# Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization

arXiv: https://arxiv.org/abs/2301.13314 · HTML: https://mlsanity.com/paper/2301.13314/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 92 references that are in the map.
- First posted on arXiv 30 Jan 2023.

## Builds on (most cited first, 30 of 59)

- [A Reductions Approach to Fair Classification](https://mlsanity.com/paper/1803.02453/) (2018)
- [Lower Bounds for Non-Convex Stochastic Optimization](https://mlsanity.com/paper/1912.02365/) (2019)
- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [Global Convergence of ADMM in Nonconvex Nonsmooth Optimization](https://mlsanity.com/paper/1511.06324/) (2015)
- [Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems](https://mlsanity.com/paper/1410.1390/) (2014)
- [Weakly-Convex Concave Min-Max Optimization: Provable Algorithms and Applications in Machine Learning](https://mlsanity.com/paper/1810.02060/) (2018)
- [Efficiency of minimizing compositions of convex functions and smooth maps](https://mlsanity.com/paper/1605.00125/) (2016)
- [Oracle Complexity in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2104.06763/) (2021)
- [Stochastic First-order Methods for Convex and Nonconvex Functional Constrained Optimization](https://mlsanity.com/paper/1908.02734/) (2019)
- [The Complexity of Finding Stationary Points with Stochastic Gradient Descent](https://mlsanity.com/paper/1910.01845/) (2019)
- [Subgradient methods for sharp weakly convex functions](https://mlsanity.com/paper/1803.02461/) (2018)
- [Stochastic subgradient method converges at the rate O(k-1/4) on weakly convex functions](https://mlsanity.com/paper/1802.02988/) (2018)
- [On the Finite-Time Complexity and Practical Computation of Approximate Stationarity Concepts of Lipschitz Functions](https://mlsanity.com/paper/2112.09002/) (2021)
- [Proximally Guided Stochastic Subgradient Method for Nonsmooth, Nonconvex Problems](https://mlsanity.com/paper/1707.03505/) (2017)
- [Sequential Quadratic Optimization for Nonlinear Equality Constrained Stochastic Optimization](https://mlsanity.com/paper/2007.10525/) (2020)
- [An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints](https://mlsanity.com/paper/1906.11357/) (2019)
- [Quadratically Regularized Subgradient Methods for Weakly Convex Optimization with Weakly Convex Constraints](https://mlsanity.com/paper/1908.01871/) (2019)
- [An Adaptive Stochastic Sequential Quadratic Programming with Differentiable Exact Augmented Lagrangians](https://mlsanity.com/paper/2102.05320/) (2021)
- [A Stochastic Sequential Quadratic Optimization Algorithm for Nonlinear Equality Constrained Optimization with Rank-Deficient Jacobians](https://mlsanity.com/paper/2106.13015/) (2021)
- [Rate-improved Inexact Augmented Lagrangian Method for Constrained Nonconvex Optimization](https://mlsanity.com/paper/2007.01284/) (2020)
- [Worst-Case Complexity of an SQP Method for Nonlinear Equality Constrained Stochastic Optimization](https://mlsanity.com/paper/2112.14799/) (2021)
- [On the Convergence Rate of Stochastic Mirror Descent for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/1806.04781/) (2018)
- [First-Order Methods for Nonsmooth Nonconvex Functional Constrained Optimization with or without Slater Points](https://mlsanity.com/paper/2212.00927/) (2022)
- [Faster Gradient-Free Algorithms for Nonsmooth Nonconvex Stochastic Optimization](https://mlsanity.com/paper/2301.06428/) (2023)
- [RSG: Beating Subgradient Method without Smoothness and Strong Convexity](https://mlsanity.com/paper/1512.03107/) (2015)
- [Complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs](https://mlsanity.com/paper/1802.03504/) (2018)
- [A Proximal Alternating Direction Method of Multiplier for Linearly Constrained Nonconvex Minimization](https://mlsanity.com/paper/1812.10229/) (2018)
- [Structured Nonconvex and Nonsmooth Optimization: Algorithms and Iteration Complexity Analysis](https://mlsanity.com/paper/1605.02408/) (2016)
- [Complexity of Finding Stationary Points of Nonsmooth Nonconvex Functions](https://mlsanity.com/paper/2002.04130/) (2020)
- [A global dual error bound and its application to the analysis of linearly constrained nonconvex optimization](https://mlsanity.com/paper/2006.16440/) (2020)

## Built on by (most cited first, 11 of 11)

- [A single-loop SPIDER-type stochastic subgradient method for expectation-constrained nonconvex nonsmooth optimization](https://mlsanity.com/paper/2501.19214/) (2025)
- [Single-loop Algorithms for Stochastic Non-convex Optimization with Weakly-Convex Constraints](https://mlsanity.com/paper/2504.15243/) (2025)
- [Lower Bounds and Proximally Anchored SGD for Non-Convex Minimization Under Unbounded Variance](https://mlsanity.com/paper/2604.16620/) (2026)
- [Primal Methods for Variational Inequality Problems with Functional Constraints](https://mlsanity.com/paper/2403.12859/) (2024)
- [Enforcing Fair Predicted Scores on Intervals of Percentiles by Difference-of-Convex Constraints](https://mlsanity.com/paper/2505.12530/) (2025)
- [Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions](https://mlsanity.com/paper/2502.19764/) (2025)
- [humancompatible.train: Implementing Optimization Algorithms for Stochastically-Constrained Stochastic Optimization Problems](https://mlsanity.com/paper/2509.21254/) (2025)
- [First-Order Softmax Weighted Switching Gradient Method for Distributed Stochastic Minimax Optimization with Stochastic Constraints](https://mlsanity.com/paper/2603.05774/) (2026)
- [Stochastic Momentum Methods for Non-smooth Non-Convex Finite-Sum Coupled Compositional Optimization](https://mlsanity.com/paper/2506.02504/) (2025)
- [Benchmarking Stochastic Approximation Algorithms for Fairness-Constrained Training of Deep Neural Networks](https://mlsanity.com/paper/2507.04033/) (2025)
- [Global Optimality for Constrained Exploration via Penalty Regularization](https://mlsanity.com/paper/2604.28144/) (2026)

## Abstract (arXiv)

We consider a non-convex constrained optimization problem, where the objective function is weakly convex and the constraint function is either convex or weakly convex. To solve this problem, we consider the classical switching subgradient method, which is an intuitive and easily implementable first-order method whose oracle complexity was only known for convex problems. This paper provides the first analysis on the oracle complexity of the switching subgradient method for finding a nearly stationary point of non-convex problems. Our results are derived separately for convex and weakly convex constraints. Compared to existing approaches, especially the double-loop methods, the switching gradient method can be applied to non-smooth problems and achieves the same complexity using only a single loop, which saves the effort on tuning the number of inner iterations.