A simple adaptive accelerated optimistic dual averaging method for unconstrained convex optimization in p-norm geometry

Efficient preconditioning plays an important role in large-scale first-order optimization. In this paper, we propose an adaptive accelerated optimistic dual averaging (A²ODA) method for unconstrained convex optimization that adopts low-cost preconditioning based on p-norm geometry for p∈[2,∞]. The method is line-search-free and employs a new stepsize selection technique based on an aggregated affine lower model. Its update structure is simple and unified across p∈[2,∞], with the case $p>2$ requiring no more elaborate updates than the Euclidean case $p=2$. Under a generalized local smoothness condition, we derive a convergence bound that separates the effects of constant and gap-dependent curvature, with the latter potentially yielding faster-decaying terms. In the special case of local smoothness, this bound recovers a nearly optimal convergence rate.