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.

翻译：我们提出了面向对齐研究中心提出的矩阵补全问题的解决方案，其求解精度$\varepsilon$具有多项式依赖关系。这些问题的提出旨在促使能够高效计算启发式估计量，从而以形式化方式评估和推理深度神经网络中的不同量化指标，服务于人工智能对齐研究。我们的解决方案将矩阵补全问题重新表述为半定规划（SDP），并利用谱束方法的最新进展，实现快速、高效且可扩展的SDP求解。