Tianxiang Gao

Tianxiang Gao, Hongyang Gao

Implicit neural networks have become increasingly attractive in the machine learning community since they can achieve competitive performance but use much less computational resources. Recently, a line of theoretical works established the global convergences for first-order methods such as gradient descent if the implicit networks are over-parameterized. However, as they train all layers together, their analyses are equivalent to only studying the evolution of the output layer. It is unclear how the implicit layer contributes to the training. Thus, in this paper, we restrict ourselves to only training the implicit layer. We show that global convergence is guaranteed, even if only the implicit layer is trained. On the other hand, the theoretical understanding of when and how the training performance of an implicit neural network can be generalized to unseen data is still under-explored. Although this problem has been studied in standard feed-forward networks, the case of implicit neural networks is still intriguing since implicit networks theoretically have infinitely many layers. Therefore, this paper investigates the generalization error for implicit neural networks. Specifically, we study the generalization of an implicit network activated by the ReLU function over random initialization. We provide a generalization bound that is initialization sensitive. As a result, we show that gradient flow with proper random initialization can train a sufficient over-parameterized implicit network to achieve arbitrarily small generalization errors.

Tianxiang Gao, Hongyang Gao

Implicit deep learning has recently become popular in the machine learning community since these implicit models can achieve competitive performance with state-of-the-art deep networks while using significantly less memory and computational resources. However, our theoretical understanding of when and how first-order methods such as gradient descent (GD) converge on \textit{nonlinear} implicit networks is limited. Although this type of problem has been studied in standard feed-forward networks, the case of implicit models is still intriguing because implicit networks have \textit{infinitely} many layers. The corresponding equilibrium equation probably admits no or multiple solutions during training. This paper studies the convergence of both gradient flow (GF) and gradient descent for nonlinear ReLU activated implicit networks. To deal with the well-posedness problem, we introduce a fixed scalar to scale the weight matrix of the implicit layer and show that there exists a small enough scaling constant, keeping the equilibrium equation well-posed throughout training. As a result, we prove that both GF and GD converge to a global minimum at a linear rate if the width $m$ of the implicit network is \textit{linear} in the sample size $N$, i.e., $m=Ω(N)$.

Xiaocong Du, Bhargav Bhushanam, Jiecao Yu et al.

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.

Xuqiang Shao, Yuqi Zhang, Di Zhang et al.

Constructing efficient and diverse datasets is essential for the development of accurate machine learning potentials (MLPs) in atomistic simulations. However, existing approaches often suffer from data redundancy and high computational costs. Herein, we propose a new method--Difference Vectors based on Local Atomic Environments (DV-LAE)--that encodes structural differences via histogram-based descriptors and enables visual analysis through t-SNE dimensionality reduction. This approach facilitates redundancy detection and dataset optimization while preserving structural diversity. We demonstrate that DV-LAE significantly reduces dataset size and training time across various materials systems, including high-pressure hydrogen, iron-hydrogen binaries, magnesium hydrides, and carbon allotropes, with minimal compromise in prediction accuracy. For instance, in the $α$-Fe/H system, maintaining a highly similar MLP accuracy, the dataset size was reduced by 56%, and the training time per iteration dropped by over 50%. Moreover, we show how visualizing the DV-LAE representation aids in identifying out-of-distribution data by examining the spatial distribution of high-error prediction points, providing a robust reliability metric for new structures during simulations. Our results highlight the utility of local environment visualization not only as an interpretability tool but also as a practical means for accelerating MLP development and ensuring data efficiency in large-scale atomistic modeling.

Yang Li, Tianxiang Gao, Junier B. Oliva

In this work, we propose to learn a generative model using both learned features (through a latent space) and memories (through neighbors). Although human learning makes seamless use of both learned perceptual features and instance recall, current generative learning paradigms only make use of one of these two components. Take, for instance, flow models, which learn a latent space of invertible features that follow a simple distribution. Conversely, kernel density techniques use instances to shift a simple distribution into an aggregate mixture model. Here we propose multiple methods to enhance the latent space of a flow model with neighborhood information. Not only does our proposed framework represent a more human-like approach by leveraging both learned features and memories, but it may also be viewed as a step forward in non-parametric methods. The efficacy of our model is shown empirically with standard image datasets. We observe compelling results and a significant improvement over baselines.

Tianxiang Gao, Vladimir Jojic

In this paper, we explore degrees of freedom in deep sigmoidal neural networks. We show that the degrees of freedom in these models is related to the expected optimism, which is the expected difference between test error and training error. We provide an efficient Monte-Carlo method to estimate the degrees of freedom for multi-class classification methods. We show degrees of freedom are lower than the parameter count in a simple XOR network. We extend these results to neural nets trained on synthetic and real data, and investigate impact of network's architecture and different regularization choices. The degrees of freedom in deep networks are dramatically smaller than the number of parameters, in some real datasets several orders of magnitude. Further, we observe that for fixed number of parameters, deeper networks have less degrees of freedom exhibiting a regularization-by-depth.