Convex Optimization with Nested Evolving Feasible Sets
For online optimization with time-varying constraints, this work provides optimal trade-offs between regret and movement cost, generalizing nested convex body chasing.
The paper introduces CONES, a convex optimization problem with nested evolving feasible sets, and proposes algorithms achieving simultaneous regret and movement cost bounds: O(T^{1-β}) and O(T^β) for convex losses, and zero regret with O(log T) movement cost for strongly convex or α-sharp losses, with matching lower bounds.
Convex Optimization with Nested Evolving Feasible Sets (CONES)} is considered where the objective function $f$ remains fixed but the feasible region evolves over time as a nested sequence $S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T$. The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known nested convex body chasing problem. When the loss function is convex, we propose a lazy-algorithm and show that it achieves $O(T^{1-β}), O(T^β)$ simultaneous regret and movement cost for any $β\in (0,1]$, over a time horizon of $T$. When the loss function is strongly convex or $α$-sharp, we propose an algorithm Frugal that simultaneously achieves zero regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with $o(T)$ regret has a movement cost of $Ω(\log{T})$ for both cases, proving optimality of Frugal.