AdaOGD-PFS is proposed, an adaptive-step-size method that achieves O(sqrt(G_T) regret with per-round feasibility while preserving per-round feasibility and identifies a nonnegative Polyak correction P_T that enters the regret bound with a negative sign.
Abstract
Constrained online convex optimization requires minimizing regret against adversarial convex costs while satisfying a convex constraint at every round, as needed in safety-critical applications. A computationally efficient method combines online gradient descent with a Polyak feasibility step, using one constraint evaluation and one subgradient per round. Although this method achieves O(sqrt(T)) regret with per-round feasibility, we derive a tighter, data-dependent analysis by retaining two quantities omitted by the standard worst-case argument. First, we replace the gradient envelope G_f^2 T with the observed accumulation G_T = sum_t ||grad f_t(x_t)||^2. Second, we identify a nonnegative Polyak correction P_T that measures the cumulative squared displacement caused by feasibility projections and enters the regret bound with a negative sign. The resulting improvement, Delta_T = (eta/2)(G_f^2 T - G_T) + P_T/(2 eta), is always nonnegative. We further propose AdaOGD-PFS, an adaptive-step-size method that achieves O(sqrt(G_T)) regret while preserving per-round feasibility. Experiments on ball- and halfspace-constrained problems improve the regret bound by 38 to 43 percent, with both data-dependent gradients and Polyak corrections contributing substantially.
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 $\Omega(\min\{\sigma,1\}T/\log^2 T)$, 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 online convex optimization with dueling (pairwise comparison) feedback, where the learner observes only a binary preference between two queried points. While dueling feedback is well understood in discrete or stochastic settings, the adversarial convex setting has remained unexplored. We propose a simple reduction that converts dueling feedback into approximate gradients, enabling the use of standard first-order methods. We show that regret guarantees transfer under this reduction, yielding the first results for this setting, including $\mathcal{O}(T^{3/4})$ static, adaptive, and dynamic regret. Under additional structure, we obtain improved rates of $\mathcal{O}(T^{2/3})$ for smooth objectives and $\mathcal{O}(\sqrt{T \log T})$ for strongly convex functions.
Yiyang Lu, Hareshkumar Jadav, M. Pedramfar et al.· 1 citation
We study online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the $d$-dimensional unit cube. The best known constructive offline approximation factor is $0.401$ under the corresponding meta-solvability assumptions, whereas comparable adversarial online guarantees had remained at $1/e$. We show that this factor is also achievable online. In the post-decision full-information value-oracle model, our algorithm attains factor $0.401$ with sublinear approximate regret when oracle feedback is conditionally unbiased and bounded. The online algorithm does not run the offline construction on a changing objective. Instead, it replaces the offline objective-dependent box step by a weighted online learner that controls the required residual terms cumulatively. An exact asymmetric balance theorem preserves the offline coefficients despite adversarial variation. The direct implementation has $O(T^{3/4})$ regret and uses $O(dT^{1/4})$ oracle calls per round. More generally, for every $\delta\in[0,1/4]$, batching gives $O(T^\delta)$ calls per round and $O(T^{4/5-\delta/5})$ regret, including a one-call $O(T^{4/5})$ endpoint. Under a positive-anchor condition, randomized blocking retains factor $0.401$ with $O(T^{5/6})$ one-point bandit regret.
We study first-order black-box convex optimization over an $\ell_p$-ball for objectives Lipschitz in the $\ell_q$-norm, solving in the affirmative the nonsmooth version of the COLT open question (Guz15b) on whether the geometry of a smaller feasible set ($p<q$) can improve convergence rates in convex optimization, and matching prior lower bounds up to logarithmic factors. Our rates include \(\widetilde O(1/T)\) for convex Euclidean-Lipschitz optimization over the $\ell_1$-ball, improving on the $O(1/\sqrt{T})$ classical rate under general assumptions. The key technical device is a new online learning game, where the comparator is evaluated using the maximum of affine losses observed so far. We bound the value of this game above and below in terms of a combinatorial online learning quantity: the sequential fat-shattering dimension, which we characterize for the $\ell_p / \ell_q$ case. Our results generally apply when the feasible set $X$ and the set of possible subgradients $H$ are convex, centrally symmetric, and admit a type of minmax theorem, advancing on a fundamental question by Sridharan [Sri12, Section 10.1.2, Q3]. As a geometric consequence of our analysis, of independent interest, we obtain estimates for the expected distance of a convex hull of samples to their mean in several Banach geometries, a version of the celebrated Wendel's theorem (Wen62), but quantitative and for bounded general distributions as opposed to centrally symmetric ones.
David Martínez-Rubio, Brian Bullins, Cristóbal Guzmán et al.· 1 citation
We give a deterministic algorithm for online inverse linear optimization with regret $O(d)$, uniform in the horizon and $O(d^{2})$ time per round. A bound of this order was obtained recently by Dewasurendra, settling a question of Gollapudi et al.\ and of Oki and Sakaue, but by an improper rule that enumerates covers at every scale and costs $T^{\Theta(d)}$ a round; ours is the first efficient such bound and the first proper one. We build on the variable-metric framework of Sakaue et al., adding a self-normalized rank-one update, and we replace the $\log\det$ potential by the trace power $\tr(H^{-1/2})$, which is bounded outright and removes the $\ln T$. The bound also holds against an expert that does not optimize, and we give corruption-robust and rank-adaptive variants, and an application to convex minimization.
Yang Cai, Anupam Gupta, Vineet Gupta et al.· 0 citations
This work addresses decentralized online Riemannian optimization on Hadamard manifolds. Prior work under geodesic convexity (g-convexity) may require curvature information in the optimization analysis, typically through a finite lower bound on the sectional curvature. Curvature may also enter the step size or contraction factor of tangent-space Riemannian consensus schemes. In this work, we relax the curvature dependence for a narrower class of horospherical convex (h-convex) functions. We study Distributed Riemannian Online Gradient Descent (D-ROGD), which combines local Riemannian h-subgradient updates with an implicit Fr\'echet-mean consensus. For h-convex and strongly h-convex local objectives, we establish $O(\sqrt{T})$ and $O(\log T)$ static regret, respectively, matching the corresponding Euclidean rates with respect to $T$, with network dependence governed solely by the spectral gap. To our knowledge, these are the first curvature-independent regret guarantees for decentralized online optimization on Hadamard manifolds. Experiments on hyperbolic embeddings corroborate the predicted rates, with no observable degradation due to curvature.
