Research map / Optimization / Implicit bias and generalization
Edge-of-stability dynamics
Characterizes gradient descent at large step sizes where sharpness hovers at the stability threshold.
116 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2016–2026) · an approach in Implicit bias and generalization, keyword cluster “sgd · sharpness · noise”
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.
26 papers in 2023–24 and 45 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic grew, from 11.1% to 17.2%.
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 · Improving Predictive State Representations via Gradient Descent (AAAI 2016 · 9 citations)
- 2018 · Visualizing the Loss Landscape of Neural Nets (NeurIPS 2018 · 606 citations)
- 2019 · An Investigation into Neural Net Optimization via Hessian Eigenvalue Density (ICML 2019 · 93 citations)
- 2020 · The Break-Even Point on Optimization Trajectories of Deep Neural Networks (ICLR 2020 · 28 citations)
- 2021 · Gradient Starvation: A Learning Proclivity in Neural Networks (NeurIPS 2021 · 40 citations)
- 2022 · Can Neural Nets Learn the Same Model Twice? Investigating Reproducibility and Double Descent From the Decision Boundary Perspective (CVPR 2022 · 34 citations)
- 2023 · Dropout Reduces Underfitting (ICML 2023 · 15 citations)
- 2024 · Why Warmup the Learning Rate? Underlying Mechanisms and Improvements (NeurIPS 2024 · 15 citations)
- 2025 · The Butterfly Effect: Neural Network Training Trajectories Are Highly Sensitive to Initial Conditions (ICML 2025 · 1 citation)
- 2026 · Gradient Descent with Large Step Size Restores Symmetry in Deep Linear Networks with Multi-Pathway (ICML 2026 · 0 citations)
Earliest papers here
- Improving Predictive State Representations via Gradient Descent (AAAI 2016 · 9 citations)
- Visualizing the Loss Landscape of Neural Nets (NeurIPS 2018 · 606 citations)
- Essentially No Barriers in Neural Network Energy Landscape (ICML 2018 · 178 citations)
What has been tried
88 papers here report 764 measured changes (a component added, removed or swapped, compared with a baseline): helped 55%, no clear effect 12%, hurt 26%, mixed 8%. Papers mostly publish changes that helped, so "helped" is over-represented.
Ablations in recent papers
| What was changed | Measured on | Result | Effect | Paper |
|---|---|---|---|---|
| initialize the second moment randomly with the same norm as the initial gradients | reported training settings · performance | Helped | Why Warmup the Learning Rate? Underlying Mechanisms and Improvements (NeurIPS 2024 · 15 citations) | |
| Use GeLU instead of ReLU | MNIST, varying hidden widths · Koopman spectral similarity | No clear effect | Identifying Equivalent Training Dynamics (NeurIPS 2024 · 6 citations) | |
| Set CDAT scaling near 2 instead of greedy scaling 1 | Full-batch deep-learning experiments · training loss | Helped | Stepping on the Edge: Curvature Aware Learning Rate Tuners (NeurIPS 2024 · 1 citation) | |
| Add bias terms to neurons | Modified 2D Rosenbrock regression · frequency of plateau saddles | Hurt | Flat Channels to Infinity in Neural Loss Landscapes (NeurIPS 2025 · 0 citations) | |
| increase weight decay | teacher-student representation alignment · representation alignment | Hurt | Neural Thermodynamics: Entropic Forces in Deep and Universal Representation Learning (NeurIPS 2025 · 0 citations) | |
| Increase the fraction of rule-flipped readout neurons | Convergence speed as flipped-neuron fraction varies · convergence speed | Hurt | Curl Descent : Non-Gradient Learning Dynamics with Sign-Diverse Plasticity (NeurIPS 2025 · 0 citations) | |
| Lower the base learning rate from 10^{-3} to 10^{-4} | Three input and three output neurons · Frobenius error to identity | Helped | Spike-timing-dependent Hebbian learning as noisy gradient descent (NeurIPS 2025 · 0 citations) | |
| Use the FSL-optimal learning-rate schedule instead of WSD schedules with stable-to-decay ratios 3:1, 4:1, or 5:1 | 20B tokens · final loss | approximately 0.01 | Helped | Functional Scaling Laws in Kernel Regression: Loss Dynamics and Learning Rate Schedules (NeurIPS 2025 · 0 citations) |
| Remove logits from the mode-connectivity feature vectors | CIFAR-10/100, FGSM, BIM, and PGD · accuracy/AUC (%) | Hurt | Input Space Mode Connectivity in Deep Neural Networks (ICLR 2025 · 0 citations) | |
| Use Beta (2, 2) sampling instead of uniform interpolation sampling | CIFAR-10, ResNet-18 · star-regular loss barrier | 0.084 ± 0.012 -> 0.069 ± 0.010 | No clear effect | Do Deep Neural Network Solutions Form a Star Domain? (ICLR 2025 · 0 citations) |
| project each SGD update onto the dominant subspace | MNIST-5k, CIFAR10-5k, and SST2-1k · training loss | Hurt | Does SGD really happen in tiny subspaces? (ICLR 2025 · 0 citations) | |
| Remove teleportation and learn only the local update | Two-layer LeakyReLU network optimization · convergence rate | Hurt | Improving Convergence and Generalization Using Parameter Symmetries (ICLR 2024 · 0 citations) |
What reviewers will expect
Datasets and baselines named in the 30 papers of this approach from 2025–2026 that have a full-text summary. Leaving out the common ones is a frequent review question.
Datasets
- CIFAR-10 · 12 of 30 papers
- MNIST · 6 of 30 papers
- CIFAR-100 · 3 of 30 papers
- Fashion-MNIST · 3 of 30 papers
- ImageNet · 3 of 30 papers
- synthetic Gaussian data · 2 of 30 papers
Baselines
- gradient flow · 3 of 30 papers
- SGD · 3 of 30 papers
- gradient descent · 2 of 30 papers
- ADAM · 2 of 30 papers
Recent preprints
Loading recent arXiv preprints…
Who publishes on it
Institutions on the most papers of this approach since 2023 (of 71 papers; from OpenAlex and the papers' affiliations).
- Flatiron Institute · 6 papers
- Princeton University · 4 papers
- Amazon · 4 papers
- University of California, San Diego · 3 papers
- Google · 3 papers
- Carnegie Mellon University · 3 papers
- New York University · 3 papers
- EPFL · 3 papers
- Massachusetts Institute of Technology · 3 papers
- Northeastern University · 2 papers
Use it
Download BibTeX of 39 key papersOpen the topic in the interactive map
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Related approaches
Same keyword cluster
- SGD noise and flat minima (112 papers)
- Stability-based generalization analysis (88 papers)
- Sharpness-aware minimization theory (37 papers)
Elsewhere in the topic
- Shallow ReLU network convergence (132 papers)
- Neural tangent kernel analysis (114 papers)
- Gradient flow and mean-field dynamics (77 papers)
- Low-rank and single-index recovery (70 papers)
- Explicit and implicit regularization interplay (63 papers)
- Implicit regularization of SGD (55 papers)
- Implicit bias in linear networks (53 papers)
- In-context learning as gradient descent (41 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).