Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Polynomial Precision Dependence Solutions to Alignment Research Center Matrix Completion Problems

Rico Angell

TL;DR

This work tackles two matrix completion questions from the Alignment Research Center that require polynomial precision in the solver error $\varepsilon$. The authors recast both problems as semidefinite programs and solve them via a spectral bundle approach, enabling scalable, fast procedures with explicit complexity bounds. Key results include a decision procedure for Question 1 with time $\tilde{O}(\min\{ nm/\varepsilon, m/\varepsilon^{3/2} \})$ and constructive solutions for Question 2 with $\tilde{O}(n^2/\varepsilon^2 + \min\{ nm/\varepsilon^2, m/\varepsilon^{5/2} \})$ (no sketch) or $\tilde{O}(n\sqrt{m}/\varepsilon^2 + \min\{ nm/\varepsilon^2, m/\varepsilon^{5/2} \})$ (with sketch). The method also yields a PSD matrix and, in infeasible cases, an infeasibility certificate or upper bound. The results provide scalable tools for evaluating heuristic estimators in AI alignment contexts and demonstrate the practical viability of SDP-based approaches for structured matrix completion.

Abstract

We present solutions to the matrix completion problems proposed by the Alignment Research Center that have a polynomial dependence on the precision $\varepsilon$. The motivation for these problems is to enable efficient computation of heuristic estimators to formally evaluate and reason about different quantities of deep neural networks in the interest of AI alignment. Our solutions involve reframing the matrix completion problems as a semidefinite program (SDP) and using recent advances in spectral bundle methods for fast, efficient, and scalable SDP solving.

Polynomial Precision Dependence Solutions to Alignment Research Center Matrix Completion Problems

TL;DR

This work tackles two matrix completion questions from the Alignment Research Center that require polynomial precision in the solver error . The authors recast both problems as semidefinite programs and solve them via a spectral bundle approach, enabling scalable, fast procedures with explicit complexity bounds. Key results include a decision procedure for Question 1 with time and constructive solutions for Question 2 with (no sketch) or (with sketch). The method also yields a PSD matrix and, in infeasible cases, an infeasibility certificate or upper bound. The results provide scalable tools for evaluating heuristic estimators in AI alignment contexts and demonstrate the practical viability of SDP-based approaches for structured matrix completion.

Abstract

We present solutions to the matrix completion problems proposed by the Alignment Research Center that have a polynomial dependence on the precision . The motivation for these problems is to enable efficient computation of heuristic estimators to formally evaluate and reason about different quantities of deep neural networks in the interest of AI alignment. Our solutions involve reframing the matrix completion problems as a semidefinite program (SDP) and using recent advances in spectral bundle methods for fast, efficient, and scalable SDP solving.
Paper Structure (9 sections, 2 theorems, 14 equations)

This paper contains 9 sections, 2 theorems, 14 equations.

Table of Contents

  1. Question 1 (existence of positive semidefinite completions):
  2. Question 2 (fast "approximate squaring"):
  3. Semidefinite programming formulation
  4. Spectral bundle method
  5. Proximal bundle method
  6. Model Requirements.
  7. Existence of positive semidefinite completions
  8. Per-iteration analysis
  9. Fast approximate squaring

Key Result

Lemma 4

Fix a minimizer $y_\star \in \mathcal{Y}_\star$ of $f$ and let $y_t \in \mathbb{R}^m \setminus \{y_\star\}$. Then, the proximal gap is lower bounded by

Theorems & Definitions (2)

  • Lemma 4
  • Lemma 5