Goel, Gautam and Lin, Yiheng and Sun, Haoyuan and Wierman, Adam (2019) Beyond Online Balanced Descent: An Optimal Algorithm for Smoothed Online Optimization. In: 33rd Conference on Neural Information Processing Systems. Neural Information Processing Systems Foundation, Inc. , Art. No. 8463. https://resolver.caltech.edu/CaltechAUTHORS:20190626-100003384
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Abstract
We study online convex optimization in a setting where the learner seeks to minimize the sum of a per-round hitting cost and a movement cost which is incurred when changing decisions between rounds. We prove a new lower bound on the competitive ratio of any online algorithm in the setting where the costs are m-strongly convex and the movement costs are the squared ℓ₂ norm. This lower bound shows that no algorithm can achieve a competitive ratio that is o(m^(−1/2)) as m tends to zero. No existing algorithms have competitive ratios matching this bound, and we show that the state-of-the-art algorithm, Online Balanced Decent (OBD), has a competitive ratio that is Ω(m^(−2/3)). We additionally propose two new algorithms, Greedy OBD (G-OBD) and Regularized OBD (R-OBD) and prove that both algorithms have an O(m^(−1/2)) competitive ratio. The result for G-OBD holds when the hitting costs are quasiconvex and the movement costs are the squared ℓ₂ norm, while the result for R-OBD holds when the hitting costs are m-strongly convex and the movement costs are Bregman Divergences. Further, we show that R-OBD simultaneously achieves constant, dimension-free competitive ratio and sublinear regret when hitting costs are strongly convex.
Item Type: | Book Section | ||||||||||
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Additional Information: | © 2020 Neural Information Processing Systems Foundation, Inc. Gautam Goel, Yiheng Lin, and Haoyuan Sun contributed equally to this work. This work was supported by NSF grants AitF-1637598 and CNS-1518941, with additional support for Gautam Goel provided by an Amazon AWS AI Fellowship. | ||||||||||
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DOI: | 10.48550/arXiv.1905.12776 | ||||||||||
Record Number: | CaltechAUTHORS:20190626-100003384 | ||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20190626-100003384 | ||||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||
ID Code: | 96721 | ||||||||||
Collection: | CaltechAUTHORS | ||||||||||
Deposited By: | Tony Diaz | ||||||||||
Deposited On: | 26 Jun 2019 17:07 | ||||||||||
Last Modified: | 02 Jun 2023 00:57 |
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