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<p>In this paper, we consider a distributed stochastic composite optimization problem with infinite variance, taking the form ‘smooth/nonsmooth + nonsmooth’, where (sub-)gradient information may be unavailable. We present a mini-batch zeroth-order proximal clipped gradient algorithm with shifts, which utilizes the well-known Gaussian smoothing technique to yield unbiased zeroth-order gradient estimators of the surrogate problem. The proposed algorithm employs the clipping gradient to tackle the infinite variance noise, which is one of the keys to deriving good high-probability guarantees. Under the convexity assumption on both terms and other moderate conditions, we derive high probability bounds for the gap between the iterative loss function values and optimal objective values for the proposed algorithm. Furthermore, under only the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3066_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\((\mu , x^*)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-quasi-strong convexity (may be nonconvex) assumption on the first term, we established high probability bounds for the distance between iteration points and the stationary point. Finally, the proposed scheme is numerically compared with some existing zeroth-order methods as well as a stochastic subgradient algorithm on a phase retrieval problem and an online absolute linear regression task.</p>

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Zeroth-order Proximal Clipped Gradient Method with Shifts for Distributed Stochastic Composite Optimization Problems with Infinite Variance

  • 期刊论文

  • 发表日期：

    2025年9月22日

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  • Zhen-Ping Yang,
  • Pin-Bo Chen,
  • Yong Zhao,
  • Lin Chen

摘要

In this paper, we consider a distributed stochastic composite optimization problem with infinite variance, taking the form ‘smooth/nonsmooth + nonsmooth’, where (sub-)gradient information may be unavailable. We present a mini-batch zeroth-order proximal clipped gradient algorithm with shifts, which utilizes the well-known Gaussian smoothing technique to yield unbiased zeroth-order gradient estimators of the surrogate problem. The proposed algorithm employs the clipping gradient to tackle the infinite variance noise, which is one of the keys to deriving good high-probability guarantees. Under the convexity assumption on both terms and other moderate conditions, we derive high probability bounds for the gap between the iterative loss function values and optimal objective values for the proposed algorithm. Furthermore, under only the \((\mu , x^*)\) ( μ , x ∗ ) -quasi-strong convexity (may be nonconvex) assumption on the first term, we established high probability bounds for the distance between iteration points and the stationary point. Finally, the proposed scheme is numerically compared with some existing zeroth-order methods as well as a stochastic subgradient algorithm on a phase retrieval problem and an online absolute linear regression task.