Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond

Dheeraj Baby, Yu-Xiang Wang
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:1805-1845, 2022.

Abstract

We study the framework of universal dynamic regret minimization with strongly convex losses. We answer an open problem in Baby and Wang 2021 by showing that in a proper learning setup, Strongly Adaptive algorithms can achieve the near optimal dynamic regret of $\tilde O(d^{1/3} n^{1/3}\text{TV}[u_{1:n}]^{2/3} \vee d)$ against any comparator sequence $u_1,\ldots,u_n$ simultaneously, where $n$ is the time horizon and $\text{TV}[u_{1:n}]$ is the Total Variation of comparator. These results are facilitated by exploiting a number of new structures imposed by the KKT conditions that were not considered in Baby and Wang 2021 which also lead to other improvements over their results such as: (a) handling non-smooth losses and (b) improving the dimension dependence on regret. Further, we also derive near optimal dynamic regret rates for the special case of proper online learning with exp-concave losses and an $L_\infty$ constrained decision set.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-baby22a, title = { Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond }, author = {Baby, Dheeraj and Wang, Yu-Xiang}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {1805--1845}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/baby22a/baby22a.pdf}, url = {https://proceedings.mlr.press/v151/baby22a.html}, abstract = { We study the framework of universal dynamic regret minimization with strongly convex losses. We answer an open problem in Baby and Wang 2021 by showing that in a proper learning setup, Strongly Adaptive algorithms can achieve the near optimal dynamic regret of $\tilde O(d^{1/3} n^{1/3}\text{TV}[u_{1:n}]^{2/3} \vee d)$ against any comparator sequence $u_1,\ldots,u_n$ simultaneously, where $n$ is the time horizon and $\text{TV}[u_{1:n}]$ is the Total Variation of comparator. These results are facilitated by exploiting a number of new structures imposed by the KKT conditions that were not considered in Baby and Wang 2021 which also lead to other improvements over their results such as: (a) handling non-smooth losses and (b) improving the dimension dependence on regret. Further, we also derive near optimal dynamic regret rates for the special case of proper online learning with exp-concave losses and an $L_\infty$ constrained decision set. } }
Endnote
%0 Conference Paper %T Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond %A Dheeraj Baby %A Yu-Xiang Wang %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-baby22a %I PMLR %P 1805--1845 %U https://proceedings.mlr.press/v151/baby22a.html %V 151 %X We study the framework of universal dynamic regret minimization with strongly convex losses. We answer an open problem in Baby and Wang 2021 by showing that in a proper learning setup, Strongly Adaptive algorithms can achieve the near optimal dynamic regret of $\tilde O(d^{1/3} n^{1/3}\text{TV}[u_{1:n}]^{2/3} \vee d)$ against any comparator sequence $u_1,\ldots,u_n$ simultaneously, where $n$ is the time horizon and $\text{TV}[u_{1:n}]$ is the Total Variation of comparator. These results are facilitated by exploiting a number of new structures imposed by the KKT conditions that were not considered in Baby and Wang 2021 which also lead to other improvements over their results such as: (a) handling non-smooth losses and (b) improving the dimension dependence on regret. Further, we also derive near optimal dynamic regret rates for the special case of proper online learning with exp-concave losses and an $L_\infty$ constrained decision set.
APA
Baby, D. & Wang, Y.. (2022). Optimal Dynamic Regret in Proper Online Learning with Strongly Convex Losses and Beyond . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:1805-1845 Available from https://proceedings.mlr.press/v151/baby22a.html.

Related Material