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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.

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].

Leverage scores have become essential in statistics and machine learning, aiding regression analysis, randomized matrix computations, and various other tasks. This paper delves into the inverse problem, aiming to recover the intrinsic model parameters given the leverage scores gradient. This endeavor not only enriches the theoretical understanding of models trained with leverage score techniques but also has substantial implications for data privacy and adversarial security. We specifically scrutinize the inversion of the leverage score gradient, denoted as $g(x)$. An innovative iterative algorithm is introduced for the approximate resolution of the regularized least squares problem stated as $\min_{x \in \mathbb{R}^d} 0.5 \|g(x) - c\|_2^2 + 0.5\|\mathrm{diag}(w)Ax\|_2^2$. Our algorithm employs subsampled leverage score distributions to compute an approximate Hessian in each iteration, under standard assumptions, considerably mitigating the time complexity. Given that a total of $T = \log(\| x_0 - x^* \|_2/ \epsilon)$ iterations are required, the cost per iteration is optimized to the order of $O( (\mathrm{nnz}(A) + d^{\omega} ) \cdot \mathrm{poly}(\log(n/\delta))$, where $\mathrm{nnz}(A)$ denotes the number of non-zero entries of $A$., Comment: arXiv admin note: text overlap with arXiv:2404.13785

Determining the John ellipsoid - the largest volume ellipsoid contained within a convex polytope - is a fundamental problem with applications in machine learning, optimization, and data analytics. Recent work has developed fast algorithms for approximating the John ellipsoid using sketching and leverage score sampling techniques. However, these algorithms do not provide privacy guarantees for sensitive input data. In this paper, we present the first differentially private algorithm for fast John ellipsoid computation. Our method integrates noise perturbation with sketching and leverage score sampling to achieve both efficiency and privacy. We prove that (1) our algorithm provides $(\epsilon,\delta)$-differential privacy, and the privacy guarantee holds for neighboring datasets that are $\epsilon_0$-close, allowing flexibility in the privacy definition; (2) our algorithm still converges to a $(1+\xi)$-approximation of the optimal John ellipsoid in $O(\xi^{-2}(\log(n/\delta_0) + (L\epsilon_0)^{-2}))$ iterations where $n$ is the number of data point, $L$ is the Lipschitz constant, $\delta_0$ is the failure probability, and $\epsilon_0$ is the closeness of neighboring input datasets. Our theoretical analysis demonstrates the algorithm's convergence and privacy properties, providing a robust approach for balancing utility and privacy in John ellipsoid computation. This is the first differentially private algorithm for fast John ellipsoid computation, opening avenues for future research in privacy-preserving optimization techniques.

Cross-attention has become a fundamental module nowadays in many important artificial intelligence applications, e.g., retrieval-augmented generation (RAG), system prompt, guided stable diffusion, and many so on. Ensuring cross-attention privacy is crucial and urgently needed because its key and value matrices may contain sensitive information about companies and their users, many of which profit solely from their system prompts or RAG data. In this work, we design a novel differential privacy (DP) data structure to address the privacy security of cross-attention with a theoretical guarantee. In detail, let $n$ be the input token length of system prompt/RAG data, $d$ be the feature dimension, $0 < \alpha \le 1$ be the relative error parameter, $R$ be the maximum value of the query and key matrices, $R_w$ be the maximum value of the value matrix, and $r,s,\epsilon_s$ be parameters of polynomial kernel methods. Then, our data structure requires $\widetilde{O}(ndr^2)$ memory consumption with $\widetilde{O}(nr^2)$ initialization time complexity and $\widetilde{O}(\alpha^{-1} r^2)$ query time complexity for a single token query. In addition, our data structure can guarantee that the user query is $(\epsilon, \delta)$-DP with $\widetilde{O}(n^{-1} \epsilon^{-1} \alpha^{-1/2} R^{2s} R_w r^2)$ additive error and $n^{-1} (\alpha + \epsilon_s)$ relative error between our output and the true answer. Furthermore, our result is robust to adaptive queries in which users can intentionally attack the cross-attention system. To our knowledge, this is the first work to provide DP for cross-attention. We believe it can inspire more privacy algorithm design in large generative models (LGMs).

Training data privacy is a fundamental problem in modern Artificial Intelligence (AI) applications, such as face recognition, recommendation systems, language generation, and many others, as it may contain sensitive user information related to legal issues. To fundamentally understand how privacy mechanisms work in AI applications, we study differential privacy (DP) in the Neural Tangent Kernel (NTK) regression setting, where DP is one of the most powerful tools for measuring privacy under statistical learning, and NTK is one of the most popular analysis frameworks for studying the learning mechanisms of deep neural networks. In our work, we can show provable guarantees for both differential privacy and test accuracy of our NTK regression. Furthermore, we conduct experiments on the basic image classification dataset CIFAR10 to demonstrate that NTK regression can preserve good accuracy under a modest privacy budget, supporting the validity of our analysis. To our knowledge, this is the first work to provide a DP guarantee for NTK regression.

We investigate the statistical and computational limits of latent \textbf{Di}ffusion \textbf{T}ransformers (\textbf{DiT}s) under the low-dimensional linear latent space assumption. Statistically, we study the universal approximation and sample complexity of the DiTs score function, as well as the distribution recovery property of the initial data. Specifically, under mild data assumptions, we derive an approximation error bound for the score network of latent DiTs, which is sub-linear in the latent space dimension. Additionally, we derive the corresponding sample complexity bound and show that the data distribution generated from the estimated score function converges toward a proximate area of the original one. Computationally, we characterize the hardness of both forward inference and backward computation of latent DiTs, assuming the Strong Exponential Time Hypothesis (SETH). For forward inference, we identify efficient criteria for all possible latent DiTs inference algorithms and showcase our theory by pushing the efficiency toward almost-linear time inference. For backward computation, we leverage the low-rank structure within the gradient computation of DiTs training for possible algorithmic speedup. Specifically, we show that such speedup achieves almost-linear time latent DiTs training by casting the DiTs gradient as a series of chained low-rank approximations with bounded error. Under the low-dimensional assumption, we show that the convergence rate and the computational efficiency are both dominated by the dimension of the subspace, suggesting that latent DiTs have the potential to bypass the challenges associated with the high dimensionality of initial data., Comment: v2 fixed typos, added Fig. 1 and added clarifications

Prompting and contextual-based fine-tuning methods, which we call Prefix Learning, have been proposed to enhance the performance of language models on various downstream tasks that can match full parameter fine-tuning. There remains a limited theoretical understanding of how these methods work. In this paper, we aim to relieve this limitation by studying the learning ability of Prefix Learning from the perspective of prefix length. In particular, we approximate the infinite-long Prefix Learning optimization process by the Neural Tangent Kernel (NTK) technique. We formulate and solve it as a learning problem of the infinite-long prefix in a one-layer attention network. Our results confirm the over-parameterization property and arbitrary small loss convergence guarantee of the infinite-long Prefix Learning in attention. To the implementation end, we propose our NTK-Attention method, which is "equivalent" to attention computation with arbitrary prefix length efficiently. Its time complexity mainly depends on the sub-quadratic of input length (without prefix), and our method only requires $d^2 + d$ extra parameters for representation, where $d$ is the feature dimension. In addition, we conducted experiments that compare our NTK-Attention with full parameters fine-tuning, LoRA, and P-Tuning V2 methods across vision or natural language datasets. The results indicate our approach may be a promising parameter-efficient-fine-tuning method since it has demonstrated superior performance in numerous scenarios. Our code can be found at \url{https://github.com/ChristianYang37/chiwun/tree/main/src/NTK-Attention}.

We study the computational limits of Low-Rank Adaptation (LoRA) update for finetuning transformer-based models using fine-grained complexity theory. Our key observation is that the existence of low-rank decompositions within the gradient computation of LoRA adaptation leads to possible algorithmic speedup. This allows us to (i) identify a phase transition behavior and (ii) prove the existence of nearly linear algorithms by controlling the LoRA update computation term by term, assuming the Strong Exponential Time Hypothesis (SETH). For the former, we identify a sharp transition in the efficiency of all possible rank-$r$ LoRA update algorithms for transformers, based on specific norms resulting from the multiplications of the input sequence $\mathbf{X}$, pretrained weights $\mathbf{W^\star}$, and adapter matrices $\alpha \mathbf{B} \mathbf{A} / r$. Specifically, we derive a shared upper bound threshold for such norms and show that efficient (sub-quadratic) approximation algorithms of LoRA exist only below this threshold. For the latter, we prove the existence of nearly linear approximation algorithms for LoRA adaptation by utilizing the hierarchical low-rank structures of LoRA gradients and approximating the gradients with a series of chained low-rank approximations. To showcase our theory, we consider two practical scenarios: partial (e.g., only $\mathbf{W}_V$ and $\mathbf{W}_Q$) and full adaptations (e.g., $\mathbf{W}_Q$, $\mathbf{W}_V$, and $\mathbf{W}_K$) of weights in attention heads.

Diffusion models have made rapid progress in generating high-quality samples across various domains. However, a theoretical understanding of the Lipschitz continuity and second momentum properties of the diffusion process is still lacking. In this paper, we bridge this gap by providing a detailed examination of these smoothness properties for the case where the target data distribution is a mixture of Gaussians, which serves as a universal approximator for smooth densities such as image data. We prove that if the target distribution is a $k$-mixture of Gaussians, the density of the entire diffusion process will also be a $k$-mixture of Gaussians. We then derive tight upper bounds on the Lipschitz constant and second momentum that are independent of the number of mixture components $k$. Finally, we apply our analysis to various diffusion solvers, both SDE and ODE based, to establish concrete error guarantees in terms of the total variation distance and KL divergence between the target and learned distributions. Our results provide deeper theoretical insights into the dynamics of the diffusion process under common data distributions.

Tensor Attention, a multi-view attention that is able to capture high-order correlations among multiple modalities, can overcome the representational limitations of classical matrix attention. However, the $\Omega(n^3)$ time complexity of tensor attention poses a significant obstacle to its practical implementation in transformers, where $n$ is the input sequence length. In this work, we prove that the backward gradient of tensor attention training can be computed in almost linear $n^{1+o(1)}$ time, the same complexity as its forward computation under a bounded entries assumption. We provide a closed-form solution for the gradient and propose a fast computation method utilizing polynomial approximation methods and tensor algebraic tricks. Furthermore, we prove the necessity and tightness of our assumption through hardness analysis, showing that slightly weakening it renders the gradient problem unsolvable in truly subcubic time. Our theoretical results establish the feasibility of efficient higher-order transformer training and may facilitate practical applications of tensor attention architectures.

Softmax distributions are widely used in machine learning, including Large Language Models (LLMs) where the attention unit uses softmax distributions. We abstract the attention unit as the softmax model, where given a vector input, the model produces an output drawn from the softmax distribution (which depends on the vector input). We consider the fundamental problem of binary hypothesis testing in the setting of softmax models. That is, given an unknown softmax model, which is known to be one of the two given softmax models, how many queries are needed to determine which one is the truth? We show that the sample complexity is asymptotically $O(\epsilon^{-2})$ where $\epsilon$ is a certain distance between the parameters of the models. Furthermore, we draw analogy between the softmax model and the leverage score model, an important tool for algorithm design in linear algebra and graph theory. The leverage score model, on a high level, is a model which, given vector input, produces an output drawn from a distribution dependent on the input. We obtain similar results for the binary hypothesis testing problem for leverage score models.

Large Language Models (LLMs) have profoundly changed the world. Their self-attention mechanism is the key to the success of transformers in LLMs. However, the quadratic computational cost $O(n^2)$ to the length $n$ input sequence is the notorious obstacle for further improvement and scalability in the longer context. In this work, we leverage the convolution-like structure of attention matrices to develop an efficient approximation method for attention computation using convolution matrices. We propose a $\mathsf{conv}$ basis system, "similar" to the rank basis, and show that any lower triangular (attention) matrix can always be decomposed as a sum of $k$ structured convolution matrices in this basis system. We then design an algorithm to quickly decompose the attention matrix into $k$ convolution matrices. Thanks to Fast Fourier Transforms (FFT), the attention {\it inference} can be computed in $O(knd \log n)$ time, where $d$ is the hidden dimension. In practice, we have $ d \ll n$, i.e., $d=3,072$ and $n=1,000,000$ for Gemma. Thus, when $kd = n^{o(1)}$, our algorithm achieve almost linear time, i.e., $n^{1+o(1)}$. Furthermore, the attention {\it training forward} and {\it backward gradient} can be computed in $n^{1+o(1)}$ as well. Our approach can avoid explicitly computing the $n \times n$ attention matrix, which may largely alleviate the quadratic computational complexity. Furthermore, our algorithm works on any input matrices. This work provides a new paradigm for accelerating attention computation in transformers to enable their application to longer contexts., Comment: 55 pages

The softmax activation function plays a crucial role in the success of large language models (LLMs), particularly in the self-attention mechanism of the widely adopted Transformer architecture. However, the underlying learning dynamics that contribute to the effectiveness of softmax remain largely unexplored. As a step towards better understanding, this paper provides a theoretical study of the optimization and generalization properties of two-layer softmax neural networks, providing theoretical insights into their superior performance as other activation functions, such as ReLU and exponential. Leveraging the Neural Tangent Kernel (NTK) framework, our analysis reveals that the normalization effect of the softmax function leads to a good perturbation property of the induced NTK matrix, resulting in a good convex region of the loss landscape. Consequently, softmax neural networks can learn the target function in the over-parametrization regime. To demonstrate the broad applicability of our theoretical findings, we apply them to the task of learning score estimation functions in diffusion models, a promising approach for generative modeling. Our analysis shows that gradient-based algorithms can learn the score function with a provable accuracy. Our work provides a deeper understanding of the effectiveness of softmax neural networks and their potential in various domains, paving the way for further advancements in natural language processing and beyond., Comment: 53 pages

Leverage score is a fundamental problem in machine learning and theoretical computer science. It has extensive applications in regression analysis, randomized algorithms, and neural network inversion. Despite leverage scores are widely used as a tool, in this paper, we study a novel problem, namely the inverting leverage score problem. We analyze to invert the leverage score distributions back to recover model parameters. Specifically, given a leverage score $\sigma \in \mathbb{R}^n$, the matrix $A \in \mathbb{R}^{n \times d}$, and the vector $b \in \mathbb{R}^n$, we analyze the non-convex optimization problem of finding $x \in \mathbb{R}^d$ to minimize $\| \mathrm{diag}( \sigma ) - I_n \circ (A(x) (A(x)^\top A(x) )^{-1} A(x)^\top ) \|_F$, where $A(x):= S(x)^{-1} A \in \mathbb{R}^{n \times d} $, $S(x) := \mathrm{diag}(s(x)) \in \mathbb{R}^{n \times n}$ and $s(x) : = Ax - b \in \mathbb{R}^n$. Our theoretical studies include computing the gradient and Hessian, demonstrating that the Hessian matrix is positive definite and Lipschitz, and constructing first-order and second-order algorithms to solve this regression problem. Our work combines iterative shrinking and the induction hypothesis to ensure global convergence rates for the Newton method, as well as the properties of Lipschitz and strong convexity to guarantee the performance of gradient descent. This important study on inverting statistical leverage opens up numerous new applications in interpretation, data recovery, and security.

The computational intensity of Large Language Models (LLMs) is a critical bottleneck, primarily due to the $O(n^2)$ complexity of the attention mechanism in transformer architectures. Addressing this, sparse attention emerges as a key innovation, aiming to reduce computational load while maintaining model performance. This study presents a rigorous theoretical analysis of the sparsity in attention scores within LLMs, particularly under the framework of Gaussian inputs. By establishing a set of foundational assumptions and employing a methodical theoretical approach, we unravel the intrinsic characteristics of attention score sparsity and its implications on computational efficiency. Our main contribution lies in providing a detailed theoretical examination of how sparsity manifests in attention mechanisms, offering insights into the potential trade-offs between computational savings and model effectiveness. This work not only advances our understanding of sparse attention but also provides a scaffold for future research in optimizing the computational frameworks of LLMs, paving the way for more scalable and efficient AI systems.

In the evolving landscape of machine learning, a pivotal challenge lies in deciphering the internal representations harnessed by neural networks and Transformers. Building on recent progress toward comprehending how networks execute distinct target functions, our study embarks on an exploration of the underlying reasons behind networks adopting specific computational strategies. We direct our focus to the complex algebraic learning task of modular addition involving $k$ inputs. Our research presents a thorough analytical characterization of the features learned by stylized one-hidden layer neural networks and one-layer Transformers in addressing this task. A cornerstone of our theoretical framework is the elucidation of how the principle of margin maximization shapes the features adopted by one-hidden layer neural networks. Let $p$ denote the modulus, $D_p$ denote the dataset of modular arithmetic with $k$ inputs and $m$ denote the network width. We demonstrate that a neuron count of $ m \geq 2^{2k-2} \cdot (p-1) $, these networks attain a maximum $ L_{2,k+1} $-margin on the dataset $ D_p $. Furthermore, we establish that each hidden-layer neuron aligns with a specific Fourier spectrum, integral to solving modular addition problems. By correlating our findings with the empirical observations of similar studies, we contribute to a deeper comprehension of the intrinsic computational mechanisms of neural networks. Furthermore, we observe similar computational mechanisms in the attention matrix of the one-layer Transformer. This research stands as a significant stride in unraveling their operation complexities, particularly in the realm of complex algebraic tasks., Comment: Update Section 5.3; clean up problem setup

Given its widespread application in machine learning and optimization, the Kronecker product emerges as a pivotal linear algebra operator. However, its computational demands render it an expensive operation, leading to heightened costs in spectral approximation of it through traditional computation algorithms. Existing classical methods for spectral approximation exhibit a linear dependency on the matrix dimension denoted by $n$, considering matrices of size $A_1 \in \mathbb{R}^{n \times d}$ and $A_2 \in \mathbb{R}^{n \times d}$. Our work introduces an innovative approach to efficiently address the spectral approximation of the Kronecker product $A_1 \otimes A_2$ using quantum methods. By treating matrices as quantum states, our proposed method significantly reduces the time complexity of spectral approximation to $O_{d,\epsilon}(\sqrt{n})$., Comment: arXiv admin note: text overlap with arXiv:2311.03215 by other authors

We investigate the computational limits of the memory retrieval dynamics of modern Hopfield models from the fine-grained complexity analysis. Our key contribution is the characterization of a phase transition behavior in the efficiency of all possible modern Hopfield models based on the norm of patterns. Specifically, we establish an upper bound criterion for the norm of input query patterns and memory patterns. Only below this criterion, sub-quadratic (efficient) variants of the modern Hopfield model exist, assuming the Strong Exponential Time Hypothesis (SETH). To showcase our theory, we provide a formal example of efficient constructions of modern Hopfield models using low-rank approximation when the efficient criterion holds. This includes a derivation of a lower bound on the computational time, scaling linearly with $\max\{$# of stored memory patterns, length of input query sequence$\}$. In addition, we prove its memory retrieval error bound and exponential memory capacity., Comment: Accepted at ICML 2024; v2 corrected typos; v3 added clarifications and references; v4,5 updated to camera-ready version

Large language models (LLMs) have made fundamental contributions over the last a few years. To train an LLM, one needs to alternatingly run `forward' computations and `backward' computations. The forward computation can be viewed as attention function evaluation, and the backward computation can be viewed as a gradient computation. In previous work by [Alman and Song, NeurIPS 2023], it was proved that the forward step can be performed in almost-linear time in certain parameter regimes, but that there is no truly sub-quadratic time algorithm in the remaining parameter regimes unless the popular hypothesis SETH is false. In this work, we show nearly identical results for the harder-seeming problem of computing the gradient of loss function of one layer attention network, and thus for the entire process of LLM training. This completely characterizes the fine-grained complexity of every step of LLM training.

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.