Constrained Online Convex Optimization without Slater's Condition

We study high-probability regret bounds for online convex optimization (OCO) with strongly convex losses and establish three results that resolve open questions at the intersection of noise adaptivity, feedback structure, and constraint satisfaction. For the full-information setting with sub-Gaussian stochastic gradients, we prove a noise-adaptive high-probability regret bound in which the martingale deviation term scales with the noise level $σ$ rather than the gradient bound $G$, yielding a multiplicative improvement of $G/σ$ over the classical Azuma-Hoeffding baseline. Our analysis introduces an exponential supermartingale argument that bypasses the bounded-difference requirement of Freedman's inequality, enabling direct treatment of unbounded sub-Gaussian noise without truncation artifacts. For bandit feedback, we prove a minimax lower bound: the high-probability regret scales linearly in $\log(1/δ)$, in contrast to the $\sqrt{\log(1/δ)}$ confidence cost under full information. This constitutes a formal separation in the confidence cost of strongly convex OCO across feedback models. Regarding constrained OCO with stochastic constraints satisfying a Slater condition, we provide simultaneous high-probability guarantees for both cumulative regret and long-run constraint violation, achieving $\mathcal{O}(\sqrt{T\log(m/δ)})$ regret and $\mathcal{O}(\sqrt{T}/(ζδ) + m\sqrt{T\log(m/δ)})$ violation. Synthetic experiments corroborate all theoretical predictions.

We study constrained online convex optimization with adversarial constraints and conditionally unbiased, finite-variance observations of constraint values and gradients. Under common feasibility, our \LEDGER\ algorithm attains $O(\sqrt T)$ expected regret and $O(\sqrt{T\log(eT)})$ expected budget violation, the largest cumulative overspend over any window. It uses a reflected exponential potential, clipped signed observations, and predictable adaptive regularization, with one feedback triple and one projection per round. Neither a Slater condition, independence between feedback channels, nor an absolute constraint-value bound is needed. A Gaussian testing lower bound proves that the budget rate has optimal horizon dependence under square-root regret at fixed positive noise, including the logarithm. The same obstruction holds for terminal violation, so the logarithm is not a cost of maximizing over windows; an $O(\sqrt T)$ budget bound instead forces linear regret. In contrast, fixed positive Gaussian value noise yields a joint regret--hard-violation lower bound of $Ω(\min\{σ,1\}T/\log^2 T)$, even with exact gradients in one dimension. The hard-violation construction matches arbitrarily many moments while preserving a feasible-endpoint gap and constant endpoint probabilities. Together, the bounds separate uncertainty about hard feasibility from learnable signed budgets. Deterministic restarts remove the horizon input without changing either upper rate.

We study constrained online convex optimization with adversarial time-varying constraints. At each round the learner acts before observing the loss and constraint, and is compared with the best fixed action satisfying all constraints in hindsight. The goal is to obtain minimax-optimal regret while controlling cumulative constraint violation (CCV). Prior algorithms achieved $O(\log T)$ regret with $O(\sqrt{T\log T})$ CCV for strongly convex losses, and $O(\sqrt{T})$ regret with $O(\sqrt{T}\log T)$ CCV for convex losses. We propose NP-OGD, an iterated nested-projection algorithm. For strongly convex losses it achieves $O(\log T)$ regret and $O(\log T)$ CCV; for convex losses it achieves $O(\sqrt{T})$ regret and $O(\sqrt{T})$ CCV. The analysis relies on a geometric movement bound: after lifting the nested projected-gradient trajectory to one higher dimension, the lifted path is self-contracted under a nonstandard norm, so a finite-length theorem for self-contracted curves controls the total projection movement. We also prove complementary lower bounds using layered sphere packings. For strongly convex losses, any online algorithm with polynomially sublinear regret can incur CCV at least $Ω((\log T)^{(d-1)/(d+1)}/\log\log T)$. For convex losses, we prove CCV lower bounds $Ω(T^{(d-1)/(2(d+3))})$ for weakly adaptive algorithms and $Ω(T^{(d-1)/(2d)})$ for NP-OGD. Finally, for the constrained experts special case over $N$ experts, an active Hedge algorithm attains $O(\sqrt{T\log N})$ regret and $O(N)$ CCV, with a matching minimax CCV lower bound for sufficiently large horizons.