  Preconditioning is a crucial operation in gradient-based numerical
optimisation. It helps decrease the local condition number of a function by
appropriately transforming its gradient. For a convex function, where the
gradient can be computed exactly, the optimal linear transformation corresponds
to the inverse of the Hessian operator, while the optimal convex transformation
is the convex conjugate of the function. Different conditions result in
variations of these dependencies. Practical algorithms often employ low-rank or
stochastic approximations of the inverse Hessian matrix for preconditioning.
However, theoretical guarantees for these algorithms typically lack a
justification for the defining property of preconditioning. This paper presents
a simple theoretical framework that demonstrates, given a smooth function and
an available unbiased stochastic approximation of its gradient, that it is
possible to refine the dependency of the convergence rate on the Lipschitz
constant of the gradient.

Optimization and Control