The Price of Hidden Curvature: Improved Lower Bounds for Bandit Convex Optimization
We establish improved lower bounds on the minimax expected regret of stochastic bandit convex optimization for $1$-Lipschitz functions on the $d$-dimensional Euclidean ball. For time horizons $n\ge d^{10/3}$, we prove a lower bound of $\Omega(d^{4/3}\sqrt{n})$, the first nontrivial bound that exceeds the $d\sqrt{n}$ dependence of linear bandits, showing that stochastic bandit convex optimization is fundamentally harder than linear bandits. For $d^2\le n\le d^{10/3}$, we obtain a lower bound of $\Omega(\sqrt{d}n^{3/4})$, matching the regret of the algorithm of Flaxman et al. (2005), establishing its optimality in this regime. The hard class of convex functions we construct takes the following form in dimension $2d$: for an action $a=(a^1,a^2)\in \mathbb{B}^{2d}$, each function is the scaled soft maximum of a"tube", $r^{-1}\|W^\star a^1-\frac{r}{8\varepsilon}a^2 \|$ (hyperparameterized by $\varepsilon,r$), and a squared distance function, $\frac12\|a^1-u^\star\|^2-\frac12\|u^\star\|^2$. Here $u^\star\in\mathbb{R}^d$ is the unknown target determining the minimizer, while $W^\star\in\mathbb{R}^{d\times d}$ hides the region in which the quadratic curvature is observable. Indeed, observations reveal substantial information about $u^\star$ only when the learner acts near the hidden tube $a^2\approx \frac{8\varepsilon}{r}W^\star a^1$; away from it, the tube branch masks the quadratic branch. Thus the learner must pay to uncover the geometry encoded by $W^\star$ before it can effectively exploit the curvature that identifies $u^\star$. Formalizing this tradeoff yields a sample complexity lower bound of $\Omega(\frac{d^{5/2}}{\varepsilon^2}\wedge\frac{d^2}{\varepsilon^4})$ for finding an $\varepsilon$-optimal action, and ultimately the $\Omega(d^{4/3}\sqrt{n}\wedge\sqrt{d}n^{3/4})$ regret lower bound. The proof was developed by GPT-5.5 Pro and GPT-5.6 Sol Pro under the authors'guidance.