Smoothed Online Convex Optimization in High Dimensions via Online Balanced Descent

Niangjun Chen, Gautam Goel, Adam Wierman
Proceedings of the 31st Conference On Learning Theory, PMLR 75:1574-1594, 2018.

Abstract

We study \emph{smoothed online convex optimization}, a version of online convex optimization where the learner incurs a penalty for changing her actions between rounds. Given a $\Omega(\sqrt{d})$ lower bound on the competitive ratio of any online algorithm, where $d$ is the dimension of the action space, we ask under what conditions this bound can be beaten. We introduce a novel algorithmic framework for this problem, Online Balanced Descent (OBD), which works by iteratively projecting the previous point onto a carefully chosen level set of the current cost function so as to balance the switching costs and hitting costs. We demonstrate the generality of the OBD framework by showing how, with different choices of “balance,” OBD can improve upon state-of-the-art performance guarantees for both competitive ratio and regret; in particular, OBD is the first algorithm to achieve a dimension-free competitive ratio, $3 + O(1/\alpha)$, for locally polyhedral costs, where $\alpha$ measures the “steepness” of the costs. We also prove bounds on the dynamic regret of OBD when the balance is performed in the dual space that are dimension-free and imply that OBD has sublinear static regret.

Cite this Paper

BibTeX
@InProceedings{pmlr-v75-chen18b, title = {Smoothed Online Convex Optimization in High Dimensions via Online Balanced Descent}, author = {Chen, Niangjun and Goel, Gautam and Wierman, Adam}, booktitle = {Proceedings of the 31st Conference On Learning Theory}, pages = {1574--1594}, year = {2018}, editor = {Bubeck, Sébastien and Perchet, Vianney and Rigollet, Philippe}, volume = {75}, series = {Proceedings of Machine Learning Research}, month = {06--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v75/chen18b/chen18b.pdf}, url = {https://proceedings.mlr.press/v75/chen18b.html}, abstract = {We study \emph{smoothed online convex optimization}, a version of online convex optimization where the learner incurs a penalty for changing her actions between rounds. Given a $\Omega(\sqrt{d})$ lower bound on the competitive ratio of any online algorithm, where $d$ is the dimension of the action space, we ask under what conditions this bound can be beaten. We introduce a novel algorithmic framework for this problem, Online Balanced Descent (OBD), which works by iteratively projecting the previous point onto a carefully chosen level set of the current cost function so as to balance the switching costs and hitting costs. We demonstrate the generality of the OBD framework by showing how, with different choices of “balance,” OBD can improve upon state-of-the-art performance guarantees for both competitive ratio and regret; in particular, OBD is the first algorithm to achieve a dimension-free competitive ratio, $3 + O(1/\alpha)$, for locally polyhedral costs, where $\alpha$ measures the “steepness” of the costs. We also prove bounds on the dynamic regret of OBD when the balance is performed in the dual space that are dimension-free and imply that OBD has sublinear static regret.} }
Endnote
%0 Conference Paper %T Smoothed Online Convex Optimization in High Dimensions via Online Balanced Descent %A Niangjun Chen %A Gautam Goel %A Adam Wierman %B Proceedings of the 31st Conference On Learning Theory %C Proceedings of Machine Learning Research %D 2018 %E Sébastien Bubeck %E Vianney Perchet %E Philippe Rigollet %F pmlr-v75-chen18b %I PMLR %P 1574--1594 %U https://proceedings.mlr.press/v75/chen18b.html %V 75 %X We study \emph{smoothed online convex optimization}, a version of online convex optimization where the learner incurs a penalty for changing her actions between rounds. Given a $\Omega(\sqrt{d})$ lower bound on the competitive ratio of any online algorithm, where $d$ is the dimension of the action space, we ask under what conditions this bound can be beaten. We introduce a novel algorithmic framework for this problem, Online Balanced Descent (OBD), which works by iteratively projecting the previous point onto a carefully chosen level set of the current cost function so as to balance the switching costs and hitting costs. We demonstrate the generality of the OBD framework by showing how, with different choices of “balance,” OBD can improve upon state-of-the-art performance guarantees for both competitive ratio and regret; in particular, OBD is the first algorithm to achieve a dimension-free competitive ratio, $3 + O(1/\alpha)$, for locally polyhedral costs, where $\alpha$ measures the “steepness” of the costs. We also prove bounds on the dynamic regret of OBD when the balance is performed in the dual space that are dimension-free and imply that OBD has sublinear static regret.
APA
Chen, N., Goel, G. & Wierman, A.. (2018). Smoothed Online Convex Optimization in High Dimensions via Online Balanced Descent. Proceedings of the 31st Conference On Learning Theory, in Proceedings of Machine Learning Research 75:1574-1594 Available from https://proceedings.mlr.press/v75/chen18b.html.

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