Lazy Lagrangians with Predictions for Online Learning

TL;DR

This paper introduces a new primal-dual algorithm for online convex optimization with time-varying constraints, leveraging predictions to improve regret and constraint violation bounds, and extending the FTRL framework.

Contribution

The paper develops a novel prediction-aware primal-dual algorithm that achieves tunable regret and constraint violation bounds, outperforming existing greedy solutions without restrictive assumptions.

Findings

Achieves $ ext{O}(T^{(3-eta)/4})$ regret bounds.

Attains $ ext{O}(T^{(1+eta)/2})$ constraint violation bounds.

Bounds improve with better prediction quality, reaching $ ext{O}(1)$ for perfect predictions.

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.