Double descent

For the concept of double descent in anthropology, see Kinship § Descent rules.

Double descent in statistics and machine learning is the phenomenon where a model with a small number of parameters and a model with an extremely large number of parameters both have a small training error, but a model whose number of parameters is about the same as the number of data points used to train the model will have a much greater test error than one with a much larger number of parameters.^[2] This phenomenon has been considered surprising, as it contradicts assumptions about overfitting in classical machine learning.^[3]

An example of the double descent phenomenon in a two-layer neural network: as the ratio of parameters to data points increases, the test error first falls, then rises, then falls again.^[1] The vertical line marks the "interpolation threshold" boundary between the underparametrized region (more data points than parameters) and the overparameterized region (more parameters than data points).

History

Early observations of what would later be called double descent in specific models date back to 1989.^[4][5]

The term "double descent" was coined by Belkin et. al.^[6] in 2019,^[3] when the phenomenon gained popularity as a broader concept exhibited by many models.^[7][8] The latter development was prompted by a perceived contradiction between the conventional wisdom that too many parameters in the model result in a significant overfitting error (an extrapolation of the bias–variance tradeoff),^[9] and the empirical observations in the 2010s that some modern machine learning techniques tend to perform better with larger models.^[6][10]

Theoretical models

Double descent occurs in linear regression with isotropic Gaussian covariates and isotropic Gaussian noise.^[11]

A model of double descent at the thermodynamic limit has been analyzed using the replica trick, and the result has been confirmed numerically.^[12]

Empirical examples

The scaling behavior of double descent has been found to follow a broken neural scaling law^[13] functional form.

See also

• Grokking (machine learning)