Reviews: Online convex optimization for cumulative constraints

In this setting our objective is to provide low regret with respect to the sequence of losses as well as with respect to the constraints. This setting is useful when one would like to avoid projections which might be costly. Previous works focused on the regret with the respect to the cumulative constraints, where constraint violation may be compensated by constrained satisfaction. In this work the authors consider a more appropriate measure of regret which accounts only for constraints violation. In this case, the authors come up with a new algorithm (similar in spirit to previous approaches), which provides regret guarantees with respect to the sum of squared constrained violations. They also extend their approach to the strongly convex setting.