WHAT SHAPE SHOULD A RULE BE? A GAUGE-THEORETIC FOUNDATION FOR ADAPTIVE ONLINE OPTIMIZATION

Adaptive first-order methods commit, each round, to a set of admissible step directions revised from past gradients; we ask what shape that set should take, and whether it must be an ellipsoid. We cast the proximal function as the squared gauge of a convex absorbing set Ct and prove a regret bound for any monotone sequence of such sets with strongly convex squared gauge. We give a rotated- ellipsoidal instance with an AdaGrad-type O(√T ) guarantee and a consistent data-driven basis estimator, and study polytopal methods (rotated hyperrectan- gles, adaptive facets) empirically. Rotated geometries converge markedly faster than axis-aligned baselines with a line search and early in neural-network training.