Research map / Optimization / Nonconvex and smooth optimization
Saddle-point escape algorithms
Designs gradient and zeroth-order methods that escape saddle points to second-order stationary points.
23 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2017–2025) · an approach in Nonconvex and smooth optimization, keyword cluster “complexity · bilevel · stationary”
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Where it stands
Not enough benchmark data Fewer than four papers report the same benchmark, so there is no record to follow.
5 papers in 2023–24 and 3 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic shrank, from 1.6% to 0.9%.
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
- 2017 · How to Escape Saddle Points Efficiently (ICML 2017 · 219 citations)
- 2018 · Escaping Saddles with Stochastic Gradients (ICML 2018 · 79 citations)
- 2019 · Efficiently escaping saddle points on manifolds (NeurIPS 2019 · 34 citations)
- 2020 · Escaping Saddle-Point Faster under Interpolation-like Conditions (NeurIPS 2020 · 1 citation)
- 2021 · Escape saddle points by a simple gradient-descent based algorithm (NeurIPS 2021 · 2 citations)
- 2022 · Zeroth-Order Negative Curvature Finding: Escaping Saddle Points without Gradients (NeurIPS 2022 · 0 citations)
- 2023 · Projection-Free Methods for Solving Nonconvex-Concave Saddle Point Problems (NeurIPS 2023 · 1 citation)
- 2024 · Convergence and Complexity Guarantee for Inexact First-order Riemannian Optimization Algorithms (ICML 2024 · 0 citations)
- 2025 · Hessian-guided Perturbed Wasserstein Gradient Flows for Escaping Saddle Points (NeurIPS 2025 · 0 citations)
Earliest papers here
- How to Escape Saddle Points Efficiently (ICML 2017 · 219 citations)
- Escaping Saddles with Stochastic Gradients (ICML 2018 · 79 citations)
- Escaping Saddle Points in Constrained Optimization (NeurIPS 2018 · 28 citations)
What has been tried
22 papers here report 98 measured changes (a component added, removed or swapped, compared with a baseline): helped 87%, no clear effect 5%, hurt 4%, mixed 4%. Papers mostly publish changes that helped, so "helped" is over-represented.
Ablations in recent papers
| What was changed | Measured on | Result | Effect | Paper |
|---|---|---|---|---|
| set m=1, removing lazy Hessian reuse | C-C minimax · computational complexity | \gO((N+d^2) d \epsilon^{-2/3}) | Hurt | Second-Order Min-Max Optimization with Lazy Hessians (ICLR 2025 · 0 citations) |
Recent preprints
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Related approaches
Same keyword cluster
- Nonconvex minimax and constrained methods (100 papers)
- Stochastic convex optimization complexity (94 papers)
- Complexity of nonconvex first-order methods (79 papers)
- Penalty and implicit-gradient bilevel methods (65 papers)
Elsewhere in the topic
- Adaptive first-order convex methods (190 papers)
- Coordinate and Adam convergence analysis (188 papers)
- Stochastic gradient last-iterate analysis (178 papers)
- Acceleration and Nesterov dynamics (91 papers)
- Equilibrium learning in zero-sum games (88 papers)
- Stochastic variance reduction (86 papers)
- Langevin sampling for nonconvex problems (70 papers)
- Proximal finite-sum methods (47 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).