Nested Convex-Body Chasing for Online Optimization with Evolving Feasible Sets
Artificial Intelligence
Summary
The authors study online optimization problems where the set of allowed solutions keeps shrinking over time. They design algorithms that balance minimizing losses with controlling how much the solution moves, using geometric methods to keep movement low while ensuring good performance. Their results provide strong guarantees on regret (a measure of how well the algorithm performs) and movement costs, improving upon previous work especially in higher dimensions and under strong convexity assumptions. They also show fundamental limits on movement complexity in two dimensions and extend their techniques to handle constraints adversarially over time.
Authors
Dhruv Sarkar, Aprameyo Chakrabartty
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, $μ$-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 $O(\sqrt{d\log(1+d)\,GD\log(eT)/μ})$. 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 $O(T^β)$, $β<1$, suffers $Ω(\sqrt{\log T})$ 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.