This paper introduces Force Matching (ForM), a novel framework for generative modeling that represents an initial exploration into leveraging special relativistic mechanics to enhance the stability of the sampling process. By incorporating the Lorentz factor, ForM imposes a velocity constraint, ensuring that sample velocities remain bounded within a constant limit. This constraint serves as a fundamental mechanism for stabilizing the generative dynamics, leading to a more robust and controlled sampling process. We provide a rigorous theoretical analysis demonstrating that the velocity constraint is preserved throughout the sampling procedure within the ForM framework. To validate the effectiveness of our approach, we conduct extensive empirical evaluations. On the half-moons dataset, ForM significantly outperforms baseline methods, achieving the lowest Euclidean distance loss of 0.714, in contrast to vanilla first-order flow matching (5.853) and first- and second-order flow matching (5.793). Additionally, we perform an ablation study to further investigate the impact of our velocity constraint, reaffirming the superiority of ForM in stabilizing the generative process. The theoretical guarantees and empirical results underscore the potential of integrating special relativity principles into generative modeling. Our findings suggest that ForM provides a promising pathway toward achieving stable, efficient, and flexible generative processes. This work lays the foundation for future advancements in high-dimensional generative modeling, opening new avenues for the application of physical principles in machine learning.

The multiplicative weight update method is a well-known algorithm commonly used for decision making based on recommendations from n experts. Despite being widely used, the classical MWU algorithm suffers from a linear runtime complexity, which limits its efficiency on large problems. Additionally, since expert data may be sensitive, it is important to develop privacy-preserving methods. In this paper, we propose the first data structure that approximates the multiplicative weight update in sublinear time (o(n)) while providing differential privacy guarantees, significantly improving efficiency for large-scale problems. Our data structure both enhances computational efficiency and ensures privacy protection, with theoretical guarantees on both utility and privacy, enabling fast and private decision making.

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. However, the quadratic complexity of attention in transformer architectures makes it impractical to train very large models on lengthy texts or use them efficiently during inference. While a recent study by [Kacham, Mirrokni and Zhong 2023] 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 D 0 and D 1, where D 1 includes a feature with a significantly larger value compared to D 0. We demonstrate that with a sufficiently high degree β, a single-layer polynomial attention network can distinguish between D 0 and D 1. However, with a low degree β, 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.

Training neural networks is a fundamental problem in theoretical machine learning. Second-order methods are rarely used in practice due to their high computational cost, even they converge much faster than first-order methods. The state-of-the-art result for the second-order method to train an over-parameterized neural network can run in O(log(1/ϵ)) iterations and each iteration has O(mnd + n3) running time, where n represent the number of data points, d is the dimension of the feature space of each data and m is the width of the neural network [Brand, Peng, Song and Weinstein ITCS 2021]. In this work, we further improve the convergence rate and the running time for each iteration to O(loglog(1/ϵ)) and O(mdτ2) respectively, where m = Ω(n4) and τ is the treewidth of the data matrix and usually very small. Our algorithm has a quadratic convergence rate, so it can be regarded as a truly second-order algorithm. To the best of our knowledge, our algorithm is the first to achieve a quadratic convergence rate for neural network training, with a per-iteration running time of O(mdτ2).

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.

Sampling is a fundamental problem in optimization, machine learning and theoretical computer science. A common region of interest for sampling is the polytope, which is defined by a set of linear inequalities. The algorithm that is sampling from polytopes usually requires heavy matrix algebra, including matrix multiplication, matrix inversion and matrix determinant. In this work, we show how to implement the heavy matrix algebra in the area of sampling in nearly linear time for the polytope that has small treewidth. In particular, given a polytope defined by a matrix A ∈ ℝn×dwith treewidth τ, we improve the running time of each iteration for three typical sampling algorithms for polytopes such as Dikin Walk, Soft-Threshold Dikin Walk and Vaidya Walk from O(nd2) to O(nτ2) by exploiting the small treewidth structures of the matrices.

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(M2/ϵ2) time with network size quadratic in the coefficient norm upper bound M and error term ϵ.

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{equation*}A_{a,b,c}=\sum_{i=1J^{=1}}^{k}\sum_{\prime}^{k}\sum_{l=1}^{k}U_{a,i+k\left(J-1\right)}\prime\otimes V_{b_{J}'+k\left(l-1\right)}\otimes W_{c,l+k\left(i-1\right)}\end{equation*}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 [Song, Woodruff, Zhong SODA 2019] and show an input sparsity time algorithm for cycle rank.

In this paper, we consider a heavy inner product identification problem, which generalizes the Light Bulb problem ([1]): 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., $\{\left(a_{1}, b_{1}\right), \cdots,\left(a_{k}, b_{k}\right)\} \subset A \times B$ such that $\forall i \in[k],\left\langle a_{i}, b_{i}\right\rangle \geq \rho \cdot d$, 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.

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 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.

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.

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 ⊂ ℝd, a binary function f : ℝd× ℝd→ ℝ, and a point y ∈ ℝd, the Pairwise Summation Estimate $PS{E_X}(y): = \frac{1}{{\left| X \right|}}\sum\nolimits_{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 ∈ ℝd, the data-structure approximately estimates PSEX(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 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 qj∈ ℝddepending on the output from previous queries q1, q2, …, qj–1.

There has been a recent effort in applying differential privacy on memory access patterns to enhance data privacy. This is called differential obliviousness. Differential obliviousness is a promising direction because it provides a principled trade-off between performance and desired level of privacy. To date, it is still an open question whether differential obliviousness can speed up database processing with respect to full obliviousness. In this paper, we present the design and implementation of Adore: A set of D ifferentially O blivious RE lational database operators. Adore includes selection with projection, grouping with aggregation, and foreign key join. We prove that they satisfy the notion of differential obliviousness. Our differentially oblivious operators have reduced cache complexity, runtime complexity, and output size compared to their state-of-the-art fully oblivious counterparts. We also demonstrate that our implementation of these differentially oblivious operators can outperform their state-of-the-art fully oblivious counterparts by up to 7.4X.

In recent years, social media websites, such as Epinions, Twitter, and Google+, have gained in popularity and have become ubiquitous in our daily lives, where rich user-generated texts are propagated through social networks. Topic models, such as Latent Dirichlet Allocation (LDA), have been proposed and shown to be useful for text analysis. The existing topic models focus on traditional document collections, which consist of a relatively small number of long and high-quality documents. However, user-generated texts tend to be shorter and noisier than traditional content. Besides, the social networks have two novel features: context information on nodes, such as user features, and edges, such as relationship, which have not been considered by the existing topic models. In this paper, we pose the problem of finding user topics in large-scale collection of documents from online social networks. We propose a comprehensive Feature based and a Social based Topic model, taking into account the user features and social networks. We demonstrate that our models have better performance than a baseline LDA in the Epinions, Twitter, and Google+ data sets.