Chiwun Yang

Timothy Chu, Zhao Song, Chiwun Yang

Large language models (LLMs) and generative AI have played a transformative role in computer research and applications. Controversy has arisen as to whether these models output copyrighted data, which can occur if the data the models are trained on is copyrighted. LLMs are built on the transformer neural network architecture, which in turn relies on a mathematical computation called Attention that uses the softmax function. In this paper, we show that large language model training and optimization can be seen as a softmax regression problem. We then establish a method of efficiently performing softmax regression, in a way that prevents the regression function from generating copyright data. This establishes a theoretical method of training large language models in a way that avoids generating copyright data.

Timothy Chu, Zhao Song, Chiwun Yang

In-context learning (ICL) is an astonishing emergent ability of large language models (LLMs). By presenting a prompt that includes multiple input-output pairs as examples and introducing a new query input, models can generate the corresponding output. However, the performance of models heavily relies on the quality of the input prompt when implementing in-context learning. Biased or imbalanced input prompts can significantly degrade the performance of language models. To address this issue, we introduce a reweighted algorithm called RICL (Reweighted In-context Learning). This algorithm fine-tunes language models using an unbiased validation set to determine the optimal weight for each input-output example to approximate unbiased in-context learning. Furthermore, we also introduce a low-cost reweighted algorithm, a linear optimal weight approximation algorithm called LARICL (Linear Approximation of Reweighted In-context Learning). This algorithm requires minimal training cost while providing effective results. We prove the convergence of our algorithm and validate its performance through experiments conducted on a numerical dataset. The experimental findings reveal a substantial improvement in comparison to benchmarks including the performance of casual prompt-based in-context learning and the performance of a classic fine-tuning method.

Zhao Song, Chiwun Yang

The delta-bar-delta algorithm is recognized as a learning rate adaptation technique that enhances the convergence speed of the training process in optimization by dynamically scheduling the learning rate based on the difference between the current and previous weight updates. While this algorithm has demonstrated strong competitiveness in full data optimization when compared to other state-of-the-art algorithms like Adam and SGD, it may encounter convergence issues in mini-batch optimization scenarios due to the presence of noisy gradients. In this study, we thoroughly investigate the convergence behavior of the delta-bar-delta algorithm in real-world neural network optimization. To address any potential convergence challenges, we propose a novel approach called RDBD (Regrettable Delta-Bar-Delta). Our approach allows for prompt correction of biased learning rate adjustments and ensures the convergence of the optimization process. Furthermore, we demonstrate that RDBD can be seamlessly integrated with any optimization algorithm and significantly improve the convergence speed. By conducting extensive experiments and evaluations, we validate the effectiveness and efficiency of our proposed RDBD approach. The results showcase its capability to overcome convergence issues in mini-batch optimization and its potential to enhance the convergence speed of various optimization algorithms. This research contributes to the advancement of optimization techniques in neural network training, providing practitioners with a reliable automatic learning rate scheduler for achieving faster convergence and improved optimization outcomes.

Yichuan Deng, Zhao Song, Shenghao Xie et al.

In the realm of deep learning, transformers have emerged as a dominant architecture, particularly in natural language processing tasks. However, with their widespread adoption, concerns regarding the security and privacy of the data processed by these models have arisen. In this paper, we address a pivotal question: Can the data fed into transformers be recovered using their attention weights and outputs? We introduce a theoretical framework to tackle this problem. Specifically, we present an algorithm that aims to recover the input data $X \in \mathbb{R}^{d \times n}$ from given attention weights $W = QK^\top \in \mathbb{R}^{d \times d}$ and output $B \in \mathbb{R}^{n \times n}$ by minimizing the loss function $L(X)$. This loss function captures the discrepancy between the expected output and the actual output of the transformer. Our findings have significant implications for the Localized Layer-wise Mechanism (LLM), suggesting potential vulnerabilities in the model's design from a security and privacy perspective. This work underscores the importance of understanding and safeguarding the internal workings of transformers to ensure the confidentiality of processed data.

Chenyang Li, Zhao Song, Weixin Wang et al.

The Deep Leakage from Gradient (DLG) attack has emerged as a prevalent and highly effective method for extracting sensitive training data by inspecting exchanged gradients. This approach poses a substantial threat to the privacy of individuals and organizations alike. This research presents a comprehensive analysis of the gradient leakage method when applied specifically to transformer-based models. Through meticulous examination, we showcase the capability to accurately recover data solely from gradients and rigorously investigate the conditions under which gradient attacks can be executed, providing compelling evidence. Furthermore, we reevaluate the approach of introducing additional noise on gradients as a protective measure against gradient attacks. To address this, we outline a theoretical proof that analyzes the associated privacy costs within the framework of differential privacy. Additionally, we affirm the convergence of the Stochastic Gradient Descent (SGD) algorithm under perturbed gradients. The primary objective of this study is to augment the understanding of gradient leakage attack and defense strategies while actively contributing to the development of privacy-preserving techniques specifically tailored for transformer-based models. By shedding light on the vulnerabilities and countermeasures associated with gradient leakage, this research aims to foster advancements in safeguarding sensitive data and upholding privacy in the context of transformer-based models.

Raghav Addanki, Chenyang Li, Zhao Song et al.

Attention computation takes both the time complexity of $O(n^2)$ and the space complexity of $O(n^2)$ simultaneously, which makes deploying Large Language Models (LLMs) in streaming applications that involve long contexts requiring substantial computational resources. In recent OpenAI DevDay (Nov 6, 2023), OpenAI released a new model that is able to support a 128K-long document, in our paper, we focus on the memory-efficient issue when context length $n$ is much greater than 128K ($n \gg 2^d$). Considering a single-layer self-attention with Query, Key, and Value matrices $Q, K, V \in \mathbb{R}^{n \times d}$, the polynomial method approximates the attention output $T \in \mathbb{R}^{n \times d}$. It accomplishes this by constructing $U_1, U_2 \in \mathbb{R}^{n \times t}$ to expedite attention ${\sf Attn}(Q, K, V)$ computation within $n^{1+o(1)}$ time executions. Despite this, computing the approximated attention matrix $U_1U_2^\top \in \mathbb{R}^{n \times n}$ still necessitates $O(n^2)$ space, leading to significant memory usage. In response to these challenges, we introduce a new algorithm that only reads one pass of the data in a streaming fashion. This method employs sublinear space $o(n)$ to store three sketch matrices, alleviating the need for exact $K, V$ storage. Notably, our algorithm exhibits exceptional memory-efficient performance with super-long tokens. As the token length $n$ increases, our error guarantee diminishes while the memory usage remains nearly constant. This unique attribute underscores the potential of our technique in efficiently handling LLMs in streaming applications.

Jing Xiong, Jianghan Shen, Chuanyang Zheng et al.

Extrapolating ultra-long contexts (text length >128K) remains a major challenge for large language models (LLMs), as most training-free extrapolation methods are not only severely limited by memory bottlenecks, but also suffer from the attention sink, which restricts their scalability and effectiveness in practice. In this work, we propose ParallelComp, a parallel long-context compression method that effectively overcomes the memory bottleneck, enabling 8B-parameter LLMs to extrapolate from 8K to 128K tokens on a single A100 80GB GPU in a training-free setting. ParallelComp splits the input into chunks, dynamically evicting redundant chunks and irrelevant tokens, supported by a parallel KV cache eviction mechanism. Importantly, we present a systematic theoretical and empirical analysis of attention biases in parallel attention-including the attention sink, recency bias, and middle bias-and reveal that these biases exhibit distinctive patterns under ultra-long context settings. We further design a KV cache eviction technique to mitigate this phenomenon. Experimental results show that ParallelComp enables an 8B model (trained on 8K context) to achieve 91.17% of GPT-4's performance under ultra-long contexts, outperforming closed-source models such as Claude-2 and Kimi-Chat. We achieve a 1.76x improvement in chunk throughput, thereby achieving a 23.50x acceleration in the prefill stage with negligible performance loss and pave the way for scalable and robust ultra-long contexts extrapolation in LLMs. We release the code at https://github.com/menik1126/ParallelComp.

Chiwun Yang

The scaling law, a cornerstone of Large Language Model (LLM) development, predicts improvements in model performance with increasing computational resources. Yet, while empirically validated, its theoretical underpinnings remain poorly understood. This work formalizes the learning dynamics of transformer-based language models as an ordinary differential equation (ODE) system, then approximates this process to kernel behaviors. Departing from prior toy-model analyses, we rigorously analyze stochastic gradient descent (SGD) training for multi-layer transformers on sequence-to-sequence data with arbitrary data distribution, closely mirroring real-world conditions. Our analysis characterizes the convergence of generalization error to the irreducible risk as computational resources scale with data, especially during the optimization process. We establish a theoretical upper bound on excess risk characterized by a distinct phase transition. In the initial optimization phase, the excess risk decays exponentially relative to the computational cost ${\sf C}$. However, once a specific resource allocation threshold is crossed, the system enters a statistical phase, where the generalization error follows a power-law decay of $Θ(\mathsf{C}^{-1/6})$. Beyond this unified framework, our theory derives isolated scaling laws for model size, training time, and dataset size, elucidating how each variable independently governs the upper bounds of generalization.

Yingyu Liang, Zhenmei Shi, Zhao Song et al.

Prompting and context-based fine-tuning methods, which we call Prefix Learning, have been proposed to enhance the performance of language models on various downstream tasks. They are empirically efficient and effective, matching the performance of full parameter fine-tuning, but the theoretical understandings are limited. In this paper, we aim to address this limitation by studying their ability from the perspective of prefix length. In particular, we provide a convergence guarantee for training an ultra-long prefix in a stylized setting using the Neural Tangent Kernel (NTK) framework. Based on this strong theoretical guarantee, we design and implement an algorithm that only needs to introduce and fine-tune a few extra trainable parameters instead of an infinite-long prefix in each layer of a transformer, and can approximate the prefix attention to a guaranteed polynomial-small error. Preliminary experimental results on vision, natural language, and math data show that our method achieves superior or competitive performance compared to existing methods like full parameters fine-tuning, P-Tuning V2, and LoRA. This demonstrates our method is promising for parameter-efficient fine-tuning. Our code can be found at \url{https://github.com/ChristianYang37/chiwun/tree/main/src/NTK-Attention}.

Yekun Ke, Yingyu Liang, Zhenmei Shi et al.

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.

Yang Cao, Zhao Song, Chiwun Yang

This paper considers an efficient video modeling process called Video Latent Flow Matching (VLFM). Unlike prior works, which randomly sampled latent patches for video generation, our method relies on current strong pre-trained image generation models, modeling a certain caption-guided flow of latent patches that can be decoded to time-dependent video frames. We first speculate multiple images of a video are differentiable with respect to time in some latent space. Based on this conjecture, we introduce the HiPPO framework to approximate the optimal projection for polynomials to generate the probability path. Our approach gains the theoretical benefits of the bounded universal approximation error and timescale robustness. Moreover, VLFM processes the interpolation and extrapolation abilities for video generation with arbitrary frame rates. We conduct experiments on several text-to-video datasets to showcase the effectiveness of our method.

Yichuan Deng, Zhao Song, Jing Xiong et al.

Sparse Attention is a technique that approximates standard attention computation with sub-quadratic complexity. This is achieved by selectively ignoring smaller entries in the attention matrix during the softmax function computation. Variations of this technique, such as pruning KV cache, sparsity-based fast attention, and Sparse Transformer, have been extensively utilized for efficient Large Language Models (LLMs) deployment. Despite its widespread use, a theoretical understanding of the conditions under which sparse attention performs on par with traditional attention remains elusive. This work aims to $\textbf{bridge this gap by examining the inherent sparsity of standard attention processes}$. Our theoretical framework reveals several brand-new key insights: $\bullet$ Attention is $n^{C}$-sparse, implying that considering only the largest $Ω(n^{C})$ entries out of all $n$ entries is sufficient for sparse attention to approximate the exact attention matrix with decreasing loss. Here, $n$ represents the input length and $C \in (0, 1)$ is a constant. $\bullet$ Stable $o(\log(n))$-sparse attention, which approximates attention computation with $\log(n)$ or fewer entries, may not be feasible since the error will persist at a minimum of $O(1)$. $\bullet$ An adaptive strategy ($α\cdot n^C, α\in \mathbb{R}$) for the window size of efficient attention methods rather than a fixed one is guaranteed to perform more accurately and efficiently in a task for inference on flexible context lengths.

Majid Daliri, Zhao Song, Chiwun Yang

Recently, 1-bit Large Language Models (LLMs) have emerged, showcasing an impressive combination of efficiency and performance that rivals traditional LLMs. Research by Wang et al. (2023); Ma et al. (2024) indicates that the performance of these 1-bit LLMs progressively improves as the number of parameters increases, hinting at the potential existence of a Scaling Law for 1-bit Neural Networks. In this paper, we present the first theoretical result that rigorously establishes this scaling law for 1-bit models. We prove that, despite the constraint of weights restricted to $\{-1, +1\}$, the dynamics of model training inevitably align with kernel behavior as the network width grows. This theoretical breakthrough guarantees convergence of the 1-bit model to an arbitrarily small loss as width increases. Furthermore, we introduce the concept of the generalization difference, defined as the gap between the outputs of 1-bit networks and their full-precision counterparts, and demonstrate that this difference maintains a negligible level as network width scales. Building on the work of Kaplan et al. (2020), we conclude by examining how the training loss scales as a power-law function of the model size, dataset size, and computational resources utilized for training. Our findings underscore the promising potential of scaling 1-bit neural networks, suggesting that int1 could become the standard in future neural network precision.

Jiangxuan Long, Zhao Song, Chiwun Yang

Recent studies suggest utilizing generative models instead of traditional auto-regressive algorithms for time series forecasting (TSF) tasks. These non-auto-regressive approaches involving different generative methods, including GAN, Diffusion, and Flow Matching for time series, have empirically demonstrated high-quality generation capability and accuracy. However, we still lack an appropriate understanding of how it processes approximation and generalization. This paper presents the first theoretical framework from the perspective of flow-based generative models to relieve the knowledge of limitations. In particular, we provide our insights with strict guarantees from three perspectives: $\textbf{Approximation}$, $\textbf{Generalization}$ and $\textbf{Efficiency}$. In detail, our analysis achieves the contributions as follows: $\bullet$ By assuming a general data model, the fitting of the flow-based generative models is confirmed to converge to arbitrary error under the universal approximation of Diffusion Transformer (DiT). $\bullet$ Introducing a polynomial-based regularization for flow matching, the generalization error thus be bounded since the generalization of polynomial approximation. $\bullet$ The sampling for generation is considered as an optimization process, we demonstrate its fast convergence with updating standard first-order gradient descent of some objective.

Yichuan Deng, Zhao Song, Chiwun Yang

Based on SGD, previous works have proposed many algorithms that have improved convergence speed and generalization in stochastic optimization, such as SGDm, AdaGrad, Adam, etc. However, their convergence analysis under non-convex conditions is challenging. In this work, we propose a unified framework to address this issue. For any first-order methods, we interpret the updated direction $g_t$ as the sum of the stochastic subgradient $\nabla f_t(x_t)$ and an additional acceleration term $\frac{2|\langle v_t, \nabla f_t(x_t) \rangle|}{\|v_t\|_2^2} v_t$, thus we can discuss the convergence by analyzing $\langle v_t, \nabla f_t(x_t) \rangle$. Through our framework, we have discovered two plug-and-play acceleration methods: \textbf{Reject Accelerating} and \textbf{Random Vector Accelerating}, we theoretically demonstrate that these two methods can directly lead to an improvement in convergence rate.