# Using reinforcement learning to find a preconditioner for linear systems of the form Ax = b

Sparse linear system are normally solved by using solvers like MINRES, Conjugate gradient, GMRES.

Efficient preconditioning, i.e., finding a matrix P such that PAx = Pb is easier to solve then the original problem, can drastically reduce the computational effort to solve for x. However, preconditioning is normally problem specific and there is not ONE preconditioner that works well for every problem.

I thought this would be an interesting problem to apply RL, since there are certain norms (e.g. condition number of matrix PA) to measure if P is a good preconditioner, but I could not find any research in this field.

Is there a specific problem why RL could not be applied?

• Why would you do that? – Martin Thoma Oct 13 '18 at 8:48
• You are right, I was wondering about a different question and went to far. I will restate my question. – timudk Oct 13 '18 at 18:19

In contrast, the problem you describe is to solve a linear system of equations, which is to say, to learn some hidden values $$x$$ such that $$Ax = b$$ for known $$A$$ and $$b$$. Gradient decent is a natural way to solve this problem because it is easy to calculate the gradient, and, since the matrices and vectors have the same dimensionality, it is reasonable to expect that the optimization surface is smooth with a single global minima.