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Solving Regularized Exp, Cosh and Sinh Regression Problems

Zhihang Li, Zhao Song, Tianyi Zhou

TL;DR

The paper addresses the efficiency of solving regularized exponential regression problems motivated by attention in large language models. By proving convexity and Lipschitz properties for the regualrized loss with $f\in\{\exp,\cosh,\sinh\}$, it lays a foundation for a fast inexact Newton method. The authors propose a randomized, input-sparsity-time algorithm with iteration complexity $O(\log(\|x_0-x^*\|_2/\epsilon))$ and per-iteration cost $\widetilde{O}(\mathrm{nnz}(A)+d^{\omega})$, achieving high-probability convergence to $x^*$ within error $\epsilon$. This approach offers a theoretically grounded, scalable path to solving attention-inspired regression problems in sparse regimes, with potential impact on efficiency of attention computations in practice.

Abstract

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.

Solving Regularized Exp, Cosh and Sinh Regression Problems

TL;DR

The paper addresses the efficiency of solving regularized exponential regression problems motivated by attention in large language models. By proving convexity and Lipschitz properties for the regualrized loss with , it lays a foundation for a fast inexact Newton method. The authors propose a randomized, input-sparsity-time algorithm with iteration complexity and per-iteration cost , achieving high-probability convergence to within error . This approach offers a theoretically grounded, scalable path to solving attention-inspired regression problems in sparse regimes, with potential impact on efficiency of attention computations in practice.

Abstract

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 , , and any of functions and denoted as . The goal is to find the optimal that minimize . The straightforward method is to use the naive Newton's method. Let denote the number of non-zeros entries in matrix . Let denote the exponent of matrix multiplication. Currently, . Let denote the accuracy error. In this paper, we make use of the input sparsity and purpose an algorithm that use iterations and per iteration time to solve the problem.
Paper Structure (45 sections, 28 theorems, 109 equations, 1 algorithm)

This paper contains 45 sections, 28 theorems, 109 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Input sparsity time algorithms
  5. Algorithmic Regularization
  6. Attention Theory
  7. Newton Method and Hessian Computation
  8. Roadmap.
  9. Preliminary
  10. Notations
  11. Basic Algebras
  12. Standard Gradient and Hessian Computation
  13. Regularization term
  14. Exponential Regression
  15. Definitions
  16. ...and 30 more sections

Key Result

Theorem 1.1

Given matrix $A \in \mathbb{R}^{n \times d}$, $b \in \mathbb{R}^n$, and $w \in \mathbb{R}^n$. Let $f$ be any of functions $\exp, \cosh$ and $\sinh$. Let $g$ denote the gradient of function $f$. Let $x^*$ denote the optimal solution of that $g(x^*) = {\bf 0}_d$ and $\| x^* \|_2 \leq R$. Let $\| A \| \leq R, \| b \|_2 \leq R$. Let $w_{i}^2 > 0.5b_{i}^2 + 1 + l/\sigma_{\min}(A)^2$ for all $i \in [n]

Theorems & Definitions (67)

  • Theorem 1.1: Main result, Informal version of Theorem \ref{['thm:main_formal']}
  • proof
  • proof
  • Lemma 2.7: Standard Gradient and Hessian Computation
  • proof
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • proof
  • Definition 3.1: Loss function for Exp Regression
  • ...and 57 more