An official website of the United States government
This work examines the deep disconnect between existing theoretical analyses of gradient-based algorithms and the practice of training deep neural networks. Specifically, we provide numerical evidence that in large-scale neural network training (e.g., ImageNet + ResNet101, and WT103 + TransformerXL models), the neural network’s weights do not converge to stationary points where the gradient of the loss is zero. Remarkably, however, we observe that even though the weights do not converge to stationary points, the progress in minimizing the loss function halts and training loss stabilizes. Inspired by this observation, we propose a new perspective based on ergodic theory of dynamical systems to explain it. Rather than studying the evolution of weights, we study the evolution of the distribution of weights. We prove convergence of the distribution of weights to an approximate invariant measure, thereby explaining how the training loss can stabilize without weights necessarily converging to stationary points. We further discuss how this perspective can better align optimization theory with empirical observations in machine learning practice.
more »
« less
On the generalization of learning algorithms that do not converge.
Generalization analyses of deep learning typically assume that the training converges to a fixed point. But, recent results indicate that in practice, the weights of deep neural networks optimized with stochastic gradient descent often oscillate indefinitely. To reduce this discrepancy between theory and practice, this paper focuses on the generalization of neural networks whose training dynamics do not necessarily converge to fixed points. Our main contribution is to propose a notion of statistical algorithmic stability (SAS) that extends classical algorithmic stability to non-convergent algorithms and to study its connection to generalization. This ergodic-theoretic approach leads to new insights when compared to the traditional optimization and learning theory perspectives. We prove that the stability of the time-asymptotic behavior of a learning algorithm relates to its generalization and empirically demonstrate how loss dynamics can provide clues to generalization performance. Our findings provide evidence that networks that “train stably generalize better” even when the training continues indefinitely and the weights do not converge.
more »
« less
- Award ID(s):
- 2134108
- PAR ID:
- 10468309
- Publisher / Repository:
- Neural Information Processing Systems (NeurIPS)
- Date Published:
- Subject(s) / Keyword(s):
- machine learning, generalization, dynamical systems
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)We study how neural networks trained by gradient descent extrapolate, i.e., what they learn outside the support of the training distribution. Previous works report mixed empirical results when extrapolating with neural networks: while feedforward neural networks, a.k.a. multilayer perceptrons (MLPs), do not extrapolate well in certain simple tasks, Graph Neural Networks (GNNs) – structured networks with MLP modules – have shown some success in more complex tasks. Working towards a theoretical explanation, we identify conditions under which MLPs and GNNs extrapolate well. First, we quantify the observation that ReLU MLPs quickly converge to linear functions along any direction from the origin, which implies that ReLU MLPs do not extrapolate most nonlinear functions. But, they can provably learn a linear target function when the training distribution is sufficiently “diverse”. Second, in connection to analyzing the successes and limitations of GNNs, these results suggest a hypothesis for which we provide theoretical and empirical evidence: the success of GNNs in extrapolating algorithmic tasks to new data (e.g., larger graphs or edge weights) relies on encoding task-specific non-linearities in the architecture or features. Our theoretical analysis builds on a connection of over-parameterized networks to the neural tangent kernel. Empirically, our theory holds across different training settings.more » « less
-
We study how neural networks trained by gradient descent extrapolate, i.e., what they learn outside the support of the training distribution. Previous works report mixed empirical results when extrapolating with neural networks: while feedforward neural networks, a.k.a. multilayer perceptrons (MLPs), do not extrapolate well in certain simple tasks, Graph Neural Networks (GNNs) – structured networks with MLP modules – have shown some success in more complex tasks. Working towards a theoretical explanation, we identify conditions under which MLPs and GNNs extrapolate well. First, we quantify the observation that ReLU MLPs quickly converge to linear functions along any direction from the origin, which implies that ReLU MLPs do not extrapolate most nonlinear functions. But, they can provably learn a linear target function when the training distribution is sufficiently “diverse”. Second, in connection to analyzing the successes and limitations of GNNs, these results suggest a hypothesis for which we provide theoretical and empirical evidence: the success of GNNs in extrapolating algorithmic tasks to new data (e.g., larger graphs or edge weights) relies on encoding task-specific non-linearities in the architecture or features. Our theoretical analysis builds on a connection of over-parameterized networks to the neural tangent kernel. Empirically, our theory holds across different training settings.more » « less
-
The prosperity of deep learning and automated machine learning (AutoML) is largely rooted in the development of novel neural networks -- but what defines and controls the "goodness" of networks in an architecture space? Test accuracy, a golden standard in AutoML, is closely related to three aspects: (1) expressivity (how complicated functions a network can approximate over the training data); (2) convergence (how fast the network can reach low training error under gradient descent); (3) generalization (whether a trained network can be generalized from the training data to unseen samples with low test error). However, most previous theory papers focus on fixed model structures, largely ignoring sophisticated networks used in practice. To facilitate the interpretation and understanding of the architecture design by AutoML, we target connecting a bigger picture: how does the architecture jointly impact its expressivity, convergence, and generalization? We demonstrate the "no free lunch" behavior in networks from an architecture space: given a fixed budget on the number of parameters, there does not exist a single architecture that is optimal in all three aspects. In other words, separately optimizing expressivity, convergence, and generalization will achieve different networks in the architecture space. Our analysis can explain a wide range of observations in AutoML. Experiments on popular benchmarks confirm our theoretical analysis. Our codes are attached in the supplement.more » « less
-
Neural Collapse refers to the curious phenomenon in the end of training of a neural network, where feature vectors and classification weights converge to a very simple geometrical arrangement (a simplex). While it has been observed empirically in various cases and has been theoretically motivated, its connection with crucial properties of neural networks, like their generalization and robustness, remains unclear. In this work, we study the stability properties of these simplices. We find that the simplex structure disappears under small adversarial attacks, and that perturbed examples "leap" between simplex vertices. We further analyze the geometry of networks that are optimized to be robust against adversarial perturbations of the input, and find that Neural Collapse is a pervasive phenomenon in these cases as well, with clean and perturbed representations forming aligned simplices, and giving rise to a robust simple nearest-neighbor classifier. By studying the propagation of the amount of collapse inside the network, we identify novel properties of both robust and non-robust machine learning models, and show that earlier, unlike later layers maintain reliable simplices on perturbed data.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
