## Abstract

We consider the setting of online convex optimization with adversarial time-varying constraints in which actions must be feasible w.r.t. a fixed constraint set, and are also required on average to approximately satisfy additional time-varying constraints. Motivated by scenarios in which the fixed feasible set (hard constraint) is difficult to project on, we consider projection-free algorithms that access this set only through a linear optimization oracle (LOO). We present an algorithm that, on a sequence of length T and using overall T calls to the LOO, guarantees Õ(T^{3/4}) regret w.r.t. the losses and O(T^{7/8}) constraints violation (ignoring all quantities except for T). In particular, these bounds hold w.r.t. any interval of the sequence. This algorithm however also requires access to an oracle for minimizing a strongly convex nonsmooth function over a Euclidean ball. We present a more efficient algorithm that does not require the latter optimization oracle but only first-order access to the time-varying constraints, and achieves similar bounds w.r.t. the entire sequence. We extend the latter to the setting of bandit feedback and obtain similar bounds (as a function of T) in expectation.

Original language | English |
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Pages (from-to) | 14988-15005 |

Number of pages | 18 |

Journal | Proceedings of Machine Learning Research |

Volume | 235 |

State | Published - 2024 |

Event | 41st International Conference on Machine Learning, ICML 2024 - Vienna, Austria Duration: 21 Jul 2024 → 27 Jul 2024 |

## All Science Journal Classification (ASJC) codes

- Artificial Intelligence
- Software
- Control and Systems Engineering
- Statistics and Probability