Breaking Precision Walls in First-, Second-, and Zeroth-Order Optimizers: One Constant, Set in Advance

Lngel Kyo

Under power-law signal decay across directions, an optimizer can fit strong directions and miss weak ones while average loss hides it. A precision wall measures the steepest decay where the weakest direction recovers within 5% in a fixed budget. On low-rank matrix sensing and quadratics one constant sets the wall: once it dominates the denominator, Adam or grafted Shampoo is momentum descent at step $η/c$, setting the wall; in Shampoo, matching grafting's constant to damping restores the wall after three hypotheses failed. In zeroth-order Adam, $\varepsilon$ acts against $\varepsilon^\ast$ from a mean-square spectral model; below $\varepsilon^\ast$ error hits a noise floor. Set from true curvature or spectrum on new shapes, the rule raised the wall above the same-step default by a forecast amount: on two sensing shapes by $1.96, 1.15$ (Adam) and $1.59, 1.74$ (grafted Shampoo), and on two Hadamard-rotated quadratics by $1.12, 0.73$ (zeroth-order Adam, longer budget), rising on all seeds; fixed-step momentum at that step reaches those walls. A cosine-scheduled default (small constant giving weak modes large steps) is higher on one sensing shape, matched on new seeds of the other,…