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SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning

Hong Wang, Jie Wang, Minghao Ma, Haoran Shao, Haoyang Liu

TL;DR

SymMaP addresses the problem of efficiently selecting preconditioning parameters for linear solvers by marrying the reliability of traditional preconditioners with the adaptability of symbolic discovery. It learns concise symbolic expressions that map problem features to optimal parameters using a prefix-notation, RNN-based search guided by a risk-seeking reward, and then deploys these expressions directly in CPU-based solvers. Across SOR, SSOR, and AMG, SymMaP achieves consistent performance gains and superior interpretability compared to fixed constants and neural baselines, while incurring minimal runtime overhead. The work demonstrates practical CPU-friendly deployment and opens avenues for extending symbolic discovery to a broader set of preconditioners and problem domains.

Abstract

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.

SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning

TL;DR

SymMaP addresses the problem of efficiently selecting preconditioning parameters for linear solvers by marrying the reliability of traditional preconditioners with the adaptability of symbolic discovery. It learns concise symbolic expressions that map problem features to optimal parameters using a prefix-notation, RNN-based search guided by a risk-seeking reward, and then deploys these expressions directly in CPU-based solvers. Across SOR, SSOR, and AMG, SymMaP achieves consistent performance gains and superior interpretability compared to fixed constants and neural baselines, while incurring minimal runtime overhead. The work demonstrates practical CPU-friendly deployment and opens avenues for extending symbolic discovery to a broader set of preconditioners and problem domains.

Abstract

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.

Paper Structure

This paper contains 55 sections, 19 equations, 4 figures, 21 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrix Preconditioning Technique
  4. Prefix Notation
  5. Motivation
  6. Motivation for Optimizing Preconditioning Parameters
  7. Challenges in Predicting Efficient Preconditioning Parameters
  8. Symbolic Discovery to Preconditioning Parameter Selection
  9. Method
  10. Input Features and Training Data Generation
  11. The Generation of Symbolic Expressions
  12. The Reward Function
  13. The Training Algorithm
  14. Deployment in Linear System Solver
  15. Experiments
  16. ...and 40 more sections

Figures (4)

  • Figure 1: Left. Variation in iteration counts and computation times under different SOR preconditioning parameters applied to a linear system from a second-order elliptic PDE. Right. Ratio of average computation times at various tolerances to default parameter times under different SOR parameter selection schemes, evaluated on the second-order elliptic PDE dataset.
  • Figure 2: Illustration of how SymMaP discovers efficient symbolic expressions for preconditioning parameters. Part 1 demonstrates the acquisition of optimal parameters and dataset generation; Part 2 illustrates the training process of the RL-based deep symbolic discovery framework; Part 3 shows how the sequential model generates symbolic policies; Part 4 presents the deployment of symbolic expressions.
  • Figure 3: SOR optimal parameter distribution for a mixed edge-length second-order elliptic PDE dataset.
  • Figure 4: Error reduction curves for different parameter settings on the biharmonic equation dataset.