Lazy Lagrangians for Optimistic Learning With Budget Constraints

Abstract—We consider the general problem of online convex optimization with time-varying additive constraints in the presence of predictions for the next cost and constraint functions. A novel primal-dual algorithm is designed by combining a Follow-The-Regularized-Leader iteration with prediction adaptive dynamic steps. The algorithm achieves \(\mathcal{O}(T^{\frac{3-\beta}{4}})\) regret and \(\mathcal{O}(T^{\frac{1+\beta}{2}})\) constraint violation bounds that are tunable via parameter \(\beta \in [1/2, 1)\) and have constant factors that shrink with the predictions quality, achieving eventually \(\mathcal{O}(1)\) regret for perfect predictions. Our work extends the FTRL framework for this constrained OCO setting and outperforms the respective state-of-the-art greedy-based solutions, without imposing conditions on the quality of predictions, the cost functions or the geometry of constraints, beyond convexity