Optimistic Rates: A Unifying Theory for Interpolation Learning and Regularization in Linear Regression
Lijia Zhoua b, Frederic Koehlerc d, Danica J. Sutherlande f g, Nathan Srebroh i
- aDepartment of Statistics, University of Chicago
- bDepartment of Statistics, University of Chicago, Chicago, USA
- cDepartment of Computer Science, Stanford University
- dDepartment of Computer Science, Stanford University, Stanford, USA
- eUniversity of British Columbia
- fAlberta Machine Intelligence Institute
- gAlberta Machine Intelligence Institute, Vancouver, Canada
- hToyota Technological Institute at Chicago
- iToyota Technological Institute at Chicago, Chicago, USA
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Abstract
We study a localized notion of uniform convergence known as an “optimistic rate” [ 34 , 39 ] for linear regression with Gaussian data. Our refined analysis avoids the hidden constant and logarithmic factor in existing results, which are known to be crucial in high-dimensional settings, especially for understanding interpolation learning. As a special case, our analysis recovers the guarantee from Koehler et al. [ 21 ], which tightly characterizes the population risk of low-norm interpolators under the benign overfitting conditions. Our optimistic rate bound, though, also analyzes predictors with arbitrary training error. This allows us to recover some classical statistical guarantees for ridge and LASSO regression under random designs and helps us obtain a precise understanding of the excess risk of near-interpolators in the over-parameterized regime.