Graph neural networks (GNNs) have recently been demonstrated to perform well on a variety of network-based tasks such as decentralized control and resource allocation, and provide computationally efficient methods for these tasks which have traditionally been challenging in that regard. However, like many neural-network based systems, GNNs are susceptible to shifts and perturbations on their inputs, which can include both node attributes and graph structure. In order to make them more useful for real-world applications, it is important to ensure their robustness post-deployment. Motivated by controlling the Lipschitz constant of GNN filters with respect to the node attributes, we propose to constrain the frequency response of the GNN's filter banks. We extend this formulation to the dynamic graph setting using a continuous frequency response constraint, and solve a relaxed variant of the problem via the scenario approach. This allows for the use of the same computationally efficient algorithm on sampled constraints, which provides PAC-style guarantees on the stability of the GNN using results in scenario optimization. We also highlight an important connection between this setup and GNN stability to graph perturbations, and provide experimental results which demonstrate the efficacy and broadness of our approach.
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Neural implicit fields have recently emerged as a useful representation for 3D shapes. These fields are commonly represented as neural networks which map latent descriptors and 3D coordinates to implicit function values. The latent descriptor of a neural field acts as a deformation handle for the 3D shape it represents. Thus, smoothness with respect to this descriptor is paramount for performing shape-editing operations. In this work, we introduce a novel regularization designed to encourage smooth latent spaces in neural fields by penalizing the upper bound on the field's Lipschitz constant. Compared with prior Lipschitz regularized networks, ours is computationally fast, can be implemented in four lines of code, and requires minimal hyperparameter tuning for geometric applications. We demonstrate the effectiveness of our approach on shape interpolation and extrapolation as well as partial shape reconstruction from 3D point clouds, showing both qualitative and quantitative improvements over existing state-of-the-art and non-regularized baselines.
In this work, we consider the problem of jointly minimizing the average cost of sampling and transmitting status updates by users over a wireless channel subject to average Age of Information (AoI) constraints. Errors in the transmission may occur and a scheduling policy has to decide if the users sample a new packet or attempt for retransmission of the packet sampled previously. The cost consists of both sampling and transmission costs. The sampling of a new packet after a failure imposes an additional cost on the system. We formulate a stochastic optimization problem with the average cost in the objective under average AoI constraints. To solve this problem, we propose three scheduling policies; a) a dynamic policy, that is centralized and requires full knowledge of the state of the system, b) two stationary randomized policies that require no knowledge of the state of the system. We utilize tools from Lyapunov optimization theory in order to provide the dynamic policy, and we prove that its solution is arbitrary close to the optimal one. In order to provide the randomized policies, we model the system by utilizing Discrete Time Markov Chain (DTMC). We provide closed-form and approximated expressions for the average AoI and its distribution, for each randomized policy. Simulation results show the importance of providing the option to transmit an old packet in order to minimize the total average cost.
This paper proposes a new regularization technique for reinforcement learning (RL) towards making policy and value functions smooth and stable. RL is known for the instability of the learning process and the sensitivity of the acquired policy to noise. Several methods have been proposed to resolve these problems, and in summary, the smoothness of policy and value functions learned mainly in RL contributes to these problems. However, if these functions are extremely smooth, their expressiveness would be lost, resulting in not obtaining the global optimal solution. This paper therefore considers RL under local Lipschitz continuity constraint, so-called L2C2. By designing the spatio-temporal locally compact space for L2C2 from the state transition at each time step, the moderate smoothness can be achieved without loss of expressiveness. Numerical noisy simulations verified that the proposed L2C2 outperforms the task performance while smoothing out the robot action generated from the learned policy.
Generative adversarial networks (GANs) are so complex that the existing learning theories do not provide a satisfactory explanation for why GANs have great success in practice. The same situation also remains largely open for deep neural networks. To fill this gap, we introduce a Lipschitz theory to analyze generalization. We demonstrate its simplicity by analyzing generalization and consistency of overparameterized neural networks. We then use this theory to derive Lipschitz-based generalization bounds for GANs. Our bounds show that penalizing the Lipschitz constant of the GAN loss can improve generalization. This result answers the long mystery of why the popular use of Lipschitz constraint for GANs often leads to great success, empirically without a solid theory. Finally but surprisingly, we show that, when using Dropout or spectral normalization, both \emph{truly deep} neural networks and GANs can generalize well without the curse of dimensionality.
Causality can be described in terms of a structural causal model (SCM) that carries information on the variables of interest and their mechanistic relations. For most processes of interest the underlying SCM will only be partially observable, thus causal inference tries to leverage any exposed information. Graph neural networks (GNN) as universal approximators on structured input pose a viable candidate for causal learning, suggesting a tighter integration with SCM. To this effect we present a theoretical analysis from first principles that establishes a novel connection between GNN and SCM while providing an extended view on general neural-causal models. We then establish a new model class for GNN-based causal inference that is necessary and sufficient for causal effect identification. Our empirical illustration on simulations and standard benchmarks validate our theoretical proofs.
We consider the problem of knowledge transfer when an agent is facing a series of Reinforcement Learning (RL) tasks. We introduce a novel metric between Markov Decision Processes and establish that close MDPs have close optimal value functions. Formally, the optimal value functions are Lipschitz continuous with respect to the tasks space. These theoretical results lead us to a value transfer method for Lifelong RL, which we use to build a PAC-MDP algorithm with improved convergence rate. We illustrate the benefits of the method in Lifelong RL experiments.
Graph Convolutional Networks (GCNs) have recently become the primary choice for learning from graph-structured data, superseding hash fingerprints in representing chemical compounds. However, GCNs lack the ability to take into account the ordering of node neighbors, even when there is a geometric interpretation of the graph vertices that provides an order based on their spatial positions. To remedy this issue, we propose Geometric Graph Convolutional Network (geo-GCN) which uses spatial features to efficiently learn from graphs that can be naturally located in space. Our contribution is threefold: we propose a GCN-inspired architecture which (i) leverages node positions, (ii) is a proper generalisation of both GCNs and Convolutional Neural Networks (CNNs), (iii) benefits from augmentation which further improves the performance and assures invariance with respect to the desired properties. Empirically, geo-GCN outperforms state-of-the-art graph-based methods on image classification and chemical tasks.
This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.
A fundamental computation for statistical inference and accurate decision-making is to compute the marginal probabilities or most probable states of task-relevant variables. Probabilistic graphical models can efficiently represent the structure of such complex data, but performing these inferences is generally difficult. Message-passing algorithms, such as belief propagation, are a natural way to disseminate evidence amongst correlated variables while exploiting the graph structure, but these algorithms can struggle when the conditional dependency graphs contain loops. Here we use Graph Neural Networks (GNNs) to learn a message-passing algorithm that solves these inference tasks. We first show that the architecture of GNNs is well-matched to inference tasks. We then demonstrate the efficacy of this inference approach by training GNNs on a collection of graphical models and showing that they substantially outperform belief propagation on loopy graphs. Our message-passing algorithms generalize out of the training set to larger graphs and graphs with different structure.
Network Virtualization is one of the most promising technologies for future networking and considered as a critical IT resource that connects distributed, virtualized Cloud Computing services and different components such as storage, servers and application. Network Virtualization allows multiple virtual networks to coexist on same shared physical infrastructure simultaneously. One of the crucial keys in Network Virtualization is Virtual Network Embedding, which provides a method to allocate physical substrate resources to virtual network requests. In this paper, we investigate Virtual Network Embedding strategies and related issues for resource allocation of an Internet Provider(InP) to efficiently embed virtual networks that are requested by Virtual Network Operators(VNOs) who share the same infrastructure provided by the InP. In order to achieve that goal, we design a heuristic Virtual Network Embedding algorithm that simultaneously embeds virtual nodes and virtual links of each virtual network request onto physic infrastructure. Through extensive simulations, we demonstrate that our proposed scheme improves significantly the performance of Virtual Network Embedding by enhancing the long-term average revenue as well as acceptance ratio and resource utilization of virtual network requests compared to prior algorithms.
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