Neural Ordinary Differential Equations (ODEs) have been successful in various applications due to their continuous nature and parameter-sharing efficiency. However, these unique characteristics also introduce challenges in training, particularly with respect to gradient computation accuracy and convergence analysis. In this paper, we address these challenges by investigating the impact of activation functions. We demonstrate that the properties of activation functions—specifically smoothness and nonlinearity—are critical to the training dynamics. Smooth activation functions guarantee globally unique solutions for both forward and backward ODEs, while sufficient nonlinearity is essential for maintaining the spectral properties of the Neural Tangent Kernel (NTK) during training. Together, these properties enable us to establish the global convergence of Neural ODEs under gradient descent in overparameterized regimes. Our theoretical findings are validated by numerical experiments, which not only support our analysis but also provide practical guidelines for scaling Neural ODEs, potentially leading to faster training and improved performance in real-world applications.

Neural networks with wide layers have attracted significant attention due to their equivalence to Gaussian processes, enabling perfect fitting of training data while maintaining generalization performance, known as benign overfitting. However, existing results mainly focus on shallow or finite-depth networks, necessitating a comprehensive analysis of wide neural networks with infinite-depth layers, such as neural ordinary differential equations (ODEs) and deep equilibrium models (DEQs). In this paper, we specifically investigate the deep equilibrium model (DEQ), an infinite-depth neural network with shared weight matrices across layers. Our analysis reveals that as the width of DEQ layers approaches infinity, it converges to a Gaussian process, establishing what is known as the Neural Network and Gaussian Process (NNGP) correspondence. Remarkably, this convergence holds even when the limits of depth and width are interchanged, which is not observed in typical infinite-depth Multilayer Perceptron (MLP) networks. Furthermore, we demonstrate that the associated Gaussian vector remains non-degenerate for any pairwise distinct input data, ensuring a strictly positive smallest eigenvalue of the corresponding kernel matrix using the NNGP kernel. These findings serve as fundamental elements for studying the training and generalization of DEQs, laying the groundwork for future research in this area.

We propose a new randomized Bregman (block) coordinate descent (RBCD) method for minimizing a composite problem, where the objective function could be either convex or nonconvex, and the smooth part are freed from the global Lipschitz-continuous (partial) gradient assumption. Under the notion of relative smoothness based on the Bregman distance, we prove that every limit point of the generated sequence is a stationary point. Further, we show that the iteration complexity of the proposed method is $\mathcal{O}\left( {n{\varepsilon ^{ - 2}}} \right)$ to achieve ϵ-stationary point, where n is the number of blocks of coordinates. If the objective is assumed to be convex, the iteration complexity is improved to $\mathcal{O}\left( {n{ \in ^{ - 1}}} \right)$. If, in addition, the objective is strongly convex (relative to the reference function), the global linear convergence rate is recovered. We also present the accelerated version of the RBCD method, which attains an $\mathcal{O}\left( {n{\varepsilon ^{ - 1/\gamma }}} \right)$ iteration complexity for the convex case, where the scalar $\gamma \in [1,2]$ is determined by the generalized translation variant of the Bregman distance. Convergence analysis without assuming the global Lipschitz-continuous (partial) gradient sets our results apart from the existing works in the composite problems.

Deep learning recommendation systems at scale have provided remarkable gains through increasing model capacity (i.e. wider and deeper neural networks), but it comes at significant training cost and infrastructure cost. Model pruning is an effective technique to reduce computation overhead for deep neural networks by removing redundant parameters. However, modern recommendation systems are still thirsty for model capacity due to the demand for handling big data. Thus, pruning a recommendation model at scale results in a smaller model capacity and consequently lower accuracy. To reduce computation cost without sacrificing model capacity, we propose a dynamic training scheme, namely alternate model growth and pruning, to alternatively construct and prune weights in the course of training. Our method leverages structured sparsification to reduce computational cost without hurting the model capacity at the end of offline training so that a full-size model is available in the recurring training stage to learn new data in real time. To the best of our knowledge, this is the first work to provide in-depth experiments and discussion of applying structural dynamics to recommendation systems at scale to reduce training cost. The proposed method is validated with an open-source deep learning recommendation model (DLRM) and state-of-the-art industrial-scale production models.

Many known supervised deep learning methods for medical image segmentation suffer an expensive burden of data annotation for model training. Recently, few-shot segmentation methods were proposed to alleviate this burden, but such methods often showed poor adaptability to the target tasks. By prudently introducing interactive learning into the few-shot learning strategy, we develop a novel few-shot segmentation approach called Interactive Few-shot Learning (IFSL), which not only addresses the annotation burden of medical image segmentation models but also tackles the common issues of the known few-shot segmentation methods. First, we design a new few-shot segmentation structure, called Medical Prior-based Few-shot Learning Network (MPrNet), which uses only a few annotated samples (e.g., 10 samples) as support images to guide the segmentation of query images without any pre-training. Then, we propose an Interactive Learning-based Test Time Optimization Algorithm (IL-TTOA) to strengthen our MPrNet on the fly for the target task in an interactive fashion. To our best knowledge, our IFSL approach is the first to allow few-shot segmentation models to be optimized and strengthened on the target tasks in an interactive and controllable manner. Experiments on four few-shot segmentation tasks show that our IFSL approach outperforms the state-of-the-art methods by more than 20% in the DSC metric. Specifically, the interactive optimization algorithm (IL-TTOA) further contributes ~10% DSC improvement for the few-shot segmentation models.

Nonnegative matrix factorization (NMF) has attracted much attention in the last decade as a dimension reduction method in many applications. Due to the explosion in the size of data, naturally the samples are collected and stored distributively in local computational nodes. Thus, there is a growing need to develop algorithms in a distributed memory architecture. We propose a novel distributed algorithm, called distributed incremental block coordinate descent (DID), to solve the problem. By adapting the block coordinate descent framework, closed-form update rules are obtained in DID. Moreover, DID performs updates incrementally based on the most recently updated residual matrix. As a result, only one communication step per iteration is required. The correctness, efficiency, and scalability of the proposed algorithm are verified in a series of numerical experiments.