Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States

Chung-En Tsai; Hao-Chung Cheng; Yen-Huan Li

Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States

Chung-En Tsai, Hao-Chung Cheng, Yen-Huan Li

Proceedings of The 34th International Conference on Algorithmic Learning Theory, PMLR 201:1481-1483, 2023.

Abstract

Consider an online convex optimization problem where the loss functions are self-concordant barriers, smooth relative to a convex function

$h$ , and possibly non-Lipschitz. We analyze the regret of online mirror descent with

$h$ . Then, based on the result, we prove the following in a unified manner. Denote by

$T$ the time horizon and

$d$ the parameter dimension. - For online portfolio selection, the regret of

$\widetilde{\text{EG}}$ , a variant of exponentiated gradient due to Helmbold et al., is

$\tilde{O} ( T^{2/3} d^{1/3} )$ when

$T > 4 d / \log d$ . This improves on the original

$\tilde{O} ( T^{3/4} d^{1/2} )$ regret bound for

$\widetilde{\text{EG}}$ . - For online portfolio selection, the regret of online mirror descent with the logarithmic barrier is

$\tilde{O}(\sqrt{T d})$ . The regret bound is the same as that of Soft-Bayes due to Orseau et al. up to logarithmic terms. - For online learning quantum states with the logarithmic loss, the regret of online mirror descent with the log-determinant function is also

$\tilde{O} ( \sqrt{T d} )$ . Its per-iteration time is shorter than all existing algorithms we know.

Cite this Paper

BibTeX


@InProceedings{pmlr-v201-tsai23a,
  title = 	 {Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States},
  author =       {Tsai, Chung-En and Cheng, Hao-Chung and Li, Yen-Huan},
  booktitle = 	 {Proceedings of The 34th International Conference on Algorithmic Learning Theory},
  pages = 	 {1481--1483},
  year = 	 {2023},
  editor = 	 {Agrawal, Shipra and Orabona, Francesco},
  volume = 	 {201},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {20 Feb--23 Feb},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v201/tsai23a/tsai23a.pdf},
  url = 	 {https://proceedings.mlr.press/v201/tsai23a.html},
  abstract = 	 {Consider an online convex optimization problem where the loss functions are self-concordant barriers, smooth relative to a convex function $h$, and possibly non-Lipschitz. We analyze the regret of online mirror descent with $h$. Then, based on the result, we prove the following in a unified manner. Denote by $T$ the time horizon and $d$ the parameter dimension. - For online portfolio selection, the regret of $\widetilde{\text{EG}}$, a variant of exponentiated gradient due to Helmbold et al., is $\tilde{O} ( T^{2/3} d^{1/3} )$ when $T > 4 d / \log d$. This improves on the original $\tilde{O} ( T^{3/4} d^{1/2} )$ regret bound for $\widetilde{\text{EG}}$. - For online portfolio selection, the regret of online mirror descent with the logarithmic barrier is $\tilde{O}(\sqrt{T d})$. The regret bound is the same as that of Soft-Bayes due to Orseau et al. up to logarithmic terms. - For online learning quantum states with the logarithmic loss, the regret of online mirror descent with the log-determinant function is also $\tilde{O} ( \sqrt{T d} )$. Its per-iteration time is shorter than all existing algorithms we know.}
}

Endnote

%0 Conference Paper
%T Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States
%A Chung-En Tsai
%A Hao-Chung Cheng
%A Yen-Huan Li
%B Proceedings of The 34th International Conference on Algorithmic Learning Theory
%C Proceedings of Machine Learning Research
%D 2023
%E Shipra Agrawal
%E Francesco Orabona	
%F pmlr-v201-tsai23a
%I PMLR
%P 1481--1483
%U https://proceedings.mlr.press/v201/tsai23a.html
%V 201
%X Consider an online convex optimization problem where the loss functions are self-concordant barriers, smooth relative to a convex function $h$, and possibly non-Lipschitz. We analyze the regret of online mirror descent with $h$. Then, based on the result, we prove the following in a unified manner. Denote by $T$ the time horizon and $d$ the parameter dimension. - For online portfolio selection, the regret of $\widetilde{\text{EG}}$, a variant of exponentiated gradient due to Helmbold et al., is $\tilde{O} ( T^{2/3} d^{1/3} )$ when $T > 4 d / \log d$. This improves on the original $\tilde{O} ( T^{3/4} d^{1/2} )$ regret bound for $\widetilde{\text{EG}}$. - For online portfolio selection, the regret of online mirror descent with the logarithmic barrier is $\tilde{O}(\sqrt{T d})$. The regret bound is the same as that of Soft-Bayes due to Orseau et al. up to logarithmic terms. - For online learning quantum states with the logarithmic loss, the regret of online mirror descent with the log-determinant function is also $\tilde{O} ( \sqrt{T d} )$. Its per-iteration time is shorter than all existing algorithms we know.

APA


Tsai, C., Cheng, H. & Li, Y.. (2023). Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States. Proceedings of The 34th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 201:1481-1483 Available from https://proceedings.mlr.press/v201/tsai23a.html.

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