Projection-Free Online Convex Optimization with Time-Varying Constraints
<|reference_start|>Projection-Free Online Convex Optimization with Time-Varying Constraints: 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 $\tilde{O}(T^{3/4})$ 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. We also present a more efficient algorithm that requires only first-order oracle access to the soft 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.<|reference_end|>
@article{garber2024projection-free,
    title={Projection-Free Online Convex Optimization with Time-Varying Constraints},
    author={Dan Garber, Ben Kretzu},
    journal={arXiv preprint arXiv:2402.08799},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.08799},
    primaryClass={cs.LG math.OC stat.ML}
}