# CLASP: An online learning algorithm for Convex Losses And Squared Penalties Source: [CLASP: An online learning algorithm for Convex Losses And Squared Penalties]() Collection index: [llms.txt]() Data updated: 2026-01-28T05:00:08.385Z Authors: Ricardo N. Ferreira, Jo\\~ao Xavier, Cl\\'audia Soares Published: 2026-01-22T16:13:52.000Z ## 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 $O( \\log T )$. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()