Abstract 

We propose the algorithms for online convex optimization which lead to cumulative squared constraint violations of the form where  Previous literature has focused on long-term constraints of the form There, strictly feasible solutions can cancel out the effects of violated t=1 constraints. In contrast, the new form heavily penalizes large constraint violations and cancellation effects cannot occur. Furthermore, useful bounds on the single step constraint violation are derived. For convex objectives, our regret bounds generalize existing bounds, and for strongly convex objectives we give improved regret bounds. In numerical experiments, we show that our algorithm closely follows the constraint boundary leading to low cumulative violation.