Abstract
The online optimization model was first introduced in the research of machine learning problems (Zinkevich, Proceedings of ICML, 928–936, 2003). It is a powerful framework that combines the principles of optimization with the challenges of online decision-making. The present research mainly consider the case that the reveal objective functions are convex or submodular. In this paper, we focus on the online maximization problem under a special objective function \(\varPhi (x):[0,1]^n\rightarrow \mathbb {R}_{+}\) which satisfies the inequality \(\frac{1}{2}\langle u^{T}\nabla ^{2}\varPhi (x),u\rangle \le \sigma \cdot \frac{\Vert u\Vert _{1}}{\Vert x\Vert _{1}}\langle u,\nabla \varPhi (x)\rangle \) for any \(x,u\in [0,1]^n, x\ne 0\). This objective function is named as one sided \(\sigma \)-smooth (OSS) function. We achieve two conclusions here. Firstly, under the assumption that the gradient function of OSS function is L-smooth, we propose an \((1-\exp ((\theta -1)(\theta /(1+\theta ))^{2\sigma }))\)- approximation algorithm with \(O(\sqrt{T})\) regret upper bound, where T is the number of rounds in the online algorithm and \(\theta , \sigma \in \mathbb {R}_{+}\) are parameters. Secondly, if the gradient function of OSS function has no L-smoothness, we provide an \(\left( 1+((\theta +1)/\theta )^{4\sigma }\right) ^{-1}\)-approximation projected gradient algorithm, and prove that the regret upper bound of the algorithm is \(O(\sqrt{T})\). We think that this research can provide different ideas for online non-convex and non-submodular learning.
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Acknowledgements
The first, second, and fourth authors are supported by National Natural Science Foundation of China (No. 12131003). The third author is supported by National Natural Science Foundation of China (No. 11201333).
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This research was supported by NNSF of China (Grant Nos. 12131003 and 11201333).
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This is an extended version of a paper Zhang et al. (2022) presented in the the 26th International Computing and Combinatorics Conference.
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Zhang, H., Xu, D., Gai, L. et al. Online learning under one sided \(\sigma \)-smooth function. J Comb Optim 47, 76 (2024). https://doi.org/10.1007/s10878-024-01174-2
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DOI: https://doi.org/10.1007/s10878-024-01174-2