On the Benefits of Over-parameterization for Out-of-Distribution Generalization

In recent years, machine learning models have achieved success based on the independently and identically distributed assumption. However, this assumption can be easily violated in real-world applications, leading to the Out-of-Distribution (OOD) problem. Understanding how modern over-parameterized DNNs behave under non-trivial natural distributional shifts is essential, as current theoretical understanding is insufficient. Existing theoretical works often provide meaningless results for over-parameterized models in OOD scenarios or even contradict empirical findings. To this end, we are investigating the performance of the over-parameterized model in terms of OOD generalization under the general benign overfitting conditions. Our analysis focuses on a random feature model and examines non-trivial natural distributional shifts, where the benign overfitting estimators demonstrate a constant excess OOD loss, despite achieving zero excess in-distribution (ID) loss. We demonstrate that in this scenario, further increasing the model's parameterization can significantly reduce the OOD loss. Intuitively, the variance term of ID loss remains low due to orthogonality of long-tail features, meaning overfitting noise during training generally doesn't raise testing loss. However, in OOD cases, distributional shift increases the variance term. Thankfully, the inherent shift is unrelated to individual x, maintaining the orthogonality of long-tail features. Expanding the hidden dimension can additionally improve this orthogonality by mapping the features into higher-dimensional spaces, thereby reducing the variance term. We further show that model ensembles also improve OOD loss, akin to increasing model capacity. These insights explain the empirical phenomenon of enhanced OOD generalization through model ensembles, supported by consistent simulations with theoretical results.