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In today's digital world, interaction with online platforms is ubiquitous, and thus content moderation is important for protecting users from content that do not comply with pre-established community guidelines. Having a robust content moderation system throughout every stage of planning is particularly important. We study the short-term planning problem of allocating human content reviewers to different harmful content categories. We use tools from fair division and study the application of competitive equilibrium and leximin allocation rules. Furthermore, we incorporate, to the traditional Fisher market setup, novel aspects that are of practical importance. The first aspect is the forecasted workload of different content categories. We show how a formulation that is inspired by the celebrated Eisenberg-Gale program allows us to find an allocation that not only satisfies the forecasted workload, but also fairly allocates the remaining reviewing hours among all content categories. The resulting allocation is also robust as the additional allocation provides a guardrail in cases where the actual workload deviates from the predicted workload. The second practical consideration is time dependent allocation that is motivated by the fact that partners need scheduling guidance for the reviewers across days to achieve efficiency. To address the time component, we introduce new extensions of the various fair allocation approaches for the single-time period setting, and we show that many properties extend in essence, albeit with some modifications. Related to the time component, we additionally investigate how to satisfy markets' desire for smooth allocation (e.g., partners for content reviewers prefer an allocation that does not vary much from time to time, to minimize staffing switch). We demonstrate the performance of our proposed approaches through real-world data obtained from Meta.

Kernel methods are ubiquitous in statistical modeling due to their theoretical guarantees as well as their competitive empirical performance. Polynomial kernels are of particular importance as their feature maps model the interactions between the dimensions of the input data. However, the construction time of explicit feature maps scales exponentially with the polynomial degree and a naive application of the kernel trick does not scale to large datasets. In this work, we propose Complex-to-Real (CtR) random features for polynomial kernels that leverage intermediate complex random projections and can yield kernel estimates with much lower variances than their real-valued analogs. The resulting features are real-valued, simple to construct and have the following advantages over the state-of-the-art: 1) shorter construction times, 2) lower kernel approximation errors for commonly used degrees, 3) they enable us to obtain a closed-form expression for their variance.

Leveraging prior knowledge on intraclass variance due to transformations is a powerful method to improve the sample complexity of deep neural networks. This makes them applicable to practically important use-cases where training data is scarce. Rather than being learned, this knowledge can be embedded by enforcing invariance to those transformations. Invariance can be imposed using group-equivariant convolutions followed by a pooling operation. For rotation-invariance, previous work investigated replacing the spatial pooling operation with invariant integration which explicitly constructs invariant representations. Invariant integration uses monomials which are selected using an iterative approach requiring expensive pre-training. We propose a novel monomial selection algorithm based on pruning methods to allow an application to more complex problems. Additionally, we replace monomials with different functions such as weighted sums, multi-layer perceptrons and self-attention, thereby streamlining the training of invariant-integration-based architectures. We demonstrate the improved sample complexity on the Rotated-MNIST, SVHN and CIFAR-10 datasets where rotation-invariant-integration-based Wide-ResNet architectures using monomials and weighted sums outperform the respective baselines in the limited sample regime. We achieve state-of-the-art results using full data on Rotated-MNIST and SVHN where rotation is a main source of intraclass variation. On STL-10 we outperform a standard and a rotation-equivariant convolutional neural network using pooling.

In ultra-dense millimeter wave (mmWave) networks, mmWave signals suffer from severe path losses and are easily blocked by obstacles. Meanwhile, ultra-dense deployment causes excessive handovers, which reduces the data link reliability. To alleviate the above issues, the novel technology, known as multi-connectivity enabled user association (MCUA) is incorporated in this letter. We aim to jointly optimize MCUAs and downlink (DL) power allocations (PAs) to maximize the DL rate of each user simultaneously, rather than total. This is a non-convex nonlinear 0-1 mixed integer multi-objective optimization problem and quite complicated. To solve it, we first use the weighted sum method to scalarize it as a single-objective optimization problem (SOOP), and then relax the binary association variables to real ones. Considering that the relaxed SOOP is still non-convex, we perform a series of transformations upon it and make it a differential of convex programming. Finally, we develop an iterative algorithm based on the convex-concave procedure to solve the SOOP. Numerical results are presented to demonstrate the effectiveness of the proposed algorithms.

We study ROUND-UFP and ROUND-SAP, two generalizations of the classical BIN PACKING problem that correspond to the unsplittable flow problem on a path (UFP) and the storage allocation problem (SAP), respectively. We are given a path with capacities on its edges and a set of tasks where for each task we are given a demand and a subpath. In ROUND-UFP, the goal is to find a packing of all tasks into a minimum number of copies (rounds) of the given path such that for each copy, the total demand of tasks on any edge does not exceed the capacity of the respective edge. In ROUND-SAP, the tasks are considered to be rectangles and the goal is to find a non-overlapping packing of these rectangles into a minimum number of rounds such that all rectangles lie completely below the capacity profile of the edges. We show that in contrast to BIN PACKING, both the problems do not admit an asymptotic polynomial-time approximation scheme (APTAS), even when all edge capacities are equal. However, for this setting, we obtain asymptotic $(2+\varepsilon)$-approximations for both problems. For the general case, we obtain an $O(\log\log n)$-approximation algorithm and an $O(\log\log\frac{1}{\delta})$-approximation under $(1+\delta)$-resource augmentation for both problems. For the intermediate setting of the no bottleneck assumption (i.e., the maximum task demand is at most the minimum edge capacity), we obtain absolute $12$- and asymptotic $(16+\varepsilon)$-approximation algorithms for ROUND-UFP and ROUND-SAP, respectively.

Graph Neural Networks (GNNs) have been studied from the lens of expressive power and generalization. However, their optimization properties are less well understood. We take the first step towards analyzing GNN training by studying the gradient dynamics of GNNs. First, we analyze linearized GNNs and prove that despite the non-convexity of training, convergence to a global minimum at a linear rate is guaranteed under mild assumptions that we validate on real-world graphs. Second, we study what may affect the GNNs' training speed. Our results show that the training of GNNs is implicitly accelerated by skip connections, more depth, and/or a good label distribution. Empirical results confirm that our theoretical results for linearized GNNs align with the training behavior of nonlinear GNNs. Our results provide the first theoretical support for the success of GNNs with skip connections in terms of optimization, and suggest that deep GNNs with skip connections would be promising in practice.

Despite the recent success of graph neural networks (GNN), common architectures often exhibit significant limitations, including sensitivity to oversmoothing, long-range dependencies, and spurious edges, e.g., as can occur as a result of graph heterophily or adversarial attacks. To at least partially address these issues within a simple transparent framework, we consider a new family of GNN layers designed to mimic and integrate the update rules of two classical iterative algorithms, namely, proximal gradient descent and iterative reweighted least squares (IRLS). The former defines an extensible base GNN architecture that is immune to oversmoothing while nonetheless capturing long-range dependencies by allowing arbitrary propagation steps. In contrast, the latter produces a novel attention mechanism that is explicitly anchored to an underlying end-toend energy function, contributing stability with respect to edge uncertainty. When combined we obtain an extremely simple yet robust model that we evaluate across disparate scenarios including standardized benchmarks, adversarially-perturbated graphs, graphs with heterophily, and graphs involving long-range dependencies. In doing so, we compare against SOTA GNN approaches that have been explicitly designed for the respective task, achieving competitive or superior node classification accuracy.

Modeling multivariate time series has long been a subject that has attracted researchers from a diverse range of fields including economics, finance, and traffic. A basic assumption behind multivariate time series forecasting is that its variables depend on one another but, upon looking closely, it is fair to say that existing methods fail to fully exploit latent spatial dependencies between pairs of variables. In recent years, meanwhile, graph neural networks (GNNs) have shown high capability in handling relational dependencies. GNNs require well-defined graph structures for information propagation which means they cannot be applied directly for multivariate time series where the dependencies are not known in advance. In this paper, we propose a general graph neural network framework designed specifically for multivariate time series data. Our approach automatically extracts the uni-directed relations among variables through a graph learning module, into which external knowledge like variable attributes can be easily integrated. A novel mix-hop propagation layer and a dilated inception layer are further proposed to capture the spatial and temporal dependencies within the time series. The graph learning, graph convolution, and temporal convolution modules are jointly learned in an end-to-end framework. Experimental results show that our proposed model outperforms the state-of-the-art baseline methods on 3 of 4 benchmark datasets and achieves on-par performance with other approaches on two traffic datasets which provide extra structural information.

Graph Neural Networks (GNNs) for representation learning of graphs broadly follow a neighborhood aggregation framework, where the representation vector of a node is computed by recursively aggregating and transforming feature vectors of its neighboring nodes. Many GNN variants have been proposed and have achieved state-of-the-art results on both node and graph classification tasks. However, despite GNNs revolutionizing graph representation learning, there is limited understanding of their representational properties and limitations. Here, we present a theoretical framework for analyzing the expressive power of GNNs in capturing different graph structures. Our results characterize the discriminative power of popular GNN variants, such as Graph Convolutional Networks and GraphSAGE, and show that they cannot learn to distinguish certain simple graph structures. We then develop a simple architecture that is provably the most expressive among the class of GNNs and is as powerful as the Weisfeiler-Lehman graph isomorphism test. We empirically validate our theoretical findings on a number of graph classification benchmarks, and demonstrate that our model achieves state-of-the-art performance.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.