SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning
This work addresses the computational efficiency and interpretability issues in matrix preconditioning for linear systems, offering a novel hybrid approach that is incremental in combining machine learning with symbolic methods.
The paper tackles the problem of selecting preconditioning parameters for linear solvers by proposing SymMaP, a symbolic discovery framework that learns efficient symbolic expressions to predict optimal parameters, resulting in consistent outperformance over traditional strategies across various benchmarks.
Matrix preconditioning is a critical technique to accelerate the solution of linear systems, where performance heavily depends on the selection of preconditioning parameters. Traditional parameter selection approaches often define fixed constants for specific scenarios. However, they rely on domain expertise and fail to consider the instance-wise features for individual problems, limiting their performance. In contrast, machine learning (ML) approaches, though promising, are hindered by high inference costs and limited interpretability. To combine the strengths of both approaches, we propose a symbolic discovery framework-namely, Symbolic Matrix Preconditioning (SymMaP)-to learn efficient symbolic expressions for preconditioning parameters. Specifically, we employ a neural network to search the high-dimensional discrete space for expressions that can accurately predict the optimal parameters. The learned expression allows for high inference efficiency and excellent interpretability (expressed in concise symbolic formulas), making it simple and reliable for deployment. Experimental results show that SymMaP consistently outperforms traditional strategies across various benchmarks.