We introduce AutoSpec, a neural network framework for discovering iterative spectral algorithms for large-scale numerical linear algebra and numerical optimization. Our self-supervised models adapt to input operators using coarse spectral information (e.g., eigenvalue estimates and residual norms), and they predict recurrence coefficients for computing or applying a matrix polynomial tailored to a downstream task. The effectiveness of AutoSpec relies on three ingredients: an architecture whose inference pass implements short, executable numerical linear algebra recurrences; efficient training on small synthetic problems with transfer to large-scale real-world operators; and task-defined objectives that enforce the desired approximation or preconditioning behavior across the range of spectral profiles represented in the training set. We apply AutoSpec to discovering algorithms for representative numerical linear algebra tasks: accelerating matrix-function approximation; accelerating sparse linear solvers; and spectral filtering/preconditioning for eigenvalue computations. On real-world matrices, the learned procedures deliver orders-of-magnitude improvements in accuracy and/or reductions in iteration count, relative to basic baselines. We also find clear connections to classical theory: the induced polynomials often exhibit near-equiripple, near-minimax behavior characteristic of Chebyshev polynomials.

翻译：我们提出了AutoSpec，一个用于发现大规模数值线性代数与数值优化中迭代谱算法的神经网络框架。我们的自监督模型利用粗略谱信息（例如特征值估计和残差范数）自适应输入算子，并预测为下游任务定制的矩阵多项式计算或应用所需的递推系数。AutoSpec的有效性依赖于三个要素：推理过程实现简短、可执行的数值线性代数递推的架构；在小规模合成问题上高效训练并迁移至大规模实际算子的能力；以及通过训练集中所涵盖的谱分布范围强制执行期望逼近或预处理行为的任务定义目标函数。我们将AutoSpec应用于代表性数值线性代数任务的算法发现：加速矩阵函数逼近；加速稀疏线性求解器；以及特征值计算中的谱滤波/预处理。在实际矩阵上，相较于基础基线方法，学习得到的算法在精度上实现了数量级提升和/或迭代次数显著减少。我们还发现了与经典理论的明确联系：诱导多项式常表现出接近等波纹、接近极小极大值的切比雪夫多项式典型特性。