CLASP: Online learning algorithms for Convex Losses And Squared Penalties

Addressing Constrained Online Convex Optimization (COCO), we introduce CLASP (Convex Losses And Squared Penalties), a framework that minimizes cumulative loss together with squared constraint violations. We propose two variants of CLASP, CLASP-I and CLASP-F, allowing for a joint or separate handling of the static decision set and the time-varying constraints, a decoupling flexibility that affords simpler implementations when projections onto the static decision set are easy. Our theoretical analysis departs from prior work by fully leveraging the variety of \emph{cutter operators}, and contraction properties such as the strongly quasi-nonexpansiveness, a proof strategy not previously applied in this setting. For convex losses, both CLASP algorithms achieve 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, we provide the first logarithmic guarantees on both regret and cumulative squared penalty: In the strongly convex case, both CLASP algorithms guarantee that the regret is upper bounded by $O( \log T )$ and the cumulative squared penalty is also upper bounded by $O( \log T )$.