Research map / Optimization / Implicit bias and generalization
Explicit and implicit regularization interplay
Studies how weight decay, gradient regularizers, and training dynamics jointly induce implicit regularization.
63 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2018–2026) · an approach in Implicit bias and generalization, keyword cluster “implicit bias · regularization · bias gradient”
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.
13 papers in 2023–24 and 18 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic grew, from 5.5% to 6.9%.
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
- 2018 · On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization (ICML 2018 · 136 citations)
- 2019 · Uniform convergence may be unable to explain generalization in deep learning (NeurIPS 2019 · 111 citations)
- 2020 · Gradient Descent Maximizes the Margin of Homogeneous Neural Networks (ICLR 2020 · 102 citations)
- 2021 · On the Theory of Implicit Deep Learning: Global Convergence with Implicit Layers (ICLR 2021 · 16 citations)
- 2022 · An Unconstrained Layer-Peeled Perspective on Neural Collapse (ICLR 2022 · 11 citations)
- 2023 · The Implicit Regularization of Momentum Gradient Descent in Overparametrized Models (AAAI 2023 · 6 citations)
- 2024 · Why Do We Need Weight Decay in Modern Deep Learning? (NeurIPS 2024 · 10 citations)
- 2025 · Explicit and Implicit Graduated Optimization in Deep Neural Networks (AAAI 2025 · 0 citations)
- 2026 · Conflicting Biases at the Edge of Stability: Norm versus Sharpness Regularization (ICML 2026 · 0 citations)
Earliest papers here
- On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization (ICML 2018 · 136 citations)
- Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced (NeurIPS 2018 · 22 citations)
- Learning Compact Neural Networks with Regularization (ICML 2018 · 3 citations)
What has been tried
49 papers here report 396 measured changes (a component added, removed or swapped, compared with a baseline): helped 60%, no clear effect 10%, hurt 26%, 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 |
|---|---|---|---|---|
| Replace decoupled weight decay with an L2 penalty | OpenWebText · training loss | No clear effect | Why Do We Need Weight Decay in Modern Deep Learning? (NeurIPS 2024 · 10 citations) | |
| Add a high-noise training phase before low-noise fitting | Matrix completion across ε values · rank-1 recovery | Helped | Implicit bias of SGD in $L_2$-regularized linear DNNs: One-way jumps from high to low rank (ICLR 2024 · 1 citation) | |
| use EGO(nice) instead of EGO(geo) | Traditional global optimization functions, D=50 · optimum objective value | 1.80E+01 -> 4.04E-03 (Ackley's); 1.85E+00 -> 2.26E-02 (Rastrigin's) | Helped | Explicit and Implicit Graduated Optimization in Deep Neural Networks (AAAI 2025 · 0 citations) |
| increase isotropic Gaussian posterior variance | CIFAR-10 MLP · sharpness | Helped | Variational Learning Finds Flatter Solutions at the Edge of Stability (NeurIPS 2025 · 0 citations) | |
| Replace Rademacher initialization with Gaussian initialization | Full parity, uniform inputs · accuracy | perfect accuracy -> ≤ 1/2+exp(-Ω(d)) | Hurt | Learning High-Degree Parities: The Crucial Role of the Initialization (ICLR 2025 · 0 citations) |
| Use the PILoT initialization instead of the spred initialization | diagonal linear network · distance to ground truth | Helped | Mask in the Mirror: Implicit Sparsification (ICLR 2025 · 0 citations) | |
| Increase factorization depth from 2 to 3 | Fashion-MNIST, at the same test accuracy · compression ratio | up to four times sparser | Helped | Deep Weight Factorization: Sparse Learning Through the Lens of Artificial Symmetries (ICLR 2025 · 0 citations) |
| remove the learned vector field | Adult · accuracy | 84.28% -> 84.21% | Hurt | Deep Sturm–Liouville: From Sample-Based to 1D Regularization with Learnable Orthogonal Basis Functions (ICML 2025) |
| Replace L1 or nuclear-norm regularization with L2 regularization | Sparse recovery and matrix completion with N < n · generalization | Hurt | Grokking Beyond the Euclidean Norm of Model Parameters (ICML 2025) |
What reviewers will expect
Datasets and baselines named in the 12 papers of this approach from 2025–2026 that have a full-text summary. Leaving out the common ones is a frequent review question.
Datasets
- MNIST · 5 of 12 papers
- CIFAR-10 · 5 of 12 papers
- CIFAR100 · 3 of 12 papers
- ImageNet · 3 of 12 papers
Baselines
- AdamW · 2 of 12 papers
- Adam · 2 of 12 papers
Recent preprints
Loading recent arXiv preprints…
Who publishes on it
Institutions on the most papers of this approach since 2023 (of 31 papers; from OpenAlex and the papers' affiliations).
- EPFL · 4 papers
- New York 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
- Implicit regularization of SGD (55 papers)
- Implicit bias in linear networks (53 papers)
Elsewhere in the topic
- Shallow ReLU network convergence (132 papers)
- Edge-of-stability dynamics (116 papers)
- Neural tangent kernel analysis (114 papers)
- SGD noise and flat minima (112 papers)
- Stability-based generalization analysis (88 papers)
- Gradient flow and mean-field dynamics (77 papers)
- Low-rank and single-index recovery (70 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).