Overfitting — When a Model Knows the Data Better Than the World

A polynomial with enough coefficients can pass through every point in a dataset. A neural network with enough parameters can memorize every label in a training set. Both achievements look like success until the model is asked to predict something it has not seen. The polynomial produces impossible values between its fitted points. The neural network reproduces random noise.

Overfitting occurs when a model captures patterns that are specific to the data used to fit it but do not generalize to new data. The model knows the training set perfectly. It does not know the world.

The problem is structural. Adding parameters always improves fit on the data used to fit them. A model with as many parameters as data points can reproduce the data exactly. That exact fit does not mean the model has learned anything useful. It means the model has learned the training set.

The question is how to understand it, how to measure it, and how to prevent it. The answer has evolved over decades across different fields, from statistics to physics to machine learning. And the answer is not what most textbooks claim.

The bias-variance decomposition

The standard explanation for overfitting is the bias-variance decomposition. It was formalized in a 1992 paper by Stuart Geman, Armand Bienenstock, and René Doursat, who applied techniques from statistical physics to neural networks. The decomposition shows that prediction error at a new data point can be split into three components.

Bias measures how far the average prediction of a model is from the true relationship. A model with too few parameters has high bias. It cannot capture the structure in the data. It underfits.

Variance measures how much the model’s predictions change when the training data changes. A model with too many parameters has high variance. It is sensitive to the particular observations in the training set. It overfits.

The third component is irreducible error. It is the noise inherent in the data-generating process. No model can eliminate it.

The bias-variance trade-off describes the tension between the first two components. As model capacity increases, bias decreases and variance increases. The total error follows a U-shaped curve. At low capacity, bias dominates and error is high. At high capacity, variance dominates and error is high. Somewhere in the middle, the two balance and error is minimized.

This curve is one of the most widely taught concepts in machine learning. It is also, in a way that researchers did not expect, incomplete.

The double descent surprise

In 2018 and 2019, two independent research groups discovered that the U-shaped bias-variance curve does not describe what happens when model capacity continues to increase beyond the point where it can perfectly fit the training data.

Mikhail Belkin, Daniel Hsu, Siyuan Ma, and Soumik Mandal at UCLA studied polynomial regression and two-layer neural networks. They found that after the model reaches the point of zero training error (the interpolation threshold), adding more capacity does not increase test error. It continues to decrease. The curve does not go back up. It goes down again.

Preetum Nakkiran, Gal Kaplun, Yamini Bansal, Tom Ding, Daniel L. K. Yamins, and Benjamin Barak at NYU and Princeton made the same discovery in the context of deep neural networks. They trained until the model attained zero training error and continued training with more parameters. Test error continued to improve.

The phenomenon is called double descent. The test error curve has two peaks. The first peak occurs at the point where model capacity equals the number of data points. The model is just complex enough to memorize the training data. The second peak occurs at the point where the model is so large that it becomes unstable. Between the two peaks, the model generalizes better than the classical bias-variance curve predicts.

This contradicts the standard teaching. The classical curve says that a model that fits the training data perfectly is overfitting and will fail on new data. Double descent shows that a model can fit the training data perfectly and still generalize better than a simpler model.

The explanation is not settled. Several mechanisms have been proposed. One is that overparameterized models have many solutions that fit the training data, and optimization algorithms like gradient descent tend to find the simplest ones. Another is that the interpolation regime is mathematically well-behaved in high dimensions, even though it is not in low dimensions. A third is that noise in the data is structured in ways that allow perfect fit without sacrificing generalization.

The mathematics of cross-validation

Before double descent, the primary tool for detecting overfitting was cross-validation. The method was formalized by Peter Stone in a 1974 paper published in the Annals of Statistics. Stone showed that leave-one-out cross-validation is asymptotically equivalent to a resampling method that estimates the prediction error by fitting the model to all but one data point and evaluating it on the held-out point, repeated for every data point in the set.

Cross-validation works by separating fitting from evaluation. The model is trained on a subset of the data and tested on data it has not seen. If the test performance is much worse than the training performance, the model is overfitting. The gap between training and test error is the empirical measure of overfitting.

Cross-validation is computationally expensive. It requires fitting the model multiple times. It requires holding out data that could otherwise be used for training. It requires storage for multiple model versions. But it does not require assumptions about the likelihood function or the parameter space. It is a direct empirical measurement.

Information criteria such as AIC and BIC offer an alternative. They estimate generalization performance from the training data alone, using a mathematical penalty for complexity. They do not require held-out data. But they require a well-specified likelihood function and assume that the model is fit by maximum likelihood. They are faster than cross-validation but less flexible.

Overfitting beyond machine learning

Overfitting is not a problem unique to machine learning. It appears in every discipline that builds models from data.

In psychology, the replication crisis was partly an overfitting problem. Researchers tested hypotheses on the same data used to generate them. The results looked significant because the models fit the noise in the data. The phenomenon was described as p-hacking or data dredging, but the underlying mechanism is the same: a model that is too flexible relative to the amount of data captures patterns that do not exist in the population.

In physics, overfitting appears in the form of spurious correlations in experimental data. A model with too many free parameters can fit any dataset, including one dominated by measurement noise. The challenge is to distinguish genuine physical relationships from coincidental patterns.

In economics, overfitting appears in the form of forecasting models that perform well on historical data but fail in production. Financial markets have more noise than signal. A model that fits historical price movements perfectly is likely fitting noise rather than predicting future movements.

The common thread is the same. A model with too many parameters relative to the amount of data will capture patterns that do not generalize. The solution is always the same in principle: constrain the model, collect more data, or evaluate on held-out data. The solution is rarely straightforward in practice.

What prevents overfitting

Regularization is the most common method for preventing overfitting in machine learning. It adds a penalty term to the loss function that discourages large parameter values. L1 regularization (Lasso) encourages sparsity by penalizing the absolute value of the parameters. L2 regularization (Ridge) encourages small parameters by penalizing their squared values.

Both penalties are mathematically similar to the complexity terms in AIC and BIC. In fact, L2 regularization is equivalent to adding a Bayesian prior on the parameters. The connection is not coincidental. All three methods share the same goal: penalize complexity to improve generalization.

Early stopping is another form of regularization. Training is halted before the model fully converges on the training data. The training iterations serve as a proxy for model complexity, and stopping early prevents the model from memorizing noise.

Dropout is a regularization method specific to neural networks. It randomly disables a fraction of neurons during each training step. This prevents any single neuron from becoming reliant on the presence of others, forcing the network to learn redundant representations.

Each method has different assumptions about what kind of complexity is dangerous. L1 regularization assumes that the true model is sparse. L2 regularization assumes that the true model has small parameters. Early stopping assumes that the optimization process encounters good solutions before bad ones. Dropout assumes that redundancy is robust.

These assumptions are not always correct. When they are wrong, regularization can hurt performance rather than help it. The best regularization method depends on the data, the model, and the task. There is no universal solution.

What remains uncertain

The double descent phenomenon has not been fully explained. Several mechanisms have been proposed, but none has been proven to be the dominant factor in all settings. The relationship between model capacity, data size, and generalization performance is more complex than the classical bias-variance curve suggests.

The applicability of double descent to real-world machine learning is an open question. Most experiments demonstrating the phenomenon use synthetic data or simplified models. Whether it holds for large-scale models trained on real-world data with complex noise structures is still being investigated.

The relationship between regularization methods and double descent is not understood. Do regularized models exhibit double descent? If so, at what point does the second descent begin? How does the choice of regularization affect the shape of the curve?

The fundamental question remains: when does a model know the data too well? The classical answer is that it happens when the model is more complex than the data can support. The double descent discovery suggests that the answer is more nuanced. A model can know the data perfectly and still generalize well. The boundary between overfitting and generalization is not where most textbooks say it is.

Primary sources

• Stone, M. (1974). “Cross-validatory choice and assessment of statistical predictions.” Journal of the Royal Statistical Society, Series B, 36(2), 111-147. The formalization of cross-validation and its asymptotic equivalence to leave-one-out resampling.
• Geman, S., Bienenstock, E., & Doursat, R. (1992). “Neural networks and the bias/variance decomposition.” Neural Computation, 4(2), 135-152. The bias-variance decomposition applied to neural networks.
• Belkin, M., Hsu, D., Ma, S., & Mandal, S. (2019). “Reconciling modern machine-learning practice and the bias-variance trade-off.” Proceedings of the National Academy of Sciences, 117(32), 19542-19550. The discovery of double descent in polynomial regression and neural networks.
• Nakkiran, P., Kaplun, G., Bansal, Y., Ding, T., Yamins, D. L. K., & Barak, B. (2021). “Deep double descent: Where bigger models and more data hurt.” Journal of Statistical Mechanics: Theory and Experiment, 2021(12), 124003. The discovery of double descent in deep neural networks.