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How to Inverting the Leverage Score Distribution?

Zhihang Li, Zhao Song, Weixin Wang, Junze Yin, Zheng Yu

TL;DR

This work introduces and studies the novel problem of inverting leverage score distributions to recover model parameters, formulating a regularized non-convex loss that aligns the leverage-score operator with a target vector. The authors derive explicit gradient and Hessian expressions, decompose the Hessian into interpretable components, and prove that the Hessian is positive definite and Lipschitz under reasonable bounds. They then design and analyze first-order (gradient descent) and second-order (Newton) algorithms that converge with provable rates, including linear convergence guarantees for gradient descent and quadratic convergence for Newton within shrinking neighborhoods. The framework opens up potential applications in model interpretation, data recovery, and security, by enabling parameter reconstruction from leverage scores, while providing rigorous optimization guarantees to support practical deployment.

Abstract

Leverage score is a fundamental problem in machine learning and theoretical computer science. It has extensive applications in regression analysis, randomized algorithms, and neural network inversion. Despite leverage scores are widely used as a tool, in this paper, we study a novel problem, namely the inverting leverage score problem. We analyze to invert the leverage score distributions back to recover model parameters. Specifically, given a leverage score $σ\in \mathbb{R}^n$, the matrix $A \in \mathbb{R}^{n \times d}$, and the vector $b \in \mathbb{R}^n$, we analyze the non-convex optimization problem of finding $x \in \mathbb{R}^d$ to minimize $\| \mathrm{diag}( σ) - I_n \circ (A(x) (A(x)^\top A(x) )^{-1} A(x)^\top ) \|_F$, where $A(x):= S(x)^{-1} A \in \mathbb{R}^{n \times d} $, $S(x) := \mathrm{diag}(s(x)) \in \mathbb{R}^{n \times n}$ and $s(x) : = Ax - b \in \mathbb{R}^n$. Our theoretical studies include computing the gradient and Hessian, demonstrating that the Hessian matrix is positive definite and Lipschitz, and constructing first-order and second-order algorithms to solve this regression problem. Our work combines iterative shrinking and the induction hypothesis to ensure global convergence rates for the Newton method, as well as the properties of Lipschitz and strong convexity to guarantee the performance of gradient descent. This important study on inverting statistical leverage opens up numerous new applications in interpretation, data recovery, and security.

How to Inverting the Leverage Score Distribution?

TL;DR

This work introduces and studies the novel problem of inverting leverage score distributions to recover model parameters, formulating a regularized non-convex loss that aligns the leverage-score operator with a target vector. The authors derive explicit gradient and Hessian expressions, decompose the Hessian into interpretable components, and prove that the Hessian is positive definite and Lipschitz under reasonable bounds. They then design and analyze first-order (gradient descent) and second-order (Newton) algorithms that converge with provable rates, including linear convergence guarantees for gradient descent and quadratic convergence for Newton within shrinking neighborhoods. The framework opens up potential applications in model interpretation, data recovery, and security, by enabling parameter reconstruction from leverage scores, while providing rigorous optimization guarantees to support practical deployment.

Abstract

Leverage score is a fundamental problem in machine learning and theoretical computer science. It has extensive applications in regression analysis, randomized algorithms, and neural network inversion. Despite leverage scores are widely used as a tool, in this paper, we study a novel problem, namely the inverting leverage score problem. We analyze to invert the leverage score distributions back to recover model parameters. Specifically, given a leverage score , the matrix , and the vector , we analyze the non-convex optimization problem of finding to minimize , where , and . Our theoretical studies include computing the gradient and Hessian, demonstrating that the Hessian matrix is positive definite and Lipschitz, and constructing first-order and second-order algorithms to solve this regression problem. Our work combines iterative shrinking and the induction hypothesis to ensure global convergence rates for the Newton method, as well as the properties of Lipschitz and strong convexity to guarantee the performance of gradient descent. This important study on inverting statistical leverage opens up numerous new applications in interpretation, data recovery, and security.
Paper Structure (51 sections, 35 theorems, 170 equations, 2 algorithms)

This paper contains 51 sections, 35 theorems, 170 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Challenge and Contribution.
  3. Our Result.
  4. Notations.
  5. Roadmap.
  6. Related Work
  7. Leverage Score
  8. Inverting Problem
  9. Attack Problem
  10. Second-Order Method
  11. Technique Overview
  12. Gradient and Hessian Computation.
  13. Showing Hessian is Positive Definite.
  14. Showing Hessian is Lipschitz.
  15. The Use of Gradient Descent and Newton's Method Under the Support of the Positive Definite and Lipschitz Properties of the Hessian Matrix.
  16. ...and 36 more sections

Key Result

Theorem 1.2

Let $L: \mathbb{R}^d \to \mathbb{R}$ denote the loss function. Let $n$ and $d$ be defined as in Definition def:inverting_leverage_score and $R > 0$. Let $\epsilon \in (0, 0.1)$. Then, there exists a gradient descent algorithm (see Algorithm alg:first_order) that uses $O(\epsilon^{-1} \mathop{\mathrm

Theorems & Definitions (80)

  • Definition 1.1: Inverting leverage score problem
  • Theorem 1.2: Main result, informal version of Theorem \ref{['thm:gradient_formal']}
  • Theorem 1.3: Main result, informal version of Theorem \ref{['thm:newton_formal']}
  • Definition 4.1: Leverage Score
  • Definition 4.2
  • Definition 4.3
  • Claim 4.4
  • Definition 4.5
  • Definition 4.6: Regularized inverting leverage score problem
  • Lemma 5.1: Informal version of Lemma \ref{['lem:gradient_leverage']}
  • ...and 70 more