Revisiting Quantum Algorithms for Linear Regressions: Quadratic Speedups without Data-Dependent Parameters

Linear regression is one of the most fundamental linear algebra problems. Given a dense matrix $A \in \mathbb{R}^{n \times d}$ and a vector $b$, the goal is to find $x'$ such that $ \| Ax' - b \|_2^2 \leq (1+\epsilon) \min_{x} \| A x - b \|_2^2 $. The best classical algorithm takes $O(nd) + \mathrm{poly}(d/\epsilon)$ time [Clarkson and Woodruff STOC 2013, Nelson and Nguyen FOCS 2013]. On the other hand, quantum linear regression algorithms can achieve exponential quantum speedups, as shown in [Wang Phys. Rev. A 96, 012335, Kerenidis and Prakash ITCS 2017, Chakraborty, Gily{\'e}n and Jeffery ICALP 2019]. However, the running times of these algorithms depend on some quantum linear algebra-related parameters, such as $\kappa(A)$, the condition number of $A$. In this work, we develop a quantum algorithm that runs in $\widetilde{O}(\epsilon^{-1}\sqrt{n}d^{1.5}) + \mathrm{poly}(d/\epsilon)$ time. It provides a quadratic quantum speedup in $n$ over the classical lower bound without any dependence on data-dependent parameters. In addition, we also show our result can be generalized to multiple regression and ridge linear regression.

A Theoretical Insight into Attack and Defense of Gradient Leakage in Transformer

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.

Fast Heavy Inner Product Identification Between Weights and Inputs in Neural Network Training

In this paper, we consider a heavy inner product identification problem, which generalizes the Light Bulb problem~(\cite{prr89}): Given two sets $A \subset \{-1,+1\}^d$ and $B \subset \{-1,+1\}^d$ with $|A|=|B| = n$, if there are exact $k$ pairs whose inner product passes a certain threshold, i.e., $\{(a_1, b_1), \cdots, (a_k, b_k)\} \subset A \times B$ such that $\forall i \in [k], \langle a_i,b_i \rangle \geq \rho \cdot d$, for a threshold $\rho \in (0,1)$, the goal is to identify those $k$ heavy inner products. We provide an algorithm that runs in $O(n^{2 \omega / 3+ o(1)})$ time to find the $k$ inner product pairs that surpass $\rho \cdot d$ threshold with high probability, where $\omega$ is the current matrix multiplication exponent. By solving this problem, our method speed up the training of neural networks with ReLU activation function.

The Expressibility of Polynomial based Attention Scheme

Large language models (LLMs) have significantly improved various aspects of our daily lives. These models have impacted numerous domains, from healthcare to education, enhancing productivity, decision-making processes, and accessibility. As a result, they have influenced and, to some extent, reshaped people's lifestyles. However, the quadratic complexity of attention in transformer architectures poses a challenge when scaling up these models for processing long textual contexts. This issue makes it impractical to train very large models on lengthy texts or use them efficiently during inference. While a recent study by [KMZ23] introduced a technique that replaces the softmax with a polynomial function and polynomial sketching to speed up attention mechanisms, the theoretical understandings of this new approach are not yet well understood. In this paper, we offer a theoretical analysis of the expressive capabilities of polynomial attention. Our study reveals a disparity in the ability of high-degree and low-degree polynomial attention. Specifically, we construct two carefully designed datasets, namely $\mathcal{D}_0$ and $\mathcal{D}_1$, where $\mathcal{D}_1$ includes a feature with a significantly larger value compared to $\mathcal{D}_0$. We demonstrate that with a sufficiently high degree $\beta$, a single-layer polynomial attention network can distinguish between $\mathcal{D}_0$ and $\mathcal{D}_1$. However, with a low degree $\beta$, the network cannot effectively separate the two datasets. This analysis underscores the greater effectiveness of high-degree polynomials in amplifying large values and distinguishing between datasets. Our analysis offers insight into the representational capacity of polynomial attention and provides a rationale for incorporating higher-degree polynomials in attention mechanisms to capture intricate linguistic correlations.

Deja Vu: Contextual Sparsity for Efficient LLMs at Inference Time

Large language models (LLMs) with hundreds of billions of parameters have sparked a new wave of exciting AI applications. However, they are computationally expensive at inference time. Sparsity is a natural approach to reduce this cost, but existing methods either require costly retraining, have to forgo LLM's in-context learning ability, or do not yield wall-clock time speedup on modern hardware. We hypothesize that contextual sparsity, which are small, input-dependent sets of attention heads and MLP parameters that yield approximately the same output as the dense model for a given input, can address these issues. We show that contextual sparsity exists, that it can be accurately predicted, and that we can exploit it to speed up LLM inference in wall-clock time without compromising LLM's quality or in-context learning ability. Based on these insights, we propose DejaVu, a system that uses a low-cost algorithm to predict contextual sparsity on the fly given inputs to each layer, along with an asynchronous and hardware-aware implementation that speeds up LLM inference. We validate that DejaVu can reduce the inference latency of OPT-175B by over 2X compared to the state-of-the-art FasterTransformer, and over 6X compared to the widely used Hugging Face implementation, without compromising model quality. The code is available at https://github.com/FMInference/DejaVu.

Unmasking Transformers: A Theoretical Approach to Data Recovery via Attention Weights

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.

Superiority of Softmax: Unveiling the Performance Edge Over Linear Attention

Large transformer models have achieved state-of-the-art results in numerous natural language processing tasks. Among the pivotal components of the transformer architecture, the attention mechanism plays a crucial role in capturing token interactions within sequences through the utilization of softmax function. Conversely, linear attention presents a more computationally efficient alternative by approximating the softmax operation with linear complexity. However, it exhibits substantial performance degradation when compared to the traditional softmax attention mechanism. In this paper, we bridge the gap in our theoretical understanding of the reasons behind the practical performance gap between softmax and linear attention. By conducting a comprehensive comparative analysis of these two attention mechanisms, we shed light on the underlying reasons for why softmax attention outperforms linear attention in most scenarios.

An Automatic Learning Rate Schedule Algorithm for Achieving Faster Convergence and Steeper Descent

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.

How to Capture Higher-order Correlations? Generalizing Matrix Softmax Attention to Kronecker Computation

In the classical transformer attention scheme, we are given three $n \times d$ size matrices $Q, K, V$ (the query, key, and value tokens), and the goal is to compute a new $n \times d$ size matrix $D^{-1} \exp(QK^\top) V$ where $D = \mathrm{diag}( \exp(QK^\top) {\bf 1}_n )$. In this work, we study a generalization of attention which captures triple-wise correlations. This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers. The potential downside of this generalization is that it appears as though computations are even more difficult, since the straightforward algorithm requires cubic time in $n$. However, we show that in the bounded-entry setting (which arises in practice, and which is well-studied in both theory and practice), there is actually a near-linear time algorithm. More precisely, we show that bounded entries are both necessary and sufficient for quickly performing generalized computations: $\bullet$ On the positive side, if all entries of the input matrices are bounded above by $o(\sqrt[3]{\log n})$ then we show how to approximate the ``tensor-type'' attention matrix in $n^{1+o(1)}$ time. $\bullet$ On the negative side, we show that if the entries of the input matrices may be as large as $\Omega(\sqrt[3]{\log n})$, then there is no algorithm that runs faster than $n^{3-o(1)}$ (assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory). We also show that our construction, algorithms, and lower bounds naturally generalize to higher-order tensors and correlations. Interestingly, the higher the order of the tensors, the lower the bound on the entries needs to be for an efficient algorithm. Our results thus yield a natural tradeoff between the boundedness of the entries, and order of the tensor one may use for more expressive, efficient attention computation.

Fine-tune Language Models to Approximate Unbiased In-context Learning

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.