Recently, a spate of papers have provided positive theoretical results for training over-parameterized neural networks (where the network size is larger than what is needed to achieve low error). The key insight is that with sufficient over-parameterization, gradient-based methods will implicitly leave some components of the network relatively unchanged, so the optimization dynamics will behave as if those components are essentially fixed at their initial random values. In fact, fixing these \emph{explicitly} leads to the well-known approach of learning with random features (e.g.

A neural network model of a differential equation, namely neural ODE, has enabled the learning of continuous-time dynamical systems and probabilistic distributions with high accuracy. The neural ODE uses the same network repeatedly during a numerical integration. The memory consumption of the backpropagation algorithm is proportional to the number of uses times the network size. This is true even if a checkpointing scheme divides the computation graph into sub-graphs.

It is well-known that modern neural networks are vulnerable to adversarial examples. To mitigate this problem, a series of robust learning algorithms have been proposed. However, although the robust training error can be near zero via some methods, all existing algorithms lead to a high robust generalization error. In this paper, we provide a theoretical understanding of this puzzling phenomenon from the perspective of expressive power for deep neural networks. Specifically, for binary classification problems with well-separated data, we show that, for ReLU networks, while mild over-parameterization is sufficient for high robust training accuracy, there exists a constant robust generalization gap unless the size of the neural network is exponential in the data dimension $d$. This result holds even if the data is linear separable (which means achieving standard generalization is easy), and more generally for any parameterized function classes as long as their VC dimension is at most polynomial in the number of parameters. Moreover, we establish an improved upper bound of $\exp({\mathcal{O}}(k))$ for the network size to achieve low robust generalization error when the data lies on a manifold with intrinsic dimension $k$ ($k \ll d$). Nonetheless, we also have a lower bound that grows exponentially with respect to $k$ --- the curse of dimensionality is inevitable. By demonstrating an exponential separation between the network size for achieving low robust training and generalization error, our results reveal that the hardness of robust generalization may stem from the expressive power of practical models.

Most real-world networks are too large to be measured or studied directly and there is substantial interest in estimating global network properties from smaller sub-samples. One of the most important global properties is the number of vertices/nodes in the network.

This paper proposes an optimization of Quantum Key Distribution (QKD) Networks using Graph Neural Networks (GNN) framework. Today, the development of quantum computers threatens the security systems of classical cryptography. Moreover, as QKD networks are designed for protecting secret communication, they suffer from multiple operational difficulties: adaptive to dynamic conditions, optimization for multiple parameters and effective resource utilization. In order to overcome these obstacles, we propose a GNN-based framework which can model QKD networks as dynamic graphs and extracts exploitable characteristics from these networks' structure. The graph contains not only topological information but also specific characteristics associated with quantum communication (the number of edges between nodes, etc). Experimental results demonstrate that the GNN-optimized QKD network achieves a substantial increase in total key rate (from 27.1 Kbits/s to 470 Kbits/s), a reduced average QBER (from 6.6% to 6.0%), and maintains path integrity with a slight reduction in average transmission distance (from 7.13 km to 6.42 km). Furthermore, we analyze network performance across varying scales (10 to 250 nodes), showing improved link prediction accuracy and enhanced key generation rate in medium-sized networks. This work introduces a novel operation mode for QKD networks, shifting the paradigm of network optimization through adaptive and scalable quantum communication systems that enhance security and performance.

Here, we are motivated by both the Hopfield network and by strong error-correcting codes (ECCs).

Many results in recent years established polynomial time learnability of various models via neural networks algorithms (e.g.