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A regression model with more parameters than data points in the training data is overparametrized and has the capability to interpolate the training data. Based on the classical bias-variance tradeoff expressions, it is commonly assumed…

Machine Learning · Computer Science 2023-04-18 Tomas McKelvey

`Double descent' delineates the generalization behaviour of models depending on the regime they belong to: under- or over-parameterized. The current theoretical understanding behind the occurrence of this phenomenon is primarily based on…

Machine Learning · Statistics 2022-03-15 Sidak Pal Singh , Aurelien Lucchi , Thomas Hofmann , Bernhard Schölkopf

Double descent is a surprising phenomenon in machine learning, in which as the number of model parameters grows relative to the number of data, test error drops as models grow ever larger into the highly overparameterized (data…

In this expository note we describe a surprising phenomenon in overparameterized linear regression, where the dimension exceeds the number of samples: there is a regime where the test risk of the estimator found by gradient descent…

Machine Learning · Statistics 2019-12-17 Preetum Nakkiran

Modern machine learning often operates in the regime where the number of parameters is much higher than the number of data points, with zero training loss and yet good generalization, thereby contradicting the classical bias-variance…

Machine Learning · Statistics 2021-02-08 Zhu Li , Weijie Su , Dino Sejdinovic

The volume of the distribution of weight sets associated with a loss value may be the source of implicit regularization from overparameterization due to the phenomenon of contracting volume with increasing dimensions for geometric figures…

Machine Learning · Computer Science 2022-09-27 Nicholas J. Teague

We demonstrate the ability of hybrid regularization methods to automatically avoid the double descent phenomenon arising in the training of random feature models (RFM). The hallmark feature of the double descent phenomenon is a spike in the…

Machine Learning · Computer Science 2020-12-15 Kelvin Kan , James G Nagy , Lars Ruthotto

The relationship between the number of training data points, the number of parameters, and the generalization capabilities of models has been widely studied. Previous work has shown that double descent can occur in the over-parameterized…

Machine Learning · Statistics 2024-10-28 Xinyue Li , Rishi Sonthalia

In this paper, we studied two identically-trained neural networks (i.e. networks with the same architecture, trained on the same dataset using the same algorithm, but with different initialization) and found that their outputs discrepancy…

Machine Learning · Computer Science 2023-05-26 Yifan Luo , Bin Dong

The "double descent" risk curve was proposed to qualitatively describe the out-of-sample prediction accuracy of variably-parameterized machine learning models. This article provides a precise mathematical analysis for the shape of this…

Machine Learning · Computer Science 2020-12-22 Mikhail Belkin , Daniel Hsu , Ji Xu

Regularization plays a vital role in the context of deep learning by preventing deep neural networks from the danger of overfitting. This paper proposes a novel deep learning regularization method named as DL-Reg, which carefully reduces…

Machine Learning · Computer Science 2020-11-05 Maryam Dialameh , Ali Hamzeh , Hossein Rahmani

Recent works have demonstrated a double descent phenomenon in over-parameterized learning. Although this phenomenon has been investigated by recent works, it has not been fully understood in theory. In this paper, we investigate the…

Statistics Theory · Mathematics 2023-10-11 Xuran Meng , Jianfeng Yao , Yuan Cao

In their thought-provoking paper [1], Belkin et al. illustrate and discuss the shape of risk curves in the context of modern high-complexity learners. Given a fixed training sample size $n$, such curves show the risk of a learner as a…

Machine Learning · Computer Science 2022-06-08 Marco Loog , Tom Viering , Alexander Mey , Jesse H. Krijthe , David M. J. Tax

In overparameterized logistic regression, gradient descent (GD) iterates diverge in norm while converging in direction to the maximum $\ell_2$-margin solution -- a phenomenon known as the implicit bias of GD. This work investigates…

Machine Learning · Computer Science 2025-07-01 Jingfeng Wu , Peter Bartlett , Matus Telgarsky , Bin Yu

We show that a variety of modern deep learning tasks exhibit a "double-descent" phenomenon where, as we increase model size, performance first gets worse and then gets better. Moreover, we show that double descent occurs not just as a…

Machine Learning · Computer Science 2019-12-06 Preetum Nakkiran , Gal Kaplun , Yamini Bansal , Tristan Yang , Boaz Barak , Ilya Sutskever

Recently, the benefit of heavily overparameterized models has been observed in machine learning tasks: models with enough capacity to easily cross the \emph{interpolation threshold} improve in generalization error compared to the classical…

High Energy Physics - Experiment · Physics 2025-09-03 Matthias Vigl , Lukas Heinrich

We investigate the asymptotic risk of a general class of overparameterized likelihood models, including deep models. The recent empirical success of large-scale models has motivated several theoretical studies to investigate a scenario…

Machine Learning · Statistics 2021-03-16 Ryumei Nakada , Masaaki Imaizumi

Increasing the size of overparameterized neural networks has been a key in achieving state-of-the-art performance. This is captured by the double descent phenomenon, where the test loss follows a decreasing-increasing-decreasing pattern (or…

Machine Learning · Computer Science 2024-05-09 Yihao Xue , Kyle Whitecross , Baharan Mirzasoleiman

Conventional statistical wisdom established a well-understood relationship between model complexity and prediction error, typically presented as a U-shaped curve reflecting a transition between under- and overfitting regimes. However,…

Machine Learning · Statistics 2023-10-31 Alicia Curth , Alan Jeffares , Mihaela van der Schaar

The double descent curve is one of the most intriguing properties of deep neural networks. It contrasts the classical bias-variance curve with the behavior of modern neural networks, occurring where the number of samples nears the number of…

Machine Learning · Computer Science 2021-07-05 John Chen , Qihan Wang , Anastasios Kyrillidis