Architecture independent generalization bounds for overparametrized deep ReLU networks (2504.05695v3)

Abstract: We prove that overparametrized neural networks are able to generalize with a test error that is independent of the level of overparametrization, and independent of the Vapnik-Chervonenkis (VC) dimension. We prove explicit bounds that only depend on the metric geometry of the test and training sets, on the regularity properties of the activation function, and on the operator norms of the weights and norms of biases. For overparametrized deep ReLU networks with a training sample size bounded by the input space dimension, we explicitly construct zero loss minimizers without use of gradient descent, and prove that the generalization error is independent of the network architecture.

Generalization Analysis of Overparametrized Deep ReLU Networks

The paper "Architecture Independent Generalization Bounds for Overparametrized Deep ReLU Networks" by Thomas Chen et al. addresses a fundamental question in the field of machine learning: under what conditions do overparametrized neural networks generalize well, independent of their architecture? Overparametrization, a common feature in contemporary deep learning models, presents unique challenges and opportunities, specifically concerning the generalization error that persists irrespective of the model's structural complexity.

Major Contributions

The authors establish that overparametrized deep ReLU networks can achieve zero test error, independent of overparametrization level and VC dimension. The paper proposes explicit generalization bounds contingent upon the metric geometry of the training and test datasets, the regularity properties of the activation function, and the operator norms of weights and biases.

1. Zero Loss Minimization: The authors construct explicit zero loss minimizers without resorting to gradient descent for strongly overparametrized networks, presenting a state where the network achieves zero training loss. Notably, this is achievable under conditions where the sample size is limited by the input space dimension $n \leq M_0$.
2. Independent Generalization Bound: The investigation reveals that generalization error bounds are independent of the network architecture. The critical factor influencing these bounds is the unidirectional Chamfer pseudodistance between test and training datasets, measured by how well the test set can be approximated by the training set.
3. VC Dimension and Overparametrization: The paper critiques traditional bounds relying on the VC dimension, which deteriorate with increased overparametrization. It establishes that these bounds are disconnected from the specifics of the data and not necessarily reflective of practical scenarios where both training and test data might be non-identically distributed.

Implications

The implications of this paper are profound in both theoretical and practical landscapes of machine learning. The independence of generalization error from network architecture challenges conventional reliance on the VC dimension as a yardstick for capacity. It proposes a shift towards data-centric measures, such as the metric properties of data distributions.

From a practical standpoint, these insights underpin the rationale behind successful deployment of overparametrized models in tasks requiring high fidelity approximations. The paper provides a theoretical foundation for practices previously supported by empirical evidence alone, such as using minimalistic regularization techniques, like L2 regularization or weight decay.

In theoretical development, this paper urges consideration of geometric properties of the dataset when analyzing model generalization, as opposed to strictly topology-based conjectures offered by measures such as VC dimension.

Future Directions

The findings prompt several avenues for future exploration. First, extending the analysis to non-ReLU activations or different overparametrization mechanisms entails promising research. Moreover, generalizing these results to other settings, such as unsupervised learning systems or reinforcement learning architectures, could elucidate broader aspects of learning theory.

Additionally, investigating practical implications of the metric space properties suggested, especially in data-driven fields like computer vision or natural language processing, could enhance model robustness and performance. Exploring the statistical intricacies underlying probability distributions of training and testing datasets further aligns theoretical advancements with real-world applications.

The paper by Chen et al. makes significant strides in unshackling generalization performance from the confines of network architecture. It lays crucial groundwork for understanding and leveraging the strengths of overparametrized models in contemporary machine learning tasks.