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<p>The ProbAbilistic Gradient Estimator algorithm (PAGE), a stochastic algorithm introduced by Li et al. in 2021, was designed to find stationary points for the average of smooth nonconvex functions. In this work, we study PAGE within the broad framework of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-weakly convex functions, providing a continuous interpolation between the general nonconvex <i>L</i>-smooth regime (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\tau =L\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>=</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation>) and the convex regime (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\tau =0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>). We establish new convergence rates for PAGE, showing that its complexity improves as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> decreases.</p>

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Convergence Analysis of the ProbAbilistic Gradient Estimator Algorithm for Weakly Convex Finite-Sum Optimization

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月28日

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  • Laurent Condat,
  • Peter Richtárik

摘要

The ProbAbilistic Gradient Estimator algorithm (PAGE), a stochastic algorithm introduced by Li et al. in 2021, was designed to find stationary points for the average of smooth nonconvex functions. In this work, we study PAGE within the broad framework of \(\tau \) τ -weakly convex functions, providing a continuous interpolation between the general nonconvex L-smooth regime ( \(\tau =L\) τ = L ) and the convex regime ( \(\tau =0\) τ = 0 ). We establish new convergence rates for PAGE, showing that its complexity improves as \(\tau \) τ decreases.