The techniques for theℓ2-perturbation do not easily transfer to theℓ -case (Leino et al., 2021). One critical step is to measure the Lipschitz constant more precisely and efficiently.

Neural Information Processing Systems http://nips.cc/

How can local-search methods such as stochastic gradient descent (SGD) avoid bad local minima in training multi-layer neural networks? Why can they fit random labels even given non-convex and non-smooth architectures? Most existing theory only covers networks with one hidden layer, so can we go deeper? In this paper, we focus on recurrent neural networks (RNNs) which are multi-layer networks widely used in natural language processing. They are harder to analyze than feedforward neural networks, because the \emph{same} recurrent unit is repeatedly applied across the entire time horizon of length $L$, which is analogous to feedforward networks of depth $L$. We show when the number of neurons is sufficiently large, meaning polynomial in the training data size and in $L$, then SGD is capable of minimizing the regression loss in the linear convergence rate. This gives theoretical evidence of how RNNs can memorize data. More importantly, in this paper we build general toolkits to analyze multi-layer networks with ReLU activations. For instance, we prove why ReLU activations can prevent exponential gradient explosion or vanishing, and build a perturbation theory to analyze first-order approximation of multi-layer networks.

Neural Information Processing Systems http://nips.cc/

Intercepting a criminal using limited police resources presents a significant challenge in dynamic crime environments, where the criminal's location continuously changes over time. The complexity is further heightened by the vastness of the transportation network. To tackle this problem, we propose a layered graph representation, in which each time step is associated with a duplicate of the transportation network. For any given set of attacker strategies, a near-optimal defender strategy is computed using the A-Star heuristic algorithm applied to the layered graph. The defender's goal is to maximize the probability of successful interdiction. We evaluate the performance of the proposed method by comparing it with a Mixed-Integer Linear Programming (MILP) approach used for the defender. The comparison considers both computational efficiency and solution quality. The results demonstrate that our approach effectively addresses the complexity of the problem and delivers high-quality solutions within a short computation time.

Estimating the global Lipschitz constant of neural networks is crucial for understanding and improving their robustness and generalization capabilities. However, precise calculations are NP-hard, and current semidefinite programming (SDP) methods face challenges such as high memory usage and slow processing speeds. In this paper, we propose \textbf{HiQ-Lip}, a hybrid quantum-classical hierarchical method that leverages Coherent Ising Machines (CIMs) to estimate the global Lipschitz constant. We tackle the estimation by converting it into a Quadratic Unconstrained Binary Optimization (QUBO) problem and implement a multilevel graph coarsening and refinement strategy to adapt to the constraints of contemporary quantum hardware. Our experimental evaluations on fully connected neural networks demonstrate that HiQ-Lip not only provides estimates comparable to state-of-the-art methods but also significantly accelerates the computation process. In specific tests involving two-layer neural networks with 256 hidden neurons, HiQ-Lip doubles the solving speed and offers more accurate upper bounds than the existing best method, LiPopt. These findings highlight the promising utility of small-scale quantum devices in advancing the estimation of neural network robustness.

How can local-search methods such as stochastic gradient descent (SGD) avoid bad local minima in training multi-layer neural networks? Why can they fit random labels even given non-convex and non-smooth architectures? Most existing theory only covers networks with one hidden layer, so can we go deeper? In this paper, we focus on recurrent neural networks (RNNs) which are multi-layer networks widely used in natural language processing. They are harder to analyze than feedforward neural networks, because the \emph{same} recurrent unit is repeatedly applied across the entire time horizon of length L, which is analogous to feedforward networks of depth L .

Interdicting a criminal with limited police resources is a challenging task as the criminal changes location over time. The size of the large transportation network further adds to the difficulty of this scenario. To tackle this issue, we consider the concept of a layered graph. At each time stamp, we create a copy of the entire transportation network to track the possible movements of both players, the attacker and the defenders. We consider a Stackelberg game in a dynamic crime scenario where the attacker changes location over time while the defenders attempt to interdict the attacker on his escape route. Given a set of defender strategies, the optimal attacker strategy is determined by applying Dijkstra's algorithm on the layered networks. Here, the attacker aims to minimize while the defenders aim to maximize the probability of interdiction. We develop an approximation algorithm on the layered networks to find near-optimal strategy for defenders. The efficacy of the developed approach is compared with the adopted MILP approach. We compare the results in terms of computational time and solution quality. The quality of the results demonstrates the need for the developed approach, as it effectively solves the complex problem within a short amount of time.

Community detection in multi-layer networks has emerged as a crucial area of modern network analysis. However, conventional approaches often assume that nodes belong exclusively to a single community, which fails to capture the complex structure of real-world networks where nodes may belong to multiple communities simultaneously. To address this limitation, we propose novel spectral methods to estimate the common mixed memberships in the multi-layer mixed membership stochastic block model. The proposed methods leverage the eigen-decomposition of three aggregate matrices: the sum of adjacency matrices, the debiased sum of squared adjacency matrices, and the sum of squared adjacency matrices. We establish rigorous theoretical guarantees for the consistency of our methods. Specifically, we derive per-node error rates under mild conditions on network sparsity, demonstrating their consistency as the number of nodes and/or layers increases under the multi-layer mixed membership stochastic block model. Our theoretical results reveal that the method leveraging the sum of adjacency matrices generally performs poorer than the other two methods for mixed membership estimation in multi-layer networks. We conduct extensive numerical experiments to empirically validate our theoretical findings. For real-world multi-layer networks with unknown community information, we introduce two novel modularity metrics to quantify the quality of mixed membership community detection. Finally, we demonstrate the practical applications of our algorithms and modularity metrics by applying them to real-world multi-layer networks, demonstrating their effectiveness in extracting meaningful community structures.

Community detection in multi-layer networks is a crucial problem in network analysis. In this paper, we analyze the performance of two spectral clustering algorithms for community detection within the multi-layer degree-corrected stochastic block model (MLDCSBM) framework. One algorithm is based on the sum of adjacency matrices, while the other utilizes the debiased sum of squared adjacency matrices. We establish consistency results for community detection using these methods under MLDCSBM as the size of the network and/or the number of layers increases. Our theorems demonstrate the advantages of utilizing multiple layers for community detection. Moreover, our analysis indicates that spectral clustering with the debiased sum of squared adjacency matrices is generally superior to spectral clustering with the sum of adjacency matrices. Numerical simulations confirm that our algorithm, employing the debiased sum of squared adjacency matrices, surpasses existing methods for community detection in multi-layer networks. Finally, the analysis of several real-world multi-layer networks yields meaningful insights.