  The backtracking line-search is an effective technique to automatically tune
the step-size in smooth optimization. It guarantees similar performance to
using the theoretically optimal step-size. Many approaches have been developed
to instead tune per-coordinate step-sizes, also known as diagonal
preconditioners, but none of the existing methods are provably competitive with
the optimal per-coordinate stepsizes. We propose multidimensional backtracking,
an extension of the backtracking line-search to find good diagonal
preconditioners for smooth convex problems. Our key insight is that the
gradient with respect to the step-sizes, also known as hypergradients, yields
separating hyperplanes that let us search for good preconditioners using
cutting-plane methods. As black-box cutting-plane approaches like the ellipsoid
method are computationally prohibitive, we develop an efficient algorithm
tailored to our setting. Multidimensional backtracking is provably competitive
with the best diagonal preconditioner and requires no manual tuning.
