# On the Generalization Mystery in Deep Learning

arXiv: https://arxiv.org/abs/2203.10036 · HTML: https://mlsanity.com/paper/2203.10036/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 63 of its 97 references that are in the map.
- First posted on arXiv 18 Mar 2022.

## Builds on (most cited first, 30 of 63)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness](https://mlsanity.com/paper/1811.12231/) (2018)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Learning from Noisy Labels with Deep Neural Networks: A Survey](https://mlsanity.com/paper/2007.08199/) (2020)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [On Exact Computation with an Infinitely Wide Neural Net](https://mlsanity.com/paper/1904.11955/) (2019)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Computing Nonvacuous Generalization Bounds for Deep (Stochastic) Neural Networks with Many More Parameters than Training Data](https://mlsanity.com/paper/1703.11008/) (2017)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [In Search of the Real Inductive Bias: On the Role of Implicit Regularization in Deep Learning](https://mlsanity.com/paper/1412.6614/) (2014)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data](https://mlsanity.com/paper/1808.01204/) (2018)
- [Uniform convergence may be unable to explain generalization in deep learning](https://mlsanity.com/paper/1902.04742/) (2019)

## Built on by (most cited first, 25 of 25)

- [Beyond Lipschitz: Sharp Generalization and Excess Risk Bounds for Full-Batch GD](https://mlsanity.com/paper/2204.12446/) (2022)
- [Dehazing Ultrasound using Diffusion Models](https://mlsanity.com/paper/2307.11204/) (2023)
- [Information-Theoretic Generalization Bounds for Deep Neural Networks](https://mlsanity.com/paper/2404.03176/) (2024)
- [JIGMARK: A Black-Box Approach for Enhancing Image Watermarks against Diffusion Model Edits](https://mlsanity.com/paper/2406.03720/) (2024)
- [Max-Margin Works while Large Margin Fails: Generalization without Uniform Convergence](https://mlsanity.com/paper/2206.07892/) (2022)
- [Data Optimization in Deep Learning: A Survey](https://mlsanity.com/paper/2310.16499/) (2023)
- [Trustworthy Machine Learning via Memorization and the Granular Long-Tail: A Survey on Interactions, Tradeoffs, and Beyond](https://mlsanity.com/paper/2503.07501/) (2025)
- [Unveiling the Hessian's Connection to the Decision Boundary](https://mlsanity.com/paper/2306.07104/) (2023)
- [Exploring the Learning Difficulty of Data Theory and Measure](https://mlsanity.com/paper/2205.07427/) (2022)
- [On the Interpretability of Regularisation for Neural Networks Through Model Gradient Similarity](https://mlsanity.com/paper/2205.12642/) (2022)
- [The Implicit Bias of Structured State Space Models Can Be Poisoned With Clean Labels](https://mlsanity.com/paper/2410.10473/) (2024)
- [Token-Level Fitting Issues of Seq2seq Models](https://mlsanity.com/paper/2305.04493/) (2023)
- [Not All Pixels Are Equal: Learning Pixel Hardness for Semantic Segmentation](https://mlsanity.com/paper/2305.08462/) (2023)
- [The Future of Intelligent Healthcare: A Systematic Analysis and Discussion on the Integration and Impact of Robots Using Large Language Models for Healthcare](https://mlsanity.com/paper/2411.03287/) (2024)
- [Astroconformer: The Prospects of Analyzing Stellar Light Curves with Transformer-Based Deep Learning Models](https://mlsanity.com/paper/2309.16316/) (2023)
- [Hybrid Quantum-inspired Resnet and Densenet for Pattern Recognition](https://mlsanity.com/paper/2403.05754/) (2024)
- [The Interaction Bottleneck of Deep Neural Networks: Discovery, Proof, and Modulation](https://mlsanity.com/paper/2512.18607/) (2025)
- [A Curriculum View of Robust Loss Functions](https://mlsanity.com/paper/2305.02139/) (2023)
- [Catching Image Retrieval Generalization](https://mlsanity.com/paper/2306.13357/) (2023)
- [WaveCastNet: Rapid Wavefield Forecasting for Earthquake Early Warning via Deep Sequence to Sequence Learning](https://mlsanity.com/paper/2405.20516/) (2024)
- [Sharpness-Aware Minimization with Z-Score Gradient Filtering](https://mlsanity.com/paper/2505.02369/) (2025)
- [SNAP: A Benchmark for Testing the Effects of Capture Conditions on Fundamental Vision Tasks](https://mlsanity.com/paper/2505.15628/) (2025)
- [Generalization Bounds of Stochastic Gradient Descent in Homogeneous Neural Networks](https://mlsanity.com/paper/2602.22936/) (2026)
- [Activation Functions, Statistics and Learning of Higher-Order Interactions in Restricted Boltzmann Machines](https://mlsanity.com/paper/2605.19178/) (2026)
- [Distributional Extrapolation for Interactions](https://mlsanity.com/paper/2608.19849/) (2026)

## Abstract (arXiv)

The generalization mystery in deep learning is the following: Why do over-parameterized neural networks trained with gradient descent (GD) generalize well on real datasets even though they are capable of fitting random datasets of comparable size? Furthermore, from among all solutions that fit the training data, how does GD find one that generalizes well (when such a well-generalizing solution exists)? We argue that the answer to both questions lies in the interaction of the gradients of different examples during training. Intuitively, if the per-example gradients are well-aligned, that is, if they are coherent, then one may expect GD to be (algorithmically) stable, and hence generalize well. We formalize this argument with an easy to compute and interpretable metric for coherence, and show that the metric takes on very different values on real and random datasets for several common vision networks. The theory also explains a number of other phenomena in deep learning, such as why some examples are reliably learned earlier than others, why early stopping works, and why it is possible to learn from noisy labels. Moreover, since the theory provides a causal explanation of how GD finds a well-generalizing solution when one exists, it motivates a class of simple modifications to GD that attenuate memorization and improve generalization. Generalization in deep learning is an extremely broad phenomenon, and therefore, it requires an equally general explanation. We conclude with a survey of alternative lines of attack on this problem, and argue that the proposed approach is the most viable one on this basis.