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　Φ(x):[0,1]n→R+　which　satisfies　the　inequality　12⟨uT∇2Φ(x),u⟩≤σ·‖u‖1‖x‖1⟨u,∇Φ(x)⟩　for　any　x,u∈[0,1]n,x≠0.　This　objective　function　is　named　as　one　sided　σ-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((θ-1)(θ/(1+θ))2σ))-　approximation　algorithm　with　O(T)　regret　upper　bound,　where　T　is　the　number　of　rounds　in　the　online　algorithm　and　θ,σ∈R+　are　parameters.　Secondly,　if　the　gradient　function　of　OSS　function　has　no　L-smoothness,　we　provide　an　1+((θ+1)/θ)4σ-1-approximation　projected　gradient　algorithm,　and　prove　that　the　regret　upper　bound　of　the　algorithm　is　O(T).　We　think　that　this　research　can　provide　different　ideas　for　online　non-convex　and　non-submodular　learning.　©　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature　2024.