Research map / Optimization / Nonconvex and smooth optimization
Stochastic gradient last-iterate analysis
Analyzes convergence of SGD and adaptive stepsizes under noise, interpolation, and nonsmooth settings.
178 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2016–2026) · an approach in Nonconvex and smooth optimization, keyword cluster “coordinate · proximal · bounds”
State History Tried Checklist Preprints Who Export
Where it stands
Not enough benchmark data Fewer than four papers report the same benchmark, so there is no record to follow.
33 papers in 2023–24 and 47 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic grew, from 10.3% to 13.4%.
How this is decided
A benchmark is tracked when at least four papers of this approach report their own method on the same dataset and metric in at least two different years. Its record is the best number reported up to 2024 compared with the best reported in 2025–26: still improving means at least 1% better, flat means at most 0.2% better with at least two papers from 2025–26 trying. Numbers on a different scale from the rest (a fraction among percentages) and far outliers are left out. Results come from different papers with different setups, so they are signals, not head-to-head comparisons. The share of papers is counted at ICML, NeurIPS, ICLR, CVPR and AAAI.
How it got here
Most cited paper of each year
- 2016 · Stochastic Gradient Methods for Distributionally Robust Optimization with f-divergences (NeurIPS 2016 · 140 citations)
- 2017 · SGDR: Stochastic Gradient Descent with Warm Restarts (ICLR 2017 · 1,699 citations)
- 2018 · SGD and Hogwild! Convergence Without the Bounded Gradients Assumption (ICML 2018 · 100 citations)
- 2019 · AdaGrad Stepsizes: Sharp Convergence Over Nonconvex Landscapes (ICML 2019 · 170 citations)
- 2020 · Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses (NeurIPS 2020 · 18 citations)
- 2021 · Stability and Convergence of Stochastic Gradient Clipping: Beyond Lipschitz Continuity and Smoothness (ICML 2021 · 16 citations)
- 2022 · Sample Average Approximation for Stochastic Optimization with Dependent Data: Performance Guarantees and Tractability (AAAI 2022 · 6 citations)
- 2023 · High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance (ICML 2023 · 5 citations)
- 2024 · Revisiting the Last-Iterate Convergence of Stochastic Gradient Methods (ICLR 2024 · 1 citation)
- 2025 · Nonconvex Stochastic Optimization under Heavy-Tailed Noises: Optimal Convergence without Gradient Clipping (ICLR 2025 · 1 citation)
- 2026 · Bregman meets Lévy: Stochastic Mirror Descent with Heavy-Tailed Noise in Continuous and Discrete Time (ICML 2026 · 0 citations)
Earliest papers here
- Stochastic Gradient Methods for Distributionally Robust Optimization with f-divergences (NeurIPS 2016 · 140 citations)
- SDNA: Stochastic Dual Newton Ascent for Empirical Risk Minimization (ICML 2016 · 53 citations)
- Generalization of ERM in Stochastic Convex Optimization: The Dimension Strikes Back (NeurIPS 2016 · 30 citations)
What has been tried
127 papers here report 779 measured changes (a component added, removed or swapped, compared with a baseline): helped 67%, no clear effect 12%, hurt 15%, mixed 5%. Papers mostly publish changes that helped, so "helped" is over-represented.
Ablations in recent papers
| What was changed | Measured on | Result | Effect | Paper |
|---|---|---|---|---|
| remove the extragradient step | Synthetic regularized log-sum-exp · convergence speed | Helped | Stochastic Newton Proximal Extragradient Method (NeurIPS 2024 · 1 citation) | |
| Use single sampling instead of double sampling | California Housing, Parkinsons Telemonitoring, and synthetic experiments · sub-optimality | Helped | Convergence of Clipped SGD on Convex $(L_0,L_1)$-Smooth Functions (NeurIPS 2025 · 0 citations) | |
| average the PSGD iterates | Examples 1 and 2 · OT map convergence rate | \mathcal{O}(1/\sqrt{n}) | Helped | Stochastic Optimization in Semi-Discrete Optimal Transport: Convergence Analysis and Minimax Rate (NeurIPS 2025 · 0 citations) |
| Set the stepsize to η = 1/(β log₂(T)) | Realizable objectives · expected excess risk | \frac{3\beta\norm{x_1-x^\star}^2}{\eta(2-\beta\eta)T^{1-\beta\eta/2}} -> \frac{6\beta\norm{x_1-x^\star}^2 \log_2(T)}{T} | Helped | Fast Last-Iterate Convergence of SGD in the Smooth Interpolation Regime (NeurIPS 2025 · 0 citations) |
| Add dynamic proximal regularization | general recurrent sampling · expected optimality gap | O(n^{-1/2}) | Helped | Stochastic Optimization with Arbitrary Recurrent Data Sampling (ICML 2024 · 0 citations) |
| Increase the number of hard equality constraints from NH=0 to NH=8 | SNL (ℓ=1, r=1.5, ϵ=0.3) · Mahalanobis distance | 0.11 ± 0.11 -> 0.04 ± 0.02 | Helped | S-SOS: Stochastic Sum-Of-Squares for Parametric Polynomial Optimization (NeurIPS 2024 · 0 citations) |
| use adaptive sampling instead of a predetermined sublinearly growing sample size | Online sparse robust regression · distance to optimal solution | Helped | An Online Adaptive Sampling Algorithm for Stochastic Difference-of-convex Optimization with Time-varying Distributions (ICML 2025) | |
| Use SHB with step decay instead of SHB with a constant learning rate | a4a, batch size 512 · training loss | 0.86±0.55 -> 0.13±0.03 | Helped | Accelerated Convergence of Stochastic Heavy Ball Method under Anisotropic Gradient Noise (ICLR 2024) |
| Use GSGD with an affine-subspace regularizer whose projection matrix has lower rank | Synthetic quadratic with an affine-subspace regularizer, diagonal matrix types 1–4 · convergence speed | Mixed | Can Gaussian Sketching Converge Faster on a Preconditioned Landscape? (ICML 2024) |
What reviewers will expect
Datasets and baselines named in the 29 papers of this approach from 2025–2026 that have a full-text summary. Leaving out the common ones is a frequent review question.
Baselines
- Adam · 3 of 29 papers
- random reshuffling · 2 of 29 papers
- with-replacement SGD · 2 of 29 papers
- SGD · 2 of 29 papers
Recent preprints
Loading recent arXiv preprints…
Who publishes on it
Institutions on the most papers of this approach since 2023 (of 80 papers; from OpenAlex and the papers' affiliations).
- Tel Aviv University · 6 papers
- Google · 5 papers
- Northeastern University · 5 papers
- Mohamed bin Zayed University of Artificial Intelligence · 4 papers
- Boston University · 4 papers
- Korea Advanced Institute of Science and Technology · 3 papers
- Moscow Institute of Physics and Technology · 3 papers
- University of Wisconsin - Madison · 3 papers
- Hong Kong University of Science and Technology · 2 papers
- New York University · 2 papers
Use it
Download BibTeX of 40 key papersOpen the topic in the interactive map
Get alerts. Describe your idea in scime atlas and save it (free scime account): you get a weekly email when new preprints land close to it.
Related approaches
Same keyword cluster
- Adaptive first-order convex methods (190 papers)
- Coordinate and Adam convergence analysis (188 papers)
- Acceleration and Nesterov dynamics (91 papers)
- Langevin sampling for nonconvex problems (70 papers)
Elsewhere in the topic
- Nonconvex minimax and constrained methods (100 papers)
- Stochastic convex optimization complexity (94 papers)
- Equilibrium learning in zero-sum games (88 papers)
- Stochastic variance reduction (86 papers)
- Complexity of nonconvex first-order methods (79 papers)
- Penalty and implicit-gradient bilevel methods (65 papers)
- Proximal finite-sum methods (47 papers)
- Stochastic Frank-Wolfe methods (43 papers)
Benchmark numbers, ablations and baseline numbers are extracted by AI from each paper's full text and can be wrong; the paper is the authority. Data: results index of 2026-11-02, citation counts from OpenAlex (2026-10-09).