Bregman Linearized Augmented Lagrangian Method for Nonconvex Constrained Stochastic Zeroth-order Optimization

• URL: http://arxiv.org/abs/2504.09409v1
• Date: Sun, 13 Apr 2025 02:44:47 GMT
• Title: Bregman Linearized Augmented Lagrangian Method for Nonconvex Constrained Stochastic Zeroth-order Optimization
• Authors: Qiankun Shi, Xiao Wang, Hao Wang,
• Abstract summary: We propose a Bregman linearized augmented Lagrangian method that utilizes zeroth-order estimators combined with variance technique.<n>Results show that the complexity of the proposed method can achieve a dimensional dependency dependency lower than required (O(d)) without requiring additional assumptions.
• Score: 9.482573620753442
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we study nonconvex constrained stochastic zeroth-order optimization problems, for which we have access to exact information of constraints and noisy function values of the objective. We propose a Bregman linearized augmented Lagrangian method that utilizes stochastic zeroth-order gradient estimators combined with a variance reduction technique. We analyze its oracle complexity, in terms of the total number of stochastic function value evaluations required to achieve an \(\epsilon\)-KKT point in \(\ell_p\)-norm metrics with \(p \ge 2\), where \(p\) is a parameter associated with the selected Bregman distance. In particular, starting from a near-feasible initial point and using Rademacher smoothing, the oracle complexity is in order \(O(p d^{2/p} \epsilon^{-3})\) for \(p \in [2, 2 \ln d]\), and \(O(\ln d \cdot \epsilon^{-3})\) for \(p > 2 \ln d\), where \(d\) denotes the problem dimension. Those results show that the complexity of the proposed method can achieve a dimensional dependency lower than \(O(d)\) without requiring additional assumptions, provided that a Bregman distance is chosen properly. This offers a significant improvement in the high-dimensional setting over existing work, and matches the lowest complexity order with respect to the tolerance \(\epsilon\) reported in the literature. Numerical experiments on constrained Lasso and black-box adversarial attack problems highlight the promising performances of the proposed method.

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