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

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

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. 1. 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 $\widetilde{\text{EG}}$. 2. 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. 3. 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.

Suggested Citation

  • Chung-En Tsai & Hao-Chung Cheng & Yen-Huan Li, 2022. "Online Self-Concordant and Relatively Smooth Minimization, With Applications to Online Portfolio Selection and Learning Quantum States," Papers 2210.00997, arXiv.org, revised Sep 2023.
  • Handle: RePEc:arx:papers:2210.00997
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    References listed on IDEAS

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    1. Haihao Lu & Robert M. Freund & Yurii Nesterov, 2018. "Relatively smooth convex optimization by first-order methods, and applications," LIDAM Reprints CORE 2965, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, December.
    3. David P. Helmbold & Robert E. Schapire & Yoram Singer & Manfred K. Warmuth, 1998. "On‐Line Portfolio Selection Using Multiplicative Updates," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 325-347, October.
    4. Thomas M. Cover, 1991. "Universal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 1-29, January.
    5. NESTEROV, Yurii, 2011. "Barrier subgradient method," LIDAM Reprints CORE 2359, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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