SIAM Journal on Control and Optimization, Volume 59, Issue 3, Page 2301-2320, January 2021.
We propose a single time-scale stochastic subgradient method for constrained optimization of a composition of several nonsmooth and nonconvex functions. The functions are assumed to be locally Lipschitz and differentiable in a generalized sense. Only stochastic estimates of the values and generalized derivatives of the functions are used. The method is parameter-free. We prove convergence with probability one of the method, by associating with it a system of differential inclusions and devising a nondifferentiable Lyapunov function for this system. For problems with functions having Lipschitz continuous derivatives, the method finds a point satisfying an optimality measure with error of order $1/\sqrt{N}$, after executing $N$ iterations with constant stepsize.

中文翻译：

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非光滑非凸多级组合优化的一种随机次梯度方法

SIAM Journal on Control and Optimization，第 59 卷，第 3 期，第 2301-2320 页，2021 年 1 月。
我们提出了一种单一时间尺度随机次梯度方法，用于对几个非光滑和非凸函数的组合进行约束优化。这些函数被假定为局部 Lipschitz 并且在广义上是可微的。仅使用函数的值和广义导数的随机估计。该方法是无参数的。我们通过将微分包含系统与其相关联并为该系统设计不可微的李雅普诺夫函数来证明该方法的概率之一收敛。对于具有 Lipschitz 连续导数的函数的问题，该方法在以恒定步长执行 $N$ 次迭代后，找到满足最优性度量的点，误差为 $1/\sqrt{N}$ 阶。