Zhao Song

Yeqi Gao, Sridhar Mahadevan, Zhao Song

Over the past few years, there has been a significant amount of research focused on studying the ReLU activation function, with the aim of achieving neural network convergence through over-parametrization. However, recent developments in the field of Large Language Models (LLMs) have sparked interest in the use of exponential activation functions, specifically in the attention mechanism. Mathematically, we define the neural function $F: \mathbb{R}^{d \times m} \times \mathbb{R}^d \rightarrow \mathbb{R}$ using an exponential activation function. Given a set of data points with labels $\{(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\} \subset \mathbb{R}^d \times \mathbb{R}$ where $n$ denotes the number of the data. Here $F(W(t),x)$ can be expressed as $F(W(t),x) := \sum_{r=1}^m a_r \exp(\langle w_r, x \rangle)$, where $m$ represents the number of neurons, and $w_r(t)$ are weights at time $t$. It's standard in literature that $a_r$ are the fixed weights and it's never changed during the training. We initialize the weights $W(0) \in \mathbb{R}^{d \times m}$ with random Gaussian distributions, such that $w_r(0) \sim \mathcal{N}(0, I_d)$ and initialize $a_r$ from random sign distribution for each $r \in [m]$. Using the gradient descent algorithm, we can find a weight $W(T)$ such that $\| F(W(T), X) - y \|_2 \leq ε$ holds with probability $1-δ$, where $ε\in (0,0.1)$ and $m = Ω(n^{2+o(1)}\log(n/δ))$. To optimize the over-parameterization bound $m$, we employ several tight analysis techniques from previous studies [Song and Yang arXiv 2019, Munteanu, Omlor, Song and Woodruff ICML 2022].

Yichuan Deng, Sridhar Mahadevan, Zhao Song

Large language models (LLMs) have shown their power in different areas. Attention computation, as an important subroutine of LLMs, has also attracted interests in theory. Recently the static computation and dynamic maintenance of attention matrix has been studied by [Alman and Song 2023] and [Brand, Song and Zhou 2023] from both algorithmic perspective and hardness perspective. In this work, we consider the sparsification of the attention problem. We make one simplification which is the logit matrix is symmetric. Let $n$ denote the length of sentence, let $d$ denote the embedding dimension. Given a matrix $X \in \mathbb{R}^{n \times d}$, suppose $d \gg n$ and $\| X X^\top \|_{\infty} < r$ with $r \in (0,0.1)$, then we aim for finding $Y \in \mathbb{R}^{n \times m}$ (where $m\ll d$) such that \begin{align*} \| D(Y)^{-1} \exp( Y Y^\top ) - D(X)^{-1} \exp( X X^\top) \|_{\infty} \leq O(r) \end{align*} We provide two results for this problem. $\bullet$ Our first result is a randomized algorithm. It runs in $\widetilde{O}(\mathrm{nnz}(X) + n^ω ) $ time, has $1-δ$ succeed probability, and chooses $m = O(n \log(n/δ))$. Here $\mathrm{nnz}(X)$ denotes the number of non-zero entries in $X$. We use $ω$ to denote the exponent of matrix multiplication. Currently $ω\approx 2.373$. $\bullet$ Our second result is a deterministic algorithm. It runs in $\widetilde{O}(\min\{\sum_{i\in[d]}\mathrm{nnz}(X_i)^2, dn^{ω-1}\} + n^{ω+1})$ time and chooses $m = O(n)$. Here $X_i$ denote the $i$-th column of matrix $X$. Our main findings have the following implication for applied LLMs task: for any super large feature dimension, we can reduce it down to the size nearly linear in length of sentence.

Mayee F. Chen, Daniel Y. Fu, Avanika Narayan et al.

An ideal learned representation should display transferability and robustness. Supervised contrastive learning (SupCon) is a promising method for training accurate models, but produces representations that do not capture these properties due to class collapse -- when all points in a class map to the same representation. Recent work suggests that "spreading out" these representations improves them, but the precise mechanism is poorly understood. We argue that creating spread alone is insufficient for better representations, since spread is invariant to permutations within classes. Instead, both the correct degree of spread and a mechanism for breaking this invariance are necessary. We first prove that adding a weighted class-conditional InfoNCE loss to SupCon controls the degree of spread. Next, we study three mechanisms to break permutation invariance: using a constrained encoder, adding a class-conditional autoencoder, and using data augmentation. We show that the latter two encourage clustering of latent subclasses under more realistic conditions than the former. Using these insights, we show that adding a properly-weighted class-conditional InfoNCE loss and a class-conditional autoencoder to SupCon achieves 11.1 points of lift on coarse-to-fine transfer across 5 standard datasets and 4.7 points on worst-group robustness on 3 datasets, setting state-of-the-art on CelebA by 11.5 points.

Alexander Munteanu, Simon Omlor, Zhao Song et al.

A common method in training neural networks is to initialize all the weights to be independent Gaussian vectors. We observe that by instead initializing the weights into independent pairs, where each pair consists of two identical Gaussian vectors, we can significantly improve the convergence analysis. While a similar technique has been studied for random inputs [Daniely, NeurIPS 2020], it has not been analyzed with arbitrary inputs. Using this technique, we show how to significantly reduce the number of neurons required for two-layer ReLU networks, both in the under-parameterized setting with logistic loss, from roughly $γ^{-8}$ [Ji and Telgarsky, ICLR 2020] to $γ^{-2}$, where $γ$ denotes the separation margin with a Neural Tangent Kernel, as well as in the over-parameterized setting with squared loss, from roughly $n^4$ [Song and Yang, 2019] to $n^2$, implicitly also improving the recent running time bound of [Brand, Peng, Song and Weinstein, ITCS 2021]. For the under-parameterized setting we also prove new lower bounds that improve upon prior work, and that under certain assumptions, are best possible.

Yichuan Deng, Hang Hu, Zhao Song et al.

The success of deep learning comes at a tremendous computational and energy cost, and the scalability of training massively overparametrized neural networks is becoming a real barrier to the progress of artificial intelligence (AI). Despite the popularity and low cost-per-iteration of traditional backpropagation via gradient decent, stochastic gradient descent (SGD) has prohibitive convergence rate in non-convex settings, both in theory and practice. To mitigate this cost, recent works have proposed to employ alternative (Newton-type) training methods with much faster convergence rate, albeit with higher cost-per-iteration. For a typical neural network with $m=\mathrm{poly}(n)$ parameters and input batch of $n$ datapoints in $\mathbb{R}^d$, the previous work of [Brand, Peng, Song, and Weinstein, ITCS'2021] requires $\sim mnd + n^3$ time per iteration. In this paper, we present a novel training method that requires only $m^{1-α} n d + n^3$ amortized time in the same overparametrized regime, where $α\in (0.01,1)$ is some fixed constant. This method relies on a new and alternative view of neural networks, as a set of binary search trees, where each iteration corresponds to modifying a small subset of the nodes in the tree. We believe this view would have further applications in the design and analysis of deep neural networks (DNNs).

Zhihang Li, Zhao Song, Tianyi Zhou

In modern machine learning, attention computation is a fundamental task for training large language models such as Transformer, GPT-4 and ChatGPT. In this work, we study exponential regression problem which is inspired by the softmax/exp unit in the attention mechanism in large language models. The standard exponential regression is non-convex. We study the regularization version of exponential regression problem which is a convex problem. We use approximate newton method to solve in input sparsity time. Formally, in this problem, one is given matrix $A \in \mathbb{R}^{n \times d}$, $b \in \mathbb{R}^n$, $w \in \mathbb{R}^n$ and any of functions $\exp, \cosh$ and $\sinh$ denoted as $f$. The goal is to find the optimal $x$ that minimize $ 0.5 \| f(Ax) - b \|_2^2 + 0.5 \| \mathrm{diag}(w) A x \|_2^2$. The straightforward method is to use the naive Newton's method. Let $\mathrm{nnz}(A)$ denote the number of non-zeros entries in matrix $A$. Let $ω$ denote the exponent of matrix multiplication. Currently, $ω\approx 2.373$. Let $ε$ denote the accuracy error. In this paper, we make use of the input sparsity and purpose an algorithm that use $\log ( \|x_0 - x^*\|_2 / ε)$ iterations and $\widetilde{O}(\mathrm{nnz}(A) + d^ω )$ per iteration time to solve the problem.

Yuzhou Gu, Zhao Song, Junze Yin et al.

Given a matrix $M\in \mathbb{R}^{m\times n}$, the low rank matrix completion problem asks us to find a rank-$k$ approximation of $M$ as $UV^\top$ for $U\in \mathbb{R}^{m\times k}$ and $V\in \mathbb{R}^{n\times k}$ by only observing a few entries specified by a set of entries $Ω\subseteq [m]\times [n]$. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli, and Sanghavi [JNS13] showed that if $M$ has incoherent rows and columns, then alternating minimization provably recovers the matrix $M$ by observing a nearly linear in $n$ number of entries. While the sample complexity has been subsequently improved [GLZ17], alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate a moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time $\widetilde O(|Ω| k)$, which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require $\widetilde O(|Ω| k^2)$ time.

Yichuan Deng, Zhihang Li, Sridhar Mahadevan et al.

Large language models (LLMs) have brought about significant transformations in human society. Among the crucial computations in LLMs, the softmax unit holds great importance. Its helps the model generating a probability distribution on potential subsequent words or phrases, considering a series of input words. By utilizing this distribution, the model selects the most probable next word or phrase, based on the assigned probabilities. The softmax unit assumes a vital function in LLM training as it facilitates learning from data through the adjustment of neural network weights and biases. With the development of the size of LLMs, computing the gradient becomes expensive. However, Zero-th Order method can approximately compute the gradient with only forward passes. In this paper, we present a Zero-th Order algorithm specifically tailored for Softmax optimization. We demonstrate the convergence of our algorithm, highlighting its effectiveness in efficiently computing gradients for large-scale LLMs. By leveraging the Zeroth-Order method, our work contributes to the advancement of optimization techniques in the context of complex language models.

Shuai Li, Zhao Song, Yu Xia et al.

Large language models (LLMs) are known for their exceptional performance in natural language processing, making them highly effective in many human life-related or even job-related tasks. The attention mechanism in the Transformer architecture is a critical component of LLMs, as it allows the model to selectively focus on specific input parts. The softmax unit, which is a key part of the attention mechanism, normalizes the attention scores. Hence, the performance of LLMs in various NLP tasks depends significantly on the crucial role played by the attention mechanism with the softmax unit. In-context learning, as one of the celebrated abilities of recent LLMs, is an important concept in querying LLMs such as ChatGPT. Without further parameter updates, Transformers can learn to predict based on few in-context examples. However, the reason why Transformers becomes in-context learners is not well understood. Recently, several works [ASA+22,GTLV22,ONR+22] have studied the in-context learning from a mathematical perspective based on a linear regression formulation $\min_x\| Ax - b \|_2$, which show Transformers' capability of learning linear functions in context. In this work, we study the in-context learning based on a softmax regression formulation $\min_{x} \| \langle \exp(Ax), {\bf 1}_n \rangle^{-1} \exp(Ax) - b \|_2$ of Transformer's attention mechanism. We show the upper bounds of the data transformations induced by a single self-attention layer and by gradient-descent on a $\ell_2$ regression loss for softmax prediction function, which imply that when training self-attention-only Transformers for fundamental regression tasks, the models learned by gradient-descent and Transformers show great similarity.

Josh Alman, Zhao Song

In modern machine learning, inner product attention computation is a fundamental task for training large language models such as Transformer, GPT-1, BERT, GPT-2, GPT-3 and ChatGPT. Formally, in this problem, one is given as input three matrices $Q, K, V \in [-B,B]^{n \times d}$, and the goal is to construct the matrix $\mathrm{Att}(Q,K,V) := \mathrm{diag}(A {\bf 1}_n)^{-1} A V \in \mathbb{R}^{n \times d}$, where $A = \exp(QK^\top/d)$ is the `attention matrix', and $\exp$ is applied entry-wise. Straightforward methods for this problem explicitly compute the $n \times n$ attention matrix $A$, and hence require time $Ω(n^2)$ even when $d = n^{o(1)}$ is small. In this paper, we investigate whether faster algorithms are possible by implicitly making use of the matrix $A$. We present two results, showing that there is a sharp transition at $B = Θ(\sqrt{\log n})$. $\bullet$ If $d = O(\log n)$ and $B = o(\sqrt{\log n})$, there is an $n^{1+o(1)}$ time algorithm to approximate $\mathrm{Att}(Q,K,V)$ up to $1/\mathrm{poly}(n)$ additive error. $\bullet$ If $d = O(\log n)$ and $B = Θ(\sqrt{\log n})$, assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory, it is impossible to approximate $\mathrm{Att}(Q,K,V)$ up to $1/\mathrm{poly}(n)$ additive error in truly subquadratic time $n^{2 - Ω(1)}$. This gives a theoretical explanation for the phenomenon observed in practice that attention computation is much more efficient when the input matrices have smaller entries.

Hang Hu, Zhao Song, Runzhou Tao et al.

Online bipartite matching is a fundamental problem in online algorithms. The goal is to match two sets of vertices to maximize the sum of the edge weights, where for one set of vertices, each vertex and its corresponding edge weights appear in a sequence. Currently, in the practical recommendation system or search engine, the weights are decided by the inner product between the deep representation of a user and the deep representation of an item. The standard online matching needs to pay $nd$ time to linear scan all the $n$ items, computing weight (assuming each representation vector has length $d$), and then deciding the matching based on the weights. However, in reality, the $n$ could be very large, e.g. in online e-commerce platforms. Thus, improving the time of computing weights is a problem of practical significance. In this work, we provide the theoretical foundation for computing the weights approximately. We show that, with our proposed randomized data structures, the weights can be computed in sublinear time while still preserving the competitive ratio of the matching algorithm.

Anshumali Shrivastava, Zhao Song, Zhaozhuo Xu

Graph-based algorithms have demonstrated state-of-the-art performance in the nearest neighbor search (NN-Search) problem. These empirical successes urge the need for theoretical results that guarantee the search quality and efficiency of these algorithms. However, there exists a practice-to-theory gap in the graph-based NN-Search algorithms. Current theoretical literature focuses on greedy search on exact near neighbor graph while practitioners use approximate near neighbor graph (ANN-Graph) to reduce the preprocessing time. This work bridges this gap by presenting the theoretical guarantees of solving NN-Search via greedy search on ANN-Graph for low dimensional and dense vectors. To build this bridge, we leverage several novel tools from computational geometry. Our results provide quantification of the trade-offs associated with the approximation while building a near neighbor graph. We hope our results will open the door for more provable efficient graph-based NN-Search algorithms.

Josh Alman, Zhao Song

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 $Ω(\sqrt[3]{\log n})$, 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.

Yichuan Deng, Yeqi Gao, Zhao Song

Large language models have become ubiquitous in modern life, finding applications in various domains such as natural language processing, language translation, and speech recognition. Recently, a breakthrough work [Zhao, Panigrahi, Ge, and Arora Arxiv 2023] explains the attention model from probabilistic context-free grammar (PCFG). One of the central computation task for computing probability in PCFG is formulating a particular tensor low rank approximation problem, we can call it tensor cycle rank. Given an $n \times n \times n$ third order tensor $A$, we say that $A$ has cycle rank-$k$ if there exists three $n \times k^2$ size matrices $U , V$, and $W$ such that for each entry in each \begin{align*} A_{a,b,c} = \sum_{i=1}^k \sum_{j=1}^k \sum_{l=1}^k U_{a,i+k(j-1)} \otimes V_{b, j + k(l-1)} \otimes W_{c, l + k(i-1) } \end{align*} for all $a \in [n], b \in [n], c \in [n]$. For the tensor classical rank, tucker rank and train rank, it has been well studied in [Song, Woodruff, Zhong SODA 2019]. In this paper, we generalize the previous ``rotation and sketch'' technique in page 186 of [Song, Woodruff, Zhong SODA 2019] and show an input sparsity time algorithm for cycle rank.

Yichuan Deng, Zhihang Li, Zhao Song

Large language models (LLMs) have made transformed changes for human society. One of the key computation in LLMs is the softmax unit. This operation is important in LLMs because it allows the model to generate a distribution over possible next words or phrases, given a sequence of input words. This distribution is then used to select the most likely next word or phrase, based on the probabilities assigned by the model. The softmax unit plays a crucial role in training LLMs, as it allows the model to learn from the data by adjusting the weights and biases of the neural network. In the area of convex optimization such as using central path method to solve linear programming. The softmax function has been used a crucial tool for controlling the progress and stability of potential function [Cohen, Lee and Song STOC 2019, Brand SODA 2020]. In this work, inspired the softmax unit, we define a softmax regression problem. Formally speaking, given a matrix $A \in \mathbb{R}^{n \times d}$ and a vector $b \in \mathbb{R}^n$, the goal is to use greedy type algorithm to solve \begin{align*} \min_{x} \| \langle \exp(Ax), {\bf 1}_n \rangle^{-1} \exp(Ax) - b \|_2^2. \end{align*} In certain sense, our provable convergence result provides theoretical support for why we can use greedy algorithm to train softmax function in practice.

Josh Alman, Jiehao Liang, Zhao Song et al.

Over the last decade, deep neural networks have transformed our society, and they are already widely applied in various machine learning applications. State-of-art deep neural networks are becoming larger in size every year to deliver increasing model accuracy, and as a result, model training consumes substantial computing resources and will only consume more in the future. Using current training methods, in each iteration, to process a data point $x \in \mathbb{R}^d$ in a layer, we need to spend $Θ(md)$ time to evaluate all the $m$ neurons in the layer. This means processing the entire layer takes $Θ(nmd)$ time for $n$ data points. Recent work [Song, Yang and Zhang, NeurIPS 2021] reduces this time per iteration to $o(nmd)$, but requires exponential time to preprocess either the data or the neural network weights, making it unlikely to have practical usage. In this work, we present a new preprocessing method that simply stores the weight-data correlation in a tree data structure in order to quickly, dynamically detect which neurons fire at each iteration. Our method requires only $O(nmd)$ time in preprocessing and still achieves $o(nmd)$ time per iteration. We complement our new algorithm with a lower bound, proving that assuming a popular conjecture from complexity theory, one could not substantially speed up our algorithm for dynamic detection of firing neurons.

Lianke Qin, Aravind Reddy, Zhao Song et al.

In this paper, we propose Adam-Hash: an adaptive and dynamic multi-resolution hashing data-structure for fast pairwise summation estimation. Given a data-set $X \subset \mathbb{R}^d$, a binary function $f:\mathbb{R}^d\times \mathbb{R}^d\to \mathbb{R}$, and a point $y \in \mathbb{R}^d$, the Pairwise Summation Estimate $\mathrm{PSE}_X(y) := \frac{1}{|X|} \sum_{x \in X} f(x,y)$. For any given data-set $X$, we need to design a data-structure such that given any query point $y \in \mathbb{R}^d$, the data-structure approximately estimates $\mathrm{PSE}_X(y)$ in time that is sub-linear in $|X|$. Prior works on this problem have focused exclusively on the case where the data-set is static, and the queries are independent. In this paper, we design a hashing-based PSE data-structure which works for the more practical \textit{dynamic} setting in which insertions, deletions, and replacements of points are allowed. Moreover, our proposed Adam-Hash is also robust to adaptive PSE queries, where an adversary can choose query $q_j \in \mathbb{R}^d$ depending on the output from previous queries $q_1, q_2, \dots, q_{j-1}$.

Kai Wang, Zhao Song, Georgios Theocharous et al.

Smoothed online combinatorial optimization considers a learner who repeatedly chooses a combinatorial decision to minimize an unknown changing cost function with a penalty on switching decisions in consecutive rounds. We study smoothed online combinatorial optimization problems when an imperfect predictive model is available, where the model can forecast the future cost functions with uncertainty. We show that using predictions to plan for a finite time horizon leads to regret dependent on the total predictive uncertainty and an additional switching cost. This observation suggests choosing a suitable planning window to balance between uncertainty and switching cost, which leads to an online algorithm with guarantees on the upper and lower bounds of the cumulative regret. Empirically, our algorithm shows a significant improvement in cumulative regret compared to other baselines in synthetic online distributed streaming problems.

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.

Zhao Song, Yitan Wang, Zheng Yu et al.

Sketching is one of the most fundamental tools in large-scale machine learning. It enables runtime and memory saving via randomly compressing the original large problem into lower dimensions. In this paper, we propose a novel sketching scheme for the first order method in large-scale distributed learning setting, such that the communication costs between distributed agents are saved while the convergence of the algorithms is still guaranteed. Given gradient information in a high dimension $d$, the agent passes the compressed information processed by a sketching matrix $R\in \mathbb{R}^{s\times d}$ with $s\ll d$, and the receiver de-compressed via the de-sketching matrix $R^\top$ to ``recover'' the information in original dimension. Using such a framework, we develop algorithms for federated learning with lower communication costs. However, such random sketching does not protect the privacy of local data directly. We show that the gradient leakage problem still exists after applying the sketching technique by presenting a specific gradient attack method. As a remedy, we prove rigorously that the algorithm will be differentially private by adding additional random noises in gradient information, which results in a both communication-efficient and differentially private first order approach for federated learning tasks. Our sketching scheme can be further generalized to other learning settings and might be of independent interest itself.

Yeqi Gao, Zhao Song, Shenghao Xie

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.

Ritwik Sinha, Zhao Song, Tianyi Zhou

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.

Yeqi Gao, Lianke Qin, Zhao Song et al.

Adversarial training is a widely used strategy for making neural networks resistant to adversarial perturbations. For a neural network of width $m$, $n$ input training data in $d$ dimension, it takes $Ω(mnd)$ time cost per training iteration for the forward and backward computation. In this paper we analyze the convergence guarantee of adversarial training procedure on a two-layer neural network with shifted ReLU activation, and shows that only $o(m)$ neurons will be activated for each input data per iteration. Furthermore, we develop an algorithm for adversarial training with time cost $o(m n d)$ per iteration by applying half-space reporting data structure.

Xiaoxiao Li, Zhao Song, Runzhou Tao et al.

Federated learning enables a large amount of edge computing devices to learn a model without data sharing jointly. As a leading algorithm in this setting, Federated Average FedAvg, which runs Stochastic Gradient Descent (SGD) in parallel on local devices and averages the sequences only once in a while, have been widely used due to their simplicity and low communication cost. However, despite recent research efforts, it lacks theoretical analysis under assumptions beyond smoothness. In this paper, we analyze the convergence of FedAvg. Different from the existing work, we relax the assumption of strong smoothness. More specifically, we assume the semi-smoothness and semi-Lipschitz properties for the loss function, which have an additional first-order term in assumption definitions. In addition, we also assume bound on the gradient, which is weaker than the commonly used bounded gradient assumption in the convergence analysis scheme. As a solution, this paper provides a theoretical convergence study on Federated Learning.

Zhao Song, Mingquan Ye, Junze Yin et al.

Given a matrix $A\in \mathbb{R}^{n\times d}$ and a vector $b\in \mathbb{R}^n$, we consider the regression problem with $\ell_\infty$ guarantees: finding a vector $x'\in \mathbb{R}^d$ such that $ \|x'-x^*\|_\infty \leq \fracε{\sqrt{d}}\cdot \|Ax^*-b\|_2\cdot \|A^\dagger\|$ where $x^*=\arg\min_{x\in \mathbb{R}^d}\|Ax-b\|_2$. One popular approach for solving such $\ell_2$ regression problem is via sketching: picking a structured random matrix $S\in \mathbb{R}^{m\times n}$ with $m\ll n$ and $SA$ can be quickly computed, solve the ``sketched'' regression problem $\arg\min_{x\in \mathbb{R}^d} \|SAx-Sb\|_2$. In this paper, we show that in order to obtain such $\ell_\infty$ guarantee for $\ell_2$ regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with $m=ε^{-2}d\log^3(n/δ)$ such that solving the sketched regression problem gives the $\ell_\infty$ guarantee, with probability at least $1-δ$. Moreover, the matrix $SA$ can be computed in time $O(nd\log n)$. Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP'17], in which a super-linear in $d$ rows, $m=Ω(ε^{-2}d^{1+γ})$ for $γ=Θ(\sqrt{\frac{\log\log n}{\log d}})$ is required. We also develop a novel analytical framework for $\ell_\infty$ guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML'21]. Our analysis is arguably much simpler and more general than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for tensor product of vectors.

Lianke Qin, Zhao Song, Ruizhe Zhang

Matrix sensing has many real-world applications in science and engineering, such as system control, distance embedding, and computer vision. The goal of matrix sensing is to recover a matrix $A_\star \in \mathbb{R}^{n \times n}$, based on a sequence of measurements $(u_i,b_i) \in \mathbb{R}^{n} \times \mathbb{R}$ such that $u_i^\top A_\star u_i = b_i$. Previous work [ZJD15] focused on the scenario where matrix $A_{\star}$ has a small rank, e.g. rank-$k$. Their analysis heavily relies on the RIP assumption, making it unclear how to generalize to high-rank matrices. In this paper, we relax that rank-$k$ assumption and solve a much more general matrix sensing problem. Given an accuracy parameter $δ\in (0,1)$, we can compute $A \in \mathbb{R}^{n \times n}$ in $\widetilde{O}(m^{3/2} n^2 δ^{-1} )$, such that $ |u_i^\top A u_i - b_i| \leq δ$ for all $i \in [m]$. We design an efficient algorithm with provable convergence guarantees using stochastic gradient descent for this problem.

Aravind Reddy, Zhao Song, Lichen Zhang

In this work, we initiate the study of \emph{Dynamic Tensor Product Regression}. One has matrices $A_1\in \mathbb{R}^{n_1\times d_1},\ldots,A_q\in \mathbb{R}^{n_q\times d_q}$ and a label vector $b\in \mathbb{R}^{n_1\ldots n_q}$, and the goal is to solve the regression problem with the design matrix $A$ being the tensor product of the matrices $A_1, A_2, \dots, A_q$ i.e. $\min_{x\in \mathbb{R}^{d_1\ldots d_q}}~\|(A_1\otimes \ldots\otimes A_q)x-b\|_2$. At each time step, one matrix $A_i$ receives a sparse change, and the goal is to maintain a sketch of the tensor product $A_1\otimes\ldots \otimes A_q$ so that the regression solution can be updated quickly. Recomputing the solution from scratch for each round is very slow and so it is important to develop algorithms which can quickly update the solution with the new design matrix. Our main result is a dynamic tree data structure where any update to a single matrix can be propagated quickly throughout the tree. We show that our data structure can be used to solve dynamic versions of not only Tensor Product Regression, but also Tensor Product Spline regression (which is a generalization of ridge regression) and for maintaining Low Rank Approximations for the tensor product.

Jiehao Liang, Zhao Song, Zhaozhuo Xu et al.

Kernel density estimation (KDE) stands out as a challenging task in machine learning. The problem is defined in the following way: given a kernel function $f(x,y)$ and a set of points $\{x_1, x_2, \cdots, x_n \} \subset \mathbb{R}^d$, we would like to compute $\frac{1}{n}\sum_{i=1}^{n} f(x_i,y)$ for any query point $y \in \mathbb{R}^d$. Recently, there has been a growing trend of using data structures for efficient KDE. However, the proposed KDE data structures focus on static settings. The robustness of KDE data structures over dynamic changing data distributions is not addressed. In this work, we focus on the dynamic maintenance of KDE data structures with robustness to adversarial queries. Especially, we provide a theoretical framework of KDE data structures. In our framework, the KDE data structures only require subquadratic spaces. Moreover, our data structure supports the dynamic update of the dataset in sublinear time. Furthermore, we can perform adaptive queries with the potential adversary in sublinear time.

Yeqi Gao, Zhao Song, Weixin Wang et al.

Large language models (LLMs) have played a pivotal role in revolutionizing various facets of our daily existence. Solving attention regression is a fundamental task in optimizing LLMs. In this work, we focus on giving a provable guarantee for the one-layer attention network objective function $L(X,Y) = \sum_{j_0 = 1}^n \sum_{i_0 = 1}^d ( \langle \langle \exp( \mathsf{A}_{j_0} x ) , {\bf 1}_n \rangle^{-1} \exp( \mathsf{A}_{j_0} x ), A_{3} Y_{*,i_0} \rangle - b_{j_0,i_0} )^2$. Here $\mathsf{A} \in \mathbb{R}^{n^2 \times d^2}$ is Kronecker product between $A_1 \in \mathbb{R}^{n \times d}$ and $A_2 \in \mathbb{R}^{n \times d}$. $A_3$ is a matrix in $\mathbb{R}^{n \times d}$, $\mathsf{A}_{j_0} \in \mathbb{R}^{n \times d^2}$ is the $j_0$-th block of $\mathsf{A}$. The $X, Y \in \mathbb{R}^{d \times d}$ are variables we want to learn. $B \in \mathbb{R}^{n \times d}$ and $b_{j_0,i_0} \in \mathbb{R}$ is one entry at $j_0$-th row and $i_0$-th column of $B$, $Y_{*,i_0} \in \mathbb{R}^d$ is the $i_0$-column vector of $Y$, and $x \in \mathbb{R}^{d^2}$ is the vectorization of $X$. In a multi-layer LLM network, the matrix $B \in \mathbb{R}^{n \times d}$ can be viewed as the output of a layer, and $A_1= A_2 = A_3 \in \mathbb{R}^{n \times d}$ can be viewed as the input of a layer. The matrix version of $x$ can be viewed as $QK^\top$ and $Y$ can be viewed as $V$. We provide an iterative greedy algorithm to train loss function $L(X,Y)$ up $ε$ that runs in $\widetilde{O}( ({\cal T}_{\mathrm{mat}}(n,n,d) + {\cal T}_{\mathrm{mat}}(n,d,d) + d^{2ω}) \log(1/ε) )$ time. Here ${\cal T}_{\mathrm{mat}}(a,b,c)$ denotes the time of multiplying $a \times b$ matrix another $b \times c$ matrix, and $ω\approx 2.37$ denotes the exponent of matrix multiplication.

Xiaoxiao Li, Zhao Song, Jiaming Yang

Federated learning (FL) is a trending training paradigm to utilize decentralized training data. FL allows clients to update model parameters locally for several epochs, then share them to a global model for aggregation. This training paradigm with multi-local step updating before aggregation exposes unique vulnerabilities to adversarial attacks. Adversarial training is a popular and effective method to improve the robustness of networks against adversaries. In this work, we formulate a general form of federated adversarial learning (FAL) that is adapted from adversarial learning in the centralized setting. On the client side of FL training, FAL has an inner loop to generate adversarial samples for adversarial training and an outer loop to update local model parameters. On the server side, FAL aggregates local model updates and broadcast the aggregated model. We design a global robust training loss and formulate FAL training as a min-max optimization problem. Unlike the convergence analysis in classical centralized training that relies on the gradient direction, it is significantly harder to analyze the convergence in FAL for three reasons: 1) the complexity of min-max optimization, 2) model not updating in the gradient direction due to the multi-local updates on the client-side before aggregation and 3) inter-client heterogeneity. We address these challenges by using appropriate gradient approximation and coupling techniques and present the convergence analysis in the over-parameterized regime. Our main result theoretically shows that the minimum loss under our algorithm can converge to $ε$ small with chosen learning rate and communication rounds. It is noteworthy that our analysis is feasible for non-IID clients.

Xiang Chen, Zhao Song, Baocheng Sun et al.

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.

Jiehao Liang, Somdeb Sarkhel, Zhao Song et al.

$k$-means++ is an important algorithm for choosing initial cluster centers for the $k$-means clustering algorithm. In this work, we present a new algorithm that can solve the $k$-means++ problem with nearly optimal running time. Given $n$ data points in $\mathbb{R}^d$, the current state-of-the-art algorithm runs in $\widetilde{O}(k )$ iterations, and each iteration takes $\widetilde{O}(nd k)$ time. The overall running time is thus $\widetilde{O}(n d k^2)$. We propose a new algorithm \textsc{FastKmeans++} that only takes in $\widetilde{O}(nd + nk^2)$ time, in total.

Zhao Song, Ke Yang, Naiyang Guan et al.

Large-scale pre-trained transformers have demonstrated remarkable success in various computer vision tasks. However, it is still highly challenging to fully fine-tune these models for downstream tasks due to their high computational and storage costs. Recently, Parameter-Efficient Tuning (PETuning) techniques, e.g., Visual Prompt Tuning (VPT) and Low-Rank Adaptation (LoRA), have significantly reduced the computation and storage cost by inserting lightweight prompt modules into the pre-trained models and tuning these prompt modules with a small number of trainable parameters, while keeping the transformer backbone frozen. Although only a few parameters need to be adjusted, most PETuning methods still require a significant amount of downstream task training data to achieve good results. The performance is inadequate on low-data regimes, especially when there are only one or two examples per class. To this end, we first empirically identify the poor performance is mainly due to the inappropriate way of initializing prompt modules, which has also been verified in the pre-trained language models. Next, we propose a Pre-trained Visual Parameter-efficient (PVP) Tuning framework, which pre-trains the parameter-efficient tuning modules first and then leverages the pre-trained modules along with the pre-trained transformer backbone to perform parameter-efficient tuning on downstream tasks. Experiment results on five Fine-Grained Visual Classification (FGVC) and VTAB-1k datasets demonstrate that our proposed method significantly outperforms state-of-the-art PETuning methods.

Zhao Song, Mingquan Ye, Junze Yin et al.

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 non-negative weight matrix $W \in \mathbb{R}_{\geq 0}^{n \times n}$, a parameter $k$, the goal is to output two matrices $X,Y\in \mathbb{R}^{n \times k}$ such that $\| W \circ (M - X Y^\top) \|_F$ is minimized, where $\circ$ denotes the Hadamard product. It naturally generalizes the well-studied low rank matrix completion problem. Such a problem is known to be NP-hard and even hard to approximate assuming the Exponential Time Hypothesis [GG11, RSW16]. Meanwhile, alternating minimization is a good heuristic solution for weighted low rank approximation. In particular, [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 that allows the alternating updates to be computed approximately. For weighted low rank approximation, this improves the runtime of [LLR16] from $\|W\|_0k^2$ to $\|W\|_0 k$ where $\|W\|_0$ denotes the number of nonzero entries of the weight matrix. At the heart of our framework is a high-accuracy multiple response regression solver together with a robust analysis of alternating minimization.

Yuzhou Gu, Zhao Song, Lichen Zhang

Quadratic programming is a ubiquitous prototype in convex programming. Many machine learning problems can be formulated as quadratic programming, including the famous Support Vector Machines (SVMs). Linear and kernel SVMs have been among the most popular models in machine learning over the past three decades, prior to the deep learning era. Generally, a quadratic program has an input size of $Θ(n^2)$, where $n$ is the number of variables. Assuming the Strong Exponential Time Hypothesis ($\textsf{SETH}$), it is known that no $O(n^{2-o(1)})$ time algorithm exists when the quadratic objective matrix is positive semidefinite (Backurs, Indyk, and Schmidt, NeurIPS'17). However, problems such as SVMs usually admit much smaller input sizes: one is given $n$ data points, each of dimension $d$, and $d$ is oftentimes much smaller than $n$. Furthermore, the SVM program has only $O(1)$ equality linear constraints. This suggests that faster algorithms are feasible, provided the program exhibits certain structures. In this work, we design the first nearly-linear time algorithm for solving quadratic programs whenever the quadratic objective admits a low-rank factorization, and the number of linear constraints is small. Consequently, we obtain results for SVMs: * For linear SVM when the input data is $d$-dimensional, our algorithm runs in time $\widetilde O(nd^{(ω+1)/2}\log(1/ε))$ where $ω\approx 2.37$ is the fast matrix multiplication exponent; * For Gaussian kernel SVM, when the data dimension $d = {\color{black}O(\log n)}$ and the squared dataset radius is sub-logarithmic in $n$, our algorithm runs in time $O(n^{1+o(1)}\log(1/ε))$. We also prove that when the squared dataset radius is at least $Ω(\log^2 n)$, then $Ω(n^{2-o(1)})$ time is required. This improves upon the prior best lower bound in both the dimension $d$ and the squared dataset radius.

Yichuan Deng, Zhao Song, Tianyi Zhou

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.

Erzhi Liu, Jerry Yao-Chieh Hu, Alex Reneau et al.

We introduce a refined differentially private (DP) data structure for kernel density estimation (KDE), offering not only improved privacy-utility tradeoff but also better efficiency over prior results. Specifically, we study the mathematical problem: given a similarity function $f$ (or DP KDE) and a private dataset $X \subset \mathbb{R}^d$, our goal is to preprocess $X$ so that for any query $y\in\mathbb{R}^d$, we approximate $\sum_{x \in X} f(x, y)$ in a differentially private fashion. The best previous algorithm for $f(x,y) =\| x - y \|_1$ is the node-contaminated balanced binary tree by [Backurs, Lin, Mahabadi, Silwal, and Tarnawski, ICLR 2024]. Their algorithm requires $O(nd)$ space and time for preprocessing with $n=|X|$. For any query point, the query time is $d \log n$, with an error guarantee of $(1+α)$-approximation and $ε^{-1} α^{-0.5} d^{1.5} R \log^{1.5} n$. In this paper, we improve the best previous result [Backurs, Lin, Mahabadi, Silwal, and Tarnawski, ICLR 2024] in three aspects: - We reduce query time by a factor of $α^{-1} \log n$. - We improve the approximation ratio from $α$ to 1. - We reduce the error dependence by a factor of $α^{-0.5}$. From a technical perspective, our method of constructing the search tree differs from previous work [Backurs, Lin, Mahabadi, Silwal, and Tarnawski, ICLR 2024]. In prior work, for each query, the answer is split into $α^{-1} \log n$ numbers, each derived from the summation of $\log n$ values in interval tree countings. In contrast, we construct the tree differently, splitting the answer into $\log n$ numbers, where each is a smart combination of two distance values, two counting values, and $y$ itself. We believe our tree structure may be of independent interest.

Zhao Song, Weixin Wang, Junze Yin

Large language models (LLMs) have brought significant changes to human society. Softmax regression and residual neural networks (ResNet) are two important techniques in deep learning: they not only serve as significant theoretical components supporting the functionality of LLMs but also are related to many other machine learning and theoretical computer science fields, including but not limited to image classification, object detection, semantic segmentation, and tensors. Previous research works studied these two concepts separately. In this paper, we provide a theoretical analysis of the regression problem: $\| \langle \exp(Ax) + A x , {\bf 1}_n \rangle^{-1} ( \exp(Ax) + Ax ) - b \|_2^2$, where $A$ is a matrix in $\mathbb{R}^{n \times d}$, $b$ is a vector in $\mathbb{R}^n$, and ${\bf 1}_n$ is the $n$-dimensional vector whose entries are all $1$. This regression problem is a unified scheme that combines softmax regression and ResNet, which has never been done before. We derive the gradient, Hessian, and Lipschitz properties of the loss function. The Hessian is shown to be positive semidefinite, and its structure is characterized as the sum of a low-rank matrix and a diagonal matrix. This enables an efficient approximate Newton method. As a result, this unified scheme helps to connect two previously thought unrelated fields and provides novel insight into loss landscape and optimization for emerging over-parameterized neural networks, which is meaningful for future research in deep learning models.

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.

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.

Yichuan Deng, Zhao Song, Zifan Wang et al.

Principal Component Analysis (PCA) is a widely used technique in machine learning, data analysis and signal processing. With the increase in the size and complexity of datasets, it has become important to develop low-space usage algorithms for PCA. Streaming PCA has gained significant attention in recent years, as it can handle large datasets efficiently. The kernel method, which is commonly used in learning algorithms such as Support Vector Machines (SVMs), has also been applied in PCA algorithms. We propose a streaming algorithm for Kernel PCA problems based on the traditional scheme by Oja. Our algorithm addresses the challenge of reducing the memory usage of PCA while maintaining its accuracy. We analyze the performance of our algorithm by studying the conditions under which it succeeds. Specifically, we show that, when the spectral ratio $R := λ_1/λ_2$ of the target covariance matrix is lower bounded by $C \cdot \log n\cdot \log d$, the streaming PCA can be solved with $O(d)$ space cost. Our proposed algorithm has several advantages over existing methods. First, it is a streaming algorithm that can handle large datasets efficiently. Second, it employs the kernel method, which allows it to capture complex nonlinear relationships among data points. Third, it has a low-space usage, making it suitable for applications where memory is limited.

Lianke Qin, Saayan Mitra, Zhao Song et al.

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 $\forall i \in [k], \langle a_i,b_i \rangle \geq ρ\cdot d$, for a threshold $ρ\in (0,1)$, the goal is to identify those $k$ heavy inner products. We provide an algorithm that runs in $O(n^{2 ω/ 3+ o(1)})$ time to find the $k$ inner product pairs that surpass $ρ\cdot d$ threshold with high probability, where $ω$ is the current matrix multiplication exponent. By solving this problem, our method speed up the training of neural networks with ReLU activation function.

Josh Alman, Zhao Song

The transformer architecture has been widely applied to many machine learning tasks. A main bottleneck in the time to perform transformer computations is a task called attention computation. [Alman and Song, NeurIPS 2023] have shown that in the bounded entry regime, there is an almost linear time algorithm to approximate the attention computation. They also proved that the bounded entry assumption is necessary for a fast algorithm assuming the popular Strong Exponential Time Hypothesis. A new version of transformer which uses position embeddings has recently been very successful. At a high level, position embedding enables the model to capture the correlations between tokens while taking into account their position in the sequence. Perhaps the most popular and effective version is Rotary Position Embedding (RoPE), which was proposed by [Su, Lu, Pan, Murtadha, Wen, and Liu, Neurocomputing 2024]. A main downside of RoPE is that it complicates the attention computation problem, so that previous techniques for designing almost linear time algorithms no longer seem to work. In this paper, we show how to overcome this issue, and give a new algorithm to compute the RoPE attention in almost linear time in the bounded entry regime. (Again, known lower bounds imply that bounded entries are necessary.) Our new algorithm combines two techniques in a novel way: the polynomial method, which was used in prior fast attention algorithms, and the Fast Fourier Transform.

Josh Alman, Zhao Song

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.

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.

Jerry Yao-Chieh Hu, Erzhi Liu, Han Liu et al.

Given a database of bit strings $A_1,\ldots,A_m\in \{0,1\}^n$, a fundamental data structure task is to estimate the distances between a given query $B\in \{0,1\}^n$ with all the strings in the database. In addition, one might further want to ensure the integrity of the database by releasing these distance statistics in a secure manner. In this work, we propose differentially private (DP) data structures for this type of tasks, with a focus on Hamming and edit distance. On top of the strong privacy guarantees, our data structures are also time- and space-efficient. In particular, our data structure is $ε$-DP against any sequence of queries of arbitrary length, and for any query $B$ such that the maximum distance to any string in the database is at most $k$, we output $m$ distance estimates. Moreover, - For Hamming distance, our data structure answers any query in $\widetilde O(mk+n)$ time and each estimate deviates from the true distance by at most $\widetilde O(k/e^{ε/\log k})$; - For edit distance, our data structure answers any query in $\widetilde O(mk^2+n)$ time and each estimate deviates from the true distance by at most $\widetilde O(k/e^{ε/(\log k \log n)})$. For moderate $k$, both data structures support sublinear query operations. We obtain these results via a novel adaptation of the randomized response technique as a bit flipping procedure, applied to the sketched strings.

Zhao Song, Jianfei Xue, Lichen Zhang

In this paper, we study differentially private mechanisms for functions whose outputs lie in a Euclidean Jordan algebra. Euclidean Jordan algebras capture many important mathematical structures and form the foundation of linear programming, second-order cone programming, and semidefinite programming. Our main contribution is a generic Gaussian mechanism for such functions, with sensitivity measured in $\ell_2$, $\ell_1$, and $\ell_\infty$ norms. Notably, this framework includes the important case where the function outputs are symmetric matrices, and sensitivity is measured in the Frobenius, nuclear, or spectral norm. We further derive private algorithms for solving symmetric cone programs under various settings, using a combination of the multiplicative weights update method and our generic Gaussian mechanism. As an application, we present differentially private algorithms for semidefinite programming, resolving a major open question posed by [Hsu, Roth, Roughgarden, and Ullman, ICALP 2014].

Jerry Yao-Chieh Hu, Xiwen Zhang, Maojiang Su et al.

We study the computational limits of learning $k$-bit Boolean functions (specifically, $\mathrm{AND}$, $\mathrm{OR}$, and their noisy variants), using a minimalist single-head softmax-attention mechanism, where $k=Θ(d)$ relevant bits are selected from $d$ inputs. We show that these simple $\mathrm{AND}$ and $\mathrm{OR}$ functions are unsolvable with a single-head softmax-attention mechanism alone. However, with teacher forcing, the same minimalist attention is capable of solving them. These findings offer two key insights: Architecturally, solving these Boolean tasks requires only minimalist attention, without deep Transformer blocks or FFNs. Methodologically, one gradient descent update with supervision suffices and replaces the multi-step Chain-of-Thought (CoT) reasoning scheme of [Kim and Suzuki, ICLR 2025] for solving Boolean problems. Together, the bounds expose a fundamental gap between what this minimal architecture achieves under ideal supervision and what is provably impossible under standard training.

Lianke Qin, Aravind Reddy, Zhao Song

Mahalanobis metrics are widely used in machine learning in conjunction with methods like $k$-nearest neighbors, $k$-means clustering, and $k$-medians clustering. Despite their importance, there has not been any prior work on applying sketching techniques to speed up algorithms for Mahalanobis metrics. In this paper, we initiate the study of dimension reduction for Mahalanobis metrics. In particular, we provide efficient data structures for solving the Approximate Distance Estimation (ADE) problem for Mahalanobis distances. We first provide a randomized Monte Carlo data structure. Then, we show how we can adapt it to provide our main data structure which can handle sequences of \textit{adaptive} queries and also online updates to both the Mahalanobis metric matrix and the data points, making it amenable to be used in conjunction with prior algorithms for online learning of Mahalanobis metrics.

Song Bian, Zhao Song, Junze Yin

Many convex optimization problems with important applications in machine learning are formulated as empirical risk minimization (ERM). There are several examples: linear and logistic regression, LASSO, kernel regression, quantile regression, $p$-norm regression, support vector machines (SVM), and mean-field variational inference. To improve data privacy, federated learning is proposed in machine learning as a framework for training deep learning models on the network edge without sharing data between participating nodes. In this work, we present an interior point method (IPM) to solve a general ERM problem under the federated learning setting. We show that the communication complexity of each iteration of our IPM is $\tilde{O}(d^{3/2})$, where $d$ is the dimension (i.e., number of features) of the dataset.