Curse of Attention: A Kernel-Based Perspective for Why Transformers Fail to Generalize on Time Series Forecasting and Beyond

The application of transformer-based models on time series forecasting (TSF) tasks has long been popular to study. However, many of these works fail to beat the simple linear residual model, and the theoretical understanding of this issue is still limited. In this work, we propose the first theoretical explanation of the inefficiency of transformers on TSF tasks. We attribute the mechanism behind it to {\bf Asymmetric Learning} in training attention networks. When the sign of the previous step is inconsistent with the sign of the current step in the next-step-prediction time series, attention fails to learn the residual features. This makes it difficult to generalize on out-of-distribution (OOD) data, especially on the sign-inconsistent next-step-prediction data, with the same representation pattern, whereas a linear residual network could easily accomplish it. We hope our theoretical insights provide important necessary conditions for designing the expressive and efficient transformer-based architecture for practitioners.

Advancing the Understanding of Fixed Point Iterations in Deep Neural Networks: A Detailed Analytical Study

Recent empirical studies have identified fixed point iteration phenomena in deep neural networks, where the hidden state tends to stabilize after several layers, showing minimal change in subsequent layers. This observation has spurred the development of practical methodologies, such as accelerating inference by bypassing certain layers once the hidden state stabilizes, selectively fine-tuning layers to modify the iteration process, and implementing loops of specific layers to maintain fixed point iterations. Despite these advancements, the understanding of fixed point iterations remains superficial, particularly in high-dimensional spaces, due to the inadequacy of current analytical tools. In this study, we conduct a detailed analysis of fixed point iterations in a vector-valued function modeled by neural networks. We establish a sufficient condition for the existence of multiple fixed points of looped neural networks based on varying input regions. Additionally, we expand our examination to include a robust version of fixed point iterations. To demonstrate the effectiveness and insights provided by our approach, we provide case studies that looped neural networks may exist $2^d$ number of robust fixed points under exponentiation or polynomial activation functions, where $d$ is the feature dimension. Furthermore, our preliminary empirical results support our theoretical findings. Our methodology enriches the toolkit available for analyzing fixed point iterations of deep neural networks and may enhance our comprehension of neural network mechanisms.