arXiv subjects

Alejandro Ribeiro

Publications and source records attributed to Alejandro Ribeiro.

At least 163 records · Page 9Linked to original sources

arXiv2021

Stability of Neural Networks on Manifolds to Relative Perturbations

Graph Neural Networks (GNNs) show impressive performance in many practical scenarios, which can be largely attributed to their stability properties. Empirically, GNNs can scale well on large size graphs, but this is contradicted by the fact that existing stability bounds grow with the number of nodes. Graphs with well-defined limits can be seen as samples from manifolds. Hence, in this paper, we analyze the stability properties of convolutional neural networks on manifolds to understand the stability of GNNs on large graphs. Specifically, we focus on stability to relative perturbations of the Laplace-Beltrami operator. To start, we construct frequency ratio threshold filters which separate the infinite-dimensional spectrum of the Laplace-Beltrami operator. We then prove that manifold neural networks composed of these filters are stable to relative operator perturbations. As a product of this analysis, we observe that manifold neural networks exhibit a trade-off between stability and discriminability. Finally, we illustrate our results empirically in a wireless resource allocation scenario where the transmitter-receiver pairs are assumed to be sampled from a manifold.

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arXiv2021

Stable and Transferable Wireless Resource Allocation Policies via Manifold Neural Networks

We consider the problem of resource allocation in large scale wireless networks. When contextualizing wireless network structures as graphs, we can model the limits of very large wireless systems as manifolds. To solve the problem in the machine learning framework, we propose the use of Manifold Neural Networks (MNNs) as a policy parametrization. In this work, we prove the stability of MNN resource allocation policies under the absolute perturbations to the Laplace-Beltrami operator of the manifold, representing system noise and dynamics present in wireless systems. These results establish the use of MNNs in achieving stable and transferable allocation policies for large scale wireless networks. We verify our results in numerical simulations that show superior performance relative to baseline methods.

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arXiv2021

Coverage Control in Multi-Robot Systems via Graph Neural Networks

This paper develops a decentralized approach to mobile sensor coverage by a multi-robot system. We consider a scenario where a team of robots with limited sensing range must position itself to effectively detect events of interest in a region characterized by areas of varying importance. Towards this end, we develop a decentralized control policy for the robots -- realized via a Graph Neural Network -- which uses inter-robot communication to leverage non-local information for control decisions. By explicitly sharing information between multi-hop neighbors, the decentralized controller achieves a higher quality of coverage when compared to classical approaches that do not communicate and leverage only local information available to each robot. Simulated experiments demonstrate the efficacy of multi-hop communication for multi-robot coverage and evaluate the scalability and transferability of the learning-based controllers.

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arXiv2021

Multi-Robot Coverage and Exploration using Spatial Graph Neural Networks

The multi-robot coverage problem is an essential building block for systems that perform tasks like inspection or search and rescue. We discretize the coverage problem to induce a spatial graph of locations and represent robots as nodes in the graph. Then, we train a Graph Neural Network controller that leverages the spatial equivariance of the task to imitate an expert open-loop routing solution. This approach generalizes well to much larger maps and larger teams that are intractable for the expert. In particular, the model generalizes effectively to a simulation of ten quadrotors and dozens of buildings. We also demonstrate the GNN controller can surpass planning-based approaches in an exploration task.

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arXiv2021

Learning Connectivity for Data Distribution in Robot Teams

Many algorithms for control of multi-robot teams operate under the assumption that low-latency, global state information necessary to coordinate agent actions can readily be disseminated among the team. However, in harsh environments with no existing communication infrastructure, robots must form ad-hoc networks, forcing the team to operate in a distributed fashion. To overcome this challenge, we propose a task-agnostic, decentralized, low-latency method for data distribution in ad-hoc networks using Graph Neural Networks (GNN). Our approach enables multi-agent algorithms based on global state information to function by ensuring it is available at each robot. To do this, agents glean information about the topology of the network from packet transmissions and feed it to a GNN running locally which instructs the agent when and where to transmit the latest state information. We train the distributed GNN communication policies via reinforcement learning using the average Age of Information as the reward function and show that it improves training stability compared to task-specific reward functions. Our approach performs favorably compared to industry-standard methods for data distribution such as random flooding and round robin. We also show that the trained policies generalize to larger teams of both static and mobile agents.

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arXiv2021

EdgeNets:Edge Varying Graph Neural Networks

Driven by the outstanding performance of neural networks in the structured Euclidean domain, recent years have seen a surge of interest in developing neural networks for graphs and data supported on graphs. The graph is leveraged at each layer of the neural network as a parameterization to capture detail at the node level with a reduced number of parameters and computational complexity. Following this rationale, this paper puts forth a general framework that unifies state-of-the-art graph neural networks (GNNs) through the concept of EdgeNet. An EdgeNet is a GNN architecture that allows different nodes to use different parameters to weigh the information of different neighbors. By extrapolating this strategy to more iterations between neighboring nodes, the EdgeNet learns edge- and neighbor-dependent weights to capture local detail. This is a general linear and local operation that a node can perform and encompasses under one formulation all existing graph convolutional neural networks (GCNNs) as well as graph attention networks (GATs). In writing different GNN architectures with a common language, EdgeNets highlight specific architecture advantages and limitations, while providing guidelines to improve their capacity without compromising their local implementation. An interesting conclusion is the unification of GCNNs and GATs -- approaches that have been so far perceived as separate. In particular, we show that GATs are GCNNs on a graph that is learned from the features. This particularization opens the doors to develop alternative attention mechanisms for improving discriminatory power.

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arXiv2021

Algebraic Neural Networks: Stability to Deformations

We study algebraic neural networks (AlgNNs) with commutative algebras which unify diverse architectures such as Euclidean convolutional neural networks, graph neural networks, and group neural networks under the umbrella of algebraic signal processing. An AlgNN is a stacked layered information processing structure where each layer is conformed by an algebra, a vector space and a homomorphism between the algebra and the space of endomorphisms of the vector space. Signals are modeled as elements of the vector space and are processed by convolutional filters that are defined as the images of the elements of the algebra under the action of the homomorphism. We analyze stability of algebraic filters and AlgNNs to deformations of the homomorphism and derive conditions on filters that lead to Lipschitz stable operators. We conclude that stable algebraic filters have frequency responses -- defined as eigenvalue domain representations -- whose derivative is inversely proportional to the frequency -- defined as eigenvalue magnitudes. It follows that for a given level of discriminability, AlgNNs are more stable than algebraic filters, thereby explaining their better empirical performance. This same phenomenon has been proven for Euclidean convolutional neural networks and graph neural networks. Our analysis shows that this is a deep algebraic property shared by a number of architectures.

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arXiv2021

Stability of Graph Convolutional Neural Networks to Stochastic Perturbations

Graph convolutional neural networks (GCNNs) are nonlinear processing tools to learn representations from network data. A key property of GCNNs is their stability to graph perturbations. Current analysis considers deterministic perturbations but fails to provide relevant insights when topological changes are random. This paper investigates the stability of GCNNs to stochastic graph perturbations induced by link losses. In particular, it proves the expected output difference between the GCNN over random perturbed graphs and the GCNN over the nominal graph is upper bounded by a factor that is linear in the link loss probability. We perform the stability analysis in the graph spectral domain such that the result holds uniformly for any graph. This result also shows the role of the nonlinearity and the architecture width and depth, and allows identifying handle to improve the GCNN robustness. Numerical simulations on source localization and robot swarm control corroborate our theoretical findings.

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arXiv2021

Stochastic Graph Neural Networks

Graph neural networks (GNNs) model nonlinear representations in graph data with applications in distributed agent coordination, control, and planning among others. Current GNN architectures assume ideal scenarios and ignore link fluctuations that occur due to environment, human factors, or external attacks. In these situations, the GNN fails to address its distributed task if the topological randomness is not considered accordingly. To overcome this issue, we put forth the stochastic graph neural network (SGNN) model: a GNN where the distributed graph convolution module accounts for the random network changes. Since stochasticity brings in a new learning paradigm, we conduct a statistical analysis on the SGNN output variance to identify conditions the learned filters should satisfy for achieving robust transference to perturbed scenarios, ultimately revealing the explicit impact of random link losses. We further develop a stochastic gradient descent (SGD) based learning process for the SGNN and derive conditions on the learning rate under which this learning process converges to a stationary point. Numerical results corroborate our theoretical findings and compare the benefits of SGNN robust transference with a conventional GNN that ignores graph perturbations during learning.

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arXiv2021

Training Robust Graph Neural Networks with Topology Adaptive Edge Dropping

Graph neural networks (GNNs) are processing architectures that exploit graph structural information to model representations from network data. Despite their success, GNNs suffer from sub-optimal generalization performance given limited training data, referred to as over-fitting. This paper proposes Topology Adaptive Edge Dropping (TADropEdge) method as an adaptive data augmentation technique to improve generalization performance and learn robust GNN models. We start by explicitly analyzing how random edge dropping increases the data diversity during training, while indicating i.i.d. edge dropping does not account for graph structural information and could result in noisy augmented data degrading performance. To overcome this issue, we consider graph connectivity as the key property that captures graph topology. TADropEdge incorporates this factor into random edge dropping such that the edge-dropped subgraphs maintain similar topology as the underlying graph, yielding more satisfactory data augmentation. In particular, TADropEdge first leverages the graph spectrum to assign proper weights to graph edges, which represent their criticality for establishing the graph connectivity. It then normalizes the edge weights and drops graph edges adaptively based on their normalized weights. Besides improving generalization performance, TADropEdge reduces variance for efficient training and can be applied as a generic method modular to different GNN models. Intensive experiments on real-life and synthetic datasets corroborate theory and verify the effectiveness of the proposed method.

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arXiv2021

Multi-task Supervised Learning via Cross-learning

In this paper we consider a problem known as multi-task learning, consisting of fitting a set of classifier or regression functions intended for solving different tasks. In our novel formulation, we couple the parameters of these functions, so that they learn in their task specific domains while staying close to each other. This facilitates cross-fertilization in which data collected across different domains help improving the learning performance at each other task. First, we present a simplified case in which the goal is to estimate the means of two Gaussian variables, for the purpose of gaining some insights on the advantage of the proposed cross-learning strategy. Then we provide a stochastic projected gradient algorithm to perform cross-learning over a generic loss function. If the number of parameters is large, then the projection step becomes computationally expensive. To avoid this situation, we derive a primal-dual algorithm that exploits the structure of the dual problem, achieving a formulation whose complexity only depends on the number of tasks. Preliminary numerical experiments for image classification by neural networks trained on a dataset divided in different domains corroborate that the cross-learned function outperforms both the task-specific and the consensus approaches.

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arXiv2021

Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks

Network data can be conveniently modeled as a graph signal, where data values are assigned to nodes of a graph that describes the underlying network topology. Successful learning from network data is built upon methods that effectively exploit this graph structure. In this work, we leverage graph signal processing to characterize the representation space of graph neural networks (GNNs). We discuss the role of graph convolutional filters in GNNs and show that any architecture built with such filters has the fundamental properties of permutation equivariance and stability to changes in the topology. These two properties offer insight about the workings of GNNs and help explain their scalability and transferability properties which, coupled with their local and distributed nature, make GNNs powerful tools for learning in physical networks. We also introduce GNN extensions using edge-varying and autoregressive moving average graph filters and discuss their properties. Finally, we study the use of GNNs in recommender systems and learning decentralized controllers for robot swarms.

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arXiv2021

Graph and graphon neural network stability

Graph neural networks (GNNs) are learning architectures that rely on knowledge of the graph structure to generate meaningful representations of large-scale network data. GNN stability is thus important as in real-world scenarios there are typically uncertainties associated with the graph. We analyze GNN stability using kernel objects called graphons. Graphons are both limits of convergent graph sequences and generating models for deterministic and stochastic graphs. Building upon the theory of graphon signal processing, we define graphon neural networks and analyze their stability to graphon perturbations. We then extend this analysis by interpreting the graphon neural network as a generating model for GNNs on deterministic and stochastic graphs instantiated from the original and perturbed graphons. We observe that GNNs are stable to graphon perturbations with a stability bound that decreases asymptotically with the size of the graph. This asymptotic behavior is further demonstrated in an experiment of movie recommendation.

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arXiv2021

Resource Allocation via Model-Free Deep Learning in Free Space Optical Communications

This paper investigates the general problem of resource allocation for mitigating channel fading effects in Free Space Optical (FSO) communications. The resource allocation problem is modeled as the constrained stochastic optimization framework, which covers a variety of FSO scenarios involving power adaptation, relay selection and their joint allocation. Under this framework, we propose two algorithms that solve FSO resource allocation problems. We first present the Stochastic Dual Gradient (SDG) algorithm that is shown to solve the problem exactly by exploiting the strong duality but whose implementation necessarily requires explicit and accurate system models. As an alternative we present the Primal-Dual Deep Learning (PDDL) algorithm based on the SDG algorithm, which parametrizes the resource allocation policy with Deep Neural Networks (DNNs) and optimizes the latter via a primal-dual method. The parametrized resource allocation problem incurs only a small loss of optimality due to the strong representational power of DNNs, and can be moreover implemented without knowledge of system models. A wide set of numerical experiments are performed to corroborate the proposed algorithms in FSO resource allocation problems. We demonstrate their superior performance and computational efficiency compared to the baseline methods in both continuous power allocation and binary relay selection settings.

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arXiv2021

Spherical Convolutional Neural Networks: Stability to Perturbations in SO(3)

Spherical convolutional neural networks (Spherical CNNs) learn nonlinear representations from 3D data by exploiting the data structure and have shown promising performance in shape analysis, object classification, and planning among others. This paper investigates the properties that Spherical CNNs exhibit as they pertain to the rotational structure inherent in spherical signals. We build upon the rotation equivariance of spherical convolutions to show that Spherical CNNs are stable to general structure perturbations. In particular, we model arbitrary structure perturbations as diffeomorphism perturbations, and define the rotation distance that measures how far from rotations these perturbations are. We prove that the output change of a Spherical CNN induced by the diffeomorphism perturbation is bounded proportionally by the perturbation size under the rotation distance. This stability property coupled with the rotation equivariance provide theoretical guarantees that underpin the practical observations that Spherical CNNs exploit the rotational structure, maintain performance under structure perturbations that are close to rotations, and offer good generalization and faster learning.

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arXiv2021

Approximately Supermodular Scheduling Subject to Matroid Constraints

Control scheduling refers to the problem of assigning agents or actuators to act upon a dynamical system at specific times so as to minimize a quadratic control cost, such as the objective of the Linear-quadratic-Gaussian (LQG) or the Linear Quadratic Regulator (LQR). When budget or operational constraints are imposed on the schedule, this problem is in general NP-hard and its solution can therefore only be approximated even for moderately sized systems. The quality of this approximation depends on the structure of both the constraints and the objective. This work shows that greedy scheduling is near-optimal when the constraints can be represented as an intersection of matroids, algebraic structures that encode requirements such as limits on the number of agents deployed per time slot, total number of actuator uses, and duty cycle restrictions. To do so, it proves that the LQG cost function is alpha-supermodular and provides a new alpha/(alpha + P)-optimality certificates for the greedy minimization of such functions over an intersections of P matroids. These certificates are shown to approach the 1/(1+P) guarantee of supermodular functions in relevant settings. These results support the use of greedy algorithms in non-supermodular quadratic control problems as opposed to typical heuristics such as convex relaxations and surrogate figures of merit, e.g., the logdet of the controllability Gramian.

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arXiv2021

Composable Learning with Sparse Kernel Representations

We present a reinforcement learning algorithm for learning sparse non-parametric controllers in a Reproducing Kernel Hilbert Space. We improve the sample complexity of this approach by imposing a structure of the state-action function through a normalized advantage function (NAF). This representation of the policy enables efficiently composing multiple learned models without additional training samples or interaction with the environment. We demonstrate the performance of this algorithm on learning obstacle-avoidance policies in multiple simulations of a robot equipped with a laser scanner while navigating in a 2D environment. We apply the composition operation to various policy combinations and test them to show that the composed policies retain the performance of their components. We also transfer the composed policy directly to a physical platform operating in an arena with obstacles in order to demonstrate a degree of generalization.

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arXiv2021

Stability of Neural Networks on Riemannian Manifolds

Convolutional Neural Networks (CNNs) have been applied to data with underlying non-Euclidean structures and have achieved impressive successes. This brings the stability analysis of CNNs on non-Euclidean domains into notice because CNNs have been proved stable on Euclidean domains. This paper focuses on the stability of CNNs on Riemannian manifolds. By taking the Laplace-Beltrami operators into consideration, we construct an $α$-frequency difference threshold filter to help separate the spectrum of the operator with an infinite dimensionality. We further construct a manifold neural network architecture with these filters. We prove that both the manifold filters and neural networks are stable under absolute perturbations to the operators. The results also implicate a trade-off between the stability and discriminability of manifold neural networks. Finally we verify our conclusions with numerical experiments in a wireless adhoc network scenario.

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