Near-Interpolators: Rapid Norm Growth and the Trade-Off between Interpolation and Generalization

arXiv:2403.07264v1 Announce Type: cross
Abstract: We study the generalization capability of nearly-interpolating linear regressors: $\boldsymbol{\beta}$'s whose training error $\tau$ is positive but small, i.e., below the noise floor. Under a random matrix theoretic assumption on the data distribution and an eigendecay assumption on the data covariance matrix $\boldsymbol{\Sigma}$, we demonstrate that any near-interpolator exhibits rapid norm growth: for $\tau$ fixed, $\boldsymbol{\beta}$ has squared $\ell_2$-norm $\mathbb{E}[\|{\boldsymbol{\beta}}\|_{2}^{2}] = \Omega(n^{\alpha})$ where $n$ is the number of samples and $\alpha >1$ is the exponent …

abstract arxiv beta capability covariance cs.lg data distribution error growth linear matrix near noise norm positive random small stat.ml study trade trade-off training type