Fast Quantum Algorithm for Attention Computation

Large language models (LLMs) have demonstrated exceptional performance across a wide range of tasks. These models, powered by advanced deep learning techniques, have revolutionized the field of natural language processing (NLP) and have achieved remarkable results in various language-related tasks. LLMs have excelled in tasks such as machine translation, sentiment analysis, question answering, text generation, text classification, language modeling, and more. They have proven to be highly effective in capturing complex linguistic patterns, understanding context, and generating coherent and contextually relevant text. The attention scheme plays a crucial role in the architecture of large language models (LLMs). It is a fundamental component that enables the model to capture and utilize contextual information during language processing tasks effectively. Making the attention scheme computation faster is one of the central questions to speed up the LLMs computation. It is well-known that quantum machine has certain computational advantages compared to the classical machine. However, it is currently unknown whether quantum computing can aid in LLM. In this work, we focus on utilizing Grover's Search algorithm to compute a sparse attention computation matrix efficiently. We achieve a polynomial quantum speed-up over the classical method. Moreover, the attention matrix outputted by our quantum algorithm exhibits an extra low-rank structure that will be useful in obtaining a faster training algorithm for LLMs. Additionally, we present a detailed analysis of the algorithm's error analysis and time complexity within the context of computing the attention matrix.

A Nearly-Linear Time Algorithm for Structured Support Vector Machines

Quadratic programming is a fundamental problem in the field of convex optimization. Many practical tasks can be formulated as quadratic programming, for example, the support vector machine (SVM). Linear SVM is one of the most popular tools over the last three decades in machine learning before deep learning method dominating. In general, a quadratic program has input size $\Theta(n^2)$ (where $n$ is the number of variables), thus takes $\Omega(n^2)$ time to solve. Nevertheless, quadratic programs coming from SVMs has input size $O(n)$, allowing the possibility of designing nearly-linear time algorithms. Two important classes of SVMs are programs admitting low-rank kernel factorizations and low-treewidth programs. Low-treewidth convex optimization has gained increasing interest in the past few years (e.g.~linear programming [Dong, Lee and Ye 2021] and semidefinite programming [Gu and Song 2022]). Therefore, an important open question is whether there exist nearly-linear time algorithms for quadratic programs with these nice structures. In this work, we provide the first nearly-linear time algorithm for solving quadratic programming with low-rank factorization or low-treewidth, and a small number of linear constraints. Our results imply nearly-linear time algorithms for low-treewidth or low-rank SVMs.

Efficient SGD Neural Network Training via Sublinear Activated Neuron Identification

Deep learning has been widely used in many fields, but the model training process usually consumes massive computational resources and time. Therefore, designing an efficient neural network training method with a provable convergence guarantee is a fundamental and important research question. In this paper, we present a static half-space report data structure that consists of a fully connected two-layer neural network for shifted ReLU activation to enable activated neuron identification in sublinear time via geometric search. We also prove that our algorithm can converge in $O(M^2/\epsilon^2)$ time with network size quadratic in the coefficient norm upper bound $M$ and error term $\epsilon$.

In-Context Learning for Attention Scheme: from Single Softmax Regression to Multiple Softmax Regression via a Tensor Trick

Large language models (LLMs) have brought significant and transformative changes in human society. These models have demonstrated remarkable capabilities in natural language understanding and generation, leading to various advancements and impacts across several domains. We consider the in-context learning under two formulation for attention related regression in this work. Given matrices $A_1 \in \mathbb{R}^{n \times d}$, and $A_2 \in \mathbb{R}^{n \times d}$ and $B \in \mathbb{R}^{n \times n}$, the purpose is to solve some certain optimization problems: Normalized version $\min_{X} \| D(X)^{-1} \exp(A_1 X A_2^\top) - B \|_F^2$ and Rescaled version $\| \exp(A_1 X A_2^\top) - D(X) \cdot B \|_F^2$. Here $D(X) := \mathrm{diag}( \exp(A_1 X A_2^\top) {\bf 1}_n )$. Our regression problem shares similarities with previous studies on softmax-related regression. Prior research has extensively investigated regression techniques related to softmax regression: Normalized version $\| \langle \exp(Ax) , {\bf 1}_n \rangle^{-1} \exp(Ax) - b \|_2^2$ and Resscaled version $\| \exp(Ax) - \langle \exp(Ax), {\bf 1}_n \rangle b \|_2^2 $ In contrast to previous approaches, we adopt a vectorization technique to address the regression problem in matrix formulation. This approach expands the dimension from $d$ to $d^2$, resembling the formulation of the regression problem mentioned earlier. Upon completing the lipschitz analysis of our regression function, we have derived our main result concerning in-context learning.

H$_2$O: Heavy-Hitter Oracle for Efficient Generative Inference of Large Language Models

Large Language Models (LLMs), despite their recent impressive accomplishments, are notably cost-prohibitive to deploy, particularly for applications involving long-content generation, such as dialogue systems and story writing. Often, a large amount of transient state information, referred to as the KV cache, is stored in GPU memory in addition to model parameters, scaling linearly with the sequence length and batch size. In this paper, we introduce a novel approach for implementing the KV cache which significantly reduces its memory footprint. Our approach is based on the noteworthy observation that a small portion of tokens contributes most of the value when computing attention scores. We call these tokens Heavy Hitters (H$_2$). Through a comprehensive investigation, we find that (i) the emergence of H$_2$ is natural and strongly correlates with the frequent co-occurrence of tokens in the text, and (ii) removing them results in significant performance degradation. Based on these insights, we propose Heavy Hitter Oracle (H$_2$O), a KV cache eviction policy that dynamically retains a balance of recent and H$_2$ tokens. We formulate the KV cache eviction as a dynamic submodular problem and prove (under mild assumptions) a theoretical guarantee for our novel eviction algorithm which could help guide future work. We validate the accuracy of our algorithm with OPT, LLaMA, and GPT-NeoX across a wide range of tasks. Our implementation of H$_2$O with 20% heavy hitters improves the throughput over three leading inference systems DeepSpeed Zero-Inference, Hugging Face Accelerate, and FlexGen by up to 29$\times$, 29$\times$, and 3$\times$ on OPT-6.7B and OPT-30B. With the same batch size, H2O can reduce the latency by up to 1.9$\times$. The code is available at https://github.com/FMInference/H2O.

InfoPrompt: Information-Theoretic Soft Prompt Tuning for Natural Language Understanding

Soft prompt tuning achieves superior performances across a wide range of few-shot tasks. However, the performances of prompt tuning can be highly sensitive to the initialization of the prompts. We also empirically observe that conventional prompt tuning methods cannot encode and learn sufficient task-relevant information from prompt tokens. In this work, we develop an information-theoretic framework that formulates soft prompt tuning as maximizing mutual information between prompts and other model parameters (or encoded representations). This novel view helps us to develop a more efficient, accurate and robust soft prompt tuning method InfoPrompt. With this framework, we develop two novel mutual information based loss functions, to (i) discover proper prompt initialization for the downstream tasks and learn sufficient task-relevant information from prompt tokens and (ii) encourage the output representation from the pretrained language model to be more aware of the task-relevant information captured in the learnt prompt. Extensive experiments validate that InfoPrompt can significantly accelerate the convergence of the prompt tuning and outperform traditional prompt tuning methods. Finally, we provide a formal theoretical result for showing to show that gradient descent type algorithm can be used to train our mutual information loss.

Efficient Alternating Minimization with Applications to Weighted Low Rank Approximation

Weighted low rank approximation is a fundamental problem in numerical linear algebra, and it has many applications in machine learning. Given a matrix $M \in \mathbb{R}^{n \times n}$, a weight matrix $W \in \mathbb{R}_{\geq 0}^{n \times n}$, a parameter $k$, the goal is to output two matrices $U, V \in \mathbb{R}^{n \times k}$ such that $\| W \circ (M - U V) \|_F$ is minimized, where $\circ$ denotes the Hadamard product. Such a problem is known to be NP-hard and even hard to approximate [RSW16]. Meanwhile, alternating minimization is a good heuristic solution for approximating weighted low rank approximation. The work [LLR16] shows that, under mild assumptions, alternating minimization does provide provable guarantees. In this work, we develop an efficient and robust framework for alternating minimization. For weighted low rank approximation, this improves the runtime of [LLR16] from $n^2 k^2$ to $n^2k$. At the heart of our work framework is a high-accuracy multiple response regression solver together with a robust analysis of alternating minimization.

Query Complexity of Active Learning for Function Family With Nearly Orthogonal Basis

Many machine learning algorithms require large numbers of labeled data to deliver state-of-the-art results. In applications such as medical diagnosis and fraud detection, though there is an abundance of unlabeled data, it is costly to label the data by experts, experiments, or simulations. Active learning algorithms aim to reduce the number of required labeled data points while preserving performance. For many convex optimization problems such as linear regression and $p$-norm regression, there are theoretical bounds on the number of required labels to achieve a certain accuracy. We call this the query complexity of active learning. However, today's active learning algorithms require the underlying learned function to have an orthogonal basis. For example, when applying active learning to linear regression, the requirement is the target function is a linear composition of a set of orthogonal linear functions, and active learning can find the coefficients of these linear functions. We present a theoretical result to show that active learning does not need an orthogonal basis but rather only requires a nearly orthogonal basis. We provide the corresponding theoretical proofs for the function family of nearly orthogonal basis, and its applications associated with the algorithmically efficient active learning framework.

A Mathematical Abstraction for Balancing the Trade-off Between Creativity and Reality in Large Language Models

Large Language Models have become popular for their remarkable capabilities in human-oriented tasks and traditional natural language processing tasks. Its efficient functioning is attributed to the attention mechanism in the Transformer architecture, enabling it to concentrate on particular aspects of the input. LLMs are increasingly being used in domains such as generating prose, poetry or art, which require the model to be creative (e.g. Adobe firefly). LLMs possess advanced language generation abilities that enable them to generate distinctive and captivating content. This utilization of LLMs in generating narratives shows their flexibility and potential for use in domains that extend beyond conventional natural language processing duties. In different contexts, we may expect the LLM to generate factually correct answers, that match reality; e.g., question-answering systems or online assistants. In such situations, being correct is critical to LLMs being trusted in practice. The Bing Chatbot provides its users with the flexibility to select one of the three output modes: creative, balanced, and precise. Each mode emphasizes creativity and factual accuracy differently. In this work, we provide a mathematical abstraction to describe creativity and reality based on certain losses. A model trained on these losses balances the trade-off between the creativity and reality of the model.

Faster Robust Tensor Power Method for Arbitrary Order

Tensor decomposition is a fundamental method used in various areas to deal with high-dimensional data. \emph{Tensor power method} (TPM) is one of the widely-used techniques in the decomposition of tensors. This paper presents a novel tensor power method for decomposing arbitrary order tensors, which overcomes limitations of existing approaches that are often restricted to lower-order (less than $3$) tensors or require strong assumptions about the underlying data structure. We apply sketching method, and we are able to achieve the running time of $\widetilde{O}(n^{p-1})$, on the power $p$ and dimension $n$ tensor. We provide a detailed analysis for any $p$-th order tensor, which is never given in previous works.