MomentumSMoE: Integrating Momentum into Sparse Mixture of Experts

Sparse Mixture of Experts (SMoE) has become the key to unlocking unparalleled scalability in deep learning. SMoE has the potential to exponentially increase parameter count while maintaining the efficiency of the model by only activating a small subset of these parameters for a given sample. However, it has been observed that SMoE suffers from unstable training and has difficulty adapting to new distributions, leading to the model's lack of robustness to data contamination. To overcome these limitations, we first establish a connection between the dynamics of the expert representations in SMoEs and gradient descent on a multi-objective optimization problem. Leveraging our framework, we then integrate momentum into SMoE and propose a new family of SMoEs named MomentumSMoE. We theoretically prove and numerically demonstrate that MomentumSMoE is more stable and robust than SMoE. In particular, we verify the advantages of MomentumSMoE over SMoE on a variety of practical tasks including ImageNet-1K object recognition and WikiText-103 language modeling. We demonstrate the applicability of MomentumSMoE to many types of SMoE models, including those in the Sparse MoE model for vision (V-MoE) and the Generalist Language Model (GLaM). We also show that other advanced momentum-based optimization methods, such as Adam, can be easily incorporated into the MomentumSMoE framework for designing new SMoE models with even better performance, almost negligible additional computation cost, and simple implementations.

Tree-Sliced Wasserstein Distance on a System of Lines

Sliced Wasserstein (SW) distance in Optimal Transport (OT) is widely used in various applications thanks to its statistical effectiveness and computational efficiency. On the other hand, Tree Wassenstein (TW) and Tree-sliced Wassenstein (TSW) are instances of OT for probability measures where its ground cost is a tree metric. TSW also has a low computational complexity, i.e. linear to the number of edges in the tree. Especially, TSW is identical to SW when the tree is a chain. While SW is prone to loss of topological information of input measures due to relying on one-dimensional projection, TSW is more flexible and has a higher degree of freedom by choosing a tree rather than a line to alleviate the curse of dimensionality in SW. However, for practical applications, popular tree metric sampling methods are heavily built upon given supports, which limits their capacity to adapt to new supports. In this paper, we propose the Tree-Sliced Wasserstein distance on a System of Lines (TSW-SL), which brings a connection between SW and TSW. Compared to SW and TSW, our TSW-SL benefits from the higher degree of freedom of TSW while being suitable to dynamic settings as SW. In TSW-SL, we use a variant of the Radon Transform to project measures onto a system of lines, resulting in measures on a space with a tree metric, then leverage TW to efficiently compute distances between them. We empirically verify the advantages of TSW-SL over the traditional SW by conducting a variety of experiments on gradient flows, image style transfer, and generative models.

A Primal-Dual Framework for Transformers and Neural Networks

Self-attention is key to the remarkable success of transformers in sequence modeling tasks including many applications in natural language processing and computer vision. Like neural network layers, these attention mechanisms are often developed by heuristics and experience. To provide a principled framework for constructing attention layers in transformers, we show that the self-attention corresponds to the support vector expansion derived from a support vector regression problem, whose primal formulation has the form of a neural network layer. Using our framework, we derive popular attention layers used in practice and propose two new attentions: 1) the Batch Normalized Attention (Attention-BN) derived from the batch normalization layer and 2) the Attention with Scaled Head (Attention-SH) derived from using less training data to fit the SVR model. We empirically demonstrate the advantages of the Attention-BN and Attention-SH in reducing head redundancy, increasing the model's accuracy, and improving the model's efficiency in a variety of practical applications including image and time-series classification.

Elliptical Attention

Pairwise dot-product self-attention is key to the success of transformers that achieve state-of-the-art performance across a variety of applications in language and vision. This dot-product self-attention computes attention weights among the input tokens using Euclidean distance, which makes the model prone to representation collapse and vulnerable to contaminated samples. In this paper, we propose using a Mahalanobis distance metric for computing the attention weights to stretch the underlying feature space in directions of high contextual relevance. In particular, we define a hyper-ellipsoidal neighborhood around each query to increase the attention weights of the tokens lying in the contextually important directions. We term this novel class of attention Elliptical Attention. Our Elliptical Attention provides two benefits: 1) reducing representation collapse and 2) enhancing the model's robustness as the Elliptical Attention pays more attention to contextually relevant information rather than focusing on some small subset of informative features. We empirically demonstrate the advantages of Elliptical Attention over the baseline dot-product attention and state-of-the-art attention methods on various practical tasks, including object classification, image segmentation, and language modeling across different data modalities.

Unveiling the Hidden Structure of Self-Attention via Kernel Principal Component Analysis

The remarkable success of transformers in sequence modeling tasks, spanning various applications in natural language processing and computer vision, is attributed to the critical role of self-attention. Similar to the development of most deep learning models, the construction of these attention mechanisms rely on heuristics and experience. In our work, we derive self-attention from kernel principal component analysis (kernel PCA) and show that self-attention projects its query vectors onto the principal component axes of its key matrix in a feature space. We then formulate the exact formula for the value matrix in self-attention, theoretically and empirically demonstrating that this value matrix captures the eigenvectors of the Gram matrix of the key vectors in self-attention. Leveraging our kernel PCA framework, we propose Attention with Robust Principal Components (RPC-Attention), a novel class of robust attention that is resilient to data contamination. We empirically demonstrate the advantages of RPC-Attention over softmax attention on the ImageNet-1K object classification, WikiText-103 language modeling, and ADE20K image segmentation task.

PIDformer: Transformer Meets Control Theory

In this work, we address two main shortcomings of transformer architectures: input corruption and rank collapse in their output representation. We unveil self-attention as an autonomous state-space model that inherently promotes smoothness in its solutions, leading to lower-rank outputs and diminished representation capacity. Moreover, the steady-state solution of the model is sensitive to input perturbations. We incorporate a Proportional-Integral-Derivative (PID) closed-loop feedback control system with a reference point into the model to improve robustness and representation capacity. This integration aims to preserve high-frequency details while bolstering model stability, rendering it more noise-resilient. The resulting controlled state-space model is theoretically proven robust and adept at addressing the rank collapse. Motivated by this control framework, we derive a novel class of transformers, PID-controlled Transformer (PIDformer), aimed at improving robustness and mitigating the rank-collapse issue inherent in softmax transformers. We empirically evaluate the model for advantages and robustness against baseline transformers across various practical tasks, including object classification, image segmentation, and language modeling.

Mitigating Over-smoothing in Transformers via Regularized Nonlocal Functionals

Transformers have achieved remarkable success in a wide range of natural language processing and computer vision applications. However, the representation capacity of a deep transformer model is degraded due to the over-smoothing issue in which the token representations become identical when the model's depth grows. In this work, we show that self-attention layers in transformers minimize a functional which promotes smoothness, thereby causing token uniformity. We then propose a novel regularizer that penalizes the norm of the difference between the smooth output tokens from self-attention and the input tokens to preserve the fidelity of the tokens. Minimizing the resulting regularized energy functional, we derive the Neural Transformer with a Regularized Nonlocal Functional (NeuTRENO), a novel class of transformer models that can mitigate the over-smoothing issue. We empirically demonstrate the advantages of NeuTRENO over the baseline transformers and state-of-the-art methods in reducing the over-smoothing of token representations on various practical tasks, including object classification, image segmentation, and language modeling.

p-Laplacian Transformer

$p$-Laplacian regularization, rooted in graph and image signal processing, introduces a parameter $p$ to control the regularization effect on these data. Smaller values of $p$ promote sparsity and interpretability, while larger values encourage smoother solutions. In this paper, we first show that the self-attention mechanism obtains the minimal Laplacian regularization ($p=2$) and encourages the smoothness in the architecture. However, the smoothness is not suitable for the heterophilic structure of self-attention in transformers where attention weights between tokens that are in close proximity and non-close ones are assigned indistinguishably. From that insight, we then propose a novel class of transformers, namely the $p$-Laplacian Transformer (p-LaT), which leverages $p$-Laplacian regularization framework to harness the heterophilic features within self-attention layers. In particular, low $p$ values will effectively assign higher attention weights to tokens that are in close proximity to the current token being processed. We empirically demonstrate the advantages of p-LaT over the baseline transformers on a wide range of benchmark datasets.

From Coupled Oscillators to Graph Neural Networks: Reducing Over-smoothing via a Kuramoto Model-based Approach

We propose the Kuramoto Graph Neural Network (KuramotoGNN), a novel class of continuous-depth graph neural networks (GNNs) that employs the Kuramoto model to mitigate the over-smoothing phenomenon, in which node features in GNNs become indistinguishable as the number of layers increases. The Kuramoto model captures the synchronization behavior of non-linear coupled oscillators. Under the view of coupled oscillators, we first show the connection between Kuramoto model and basic GNN and then over-smoothing phenomenon in GNNs can be interpreted as phase synchronization in Kuramoto model. The KuramotoGNN replaces this phase synchronization with frequency synchronization to prevent the node features from converging into each other while allowing the system to reach a stable synchronized state. We experimentally verify the advantages of the KuramotoGNN over the baseline GNNs and existing methods in reducing over-smoothing on various graph deep learning benchmark tasks.

ARIST: An Effective API Argument Recommendation Approach

Learning and remembering to use APIs are difficult. Several techniques have been proposed to assist developers in using APIs. Most existing techniques focus on recommending the right API methods to call, but very few techniques focus on recommending API arguments. In this paper, we propose ARIST, a novel automated argument recommendation approach which suggests arguments by predicting developers' expectations when they define and use API methods. To implement this idea in the recommendation process, ARIST combines program analysis (PA), language models (LMs), and several features specialized for the recommendation task which consider the functionality of formal parameters and the positional information of code elements (e.g., variables or method calls) in the given context. In ARIST, the LMs and the recommending features are used to suggest the promising candidates identified by PA. Meanwhile, PA navigates the LMs and the features working on the set of the valid candidates which satisfy syntax, accessibility, and type-compatibility constraints defined by the programming language in use. Our evaluation on a large dataset of real-world projects shows that ARIST improves the state-of-the-art approach by 19% and 18% in top-1 precision and recall for recommending arguments of frequently-used libraries. For general argument recommendation task, i.e., recommending arguments for every method call, ARIST outperforms the baseline approaches by up to 125% top-1 accuracy. Moreover, for newly-encountered projects, ARIST achieves more than 60% top-3 accuracy when evaluating on a larger dataset. For working/maintaining projects, with a personalized LM to capture developers' coding practice, ARIST can productively rank the expected arguments at the top-1 position in 7/10 requests.