Faster Gradient-Free Algorithms for Nonsmooth Nonconvex Stochastic Optimization

• URL: http://arxiv.org/abs/2301.06428v3
• Date: Tue, 14 May 2024 10:47:26 GMT
• Title: Faster Gradient-Free Algorithms for Nonsmooth Nonconvex Stochastic Optimization
• Authors: Lesi Chen, Jing Xu, Luo Luo,
• Abstract summary: We consider the problem of the form $min_x in mathbbRd f(x) triangleq mathbbE_xi [Fxi]$inf(x)$ Lipschitz. The recently proposed gradient-free method requires at most $mathcalO( L4 d3/2 epsilon-4 + Goldstein L d3/2 delta-1 epsilon-4)$ zeroth-order complexity
• Score: 20.54801745090522
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the optimization problem of the form $\min_{x \in \mathbb{R}^d} f(x) \triangleq \mathbb{E}_{\xi} [F(x; \xi)]$, where the component $F(x;\xi)$ is $L$-mean-squared Lipschitz but possibly nonconvex and nonsmooth. The recently proposed gradient-free method requires at most $\mathcal{O}( L^4 d^{3/2} \epsilon^{-4} + \Delta L^3 d^{3/2} \delta^{-1} \epsilon^{-4})$ stochastic zeroth-order oracle complexity to find a $(\delta,\epsilon)$-Goldstein stationary point of objective function, where $\Delta = f(x_0) - \inf_{x \in \mathbb{R}^d} f(x)$ and $x_0$ is the initial point of the algorithm. This paper proposes a more efficient algorithm using stochastic recursive gradient estimators, which improves the complexity to $\mathcal{O}(L^3 d^{3/2} \epsilon^{-3}+ \Delta L^2 d^{3/2} \delta^{-1} \epsilon^{-3})$.

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