The Price of Hidden Curvature: An $\widetildeΩ (d^{5/4} \sqrt{T})$ Lower Bound for Bandit Convex Optimization

2026-07-20

AI Summary

This work investigates the minimax regret lower bound for stochastic constrained convex optimization under bandit feedback, aiming to determine whether it surpasses the well-known $d\sqrt{T}$ rate established for linear bandits. By constructing a novel class of convex functions that embed hidden curvature—built from combinations of soft-max and distance functions—and leveraging posterior diffusion analysis of the Fisher information matrix together with adaptive action sequence techniques, the study establishes the first regret lower bound of $\widetilde{\Omega}(d^{5/4} \sqrt{T})$. This result breaks the long-standing $d\sqrt{T}$ barrier and further yields a corresponding sample complexity lower bound of $\widetilde{\Omega}(d^{5/2}/\varepsilon^2)$. The findings demonstrate that bandit convex optimization is inherently more challenging than its linear counterpart and extend naturally to the unconstrained setting.

Abstract

We establish a $\widetildeΩ(d^{5/4}\sqrt T)$ lower bound on the minimax expected regret of stochastic bandit convex optimization of $1$-Lipschitz functions on the Euclidean ball. This presents the first nontrivial regret lower bound that grows faster than $d\sqrt{T}$ for this problem, establishing that stochastic bandit convex optimization is fundamentally harder than linear bandits. The hard class of convex functions we construct takes the following form in dimension $2d$: for an action $a = (a^1,a^2) \in \mathbb{B}^{2d}_2$, each function is the scaled soft maximum of a "tube", $r^{-1} \| W^\star a^1 - \frac{r}{8\varepsilon} a^2 \|_2$ (hyperparameterized by $\varepsilon,r$), and a squared distance function, $\frac12 \| a^1 - u^\star \|_2^2 - \frac12 \| u^\star \|_2^2$. Here, $W^\star \in \mathbb{R}^{d \times d}$ is an unknown linear transformation, and $u^\star \in \mathbb{R}^{d}$ is an unknown vector which must be learned to minimize the function. Observations are informative about $u^\star$ only when the learner's action lies near the tube determined by $W^\star$, satisfying $a^2 \approx \frac{8\varepsilon}{r} W^\star a^1$: thus the learner must either find this tube without knowing $W^\star$, or spend observations learning useful directions of $W^\star$. Formally, our regret analysis exploits this tradeoff by bounding the posterior spread of Fisher information matrices obtained under an adaptive sequence of actions. Together, these ingredients give a sample complexity lower bound of $\widetildeΩ(d^{5/2}/\varepsilon^2)$ to find an $\varepsilon$-optimal action, which translates to an $\widetildeΩ (d^{5/4} \sqrt{T})$ regret lower bound. We also extend this lower bound to the unconstrained setting where the action space is $\mathbb{R}^d$.

Problem

• bandit convex optimization
• minimax regret
• lower bound
• stochastic optimization
• Lipschitz functions

Innovation

• bandit convex optimization
• minimax lower bound
• hidden curvature
• Fisher information
• adaptive exploration