Research map / Learning theory and algorithms / Sample complexity
Hardness of learning neural networks
Shows computational complexity and learnability limits for training neural networks.
30 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.
8 papers in 2023–24 and 4 in 2025–26 (at ICML, NeurIPS, ICLR, CVPR and AAAI); its share of the topic shrank, from 2.4% to 1.2%.
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 · On the Depth of Deep Neural Networks: A Theoretical View (AAAI 2016 · 144 citations)
- 2017 · When and Why Are Deep Networks Better Than Shallow Ones? (AAAI 2017 · 174 citations)
- 2018 · Deep Homogeneous Mixture Models: Representation, Separation, and Approximation (NeurIPS 2018 · 9 citations)
- 2019 · Efficient Approximation of Deep ReLU Networks for Functions on Low Dimensional Manifolds (NeurIPS 2019 · 16 citations)
- 2020 · Hardness of Learning Neural Networks with Natural Weights (NeurIPS 2020 · 7 citations)
- 2021 · On the Representation Power of Set Pooling Networks (NeurIPS 2021 · 5 citations)
- 2022 · Synergy and Symmetry in Deep Learning: Interactions between the Data, Model, and Inference Algorithm (ICML 2022 · 4 citations)
- 2023 · Optimal approximation using complex-valued neural networks (NeurIPS 2023 · 3 citations)
- 2024 · Hardness of Learning Neural Networks under the Manifold Hypothesis (NeurIPS 2024 · 4 citations)
- 2025 · On the Hardness of Training Deep Neural Networks Discretely (AAAI 2025 · 1 citation)
- 2026 · Improving Set Function Approximation with Quasi-Arithmetic Neural Networks (ICLR 2026 · 0 citations)
Earliest papers here
- On the Depth of Deep Neural Networks: A Theoretical View (AAAI 2016 · 144 citations)
- When and Why Are Deep Networks Better Than Shallow Ones? (AAAI 2017 · 174 citations)
- Why Deep Neural Networks for Function Approximation? (ICLR 2017 · 146 citations)
What has been tried
15 papers here report 87 measured changes (a component added, removed or swapped, compared with a baseline): helped 55%, no clear effect 13%, hurt 22%, 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 |
|---|---|---|---|---|
| Increase ResNet width from 10 to 20 | Eq. (2), d=100 · MSE loss | 0.0102 -> 0.0093 | Helped | Characterizing ResNet's Universal Approximation Capability (ICML 2024) |
Recent preprints
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Who publishes on it
Institutions on the most papers of this approach since 2023 (of 12 papers; from OpenAlex and the papers' affiliations).
- Google · 2 papers
Use it
Download BibTeX of 25 key papersOpen the topic in the interactive map
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Related approaches
Same keyword cluster
- Neural network approximation theory (77 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).