In Defense of Uniform Convergence: Generalization via derandomization with an application to interpolating predictors
This work addresses theoretical challenges in understanding generalization for interpolating models, which is incremental as it builds on prior studies by Nagarajan and Kolter (2019) and Bartlett et al. (2019).
The paper tackles the generalization error of interpolating predictors by introducing surrogate predictors that trade empirical risk for generalization control, and demonstrates that these surrogates can belong to classes with uniformly small generalization error while bounding the risk of the original predictor.
We propose to study the generalization error of a learned predictor $\hat h$ in terms of that of a surrogate (potentially randomized) predictor that is coupled to $\hat h$ and designed to trade empirical risk for control of generalization error. In the case where $\hat h$ interpolates the data, it is interesting to consider theoretical surrogate classifiers that are partially derandomized or rerandomized, e.g., fit to the training data but with modified label noise. We also show that replacing $\hat h$ by its conditional distribution with respect to an arbitrary $σ$-field is a convenient way to derandomize. We study two examples, inspired by the work of Nagarajan and Kolter (2019) and Bartlett et al. (2019), where the learned classifier $\hat h$ interpolates the training data with high probability, has small risk, and, yet, does not belong to a nonrandom class with a tight uniform bound on two-sided generalization error. At the same time, we bound the risk of $\hat h$ in terms of surrogates constructed by conditioning and denoising, respectively, and shown to belong to nonrandom classes with uniformly small generalization error.