Circuit Complexity Bounds for RoPE-based Transformer Architecture

Characterizing the express power of the Transformer architecture is critical to understanding its capacity limits and scaling law. Recent works provide the circuit complexity bounds to Transformer-like architecture. On the other hand, Rotary Position Embedding ($\mathsf{RoPE}$) has emerged as a crucial technique in modern large language models, offering superior performance in capturing positional information compared to traditional position embeddings, which shows great potential in application prospects, particularly for the long context scenario. Empirical evidence also suggests that $\mathsf{RoPE}$-based Transformer architectures demonstrate greater generalization capabilities compared to conventional Transformer models. In this work, we establish a tighter circuit complexity bound for Transformers with $\mathsf{RoPE}$ attention. Our key contribution is that we show that unless $\mathsf{TC}^0 = \mathsf{NC}^1$, a $\mathsf{RoPE}$-based Transformer with $\mathrm{poly}(n)$-precision, $O(1)$ layers, hidden dimension $d \leq O(n)$ cannot solve the arithmetic problem or the Boolean formula value problem. This result significantly demonstrates the fundamental limitation of the expressivity of the $\mathsf{RoPE}$-based Transformer architecture, although it achieves giant empirical success. Our theoretical framework not only establishes tighter complexity bounds but also may instruct further work on the $\mathsf{RoPE}$-based Transformer.

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.

Bypassing the Exponential Dependency: Looped Transformers Efficiently Learn In-context by Multi-step Gradient Descent

In-context learning has been recognized as a key factor in the success of Large Language Models (LLMs). It refers to the model's ability to learn patterns on the fly from provided in-context examples in the prompt during inference. Previous studies have demonstrated that the Transformer architecture used in LLMs can implement a single-step gradient descent update by processing in-context examples in a single forward pass. Recent work has further shown that, during in-context learning, a looped Transformer can implement multi-step gradient descent updates in forward passes. However, their theoretical results require an exponential number of in-context examples, $n = \exp(\Omega(T))$, where $T$ is the number of loops or passes, to achieve a reasonably low error. In this paper, we study linear looped Transformers in-context learning on linear vector generation tasks. We show that linear looped Transformers can implement multi-step gradient descent efficiently for in-context learning. Our results demonstrate that as long as the input data has a constant condition number, e.g., $n = O(d)$, the linear looped Transformers can achieve a small error by multi-step gradient descent during in-context learning. Furthermore, our preliminary experiments validate our theoretical analysis. Our findings reveal that the Transformer architecture possesses a stronger in-context learning capability than previously understood, offering new insights into the mechanisms behind LLMs and potentially guiding the better design of efficient inference algorithms for LLMs.

Beyond Linear Approximations: A Novel Pruning Approach for Attention Matrix

Large Language Models (LLMs) have shown immense potential in enhancing various aspects of our daily lives, from conversational AI to search and AI assistants. However, their growing capabilities come at the cost of extremely large model sizes, making deployment on edge devices challenging due to memory and computational constraints. This paper introduces a novel approach to LLM weight pruning that directly optimizes for approximating the attention matrix, a core component of transformer architectures. Unlike existing methods that focus on linear approximations, our approach accounts for the non-linear nature of the Softmax attention mechanism. We provide theoretical guarantees for the convergence of our Gradient Descent-based optimization method to a near-optimal pruning mask solution. Our preliminary empirical results demonstrate the effectiveness of this approach in maintaining model performance while significantly reducing computational costs. This work establishes a new theoretical foundation for pruning algorithm design in LLMs, potentially paving the way for more efficient LLM inference on resource-constrained devices.

HSR-Enhanced Sparse Attention Acceleration

Large Language Models (LLMs) have demonstrated remarkable capabilities across various applications, but their performance on long-context tasks is often limited by the computational complexity of attention mechanisms. This paper introduces a novel approach to accelerate attention computation in LLMs, particularly for long-context scenarios. We leverage the inherent sparsity within attention mechanisms, both in conventional Softmax attention and ReLU attention (with $\mathsf{ReLU}^\alpha$ activation, $\alpha \in \mathbb{N}_+$), to significantly reduce the running time complexity. Our method employs a Half-Space Reporting (HSR) data structure to rapidly identify non-zero or "massively activated" entries in the attention matrix. We present theoretical analyses for two key scenarios: attention generation and full attention computation with long input context. Our approach achieves a running time of $O(mn^{4/5})$ significantly faster than the naive approach $O(mn)$ for attention generation, where $n$ is the context length, $m$ is the query length, and $d$ is the hidden dimension. We can also reduce the running time of full attention computation from $O(mn)$ to $O(mn^{1 - 1 / \lfloor d/2\rfloor} + mn^{4/5})$. Importantly, our method introduces no error for ReLU attention and only provably negligible error for Softmax attention, where the latter is supported by our empirical validation. This work represents a significant step towards enabling efficient long-context processing in LLMs, potentially broadening their applicability across various domains.

Looped ReLU MLPs May Be All You Need as Practical Programmable Computers

Previous work has demonstrated that attention mechanisms are Turing complete. More recently, it has been shown that a looped 13-layer Transformer can function as a universal programmable computer. In contrast, the multi-layer perceptrons with $\mathsf{ReLU}$ activation ($\mathsf{ReLU}$-$\mathsf{MLP}$), one of the most fundamental components of neural networks, is known to be expressive; specifically, a two-layer neural network is a universal approximator given an exponentially large number of hidden neurons. However, it remains unclear whether a $\mathsf{ReLU}$-$\mathsf{MLP}$ can be made into a universal programmable computer using a practical number of weights. In this work, we provide an affirmative answer that a looped 23-layer $\mathsf{ReLU}$-$\mathsf{MLP}$ is capable to perform the basic necessary operations, effectively functioning as a programmable computer. This indicates that simple modules have stronger expressive power than previously expected and have not been fully explored. Our work provides insights into the mechanisms of neural networks and demonstrates that complex tasks, such as functioning as a programmable computer, do not necessarily require advanced architectures like Transformers.

Fine-grained Attention I/O Complexity: Comprehensive Analysis for Backward Passes

Large Language Models (LLMs) have demonstrated remarkable capabilities in processing long-context information. However, the quadratic complexity of attention computation with respect to sequence length poses significant computational challenges, and I/O aware algorithms have been proposed. This paper presents a comprehensive analysis of the I/O complexity for attention mechanisms, focusing on backward passes by categorizing into small and large cache scenarios. Using the red-blue pebble game framework, we establish tight bounds on I/O complexity across all cache sizes. We confirm that the de facto standard I/O aware algorithm FlashAttention is optimal for both forward and backward passes for the large cache size scenario. For small cache sizes, we provide an algorithm that improves over existing methods and achieves the tight bounds. Additionally, we extend our analysis to sparse attention, a mainstream speeding-up approach, deriving fine-grained lower bounds for both forward and backward passes and both small and large caches. Our findings complete the theoretical foundation for I/O complexity in attention mechanisms, offering insights for designing efficient algorithms of LLM training and inference.

Discovering the Gems in Early Layers: Accelerating Long-Context LLMs with 1000x Input Token Reduction

Large Language Models (LLMs) have demonstrated remarkable capabilities in handling long context inputs, but this comes at the cost of increased computational resources and latency. Our research introduces a novel approach for the long context bottleneck to accelerate LLM inference and reduce GPU memory consumption. Our research demonstrates that LLMs can identify relevant tokens in the early layers before generating answers to a query. Leveraging this insight, we propose an algorithm that uses early layers of an LLM as filters to select and compress input tokens, significantly reducing the context length for subsequent processing. Our method, GemFilter, demonstrates substantial improvements in both speed and memory efficiency compared to existing techniques, such as standard attention and SnapKV/H2O. Notably, it achieves a 2.4$\times$ speedup and 30\% reduction in GPU memory usage compared to SOTA methods. Evaluation on the Needle in a Haystack task shows that GemFilter significantly outperforms standard attention, SnapKV and demonstrates comparable performance on the LongBench challenge. GemFilter is simple, training-free, and broadly applicable across different LLMs. Crucially, it provides interpretability by allowing humans to inspect the selected input sequence. These findings not only offer practical benefits for LLM deployment, but also enhance our understanding of LLM internal mechanisms, paving the way for further optimizations in LLM design and inference. Our code is available at \url{https://github.com/SalesforceAIResearch/GemFilter}.

Multi-Layer Transformers Gradient Can be Approximated in Almost Linear Time

The quadratic computational complexity in the self-attention mechanism of popular transformer architectures poses significant challenges for training and inference, particularly in terms of efficiency and memory requirements. Towards addressing these challenges, this paper introduces a novel fast computation method for gradient calculation in multi-layer transformer models. Our approach enables the computation of gradients for the entire multi-layer transformer model in almost linear time $n^{1+o(1)}$, where $n$ is the input sequence length. This breakthrough significantly reduces the computational bottleneck associated with the traditional quadratic time complexity. Our theory holds for any loss function and maintains a bounded approximation error across the entire model. Furthermore, our analysis can hold when the multi-layer transformer model contains many practical sub-modules, such as residual connection, casual mask, and multi-head attention. By improving the efficiency of gradient computation in large language models, we hope that our work will facilitate the more effective training and deployment of long-context language models based on our theoretical results.

A Tighter Complexity Analysis of SparseGPT

In this work, we improved the analysis of the running time of SparseGPT [Frantar, Alistarh ICML 2023] from $O(d^{3})$ to $O(d^{\omega} + d^{2+a+o(1)} + d^{1+\omega(1,1,a)-a})$ for any $a \in [0, 1]$, where $\omega$ is the exponent of matrix multiplication. In particular, for the current $\omega \approx 2.371$ [Alman, Duan, Williams, Xu, Xu, Zhou 2024], our running times boil down to $O(d^{2.53})$. This running time is due to the analysis of the lazy update behavior in iterative maintenance problems, such as [Deng, Song, Weinstein 2022, Brand, Song, Zhou ICML 2024].