# Interpolation, extrapolation and memorisation in neural networks

January 25, 2022 — October 30, 2024

Interpreting models in terms of their ability to interpolate and memorise, and when that becomes extrapolation. Connection to neural scaling, overparameterization, and possibly singular learning theory.

Balestriero, Pesenti, and LeCun (2021):

The notion of interpolation and extrapolation is fundamental in various fields from deep learning to function approximation. Interpolation occurs for a sample x whenever this sample falls inside or on the boundary of the given dataset’s convex hull. Extrapolation occurs when x falls outside of that convex hull. One fundamental (mis)conception is that state-of-the-art algorithms work so well because of their ability to correctly interpolate training data. A second (mis)conception is that interpolation happens throughout tasks and datasets, in fact, many intuitions and theories rely on that assumption. We empirically and theoretically argue against those two points and demonstrate that on any high-dimensional (>100) dataset, interpolation almost surely never happens. Those results challenge the validity of our current interpolation/extrapolation definition as an indicator of generalization performances.

Bubeck and Sellke (2021) argue

Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in the current practice of deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a theoretical explanation for this phenomenon. We prove that for a broad class of data distributions and model classes, overparametrization is

necessaryif one wants to interpolate the datasmoothly. Namely we show thatsmoothinterpolation requires \(d\) times more parameters than mere interpolation, where \(d\) is the ambient data dimension. We prove this universal law of robustness for any smoothly parametrized function class with polynomial size weights, and any covariate distribution verifying isoperimetry. In the case of two-layers neural networks and Gaussian covariates, this law was conjectured in prior work by Bubeck, Li and Nagaraj. We also give an interpretation of our result as an improved generalization bound for model classes consisting of smooth functions.

There are other works in this domain (Le 2018; Ma, Bassily, and Belkin 2018; Zhang et al. 2017, 2021).

I am not sure if this is a distinct thing from other double descent phenomena. Hastie et al. (2020) suggests perhaps not?

Interpolators — estimators that achieve zero training error — have attracted growing attention in machine learning, mainly because state-of-the-art neural networks appear to be models of this type. In this paper, we study minimum \(\ell_2\) norm (“ridgeless”) interpolation in high-dimensional least squares regression. We consider two different models for the feature distribution: a linear model, where the feature vectors \(x_i \in {\mathbb R}^p\) are obtained by applying a linear transform to a vector of i.i.d. entries, \(x_i = \Sigma^{1/2} z_i\) (with \(z_i \in {\mathbb R}^p\)); and a nonlinear model, where the feature vectors are obtained by passing the input through a random one-layer neural network, \(x_i = \varphi(W z_i)\) (with \(z_i \in {\mathbb R}^d\), \(W \in {\mathbb R}^{p \times d}\) a matrix of i.i.d. entries, and \(\varphi\) an activation function acting componentwise on \(W z_i\)). We recover — in a precise quantitative way — several phenomena that have been observed in large-scale neural networks and kernel machines, including the “double descent” behaviour of the prediction risk, and the potential benefits of overparametrization.

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