Benign, Tempered, or Catastrophic: A Taxonomy of Overfitting (2207.06569v3)

Abstract: The practical success of overparameterized neural networks has motivated the recent scientific study of interpolating methods, which perfectly fit their training data. Certain interpolating methods, including neural networks, can fit noisy training data without catastrophically bad test performance, in defiance of standard intuitions from statistical learning theory. Aiming to explain this, a body of recent work has studied benign overfitting, a phenomenon where some interpolating methods approach Bayes optimality, even in the presence of noise. In this work we argue that while benign overfitting has been instructive and fruitful to study, many real interpolating methods like neural networks do not fit benignly: modest noise in the training set causes nonzero (but non-infinite) excess risk at test time, implying these models are neither benign nor catastrophic but rather fall in an intermediate regime. We call this intermediate regime tempered overfitting, and we initiate its systematic study. We first explore this phenomenon in the context of kernel (ridge) regression (KR) by obtaining conditions on the ridge parameter and kernel eigenspectrum under which KR exhibits each of the three behaviors. We find that kernels with powerlaw spectra, including Laplace kernels and ReLU neural tangent kernels, exhibit tempered overfitting. We then empirically study deep neural networks through the lens of our taxonomy, and find that those trained to interpolation are tempered, while those stopped early are benign. We hope our work leads to a more refined understanding of overfitting in modern learning.

• The paper introduces a novel taxonomy that classifies overfitting in neural networks into benign, tempered, and catastrophic regimes.
• It derives spectral conditions in kernel regression linking eigenvalue decay to distinct overfitting behaviors and generalization error.
• Empirical results on synthetic data and CIFAR-10 demonstrate that DNNs often exhibit tempered overfitting, achieving bounded test performance.

An Analysis of Overfitting: Introducing a Taxonomy

The recent work by Mallinar et al., "Benign, Tempered, or Catastrophic: A Taxonomy of Overfitting," presents a nuanced perspective on the phenomenon of overfitting in the context of modern machine learning models, particularly neural networks. Overfitting is traditionally understood within classical statistical learning theory as a behavior where models, in their attempt to perfectly fit training data, end up with poor generalization on unseen data. However, with the advent of overparameterized models such as deep neural networks (DNNs), this conventional understanding is challenged. This paper introduces a novel classification system for overfitting behaviors: benign, tempered, and catastrophic.

Overview of Overfitting Taxonomy

Mallinar et al. argue that overfitting behaviors can be systematically categorized as follows:

1. Benign Overfitting: Algorithms demonstrate benign overfitting when they achieve near-optimal generalization while fully fitting the training data, even in the presence of noise. This is contrary to the classical intuition which would predict poor test performance. An example given is the Nadaraya-Watson kernel smoothing technique with specific kernel choices leading to this behavior.
2. Tempered Overfitting: The paper identifies a middle ground termed tempered overfitting. In this regime, while the model does not generalize perfectly, its generalization error remains finite and bounded away from the Bayes-optimal risk. Tempered overfitting represents a scenario where the model's ability to generalize degrades gracefully with noise, rather than collapses catastrophically. The authors provide evidence that DNNs trained to interpolation often fall into this category.
3. Catastrophic Overfitting: Finally, catastrophic overfitting is the scenario most aligned with classical theory, where overfitting results in horrendous generalization, exemplified by polynomial interpolation of data or Gaussian kernel regression in some settings.

Spectral Conditions in Kernel Regression

A significant portion of the paper is devoted to understanding these overfitting behaviors through the lens of kernel regression (KR). The authors derive conditions on the eigenspectrum of kernels that correspond to each type of overfitting. They document that kernels with eigenvalues decaying faster than any set powerlaw are prone to benign overfitting, while those with powerlaw decay demonstrate tempered overfitting. Gaussian kernels without ridge regularization typically result in catastrophic overfitting.

The theoretical analysis for KR is built on the expected test mean squared error, leveraging recent advances connecting kernel theory and high-dimensional statistical physics. Through this formulation, clear conditions are proposed under which each overfitting regime is likely to occur based on the ridge parameter and the structure of the kernel’s eigenspectrum.

Empirical Investigation and Real-world Implications

Mallinar et al. extend their theoretical findings through empirical analysis on both synthetic data and standard datasets like CIFAR-10 using DNNs. They show that many neural networks commonly used in practice exhibit tempered overfitting when trained to interpolation, characterized by a persistent but non-exploding test error as data noise increases.

The practical implications of this research are profound, particularly in understanding and optimizing the training methods of neural networks. Tempered overfitting suggests that achieving perfect training accuracy with an overparameterized model does not necessarily lead to generalized failure. This understanding can lead to improved strategies for training robust models.

Future Directions and Theoretical Insights

The introduction of tempered overfitting opens several avenues for further investigation. How architectural choices, data dimensionality, and training methods influence which overfitting regime a model falls into remains an active area of research. Additionally, understanding the transition dynamics between these regimes, especially in iterative model training, holds potential for developing more theoretically grounded methods of early stopping.

As DNNs and other machine learning paradigms become increasingly central to diverse applications, clarity on overfitting behaviors becomes critical. The taxonomy provided by this paper equips researchers with a more sophisticated framework to analyze and potentially mitigate the challenges posed by overfitting, aligning practical training outcomes with theoretical expectations.