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This paper advances the theoretical foundations of online optimization by introducing deterministic resettable nested convex-body chasers that decouple loss control from geometric movement. The research establishes optimal horizon dependence for movement bounds in constrained settings and replaces prior exponential dimension factors with polynomial dependencies via Euclidean chasing techniques. These results are critical for improving the efficiency and stability of adaptive algorithms operating within evolving feasible regions, a key challenge in dynamic AI systems.
Nested Convex-Body Chasing for Online Optimization with Evolving Feasible Sets
arXiv:2608.29074v1 Announce Type: new Abstract: We study online optimization with nested shrinking feasible regions in two settings: convex optimization with nested evolving feasible sets (CONES) and adversarial constrained online convex optimization (COCO). Our algorithms separate loss control from geometric movement: constrained minimizers and cumulative-loss tests preserve regret guarantees, while a deterministic resettable nested convex-body chaser limits movement. For CONES with a $G$-Lipschitz, $\mu$-strongly convex objective on a diameter-$D$ domain, we chase intersections of the current feasible set with adaptive objective sublevel sets. Using the Euclidean chasing ratio $O(\sqrt{d\log(1+d)})$, we obtain nonpositive regret at every prefix and movement $O(\sqrt{d\log(1+d)\,GD\log(eT)/\mu})$. The bound adapts to the increase in the constrained optimum value. In dimension two, with all other parameters fixed, every randomized algorithm with terminal expected regret $O(T^\beta)$, $\beta<1$, suffers $\Omega(\sqrt{\log T})$ expected movement on some deterministic nested sequence, proving optimal horizon dependence. Under linear growth away from the constrained minimizer set, Steiner-point tracking yields movement independent of $T$. For general convex COCO, one-step-delayed chasing with regularized-leader resets gives regret $O(G_fD\sqrt{d\log(1+d)T})$ and cumulative constraint violation $O(G_gD\sqrt{d\log(1+d)T})$. For strongly convex losses, both are $O(d\log(1+d)\log(eT))$ when other parameters are fixed. These reductions replace the $O(d^{d/2})$ projection-path factor in prior analyses by the polynomial dimension dependence of Euclidean nested convex-body chasing.
Abstract:We study online optimization with nested shrinking feasible regions in two settings: convex optimization with nested evolving feasible sets (CONES) and adversarial constrained online convex optimization (COCO). Our algorithms separate loss control from geometric movement: constrained minimizers and cumulative-loss tests preserve regret guarantees, while a deterministic resettable nested convex-body chaser limits movement. For CONES with a $G$-Lipschitz, $\mu$-strongly convex objective on a diameter-$D$ domain, we chase intersections of the current feasible set with adaptive objective sublevel sets. Using the Euclidean chasing ratio $O(\sqrt{d\log(1+d)})$, we obtain nonpositive regret at every prefix and movement $O(\sqrt{d\log(1+d)\,GD\log(eT)/\mu})$. The bound adapts to the increase in the constrained optimum value. In dimension two, with all other parameters fixed, every randomized algorithm with terminal expected regret $O(T^\beta)$, $\beta<1$, suffers $\Omega(\sqrt{\log T})$ expected movement on some deterministic nested sequence, proving optimal horizon dependence. Under linear growth away from the constrained minimizer set, Steiner-point tracking yields movement independent of $T$. For general convex COCO, one-step-delayed chasing with regularized-leader resets gives regret $O(G_fD\sqrt{d\log(1+d)T})$ and cumulative constraint violation $O(G_gD\sqrt{d\log(1+d)T})$. For strongly convex losses, both are $O(d\log(1+d)\log(eT))$ when other parameters are fixed. These reductions replace the $O(d^{d/2})$ projection-path factor in prior analyses by the polynomial dimension dependence of Euclidean nested convex-body chasing.
Artificial Intelligence (cs.AI)
arXiv:2608.29074 [cs.AI]
(or arXiv:2608.29074v1 [cs.AI] for this version)
https://doi.org/10.48550/arXiv.2608.29074 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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