Understanding Double Descent Requires a Fine-Grained Bias-Variance Decomposition
This work addresses a foundational problem in machine learning theory by providing insights into the generalization behavior of overparameterized models, which is incremental but clarifies mechanisms behind double descent.
The paper tackled the challenge of explaining double descent in deep learning by developing a fine-grained bias-variance decomposition that accounts for randomness from sampling, initialization, and labels, finding that bias decreases with network width while variance terms can diverge at the interpolation boundary due to interactions, which can be mitigated by bagging or ensemble learning.
Classical learning theory suggests that the optimal generalization performance of a machine learning model should occur at an intermediate model complexity, with simpler models exhibiting high bias and more complex models exhibiting high variance of the predictive function. However, such a simple trade-off does not adequately describe deep learning models that simultaneously attain low bias and variance in the heavily overparameterized regime. A primary obstacle in explaining this behavior is that deep learning algorithms typically involve multiple sources of randomness whose individual contributions are not visible in the total variance. To enable fine-grained analysis, we describe an interpretable, symmetric decomposition of the variance into terms associated with the randomness from sampling, initialization, and the labels. Moreover, we compute the high-dimensional asymptotic behavior of this decomposition for random feature kernel regression, and analyze the strikingly rich phenomenology that arises. We find that the bias decreases monotonically with the network width, but the variance terms exhibit non-monotonic behavior and can diverge at the interpolation boundary, even in the absence of label noise. The divergence is caused by the \emph{interaction} between sampling and initialization and can therefore be eliminated by marginalizing over samples (i.e. bagging) \emph{or} over the initial parameters (i.e. ensemble learning).