Research map / Learning theory and algorithms / Sample complexity
Neural network approximation theory
Proves approximation rates and universality for shallow and deep networks across function classes.
77 accepted papers at ICML, NeurIPS, ICLR, CVPR and AAAI (2016–2026) · an approach in Sample complexity, keyword cluster “relu · functions · activation”
State History Tried 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.
14 papers in 2023–24 and 24 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic grew, from 4.2% to 7.3%.
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 · L1-regularized Neural Networks are Improperly Learnable in Polynomial Time (ICML 2016 · 61 citations)
- 2017 · Recovery Guarantees for One-hidden-layer Neural Networks (ICML 2017 · 166 citations)
- 2018 · Bounds on the Approximation Power of Feedforward Neural Networks (ICML 2018 · 5 citations)
- 2019 · Sorting Out Lipschitz Function Approximation (ICML 2019 · 65 citations)
- 2020 · Exactly Computing the Local Lipschitz Constant of ReLU Networks (NeurIPS 2020 · 32 citations)
- 2021 · Universal approximation power of deep residual neural networks via nonlinear control theory (ICLR 2021 · 19 citations)
- 2022 · On the Finite-Time Complexity and Practical Computation of Approximate Stationarity Concepts of Lipschitz Functions (ICML 2022 · 3 citations)
- 2023 · On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds (AAAI 2023 · 9 citations)
- 2024 · Complexity of Neural Network Training and ETR: Extensions with Effectively Continuous Functions (AAAI 2024 · 3 citations)
- 2025 · Generalization Bounds and Model Complexity for Kolmogorov–Arnold Networks (ICLR 2025 · 2 citations)
- 2026 · A Noise Sensitivity Exponent Controls Large Statistical-to-Computational Gaps in Single- and Multi-Index Models (ICML 2026 · 0 citations)
Earliest papers here
- L1-regularized Neural Networks are Improperly Learnable in Polynomial Time (ICML 2016 · 61 citations)
- Conservativeness of Untied Auto-Encoders (AAAI 2016 · 3 citations)
- Recovery Guarantees for One-hidden-layer Neural Networks (ICML 2017 · 166 citations)
What has been tried
46 papers here report 270 measured changes (a component added, removed or swapped, compared with a baseline): helped 61%, no clear effect 11%, hurt 17%, mixed 10%. Papers mostly publish changes that helped, so "helped" is over-represented.
Ablations in recent papers
| What was changed | Measured on | Result | Effect | Paper |
|---|---|---|---|---|
| add dropout with rate 0.1 to the activation functions | Setup (i) simulated regression · excess loss | 0.33 to 0.03 | Helped | Generalization Bounds and Model Complexity for Kolmogorov–Arnold Networks (ICLR 2025 · 2 citations) |
| Increase bounding segments per convex section from 5 to 10 | Iris Setosa output · approximation closeness | Helped | Exact Upper and Lower Bounds for the Output Distribution of Neural Networks with Random Inputs (ICML 2025 · 0 citations) | |
| Use a one-layer hat-activation pointwise model rather than a one-layer ReLU pointwise model | Synthetic set containment · accuracy | Helped | Monotone and Separable Set Functions: Characterizations and Neural Models (NeurIPS 2025 · 0 citations) | |
| Set P=S=0 | MNIST residual network, 5-layer 32 units, ℓ2 · Lipschitz bound | 29.75 -> 33.04 | Hurt | Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activations (ICLR 2024 · 0 citations) |
Recent preprints
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Who publishes on it
Institutions on the most papers of this approach since 2023 (of 38 papers; from OpenAlex and the papers' affiliations).
- Technische Universität Wien · 2 papers
- EPFL · 2 papers
- Technion - Israel Institute of Technology · 2 papers
- Hong Kong University of Science and Technology · 2 papers
Use it
Download BibTeX of 40 key papersOpen the topic in the interactive map
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Related approaches
Same keyword cluster
- Hardness of learning neural networks (30 papers)
Elsewhere in the topic
- Minimax estimation rates (109 papers)
- Sample complexity of learning distributions (101 papers)
- Optimal mean estimation and subsampling (95 papers)
- Density and property estimation (86 papers)
- Statistical-query lower bounds (85 papers)
- Computational thresholds in graphs and structure (84 papers)
- Learning halfspaces under noise (84 papers)
- Robust and surrogate-loss learning bounds (83 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).