Online learning under one sided $$\sigma $$ -smooth function

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.