# Fitting Elephants

arXiv: https://arxiv.org/abs/2104.00526 · HTML: https://mlsanity.com/paper/2104.00526/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 68 references that are in the map.
- First posted on arXiv 31 Mar 2021.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [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)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [Rainbow: Combining Improvements in Deep Reinforcement Learning](https://mlsanity.com/paper/1710.02298/) (2017)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [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)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [The Power of Interpolation: Understanding the Effectiveness of SGD in Modern Over-parametrized Learning](https://mlsanity.com/paper/1712.06559/) (2017)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization](https://mlsanity.com/paper/2008.06786/) (2020)
- [Overfitting or perfect fitting? Risk bounds for classification and regression rules that interpolate](https://mlsanity.com/paper/1806.05161/) (2018)
- [Exploring the Landscape of Spatial Robustness](https://mlsanity.com/paper/1712.02779/) (2017)
- [Explaining the Success of AdaBoost and Random Forests as Interpolating Classifiers](https://mlsanity.com/paper/1504.07676/) (2015)
- [Consistency of Interpolation with Laplace Kernels is a High-Dimensional Phenomenon](https://mlsanity.com/paper/1812.11167/) (2018)
- [A Brief Prehistory of Double Descent](https://mlsanity.com/paper/2004.04328/) (2020)
- [Understanding overfitting peaks in generalization error: Analytical risk curves for l2 and l1 penalized interpolation](https://mlsanity.com/paper/1906.03667/) (2019)
- [Benefit of Interpolation in Nearest Neighbor Algorithms](https://mlsanity.com/paper/1909.11720/) (2019)
- [Statistical Optimality of Interpolated Nearest Neighbor Algorithms](https://mlsanity.com/paper/1810.02814/) (2018)
- [Fast Convergence for Stochastic and Distributed Gradient Descent in the Interpolation Limit](https://mlsanity.com/paper/1803.02922/) (2018)

## Abstract (arXiv)

Textbook wisdom advocates for smooth function fits and implies that interpolation of noisy data should lead to poor generalization. A related heuristic is that fitting parameters should be fewer than measurements (Occam's Razor). Surprisingly, contemporary machine learning (ML) approaches, cf. deep nets (DNNs), generalize well despite interpolating noisy data. This may be understood via Statistically Consistent Interpolation (SCI), i.e. data interpolation techniques that generalize optimally for big data. In this article we elucidate SCI using the weighted interpolating nearest neighbors (wiNN) algorithm, which adds singular weight functions to kNN (k-nearest neighbors). This shows that data interpolation can be a valid ML strategy for big data. SCI clarifies the relation between two ways of modeling natural phenomena: the rationalist approach (strong priors) of theoretical physics with few parameters and the empiricist (weak priors) approach of modern ML with more parameters than data. SCI shows that the purely empirical approach can successfully predict. However data interpolation does not provide theoretical insights, and the training data requirements may be prohibitive. Complex animal brains are between these extremes, with many parameters, but modest training data, and with prior structure encoded in species-specific mesoscale circuitry. Thus, modern ML provides a distinct epistemological approach different both from physical theories and animal brains.