Finding a stationary point of a stochastic convex problem

2026-07-09 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

A research paper from July 2026 addresses the challenge of finding stationary points for stochastic convex optimization problems, proposing a stronger notion of stationarity where the objective's subdifferential contains a small element. This approach contrasts with traditional criteria like proximity to a minimizer or small Moreau envelope gradients, which are shown to be insufficient through examples like median finding. The authors' convergence guarantees leverage dimension theory to decompose the graph of a convex function's subdifferential, demonstrating how stochastic sampling preserves these graph "pieces." This enables the effective application of proximal-point-like methods. The paper introduces a randomized resampling algorithm (Algorithm 3.1/3.2) that achieves convergence in probability for the stationary residual, even for non-single-valued or non-Lipschitz operators, without relying on smoothed surrogates.

Key takeaway

For Machine Learning Engineers developing robust optimization algorithms for stochastic convex problems, you should consider implementing the proposed randomized resampling procedure. This method directly controls the stationary residual, offering stronger guarantees than traditional approaches, especially when dealing with non-differentiable or non-Lipschitz objectives. It ensures convergence in probability without relying on smoothed surrogates, improving the reliability of your optimization outcomes.

Key insights

A novel approach to stochastic convex optimization uses dimension theory and randomized sampling for stronger stationary point guarantees.

Principles

• Subdifferentials of convex functions do not converge uniformly.
• Dimension theory can decompose subdifferential graphs into full-dimensional pieces.
• Stochastic sampling preserves key "pieces" of subdifferential graphs.

Method

A two-phase randomized resampling procedure: first, find a point near the minimum norm solution using a regularized empirical risk minimizer; then, sample in a small neighborhood and select the point minimizing the empirical subdifferential.

In practice

• Implement randomized resampling for robust stationary point finding.
• Apply proximal-point methods in stochastic convex optimization.
• Consider stronger stationarity criteria beyond proximity to minimizers.

Topics

• Stochastic Optimization
• Stationary Points
• Convex Analysis
• Monotone Operators
• Dimension Theory
• Randomized Algorithms

Best for: AI Scientist, Machine Learning Engineer

Related on AIssential

• Stochastic Gradient Methods: Bias, Stability and Generalization — A new framework analyzes stability and generalization for biased stochastic gradient methods.
• A Stabilized Path-Space Approach to Diffusion-Based Posterior Sampling — A stabilized path-space framework improves diffusion posterior sampling by connecting it to stochastic optimal control and removing bias.
• A Single-Loop Stochastic Proximal Quasi-Newton Method for Large-Scale Nonsmooth Convex Optimization — Integrating L-SVRG and stochastic L-BFGS yields a linearly convergent quasi-Newton method for nonsmooth convex optimization.
• Convergence Rates for Non-Log-Concave Sampling and Log-Partition Estimation — Optimal rates for non-log-concave sampling and log-partition estimation can sometimes surpass optimization rates.
• Decentralized Proximal Stochastic Gradient Langevin Dynamics — DE-PSGLD enables decentralized MCMC sampling from constrained log-concave distributions using proximal regularization.
• Decentralized Stochastic Subgradient-type Methods with Communication Compression for Nonsmooth Nonconvex Optimization — A unified framework ensures global convergence for decentralized nonsmooth nonconvex optimization with communication compression.

See Counsel's argued verdicts on the open AI decisions leaders are weighing →

Open in AIssential →

Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.