发表机构
Centro de Modelamiento Matemático; Universidad de Chile; Department of Mathematics, University of Alicante（数学建模中心; 智利大学; 阿利坎特大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出满足Kurdyka-Łojasiewicz条件的邻近随机次梯度方法，可处理更通用非凸随机优化问题，保证轨迹收敛性并导出多项式收敛速率。
AI 中文摘要
本工作提出一种邻近随机次梯度方法，用于最小化期望代价（其被积函数可能非光滑且非凸）与一个下半连续、邻近有界函数的和。我们针对决策变量中满足非光滑局部化下降引理变体的广泛类被积函数，该结构假设同时涵盖具有Lipschitz梯度的光滑损失以及此类损失与凸函数的差。每次迭代中，期望代价由逐步精细化的样本均值替代，邻近次梯度步长通过Armijo型线搜索选取，该搜索需满足由基于样本的近似诱导的随机误差下的充分下降性质。该框架比现有方法能处理更通用的问题形式，尤其既不需要正则项的（弱）凸性，也不需要随机神谕的方差有统一界，且我们的分析给出了甚至在光滑设置下也是新的收敛保证。具体而言，在样本量序列仅为非递减且无界、无规定增长率的宽松要求下，我们建立了函数值序列的几乎必然收敛及轨迹每一个聚点的平稳性。利用Kurdyka-Łojasiewicz（KL）性质，我们进一步将该子序列保证升级为整个轨迹收敛到单个平稳点。最后，针对指数型KL去奇异函数和多项式增长的样本量，我们导出了函数值和迭代点的显式多项式收敛速率，带有一个对数因子。
英文摘要
This work introduces a proximal stochastic subgradient method for minimizing the sum of an expected cost, whose integrand is potentially nonsmooth and nonconvex, and a lower semicontinuous, prox-bounded function. We target a broad class of integrands obeying a nonsmooth, localized variant of the descent lemma in the decision variable, a structural assumption that simultaneously covers smooth losses with Lipschitz gradient and differences of such losses with convex functions. At each iteration the expected cost is replaced by a sample average that is progressively refined, and the proximal-subgradient stepsize is selected by an Armijo-type line search enforcing a sufficient-decrease property up to stochastic errors induced by the sample-based approximation. This framework accommodates substantially more general problem formulations than existing methods, in particular, it requires neither (weak) convexity of the regularizer nor a uniform bound on the variance of the stochastic oracle, and our analysis yields convergence guarantees that are new even in the smooth setting. Specifically, we establish almost sure convergence of the sequence of function values and stationarity of every accumulation point of the trajectories under the relaxed requirement that the sample-size sequence be merely nondecreasing and unbounded, with no prescribed growth rate. Leveraging the Kurdyka-Lojasiewicz (KL) property, we further upgrade this subsequential guarantee to convergence of the whole trajectory to a single stationary point. Finally, for exponential-type KL desingularizing functions and polynomially growing sample sizes, we derive explicit polynomial convergence rates, up to a logarithmic factor, for both the function values and the iterates.
