Adversarial Bandit Optimization with Globally Bounded Perturbations to Convex Losses

We study adversarial online learning with hidden-convex losses, i.e., nonconvex losses that become convex after a nonlinear reparameterization. Ghai, Lu and Hazan (2022) proved that, under geometric and smoothness assumptions, online gradient descent (OGD) on such nonconvex losses approximately simulates online mirror descent (OMD) on the underlying convex losses with a suitable regularizer, yielding $\mathcal{O}(T^{2/3})$ regret. They left open whether the optimal $Θ(\sqrt{T})$ regret from online convex optimization can be recovered in this hidden-convex setting. We answer this question affirmatively. More specifically, via a sharper discrete-time algorithmic equivalence argument, we prove that OGD achieves $\mathcal{O}(\sqrt{T})$ regret under the same assumptions, matching the optimal worst-case rate for adversarial online convex optimization. We also address another open question of Ghai, Lu and Hazan (2022) by clarifying the geometry required for this algorithmic equivalence. We replace the diagonal-Jacobian sufficient condition with a necessary-and-sufficient Hessian compatibility condition, thereby expanding the class of admissible reparameterizations. We complement our tight regret bound with a lower bound showing that the Hessian compatibility assumption is essential for OGD; when it fails, we construct a smooth reparameterization and an adversarial sequence of hidden-convex losses for which OGD suffers $Ω(T)$ regret. Finally, we extend our analysis to one-point bandit feedback and prove a $\mathcal{O}(T^{3/4})$ expected regret bound for bandit OGD with spherical smoothing, matching its classical rate on convex losses.

This paper introduces a new problem-dependent regret measure for online convex optimization with smooth losses. The notion, which we call the $G^\star$ regret, depends on the cumulative squared gradient norm evaluated at the decision in hindsight. We show that the $G^\star$ regret strictly refines the existing $L^\star$ (small loss) regret, and that it can be arbitrarily sharper when the losses have vanishing curvature around the hindsight decision. We establish upper and lower bounds on the $G^\star$ regret and extend our results to dynamic regret and bandit settings. As a byproduct, we refine the existing convergence analysis of stochastic optimization algorithms in the interpolation regime. Some experiments validate our theoretical findings.

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 $\mathbb{R}^d$.