Adaptive Algorithms for Online Convex Optimization with Long-term Constraints
Rodolphe Jenatton, Jim Huang, Cedric Archambeau
Abstract
We present an adaptive online gradient descent algorithm to solve online convex optimization problems with long-term constraints, which are constraints that need to be satisfied when accumulated over a finite number of rounds T, but can be violated in intermediate rounds. For some user-defined trade-off parameter βin (0, 1), the proposed algorithm achieves cumulative regret bounds of O(T^maxβ,1_β) and O(T^1_β/2), respectively for the loss and the constraint violations. Our results hold for convex losses, can handle arbitrary convex constraints and rely on a single computationally efficient algorithm. Our contributions improve over the best known cumulative regret bounds of Mahdavi et al. (2012), which are respectively O(T^1/2) and O(T^3/4) for general convex domains, and respectively O(T^2/3) and O(T^2/3) when the domain is further restricted to be a polyhedral set. We supplement the analysis with experiments validating the performance of our algorithm in practice.
BibTeX
@InProceedings{pmlr-v48-jenatton16,
title = {Adaptive Algorithms for Online Convex Optimization with Long-term Constraints},
author = {Jenatton, Rodolphe and Huang, Jim and Archambeau, Cedric},
booktitle = {Proceedings of The 33rd International Conference on Machine Learning},
pages = {402--411},
year = {2016},
editor = {Balcan, Maria Florina and Weinberger, Kilian Q.},
volume = {48},
series = {Proceedings of Machine Learning Research},
address = {New York, New York, USA},
month = {20--22 Jun},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v48/jenatton16.pdf},
url = {https://proceedings.mlr.press/v48/jenatton16.html},
abstract = {We present an adaptive online gradient descent algorithm to solve online convex optimization problems with long-term constraints, which are constraints that need to be satisfied when accumulated over a finite number of rounds T, but can be violated in intermediate rounds. For some user-defined trade-off parameter βin (0, 1), the proposed algorithm achieves cumulative regret bounds of O(T^maxβ,1_β) and O(T^1_β/2), respectively for the loss and the constraint violations. Our results hold for convex losses, can handle arbitrary convex constraints and rely on a single computationally efficient algorithm. Our contributions improve over the best known cumulative regret bounds of Mahdavi et al. (2012), which are respectively O(T^1/2) and O(T^3/4) for general convex domains, and respectively O(T^2/3) and O(T^2/3) when the domain is further restricted to be a polyhedral set. We supplement the analysis with experiments validating the performance of our algorithm in practice.}
}