Memorizing without overfitting: Bias, variance, and interpolation in over-parameterized models
This addresses a fundamental problem in machine learning for researchers and practitioners by providing a holistic understanding of generalization error in over-parameterized models, though it is incremental in building on existing theories.
The paper tackles the problem of understanding bias and variance in over-parameterized models, which challenge the classical bias-variance trade-off by achieving state-of-the-art performance despite fitting training data perfectly. Using statistical physics methods on linear regression and two-layer neural networks, they find that increasing parameters leads to a phase transition where test error diverges due to variance, but beyond this, both bias and variance decrease monotonically, reducing test error.
The bias-variance trade-off is a central concept in supervised learning. In classical statistics, increasing the complexity of a model (e.g., number of parameters) reduces bias but also increases variance. Until recently, it was commonly believed that optimal performance is achieved at intermediate model complexities which strike a balance between bias and variance. Modern Deep Learning methods flout this dogma, achieving state-of-the-art performance using "over-parameterized models" where the number of fit parameters is large enough to perfectly fit the training data. As a result, understanding bias and variance in over-parameterized models has emerged as a fundamental problem in machine learning. Here, we use methods from statistical physics to derive analytic expressions for bias and variance in two minimal models of over-parameterization (linear regression and two-layer neural networks with nonlinear data distributions), allowing us to disentangle properties stemming from the model architecture and random sampling of data. In both models, increasing the number of fit parameters leads to a phase transition where the training error goes to zero and the test error diverges as a result of the variance (while the bias remains finite). Beyond this threshold, the test error of the two-layer neural network decreases due to a monotonic decrease in \emph{both} the bias and variance in contrast with the classical bias-variance trade-off. We also show that in contrast with classical intuition, over-parameterized models can overfit even in the absence of noise and exhibit bias even if the student and teacher models match. We synthesize these results to construct a holistic understanding of generalization error and the bias-variance trade-off in over-parameterized models and relate our results to random matrix theory.