TY - JOUR

T1 - An Online Semi-Definite Programming with a Generalized Log-Determinant Regularizer and Its Applications

AU - Liu, Yaxiong

AU - Moridomi, Ken Ichiro

AU - Hatano, Kohei

AU - Takimoto, Eiji

N1 - Funding Information:
Funding: This research was funded by JSPS KAKENHI Grant Numbers JP19H04174, JP19H04067 and JP20H05967, respectively.
Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/4/1

Y1 - 2022/4/1

N2 - We consider a variant of the online semi-definite programming problem (OSDP). Specif-ically, in our problem, the setting of the decision space is a set of positive semi-definite matrices constrained by two norms in parallel: the L∞ norm to the diagonal entries and the Γ-trace norm, which is a generalized trace norm with a positive definite matrix Γ. Our setting recovers the original one when Γ is an identity matrix. To solve this problem, we design a follow-the-regularized-leader algorithm with a Γ-dependent regularizer, which also generalizes the log-determinant function. Next, we focus on online binary matrix completion (OBMC) with side information and online similarity prediction with side information. By reducing to the OSDP framework and applying our proposed algorithm, we remove the logarithmic factors in the previous mistake bound of the above two prob-lems. In particular, for OBMC, our bound is optimal. Furthermore, our result implies a better offline generalization bound for the algorithm, which is similar to those of SVMs with the best kernel, if the side information is involved in advance.

AB - We consider a variant of the online semi-definite programming problem (OSDP). Specif-ically, in our problem, the setting of the decision space is a set of positive semi-definite matrices constrained by two norms in parallel: the L∞ norm to the diagonal entries and the Γ-trace norm, which is a generalized trace norm with a positive definite matrix Γ. Our setting recovers the original one when Γ is an identity matrix. To solve this problem, we design a follow-the-regularized-leader algorithm with a Γ-dependent regularizer, which also generalizes the log-determinant function. Next, we focus on online binary matrix completion (OBMC) with side information and online similarity prediction with side information. By reducing to the OSDP framework and applying our proposed algorithm, we remove the logarithmic factors in the previous mistake bound of the above two prob-lems. In particular, for OBMC, our bound is optimal. Furthermore, our result implies a better offline generalization bound for the algorithm, which is similar to those of SVMs with the best kernel, if the side information is involved in advance.

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U2 - 10.3390/math10071055

DO - 10.3390/math10071055

M3 - Article

AN - SCOPUS:85127799411

SN - 2227-7390

VL - 10

JO - Mathematics

JF - Mathematics

IS - 7

M1 - 1055

ER -