Efficient preconditioning plays an important role in large-scale first-order optimization. In this paper, we propose an adaptive accelerated optimistic dual averaging ($\text{A}^2$ODA) method for unconstrained convex optimization that adopts low-cost preconditioning based on $p$-norm geometry for $p\in[2,\infty]$. 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 $p\in[2,\infty]$, 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.
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