CLASP: An online learning algorithm for Convex Losses And Squared Penalties

TL;DR

This paper introduces CLASP, an online learning algorithm for convex optimization with constraints, achieving optimal regret and penalty bounds, especially logarithmic guarantees in strongly convex scenarios.

Contribution

The paper presents CLASP, a novel algorithm that effectively minimizes loss and constraint violations using a new proof strategy based on convex projectors.

Findings

Achieves regret of $O(T^{eta})$ for convex losses.

Provides the first logarithmic bounds for strongly convex problems.

Ensures bounded cumulative squared penalties with $O( ext{log } T)$.

Abstract

We study Constrained Online Convex Optimization (COCO), where a learner chooses actions iteratively, observes both unanticipated convex loss and convex constraint, and accumulates loss while incurring penalties for constraint violations. We introduce CLASP (Convex Losses And Squared Penalties), an algorithm that minimizes cumulative loss together with squared constraint violations. Our analysis departs from prior work by fully leveraging the firm non-expansiveness of convex projectors, a proof strategy not previously applied in this setting. For convex losses, CLASP achieves regret $O\left(T^{\max\{\beta,1-\beta\}}\right)$ and cumulative squared penalty $O\left(T^{1-\beta}\right)$ for any $\beta \in (0,1)$. Most importantly, for strongly convex problems, CLASP provides the first logarithmic guarantees on both regret and cumulative squared penalty. In the strongly convex case, the regret is upper bounded by $O( \log T )$ and the cumulative squared penalty is also upper bounded by $O( \log T )$.