Online Learning Quantum States with the Logarithmic Loss via VB-FTRL
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910822562791424 |
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| author | Tseng, Wei-Fu Chen, Kai-Chun Xiao, Zi-Hong Li, Yen-Huan |
| author_facet | Tseng, Wei-Fu Chen, Kai-Chun Xiao, Zi-Hong Li, Yen-Huan |
| contents | Online learning of quantum states with the logarithmic loss (LL-OLQS) is a quantum generalization of online portfolio selection (OPS), a classic open problem in online learning for over three decades. This problem also emerges in designing stochastic optimization algorithms for maximum-likelihood quantum state tomography. Recently, Jezequel et al. (arXiv:2209.13932) proposed the VB-FTRL algorithm, the first regret-optimal algorithm for OPS with moderate computational complexity. In this paper, we generalize VB-FTRL for LL-OLQS. Let $d$ denote the dimension and $T$ the number of rounds. The generalized algorithm achieves a regret rate of $O ( d^2 \log ( d + T ) )$ for LL-OLQS. Each iteration of the algorithm consists of solving a semidefinite program that can be implemented in polynomial time by, for example, cutting-plane methods. For comparison, the best-known regret rate for LL-OLQS is currently $O ( d^2 \log T )$, achieved by an exponential weight method. However, no explicit implementation is available for the exponential weight method for LL-OLQS. To facilitate the generalization, we introduce the notion of VB-convexity. VB-convexity is a sufficient condition for the volumetric barrier associated with any function to be convex and is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04237 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Online Learning Quantum States with the Logarithmic Loss via VB-FTRL Tseng, Wei-Fu Chen, Kai-Chun Xiao, Zi-Hong Li, Yen-Huan Quantum Physics Machine Learning Optimization and Control Online learning of quantum states with the logarithmic loss (LL-OLQS) is a quantum generalization of online portfolio selection (OPS), a classic open problem in online learning for over three decades. This problem also emerges in designing stochastic optimization algorithms for maximum-likelihood quantum state tomography. Recently, Jezequel et al. (arXiv:2209.13932) proposed the VB-FTRL algorithm, the first regret-optimal algorithm for OPS with moderate computational complexity. In this paper, we generalize VB-FTRL for LL-OLQS. Let $d$ denote the dimension and $T$ the number of rounds. The generalized algorithm achieves a regret rate of $O ( d^2 \log ( d + T ) )$ for LL-OLQS. Each iteration of the algorithm consists of solving a semidefinite program that can be implemented in polynomial time by, for example, cutting-plane methods. For comparison, the best-known regret rate for LL-OLQS is currently $O ( d^2 \log T )$, achieved by an exponential weight method. However, no explicit implementation is available for the exponential weight method for LL-OLQS. To facilitate the generalization, we introduce the notion of VB-convexity. VB-convexity is a sufficient condition for the volumetric barrier associated with any function to be convex and is of independent interest. |
| title | Online Learning Quantum States with the Logarithmic Loss via VB-FTRL |
| topic | Quantum Physics Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2311.04237 |