Feature-Robustness, Flatness and Generalization Error for Deep Neural Networks

The performance of deep neural networks is often attributed to their automated, task-related feature construction. It remains an open question, though, why this leads to solutions with good generalization, even in cases where the number of parameters is larger than the number of samples. Back in the 90s, Hochreiter and Schmidhuber observed that flatness of the loss surface around a local minimum correlates with low generalization error. For several flatness measures, this correlation has been empirically validated. However, it has recently been shown that existing measures of flatness cannot theoretically be related to generalization: if a network uses ReLU activations, the network function can be reparameterized without changing its output in such a way that flatness is changed almost arbitrarily. This paper proposes a natural modification of existing flatness measures that results in invariance to reparameterization. The proposed measures imply a robustness of the network to changes in the input and the hidden layers. Connecting this feature robustness to generalization leads to a generalized definition of the representativeness of data. With this, the generalization error of a model trained on representative data can be bounded by its feature robustness which depends on our novel flatness measure.