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Published December 2019 | Published + Submitted + Supplemental Material
Book Section - Chapter Open

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

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.

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.

Attached Files

Published - 8463-beyond-online-balanced-descent-an-optimal-algorithm-for-smoothed-online-optimization.pdf

Submitted - 1905.12776.pdf

Supplemental Material - 8463-beyond-online-balanced-descent-an-optimal-algorithm-for-smoothed-online-optimization-supplemental.zip

Files

8463-beyond-online-balanced-descent-an-optimal-algorithm-for-smoothed-online-optimization-supplemental.zip

Additional details

Created:
August 19, 2023
Modified:
October 20, 2023