When do Models Generalize? A Perspective from Data-Algorithm Compatibility

One of the major open problems in machine learning is to characterize generalization in the overparameterized regime, where most traditional generalization bounds become inconsistent (Nagarajan and Kolter, 2019). In many scenarios, their failure can be attributed to obscuring the crucial interplay between the training algorithm and the underlying data distribution. To address this issue, we propose a concept named compatibility, which quantitatively characterizes generalization in a both data-relevant and algorithm-relevant manner. By considering the entire training trajectory and focusing on early-stopping iterates, compatibility exploits the data and the algorithm information and is therefore a more suitable notion for generalization. We validate this by theoretically studying compatibility under the setting of solving overparameterized linear regression with gradient descent. Specifically, we perform a data-dependent trajectory analysis and derive a sufficient condition for compatibility in such a setting. Our theoretical results demonstrate that in the sense of compatibility, generalization holds with significantly weaker restrictions on the problem instance than the previous last iterate analysis

Generalization Error Bounds of Gradient Descent for Learning Over-parameterized Deep ReLU Networks

Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing generalization bounds is that they are based on uniform convergence and are independent of the training algorithm. In this work, we derive an algorithm-dependent generalization error bound for deep ReLU networks, and show that under certain assumptions on the data distribution, gradient descent (GD) with proper random initialization is able to train a sufficiently over-parameterized DNN to achieve arbitrarily small generalization error. Our work sheds light on explaining the good generalization performance of over-parameterized deep neural networks.Comment: 27 pages. This version simplifies the proof and improves the presentation in Version 3. In AAAI 202