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<p>Nonconvex minimax problems frequently arise in machine learning, distributionally robust optimization, and many other research fields. In this paper, we propose a Stochastic Alternating Mirror Descent Ascent with Momentum (SAMDAM) algorithm to solve nonconvex-strongly concave minimax optimization problems. SAMDAM employs simple mirror descent ascent steps along with momentum acceleration to update the variables <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2563_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(x\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2563_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(y\)</EquationSource> </InlineEquation> alternately at each iteration. We further prove that SAMDAM achieves a gradient complexity of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2563_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal{O}(\kappa^{3}\epsilon^{-4})\)</EquationSource> </InlineEquation> for finding an <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2563_Article_IEq4.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">\(\epsilon\)</EquationSource> </InlineEquation>-stationary point in stochastic nonconvex settings, where <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2563_Article_IEq5.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\kappa\)</EquationSource> </InlineEquation> represents the condition number of the problem. Finally, computational experiments demonstrate that SAMDAM outperforms several state-of-the-art algorithms in distributionally robust optimization and fair classification tasks.</p>

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Accelerated stochastic alternating mirror descent ascent algorithm for nonconvex-strongly concave minimax problems

  • 期刊论文

  • 发表日期：

    2025年6月22日

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  • Lulu Zhao,
  • Yue Liu,
  • Yong-Jin Liu

摘要

Nonconvex minimax problems frequently arise in machine learning, distributionally robust optimization, and many other research fields. In this paper, we propose a Stochastic Alternating Mirror Descent Ascent with Momentum (SAMDAM) algorithm to solve nonconvex-strongly concave minimax optimization problems. SAMDAM employs simple mirror descent ascent steps along with momentum acceleration to update the variables \(x\) and \(y\) alternately at each iteration. We further prove that SAMDAM achieves a gradient complexity of \(\mathcal{O}(\kappa^{3}\epsilon^{-4})\) for finding an \(\epsilon\) -stationary point in stochastic nonconvex settings, where \(\kappa\) represents the condition number of the problem. Finally, computational experiments demonstrate that SAMDAM outperforms several state-of-the-art algorithms in distributionally robust optimization and fair classification tasks.