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Beyond Online Balanced Descent: An Optimal Algorithm for Smoothed Online Optimization

Goel, Gautam and Lin, Yiheng and Sun, Haoyuan and Wierman, Adam (2019) Beyond Online Balanced Descent: An Optimal Algorithm for Smoothed Online Optimization. . (Unpublished) http://resolver.caltech.edu/CaltechAUTHORS:20190626-100003384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 ℓ_2 norm. This lower bound shows that no algorithm can achieve a competitive ratio that is o(m^(−1/2)) as mtends 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 ℓ_2 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:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1905.12776arXivDiscussion Paper
Additional Information:Gautam Goel, Yiheng Lin, and Haoyuan Sun contributed equally to this work.
Record Number:CaltechAUTHORS:20190626-100003384
Persistent URL:http://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:26 Jun 2019 17:07

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