  Modern machine learning methods are often overparametrized, allowing
adaptation to the data at a fine level. This can seem puzzling; in the worst
case, such models do not need to generalize. This puzzle inspired a great
amount of work, arguing when overparametrization reduces test error, in a
phenomenon called "double descent". Recent work aimed to understand in greater
depth why overparametrization is helpful for generalization. This leads to
discovering the unimodality of variance as a function of the level of
parametrization, and to decomposing the variance into that arising from label
noise, initialization, and randomness in the training data to understand the
sources of the error.
  In this work we develop a deeper understanding of this area. Specifically, we
propose using the analysis of variance (ANOVA) to decompose the variance in the
test error in a symmetric way, for studying the generalization performance of
certain two-layer linear and non-linear networks. The advantage of the analysis
of variance is that it reveals the effects of initialization, label noise, and
training data more clearly than prior approaches. Moreover, we also study the
monotonicity and unimodality of the variance components. While prior work
studied the unimodality of the overall variance, we study the properties of
each term in variance decomposition.
  One key insight is that in typical settings, the interaction between training
samples and initialization can dominate the variance; surprisingly being larger
than their marginal effect. Also, we characterize "phase transitions" where the
variance changes from unimodal to monotone. On a technical level, we leverage
advanced deterministic equivalent techniques for Haar random matrices, that --
to our knowledge -- have not yet been used in the area. We also verify our
results in numerical simulations and on empirical data examples.
