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We establish matching polynomial query bounds for low-rank approximation from exact matrix--vector products. Given an unknown matrix $A\in\mathbb{R}^{m\times n}$, at each step a randomized algorithm chooses either $v\in\mathbb{R}^n$ and receives $Av$, or $u\in\mathbb{R}^m$ and receives $A^\top u$. The choice may depend...
The same accuracy exponent holds beyond tensor update rules, even for randomized queries and arbitrary feasible outputs, for smooth convex--concave minimax optimization.
A matching $\Omega_p(Q^{2/(3p-1)})$ lower bound is proved for arbitrary adaptive deterministic algorithms and randomized algorithms with per-instance success probability at least $2/3$, without span or tensor-update restrictions.
A single fixed smooth convex hard instance is constructed using a Moreau-smoothed biased max chain, an exact prefix-shielding mechanism, and batched delayed rotations to preserve consistency with the full adaptive transcript and establish the optimality of the square-root complexity branch for deterministic bounded-que...
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