n})$ for normalized valuations, which is unknown in the previous literature.

In this paper, we study nonparametric estimation of instrumental variable (IV) regressions. Recently, many flexible machine learning methods have been developed for instrumental variable estimation. However, these methods have at least one of the following limitations: (1) restricting the IV regression to be uniquely identified; (2) only obtaining estimation error rates in terms of pseudometrics (\emph{e.g.,} projected norm) rather than valid metrics (\emph{e.g.,} $L_2$ norm); or (3) imposing the so-called closedness condition that requires a certain conditional expectation operator to be sufficiently smooth. In this paper, we present the first method and analysis that can avoid all three limitations, while still permitting general function approximation. Specifically, we propose a new penalized minimax estimator that can converge to a fixed IV solution even when there are multiple solutions, and we derive a strong $L_2$ error rate for our estimator under lax conditions. Notably, this guarantee only needs a widely-used source condition and realizability assumptions, but not the so-called closedness condition. We argue that the source condition and the closedness condition are inherently conflicting, so relaxing the latter significantly improves upon the existing literature that requires both conditions. Our estimator can achieve this improvement because it builds on a novel formulation of the IV estimation problem as a constrained optimization problem.

In this work, we present an alternative formulation of the higher eigenvalue problem associated to the infinity Laplacian, which opens the door for numerical approximation of eigenfunctions. A rigorous analysis is performed to show the equivalence of the new formulation to the traditional one. Subsequently, we present consistent monotone schemes to approximate infinity ground states and higher eigenfunctions on grids. We prove that our method converges (up to a subsequence) to a viscosity solution of the eigenvalue problem, and perform numerical experiments which investigate theoretical conjectures and compute eigenfunctions on a variety of different domains.

In recent years, recommender systems have advanced rapidly, where embedding learning for users and items plays a critical role. A standard method learns a unique embedding vector for each user and item. However, such a method has two important limitations in real-world applications: 1) it is hard to learn embeddings that generalize well for users and items with rare interactions on their own; and 2) it may incur unbearably high memory costs when the number of users and items scales up. Existing approaches either can only address one of the limitations or have flawed overall performances. In this paper, we propose Clustered Embedding Learning (CEL) as an integrated solution to these two problems. CEL is a plug-and-play embedding learning framework that can be combined with any differentiable feature interaction model. It is capable of achieving improved performance, especially for cold users and items, with reduced memory cost. CEL enables automatic and dynamic clustering of users and items in a top-down fashion, where clustered entities jointly learn a shared embedding. The accelerated version of CEL has an optimal time complexity, which supports efficient online updates. Theoretically, we prove the identifiability and the existence of a unique optimal number of clusters for CEL in the context of nonnegative matrix factorization. Empirically, we validate the effectiveness of CEL on three public datasets and one business dataset, showing its consistently superior performance against current state-of-the-art methods. In particular, when incorporating CEL into the business model, it brings an improvement of $+0.6\%$ in AUC, which translates into a significant revenue gain; meanwhile, the size of the embedding table gets $2650$ times smaller.

In recent years, Graph Neural Networks (GNNs), which can naturally integrate node information and topological structure, have been demonstrated to be powerful in learning on graph data. These advantages of GNNs provide great potential to advance social recommendation since data in social recommender systems can be represented as user-user social graph and user-item graph; and learning latent factors of users and items is the key. However, building social recommender systems based on GNNs faces challenges. For example, the user-item graph encodes both interactions and their associated opinions; social relations have heterogeneous strengths; users involve in two graphs (e.g., the user-user social graph and the user-item graph). To address the three aforementioned challenges simultaneously, in this paper, we present a novel graph neural network framework (GraphRec) for social recommendations. In particular, we provide a principled approach to jointly capture interactions and opinions in the user-item graph and propose the framework GraphRec, which coherently models two graphs and heterogeneous strengths. Extensive experiments on two real-world datasets demonstrate the effectiveness of the proposed framework GraphRec. Our code is available at \url{//github.com/wenqifan03/GraphRec-WWW19}

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.