A recent work by [Larsen, SODA 2023] introduced a faster combinatorial alternative to Bansal’s SDP algorithm for finding a coloring $x \in \\{-1, 1\\}^n$ that approximately minimizes the discrepancy $\mathrm{disc}(A, x) := \\|{A} x \\|_{\infty}$ of a real-valued $m \times n$ matrix $A$. Larsen’s algorithm runs in $\widetilde{O}(mn^2)$ time compared to Bansal’s $\widetilde{O}(mn^{4.5})$-time algorithm, with a slightly weaker logarithmic approximation ratio in terms of the hereditary discrepancy of $A$ [Bansal, FOCS 2010]. We present a combinatorial $\widetilde{O}(\mathrm{nnz}(A) + n^3)$-time algorithm with the same approximation guarantee as Larsen’s, optimal for tall matrices where $m = \mathrm{poly}(n)$. Using a more intricate analysis and fast matrix multiplication, we further achieve a runtime of $\widetilde{O}(\mathrm{nnz}(A) + n^{2.53})$, breaking the cubic barrier for square matrices and surpassing the limitations of linear-programming approaches [Eldan and Singh, RS&A 2018]. Our algorithm relies on two key ideas: (i) a new sketching technique for finding a projection matrix with a short $\ell_2$-basis using implicit leverage-score sampling, and (ii) a data structure for efficiently implementing the iterative Edge-Walk partial-coloring algorithm [Lovett and Meka, SICOMP 2015], and using an alternative analysis to enable “lazy” batch updates with low-rank corrections. Our results nearly close the computational gap between real-valued and binary matrices, for which input-sparsity time coloring was recently obtained by [Jain, Sah and Sawhney, SODA 2023].

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.

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 the \emph{Distance Oracle} problem, the goal is to preprocess $n$ vectors $x_1, x_2, \cdots, x_n$ in a $d$-dimensional normed space $(\mathbb{X}^d, \| \cdot \|_l)$ into a cheap data structure, so that given a query vector $q \in \mathbb{X}^d$, all distances $\| q - x_i \|_l$ to the data points $\{x_i\}_{i\in [n]}$ can be quickly approximated (faster than the trivial $\sim nd$ query time). This primitive is a basic subroutine in machine learning, data mining and similarity search applications. In the case of $\ell_p$ norms, the problem is well understood, and optimal data structures are known for most values of $p$. Our main contribution is a fast $(1\pm \varepsilon)$ distance oracle for \emph{any symmetric} norm $\|\cdot\|_l$. This class includes $\ell_p$ norms and Orlicz norms as special cases, as well as other norms used in practice, e.g. top-$k$ norms, max-mixture and sum-mixture of $\ell_p$ norms, small-support norms and the box-norm. We propose a novel data structure with $\tilde{O}(n (d + \mathrm{mmc}(l)^2 ) )$ preprocessing time and space, and $t_q = \tilde{O}(d + n \cdot \mathrm{mmc}(l)^2)$ query time, where $\mathrm{mmc}(l)$ is a complexity-measure (modulus) of the symmetric norm under consideration. When $l = \ell_{p}$ , this runtime matches the aforementioned state-of-art oracles.