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\section{Conclusion} 
\label{sec: conclusion}

In this paper, we proposed \algns, a new variance-reduction-based stochastic multi-gradient-based algorithm to achieve fast convergence and low sample complexity multi-objective optimization (MOO). 
We analyze its Pareto stationarity convergence and sample complexity under non-convex and strongly convex settings. To enhance empirical convergence, we propose \algm, which incorporates momentum. To reduce the periodic full multi-gradient evaluation in \alg and \algmns, we introduce adaptive batching versions, \algpns/\algmpns, with theoretical performance analysis. Overall, our \alg algorithm family advances MOO algorithm design and analysis.


\section{Acknowledgement}
This work is supported in part by NSF grants CAREER CNS-2110259, CNS-2112471, IIS-2324052, DARPA YFA D24AP00265, DARPA HR0011-25-2-0019, ONR grant N00014-24-1-2729, and AFRL grant PGSC-SC-111374-19s.



%We theoretically analyzed the Pareto stationarity convergence and sample complexity of our proposed \alg algorithms under non-convex and strongly convex settings.
%To further enhance the empirical Pareto stationarity convergence of \algns, we further proposed an algorithm called \algm, which incorporates momentum information to expedite convergence. 
%To alleviate the periodic full multi-gradient evaluation requirement in \alg and \algmns, we further proposed enhanced versions for both algorithms with adaptive batching called \algpns/ \algmpns. 
%Further, we provided theoretical performance analysis for all enhanced algorithms of \algns.
%In conclusion, our proposed \alg algorithm family advances the state of the art of MOO algorithm design and analysis.
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\section{Broader Impact}

In engineering disciplines and operations research, MOO algorithm plays a crucial role in solving problems where trade-offs between conflicting objectives. 
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Our paper introduces \alg family, groundbreaking algorithms that significantly enhance the field of multi-objective optimization (MOO) by improving training sample complexity. 
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Such progress lays a robust foundation for future research and practical applications, potentially guiding the development of more sophisticated machine-learning models and optimizing engineering solutions. 
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