Exploring the Limits of KV Cache Compression in Visual Autoregressive Transformers

A fundamental challenge in Visual Autoregressive models is the substantial memory overhead required during inference to store previously generated representations. Despite various attempts to mitigate this issue through compression techniques, prior works have not explicitly formalized the problem of KV-cache compression in this context. In this work, we take the first step in formally defining the KV-cache compression problem for Visual Autoregressive transformers. We then establish a fundamental negative result, proving that any mechanism for sequential visual token generation under attention-based architectures must use at least $\Omega(n^2 d)$ memory, when $d = \Omega(\log n)$, where $n$ is the number of tokens generated and $d$ is the embedding dimensionality. This result demonstrates that achieving truly sub-quadratic memory usage is impossible without additional structural constraints. Our proof is constructed via a reduction from a computational lower bound problem, leveraging randomized embedding techniques inspired by dimensionality reduction principles. Finally, we discuss how sparsity priors on visual representations can influence memory efficiency, presenting both impossibility results and potential directions for mitigating memory overhead.

On Computational Limits of FlowAR Models: Expressivity and Efficiency

The expressive power and computational complexity of deep visual generative models, such as flow-based and autoregressive (AR) models, have gained considerable interest for their wide-ranging applications in generative tasks. However, the theoretical characterization of their expressiveness through the lens of circuit complexity remains underexplored, particularly for the state-of-the-art architecture like FlowAR proposed by [Ren et al., 2024], which integrates flow-based and autoregressive mechanisms. This gap limits our understanding of their inherent computational limits and practical efficiency. In this study, we address this gap by analyzing the circuit complexity of the FlowAR architecture. We demonstrate that when the largest feature map produced by the FlowAR model has dimensions $n \times n \times c$, the FlowAR model is simulable by a family of threshold circuits $\mathsf{TC}^0$, which have constant depth $O(1)$ and polynomial width $\mathrm{poly}(n)$. This is the first study to rigorously highlight the limitations in the expressive power of FlowAR models. Furthermore, we identify the conditions under which the FlowAR model computations can achieve almost quadratic time. To validate our theoretical findings, we present efficient model variant constructions based on low-rank approximations that align with the derived criteria. Our work provides a foundation for future comparisons with other generative paradigms and guides the development of more efficient and expressive implementations.

Circuit Complexity Bounds for Visual Autoregressive Model

Understanding the expressive ability of a specific model is essential for grasping its capacity limitations. Recently, several studies have established circuit complexity bounds for Transformer architecture. Besides, the Visual AutoRegressive (VAR) model has risen to be a prominent method in the field of image generation, outperforming previous techniques, such as Diffusion Transformers, in generating high-quality images. We investigate the circuit complexity of the VAR model and establish a bound in this study. Our primary result demonstrates that the VAR model is equivalent to a simulation by a uniform $\mathsf{TC}^0$ threshold circuit with hidden dimension $d \leq O(n)$ and $\mathrm{poly}(n)$ precision. This is the first study to rigorously highlight the limitations in the expressive power of VAR models despite their impressive performance. We believe our findings will offer valuable insights into the inherent constraints of these models and guide the development of more efficient and expressive architectures in the future.

On Computational Limits and Provably Efficient Criteria of Visual Autoregressive Models: A Fine-Grained Complexity Analysis

Recently, Visual Autoregressive ($\mathsf{VAR}$) Models introduced a groundbreaking advancement in the field of image generation, offering a scalable approach through a coarse-to-fine "next-scale prediction" paradigm. However, the state-of-the-art algorithm of $\mathsf{VAR}$ models in [Tian, Jiang, Yuan, Peng and Wang, NeurIPS 2024] takes $O(n^4)$ time, which is computationally inefficient. In this work, we analyze the computational limits and efficiency criteria of $\mathsf{VAR}$ Models through a fine-grained complexity lens. Our key contribution is identifying the conditions under which $\mathsf{VAR}$ computations can achieve sub-quadratic time complexity. Specifically, we establish a critical threshold for the norm of input matrices used in $\mathsf{VAR}$ attention mechanisms. Above this threshold, assuming the Strong Exponential Time Hypothesis ($\mathsf{SETH}$) from fine-grained complexity theory, a sub-quartic time algorithm for $\mathsf{VAR}$ models is impossible. To substantiate our theoretical findings, we present efficient constructions leveraging low-rank approximations that align with the derived criteria. This work initiates the study of the computational efficiency of the $\mathsf{VAR}$ model from a theoretical perspective. Our technique will shed light on advancing scalable and efficient image generation in $\mathsf{VAR}$ frameworks.

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.