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Recent studies revealed the mathematical connection of deep neural network (DNN) and dynamic system. However, the fundamental principle of DNN has not been fully characterized with dynamic system in terms of optimization and generalization.…

Deep neural networks (DNNs) are known to be vulnerable to adversarial attacks that would trigger misclassification of DNNs but may be imperceptible to human perception. Adversarial defense has been an important way to improve the robustness…

Developing machine learning methods that are privacy preserving is today a central topic of research, with huge practical impacts. Among the numerous ways to address privacy-preserving learning, we here take the perspective of computing the…

In this work, we introduce a novel framework for privately optimizing objectives that rely on Wasserstein distances between data-dependent empirical measures. Our main theoretical contribution is, based on an explicit formulation of the…

The performance of Deep Q-Networks (DQN) is critically dependent on the ability of its underlying neural network to accurately approximate the action-value function. Standard function approximators, such as multi-layer perceptrons, may…

Given any deep fully connected neural network, initialized with random Gaussian parameters, we bound from above the quadratic Wasserstein distance between its output distribution and a suitable Gaussian process. Our explicit inequalities…

We introduce the so called DeepParticle method to learn and generate invariant measures of stochastic dynamical systems with physical parameters based on data computed from an interacting particle method (IPM). We utilize the expressiveness…

The estimation of quantum entropies and distance measures, such as von Neumann entropy, R\'{e}nyi entropy, Tsallis entropy, trace distance, and fidelity-induced distances such as the Bures distance, has been a key area of research in…

We quantify, uniformly over time and with high probability, the discrepancy between the predictions of a two-layer neural network trained by stochastic gradient descent (SGD) and their mean-field limit, for quadratic loss and ridge…

Quantifying how far the output of a learning algorithm is from its target is an essential task in machine learning. However, in quantum settings, the loss landscapes of commonly used distance metrics often produce undesirable outcomes such…

Pseudospectral approximation provides a means to approximate the dynamics of delay differential equations (DDE) by ordinary differential equations (ODE). This article develops a computer-aided algorithm to determine the distance between the…

In this paper, we consider distributed algorithms for solving the empirical risk minimization problem under the master/worker communication model. We develop a distributed asynchronous quasi-Newton algorithm that can achieve superlinear…

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate…

Hardware-friendly network quantization (e.g., binary/uniform quantization) can efficiently accelerate the inference and meanwhile reduce memory consumption of the deep neural networks, which is crucial for model deployment on…

Generative adversarial networks (GANs) have received a tremendous amount of attention in the past few years, and have inspired applications addressing a wide range of problems. Despite its great potential, GANs are difficult to train.…

Training a neural network (NN) typically relies on some type of curve-following method, such as gradient descent (GD) (and stochastic gradient descent (SGD)), ADADELTA, ADAM or limited memory algorithms. Convergence for these algorithms…

The statistical finite element method (StatFEM) is an emerging probabilistic method that allows observations of a physical system to be synthesised with the numerical solution of a PDE intended to describe it in a coherent statistical…

Adversarial examples are crafted by adding indistinguishable perturbations to normal examples in order to fool a well-trained deep learning model to misclassify. In the context of computer vision, this notion of indistinguishability is…

Consider the following distributed optimization scenario. A worker has access to training data that it uses to compute the gradients while a server decides when to stop iterative computation based on its target accuracy or delay…

This paper establishes a quantitative stability theory for one-dimensional stochastic differential equations (SDEs) with non-zero drift, driven by a symmetric $\alpha$-stable process for $\alpha\in(1,2)$. Our work generalizes the classical…