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<p>We investigate the multi-objective online convex optimization (MO-OCO), where the goal is to minimize the cumulative vector-valued loss with respect to the Pareto front, serving as the optimal reference set in the objective space. MO-OCO presents unique challenges due to concept drift and conflicting objectives over time. To address these challenges, we propose a novel reformulation of the MO-OCO problem via a quadratic distance minimizing problem, anchored at a predefined utopian point. This approach preserves the original objective structure without introducing auxiliary variables. Based on this formulation, we develop a new MO-OCO algorithm, the Online Utopian Point (OUP) algorithm, which guides sequential decision-making toward the utopian point representing idealized performance across all objectives. Under standard assumptions of convexity and boundedness, we prove that the OUP algorithm achieves an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \mathcal {O}(\sqrt{T}) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msqrt> <mi>T</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> multi-objective regret, implying that the time-averaged loss converges sublinearly to the Pareto front. Experimental results on two real-world convex learning tasks, linear regression and logistic regression, demonstrate that OUP consistently outperforms baseline methods in terms of convergence rate, adaptability, and computational efficiency.</p>

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Multi-objective online convex optimization via quadratic distance reformulation with utopian anchoring

  • 期刊论文

  • 发表日期：

    2026年2月20日

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  • Jieyuan Guo,
  • Lizhen Shao,
  • Fangyuan Zhao

摘要

We investigate the multi-objective online convex optimization (MO-OCO), where the goal is to minimize the cumulative vector-valued loss with respect to the Pareto front, serving as the optimal reference set in the objective space. MO-OCO presents unique challenges due to concept drift and conflicting objectives over time. To address these challenges, we propose a novel reformulation of the MO-OCO problem via a quadratic distance minimizing problem, anchored at a predefined utopian point. This approach preserves the original objective structure without introducing auxiliary variables. Based on this formulation, we develop a new MO-OCO algorithm, the Online Utopian Point (OUP) algorithm, which guides sequential decision-making toward the utopian point representing idealized performance across all objectives. Under standard assumptions of convexity and boundedness, we prove that the OUP algorithm achieves an \( \mathcal {O}(\sqrt{T}) \) O ( T ) multi-objective regret, implying that the time-averaged loss converges sublinearly to the Pareto front. Experimental results on two real-world convex learning tasks, linear regression and logistic regression, demonstrate that OUP consistently outperforms baseline methods in terms of convergence rate, adaptability, and computational efficiency.