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

Searching for Optimal Per-Coordinate Step-sizes with Multidimensional Backtracking