Neural Networks for Predicting the Behavior of Preconditioned Iterative Solvers

Abstract

Abstract. We evaluate the effectiveness of neural networks as a tool for predicting whether a particular combination of preconditioner and iterative method will correctly solve a given sparse linear system Ax = b. We consider several scenarios corresponding to different assumptions about the relationship between the systems used to train the neural network and those for which the neural network is expected to predict behavior. Greater similarity between those two sets leads to better accuracy, but even when the two sets are very different prediction accuracy can be improved by using additional computation.

On Using Reinforcement Learning to Solve Sparse Linear Systems

“…The reinforcement learning framework allows for many more combinations of preconditioners than earlier studies which also restrict the solver to restarted GMRES and/or the preconditioner to a variant of ILU [4,5,6,7,18]. Observe, for example, that equilibration is now optional.…”

“…Observe, for example, that equilibration is now optional. Hence a total of 576 preconditioned solvers are described by the above framework; this is notably more than used to evaluate systems based on other machine learning techniques [3,4,5]. A system for automatically selecting from amongst so many options is particularly valuable given previous work that shows the difficulty of presenting information accurately comparing different preconditioned solvers across a range of metrics [19].…”

“…These are the same features as those used in [4], which are a subset of those used in [3,18]. Since each action changed the values of some of the features, this allowed the agent to observe the changes it made to the matrix during the computation and to react to those changes.…”

“…To provide more useful problem-specific guidelines, recent work explores the use of machine learning techniques such as decision trees [3], neural networks [4], and support vector machines [5,6] for recommending preconditioned solvers. This line of research attempts to create a classifier that uses assorted structural and numerical features of a matrix in order to recommend a good preconditioned solver (with parameter settings when appropriate).…”

“…At a minimum, these techniques recommend a solver that should be likely to converge to the solution vector. However, each paper also describes assorted extensions: [3] attempts to recommend a preconditioned solver that converges within some user-defined parameter of optimal, [5] attempts to give insight into why certain solvers fail, and [4] considers different use scenarios. In addition, [7] tries to predict the efficiency of a solver in terms of its time and memory usage, and [3] describes a general framework within which many machine learning approaches could be used.…”