PDE-Based Optimal Strategy for Unconstrained Online Learning

Zhiyu Zhang, Ashok Cutkosky, Ioannis Paschalidis
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:26085-26115, 2022.

Abstract

Unconstrained Online Linear Optimization (OLO) is a practical problem setting to study the training of machine learning models. Existing works proposed a number of potential-based algorithms, but in general the design of these potential functions relies heavily on guessing. To streamline this workflow, we present a framework that generates new potential functions by solving a Partial Differential Equation (PDE). Specifically, when losses are 1-Lipschitz, our framework produces a novel algorithm with anytime regret bound C√T+||u||√2T[√log(1+||u||/C)+2], where C is a user-specified constant and u is any comparator unknown and unbounded a priori. Such a bound attains an optimal loss-regret trade-off without the impractical doubling trick. Moreover, a matching lower bound shows that the leading order term, including the constant multiplier √2, is tight. To our knowledge, the proposed algorithm is the first to achieve such optimalities.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-zhang22d, title = {{PDE}-Based Optimal Strategy for Unconstrained Online Learning}, author = {Zhang, Zhiyu and Cutkosky, Ashok and Paschalidis, Ioannis}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {26085--26115}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/zhang22d/zhang22d.pdf}, url = {https://proceedings.mlr.press/v162/zhang22d.html}, abstract = {Unconstrained Online Linear Optimization (OLO) is a practical problem setting to study the training of machine learning models. Existing works proposed a number of potential-based algorithms, but in general the design of these potential functions relies heavily on guessing. To streamline this workflow, we present a framework that generates new potential functions by solving a Partial Differential Equation (PDE). Specifically, when losses are 1-Lipschitz, our framework produces a novel algorithm with anytime regret bound $C\sqrt{T}+||u||\sqrt{2T}[\sqrt{\log(1+||u||/C)}+2]$, where $C$ is a user-specified constant and $u$ is any comparator unknown and unbounded a priori. Such a bound attains an optimal loss-regret trade-off without the impractical doubling trick. Moreover, a matching lower bound shows that the leading order term, including the constant multiplier $\sqrt{2}$, is tight. To our knowledge, the proposed algorithm is the first to achieve such optimalities.} }
Endnote
%0 Conference Paper %T PDE-Based Optimal Strategy for Unconstrained Online Learning %A Zhiyu Zhang %A Ashok Cutkosky %A Ioannis Paschalidis %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-zhang22d %I PMLR %P 26085--26115 %U https://proceedings.mlr.press/v162/zhang22d.html %V 162 %X Unconstrained Online Linear Optimization (OLO) is a practical problem setting to study the training of machine learning models. Existing works proposed a number of potential-based algorithms, but in general the design of these potential functions relies heavily on guessing. To streamline this workflow, we present a framework that generates new potential functions by solving a Partial Differential Equation (PDE). Specifically, when losses are 1-Lipschitz, our framework produces a novel algorithm with anytime regret bound $C\sqrt{T}+||u||\sqrt{2T}[\sqrt{\log(1+||u||/C)}+2]$, where $C$ is a user-specified constant and $u$ is any comparator unknown and unbounded a priori. Such a bound attains an optimal loss-regret trade-off without the impractical doubling trick. Moreover, a matching lower bound shows that the leading order term, including the constant multiplier $\sqrt{2}$, is tight. To our knowledge, the proposed algorithm is the first to achieve such optimalities.
APA
Zhang, Z., Cutkosky, A. & Paschalidis, I.. (2022). PDE-Based Optimal Strategy for Unconstrained Online Learning. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:26085-26115 Available from https://proceedings.mlr.press/v162/zhang22d.html.

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