# Finding a stationary point of a stochastic convex problem

arXiv: https://arxiv.org/abs/2607.06883 · HTML: https://mlsanity.com/paper/2607.06883/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 54 references that are in the map.
- First posted on arXiv 8 Jul 2026.

## Builds on (most cited first, 14 of 14)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [Conformal Prediction With Conditional Guarantees](https://mlsanity.com/paper/2305.12616/) (2023)
- [Efficiency of minimizing compositions of convex functions and smooth maps](https://mlsanity.com/paper/1605.00125/) (2016)
- [Simple and optimal methods for stochastic variational inequalities, I: operator extrapolation](https://mlsanity.com/paper/2011.02987/) (2020)
- [The Complexity of Making the Gradient Small in Stochastic Convex Optimization](https://mlsanity.com/paper/1902.04686/) (2019)
- [The importance of better models in stochastic optimization](https://mlsanity.com/paper/1903.08619/) (2019)
- [Proximally Guided Stochastic Subgradient Method for Nonsmooth, Nonconvex Problems](https://mlsanity.com/paper/1707.03505/) (2017)
- [Graphical Convergence of Subgradients in Nonconvex Optimization and Learning](https://mlsanity.com/paper/1810.07590/) (2018)
- [Ergodic convergence of a stochastic proximal point algorithm](https://mlsanity.com/paper/1504.05400/) (2015)
- [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)
- [Distribution free M-estimation](https://mlsanity.com/paper/2505.22807/) (2025)

## Built on by (most cited first, 2 of 2)

- [Lipschitzian SLLNs for random functions](https://mlsanity.com/paper/2607.20411/) (2026)
- [Random tilts to find stationary points in stochastic convex optimization](https://mlsanity.com/paper/2609.17798/) (2026)

## Abstract (arXiv)

We consider the problem of finding stationary points for stochastic convex optimization problems. Rather than surrogates to stationarity, such as a proximity-to-stationarity guarantee or small gradient of the Moreau envelope, we ask for a stronger notion: that the subdifferential of the objective actually contains a small element. This criterion is non-trivial, because subdifferentials of convex functions fail to converge uniformly, even in arbitrarily small neighborhoods of the optimum. Our convergence guarantees rely on dimension theory to decompose the graph of the subdifferential of a convex function, showing how stochastic sampling preserves "pieces" of these graphs, and allowing effective application of proximal-point-like methods.