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Preconditioning is a crucial operation in gradient-based numericaloptimisation. It helps decrease the local condition number of a function byappropriately transforming its gradient. For a convex function, where thegradient can be computed exactly, the optimal linear transformation correspondsto the inverse of the Hessian operator, while the optimal convex transformationis the convex conjugate of the function. Different conditions result invariations of these dependencies. Practical algorithms often employ low-rank orstochastic approximations of the inverse Hessian matrix for preconditioning.However, theoretical guarantees for these algorithms typically lack ajustification for the defining property of preconditioning. This paper presentsa simple theoretical framework that demonstrates, given a smooth function andan available unbiased stochastic approximation of its gradient, that it ispossible to refine the dependency of the convergence rate on the Lipschitzconstant of the gradient.

Convergence analysis of stochastic gradient descent with adaptive preconditioning for non-convex and convex functions