# Graphical Convergence of Subgradients in Nonconvex Optimization and Learning

arXiv: https://arxiv.org/abs/1810.07590 · HTML: https://mlsanity.com/paper/1810.07590/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2018.

## Built on by (most cited first, 17 of 17)

- [Stability and Generalization of Stochastic Optimization with Nonconvex and Nonsmooth Problems](https://mlsanity.com/paper/2206.07082/) (2022)
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- [On the Uniform Convergence of Subdifferentials in Stochastic Optimization and Learning](https://mlsanity.com/paper/2405.10289/) (2024)
- [Failure of uniform laws of large numbers for subdifferentials and beyond](https://mlsanity.com/paper/2511.16568/) (2025)
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- [Towards Sharper Risk Bounds for Minimax Problems](https://mlsanity.com/paper/2410.08497/) (2024)
- [Improved Learning Rates for Stochastic Optimization](https://mlsanity.com/paper/2107.08686/) (2021)
- [Learning Rates for Nonconvex Pairwise Learning](https://mlsanity.com/paper/2111.05232/) (2021)
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- [Loss Gradient Gaussian Width based Generalization and Optimization Guarantees](https://mlsanity.com/paper/2406.07712/) (2024)
- [Stability-based Generalization Analysis of Randomized Coordinate Descent for Pairwise Learning](https://mlsanity.com/paper/2503.01530/) (2025)
- [Unregularized limit of stochastic gradient method for Wasserstein distributionally robust optimization](https://mlsanity.com/paper/2506.04948/) (2025)
- [Stability and Generalization of Stochastic Nonconvex Optimization under Heavy-Tailed Noise: From Gradient Norm to Goldstein Stationarity](https://mlsanity.com/paper/2601.19730/) (2026)
- [Unveiling High-Probability Generalization in Decentralized SGD](https://mlsanity.com/paper/2605.10205/) (2026)
- [Lipschitzian SLLNs for random functions](https://mlsanity.com/paper/2607.20411/) (2026)

## Abstract (arXiv)

We investigate the stochastic optimization problem of minimizing population risk, where the loss defining the risk is assumed to be weakly convex. Compositions of Lipschitz convex functions with smooth maps are the primary examples of such losses. We analyze the estimation quality of such nonsmooth and nonconvex problems by their sample average approximations. Our main results establish dimension-dependent rates on subgradient estimation in full generality and dimension-independent rates when the loss is a generalized linear model. As an application of the developed techniques, we analyze the nonsmooth landscape of a robust nonlinear regression problem.