We explore the application of preconditioning in optimisation algorithms, specifically those appearing in Inverse Problems in imaging. Such problems often contain an ill-posed forward operator and are large-scale. Therefore, computationally efficient algorithms which converge quickly are desirable. To remedy these issues, learning-to-optimise leverages training data to accelerate solving particular optimisation problems. Many traditional optimisation methods use scalar hyperparameters, significantly limiting their convergence speed when applied to ill-conditioned problems. In contrast, we propose a novel approach that replaces these scalar quantities with matrices learned using data. Often, preconditioning considers only symmetric positive-definite preconditioners. However, we consider multiple parametrisations of the preconditioner, which do not require symmetry or positive-definiteness. These parametrisations include using full matrices, diagonal matrices, and convolutions. We analyse the convergence properties of these methods and compare their performance against classical optimisation algorithms. Generalisation performance of these methods is also considered, both for in-distribution and out-of-distribution data.

翻译：我们探讨了预条件技术在优化算法中的应用，特别是成像逆问题中出现的算法。这类问题通常包含不适定的前向算子且规模庞大，因此需要具有快速收敛性的计算高效算法。为解决这些问题，学习优化方法利用训练数据来加速求解特定优化问题。许多传统优化方法使用标量超参数，这在应用于病态问题时严重限制了其收敛速度。相比之下，我们提出了一种新方法，用通过数据学习得到的矩阵来替代这些标量参数。通常预条件技术仅考虑对称正定预条件子，但我们研究了预条件子的多种参数化形式，这些形式不要求对称性或正定性。这些参数化包括使用完整矩阵、对角矩阵和卷积运算。我们分析了这些方法的收敛特性，并将其性能与经典优化算法进行比较。同时考察了这些方法在分布内和分布外数据上的泛化性能。