Minimax-Optimal Small-Loss Bounds for Online Linear Optimisation on $\ell_p$ Balls

We characterize the regret attainable in online convex optimization when access to the feasible set is limited to an exact linear optimization oracle. The learner is given an inscribed ball and a diameter bound and must remain feasible on every consistent instance. For convex $G$-Lipschitz losses, diameter at most $D$, a total allowance of $Q$ oracle calls, and a strict limit of $B$ calls per round, the dimension-free minimax expected regret is $Θ(GD\max\{\sqrt T,T/(1+\min\{Q,BT\})^{1/4}\})$. The lower bound applies to arbitrary randomized learners. Universal feasibility first forces each action into the hull of the supplied ball and the preceding oracle replies. A fixed-body construction then couples fresh phase directions to a shared simplex, making useful replies costly repeatedly even though all losses have a common minimizer. A counted approximate-gradient method with interleaved blocks attains the matching rate. Total-budget and strict per-round guarantees follow as special cases, including the $T^{3/4}$ rate with one call per round and the quadratic total budget needed for $\sqrt T$ regret. For prescribed smoothness $β$, an analytic construction yields a curvature-dependent lower bound and identifies the threshold above which the general characterization remains sharp.

Can a constant number of linear minimizations per round improve on the $T^{3/4}$ regret rate of online Frank-Wolfe on general convex sets? Weibel et al. conjectured that fixed-coefficient methods cannot. We prove their conjecture and extend the lower bound to every deterministic learner in an oracle-only model. The learner receives an initial feasible point and a diameter bound, and must remain feasible on every domain consistent with its oracle replies. For $T$ rounds, at most $b$ calls between decisions, diameter bound $D$, and gradient norm bound $L$, we construct an instance in dimension $d=2b(T-1)+1$ with regret at least $2^{-1/4}LDb^{-1/4}T^{3/4}$. The adversary fixes the domain, initial point, deterministic tie rule and linear losses before play. The vertices form a path on which every point available before a decision has zero current loss, while the final vertex has negative loss on every round. For constant $b$, the result matches the known upper rate for dimension-independent guarantees. For one-call fixed schedules with a nonzero coefficient on the newest gradient, a second construction gives regret at least $3LDT^{3/4}/4$ with unique minimizers at every issued query. Exact-arithmetic certificates for the tuned schedule of Weibel et al. closely match their finite-horizon numerical worst cases, with unique oracle replies.

We give the first adversarial linear bandit algorithm based on Follow-the-Perturbed-Leader (FTPL) that achieves $\sqrt{T}$ regret over general action sets. Our algorithm attains $\widetilde{\mathcal{O}}(d^{5/4}\sqrt{T})$ regret using only linear optimization oracles, and the near-optimal $\widetilde{\mathcal{O}}(d\sqrt{T})$ regret when an optimal-design exploration distribution is further available. This improves on the $T^{2/3}$ regret of the FTPL algorithm of Dani and Hayes (2006) for general action sets, and generalizes the $\sqrt{T}$ regret of Lévy et al. (2026) over two special action sets (hypercube and Euclidean ball). Compared to previous oracle-efficient linear bandit algorithms based on continuous exponential weights, the FTPL approach generally has better computational efficiency, and is much simpler to implement. The key algorithmic element is a perturbation distribution that adapts to the past reward estimators. Its simplest form, an adaptive Gaussian perturbation, is straightforward to implement and already attains the regret bound above. We further extend it to two general constructions, random-weighting and elliptical perturbations, each covering distributions from light tails (Gaussian) to heavy tails (Student-$t_2$). On the technical side, we develop a systematic way to analyze the stability term in the FTPL regret through the Fisher information of the perturbation distribution. This builds a general framework for FTPL beyond existing ones such as the hazard rate and the FTRL-FTPL duality.