# Triple descent and the two kinds of overfitting: Where & why do they appear?

arXiv: https://arxiv.org/abs/2006.03509 · HTML: https://mlsanity.com/paper/2006.03509/

## Key facts

- Cited by 65 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2020.

## Built on by (most cited first, 30 of 65)

- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
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- [Deep networks for system identification: a Survey](https://mlsanity.com/paper/2301.12832/) (2023)
- [Benign, Tempered, or Catastrophic: A Taxonomy of Overfitting](https://mlsanity.com/paper/2207.06569/) (2022)
- [Limitations of the NTK for Understanding Generalization in Deep Learning](https://mlsanity.com/paper/2206.10012/) (2022)
- [Deterministic equivalent and error universality of deep random features learning](https://mlsanity.com/paper/2302.00401/) (2023)
- [Taxonomizing local versus global structure in neural network loss landscapes](https://mlsanity.com/paper/2107.11228/) (2021)
- [Towards an Understanding of Benign Overfitting in Neural Networks](https://mlsanity.com/paper/2106.03212/) (2021)
- [Grokking phase transitions in learning local rules with gradient descent](https://mlsanity.com/paper/2210.15435/) (2022)
- [The Onset of Variance-Limited Behavior for Networks in the Lazy and Rich Regimes](https://mlsanity.com/paper/2212.12147/) (2022)
- [Multi-scale Feature Learning Dynamics: Insights for Double Descent](https://mlsanity.com/paper/2112.03215/) (2021)
- [Contrasting random and learned features in deep Bayesian linear regression](https://mlsanity.com/paper/2203.00573/) (2022)
- [Out-of-Distribution Generalization in Kernel Regression](https://mlsanity.com/paper/2106.02261/) (2021)
- [Fluctuations, Bias, Variance & Ensemble of Learners: Exact Asymptotics for Convex Losses in High-Dimension](https://mlsanity.com/paper/2201.13383/) (2022)
- [Model, sample, and epoch-wise descents: exact solution of gradient flow in the random feature model](https://mlsanity.com/paper/2110.11805/) (2021)
- [On the interplay between data structure and loss function in classification problems](https://mlsanity.com/paper/2103.05524/) (2021)
- [Learning curves for deep structured Gaussian feature models](https://mlsanity.com/paper/2303.00564/) (2023)
- [The Interpolating Information Criterion for Overparameterized Models](https://mlsanity.com/paper/2307.07785/) (2023)
- [Beyond Scaling Laws: Understanding Transformer Performance with Associative Memory](https://mlsanity.com/paper/2405.08707/) (2024)
- [Effect of Pre-Training Scale on Intra- and Inter-Domain Full and Few-Shot Transfer Learning for Natural and Medical X-Ray Chest Images](https://mlsanity.com/paper/2106.00116/) (2021)
- [Monotonicity and Double Descent in Uncertainty Estimation with Gaussian Processes](https://mlsanity.com/paper/2210.07612/) (2022)
- [Multiple Descent in the Multiple Random Feature Model](https://mlsanity.com/paper/2208.09897/) (2022)
- [Fundamental tradeoffs between memorization and robustness in random features and neural tangent regimes](https://mlsanity.com/paper/2106.02630/) (2021)
- [Origami in N dimensions: How feed-forward networks manufacture linear separability](https://mlsanity.com/paper/2203.11355/) (2022)
- [Information bottleneck theory of high-dimensional regression: relevancy, efficiency and optimality](https://mlsanity.com/paper/2208.03848/) (2022)
- [Why Do Probabilistic Clinical Models Fail To Transport Between Sites?](https://mlsanity.com/paper/2311.04787/) (2023)

## Abstract (arXiv)

A recent line of research has highlighted the existence of a "double descent" phenomenon in deep learning, whereby increasing the number of training examples N causes the generalization error of neural networks to peak when N is of the same order as the number of parameters P. In earlier works, a similar phenomenon was shown to exist in simpler models such as linear regression, where the peak instead occurs when N is equal to the input dimension D. Since both peaks coincide with the interpolation threshold, they are often conflated in the litterature. In this paper, we show that despite their apparent similarity, these two scenarios are inherently different. In fact, both peaks can co-exist when neural networks are applied to noisy regression tasks. The relative size of the peaks is then governed by the degree of nonlinearity of the activation function. Building on recent developments in the analysis of random feature models, we provide a theoretical ground for this sample-wise triple descent. As shown previously, the nonlinear peak at N!=!P is a true divergence caused by the extreme sensitivity of the output function to both the noise corrupting the labels and the initialization of the random features (or the weights in neural networks). This peak survives in the absence of noise, but can be suppressed by regularization. In contrast, the linear peak at N!=!D is solely due to overfitting the noise in the labels, and forms earlier during training. We show that this peak is implicitly regularized by the nonlinearity, which is why it only becomes salient at high noise and is weakly affected by explicit regularization. Throughout the paper, we compare analytical results obtained in the random feature model with the outcomes of numerical experiments involving deep neural networks.