Optimal Parameter-Free Gradient Minimization in $\ell_p$ Geometry

What the math gives to ML

The paper identifies a practical obstruction in parameter-free first-order optimization: the dimensionless quantity LR/ε cannot determine a finite oracle budget unless the algorithm first obtains a nondegenerate local curvature observation. Its constructive asset is the secant scale M_0, which can bootstrap unknown smoothness in general l_p geometry, together with dimension-free complexity exponents that vary with p. This suggests an optimizer using secant-calibrated curvature and dual-norm steepest-descent directions, adapting the scale through rejected trial steps without requiring L, R, or f*. The first implementation should test whether this reduces gradient evaluations or improves stability relative to AdamW and norm-adaptive line searches.