A generalization gap estimation for overparameterized models via the Langevin functional variance
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This paper discusses the estimation of the generalization gap, the difference between a generalization error and an empirical error, for overparameterized models (e.g., neural networks). We first show that a functional variance, a key concept in defining a widely-applicable information criterion, characterizes the generalization gap even in overparameterized settings where a conventional theory cannot be applied. We also propose a computationally efficient approximation of the function variance, the Langevin approximation of the functional variance (Langevin FV). This method leverages only the 1st-order gradient of the squared loss function, without referencing the 2nd-order gradient; this ensures that the computation is efficient and the implementation is consistent with gradient-based optimization algorithms. We demonstrate the Langevin FV numerically by estimating the generalization gaps of overparameterized linear regression and non-linear neural network models.
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