Convex Optimization with Nested Evolving Feasible Sets

TL;DR

This paper introduces algorithms for convex optimization with evolving feasible sets, achieving optimal regret and movement costs, and extends the nested convex body chasing problem.

Contribution

It proposes new online algorithms with optimal regret and movement cost bounds for convex optimization with nested evolving feasible sets.

Findings

Lazy-algorithm achieves $O(T^{1-eta})$ regret and $O(T^eta)$ movement cost.

Frugal algorithm achieves zero regret and $O( ext{log } T)$ movement cost.

Any algorithm with sublinear regret incurs at least $O( ext{log } T)$ movement cost.

Abstract

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 $O(T^{1-\beta}), O(T^\beta)$ simultaneous regret and movement cost for any $\beta \in (0,1]$, over a time horizon of $T$. When the loss function is strongly convex or $\alpha$-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 $\Omega(\log{T})$ for both cases, proving optimality of Frugal.