Constrained Online Convex Optimization with Polyak Feasibility Steps

Abstract

In this work, we study online convex optimization with a fixed constraint function $g : \mathbb{R}^d \rightarrow \mathbb{R}$. Prior work on this problem has shown $O(\sqrt{T})$ regret and cumulative constraint satisfaction $\sum_{t=1}^{T} g(x_t) \leq 0$, while only accessing the constraint value and subgradient at the played actions $g(x_t), \partial g(x_t)$. Using the same constraint information, we show a stronger guarantee of anytime constraint satisfaction $g(x_t) \leq 0 \ \forall t \in [T]$, and matching $O(\sqrt{T})$ regret guarantees. These contributions are thanks to our approach of using Polyak feasibility steps to ensure constraint satisfaction, without sacrificing regret. Specifically, after each step of online gradient descent, our algorithm applies a subgradient descent step on the constraint function where the step-size is chosen according to the celebrated Polyak step-size. We further validate this approach with numerical experiments.