Towards Understanding Generalization via Decomposing Excess Risk Dynamics
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by
Jiaye Teng, Jianhao Ma, Yang Yuan
2021
Abstract
Generalization is one of the critical issues in machine learning. However,
traditional methods like uniform convergence are not powerful enough to fully
explain generalization because they may yield vacuous bounds even in
overparameterized linear regression regimes. An alternative solution is to
analyze the generalization dynamics to derive algorithm-dependent bounds, e.g.,
stability. Unfortunately, the stability-based bound is still far from
explaining the remarkable generalization ability of neural networks due to the
coarse-grained analysis of the signal and noise. Inspired by the observation
that neural networks show a slow convergence rate when fitting noise, we
propose decomposing the excess risk dynamics and applying stability-based bound
only on the variance part (which measures how the model performs on pure
noise). We provide two applications for the framework, including a linear case
(overparameterized linear regression with gradient descent) and a non-linear
case (matrix recovery with gradient flow). Under the decomposition framework,
the new bound accords better with the theoretical and empirical evidence
compared to the stability-based bound and uniform convergence bound.
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