Charles H. Martin

Hari K. Prakash, Charles H. Martin

We study the well-known grokking phenomena in neural networks (NNs) using a 3-layer MLP trained on 1 k-sample subset of MNIST, with and without weight decay, and discover a novel third phase -- \emph{anti-grokking} -- that occurs very late in training and resembles but is distinct from the familiar \emph{pre-grokking} phases: test accuracy collapses while training accuracy stays perfect. This late-stage collapse is distinct, from the known pre-grokking and grokking phases, and is not detected by other proposed grokking progress measures. Leveraging Heavy-Tailed Self-Regularization HTSR through the open-source WeightWatcher tool, we show that the HTSR layer quality metric $α$ alone delineates all three phases, whereas the best competing metrics detect only the first two. The \emph{anti-grokking} is revealed by training for $10^7$ and is invariably heralded by $α< 2$ and the appearance of \emph{Correlation Traps} -- outlier singular values in the randomized layer weight matrices that make the layer weight matrix atypical and signal overfitting of the training set. Such traps are verified by visual inspection of the layer-wise empirical spectral densities, and by using Kolmogorov--Smirnov tests on randomized spectra. Comparative metrics, including activation sparsity, absolute weight entropy, circuit complexity, and $l^2$ weight norms track pre-grokking and grokking but fail to distinguish grokking from anti-grokking. This discovery provides a way to measure overfitting and generalization collapse without direct access to the test data. These results strengthen the claim that the \emph{HTSR} $α$ provides universal layer-convergence target at $α\approx 2$ and underscore the value of using the HTSR alpha $(α)$ metric as a measure of generalization.

Yaoqing Yang, Ryan Theisen, Liam Hodgkinson et al.

Selecting suitable architecture parameters and training hyperparameters is essential for enhancing machine learning (ML) model performance. Several recent empirical studies conduct large-scale correlational analysis on neural networks (NNs) to search for effective \emph{generalization metrics} that can guide this type of model selection. Effective metrics are typically expected to correlate strongly with test performance. In this paper, we expand on prior analyses by examining generalization-metric-based model selection with the following objectives: (i) focusing on natural language processing (NLP) tasks, as prior work primarily concentrates on computer vision (CV) tasks; (ii) considering metrics that directly predict \emph{test error} instead of the \emph{generalization gap}; (iii) exploring metrics that do not need access to data to compute. From these objectives, we are able to provide the first model selection results on large pretrained Transformers from Huggingface using generalization metrics. Our analyses consider (I) hundreds of Transformers trained in different settings, in which we systematically vary the amount of data, the model size and the optimization hyperparameters, (II) a total of 51 pretrained Transformers from eight families of Huggingface NLP models, including GPT2, BERT, etc., and (III) a total of 28 existing and novel generalization metrics. Despite their niche status, we find that metrics derived from the heavy-tail (HT) perspective are particularly useful in NLP tasks, exhibiting stronger correlations than other, more popular metrics. To further examine these metrics, we extend prior formulations relying on power law (PL) spectral distributions to exponential (EXP) and exponentially-truncated power law (E-TPL) families.

Charles H. Martin, Michael W. Mahoney

To understand better good generalization performance in state-of-the-art neural network (NN) models, and in particular the success of the ALPHAHAT metric based on Heavy-Tailed Self-Regularization (HT-SR) theory, we analyze of a corpus of models that was made publicly-available for a contest to predict the generalization accuracy of NNs. These models include a wide range of qualities and were trained with a range of architectures and regularization hyperparameters. We break ALPHAHAT into its two subcomponent metrics: a scale-based metric; and a shape-based metric. We identify what amounts to a Simpson's paradox: where "scale" metrics (from traditional statistical learning theory) perform well in aggregate, but can perform poorly on subpartitions of the data of a given depth, when regularization hyperparameters are varied; and where "shape" metrics (from HT-SR theory) perform well on each subpartition of the data, when hyperparameters are varied for models of a given depth, but can perform poorly overall when models with varying depths are aggregated. Our results highlight the subtlety of comparing models when both architectures and hyperparameters are varied; the complementary role of implicit scale versus implicit shape parameters in understanding NN model quality; and the need to go beyond one-size-fits-all metrics based on upper bounds from generalization theory to describe the performance of NN models. Our results also clarify further why the ALPHAHAT metric from HT-SR theory works so well at predicting generalization across a broad range of CV and NLP models.

Charles H. Martin, Tongsu, Peng et al.

In many applications, one works with neural network models trained by someone else. For such pretrained models, one may not have access to training data or test data. Moreover, one may not know details about the model, e.g., the specifics of the training data, the loss function, the hyperparameter values, etc. Given one or many pretrained models, it is a challenge to say anything about the expected performance or quality of the models. Here, we address this challenge by providing a detailed meta-analysis of hundreds of publicly-available pretrained models. We examine norm based capacity control metrics as well as power law based metrics from the recently-developed Theory of Heavy-Tailed Self Regularization. We find that norm based metrics correlate well with reported test accuracies for well-trained models, but that they often cannot distinguish well-trained versus poorly-trained models. We also find that power law based metrics can do much better -- quantitatively better at discriminating among series of well-trained models with a given architecture; and qualitatively better at discriminating well-trained versus poorly-trained models. These methods can be used to identify when a pretrained neural network has problems that cannot be detected simply by examining training/test accuracies.

Charles H. Martin, Michael W. Mahoney

Given two or more Deep Neural Networks (DNNs) with the same or similar architectures, and trained on the same dataset, but trained with different solvers, parameters, hyper-parameters, regularization, etc., can we predict which DNN will have the best test accuracy, and can we do so without peeking at the test data? In this paper, we show how to use a new Theory of Heavy-Tailed Self-Regularization (HT-SR) to answer this. HT-SR suggests, among other things, that modern DNNs exhibit what we call Heavy-Tailed Mechanistic Universality (HT-MU), meaning that the correlations in the layer weight matrices can be fit to a power law (PL) with exponents that lie in common Universality classes from Heavy-Tailed Random Matrix Theory (HT-RMT). From this, we develop a Universal capacity control metric that is a weighted average of PL exponents. Rather than considering small toy NNs, we examine over 50 different, large-scale pre-trained DNNs, ranging over 15 different architectures, trained on ImagetNet, each of which has been reported to have different test accuracies. We show that this new capacity metric correlates very well with the reported test accuracies of these DNNs, looking across each architecture (VGG16/.../VGG19, ResNet10/.../ResNet152, etc.). We also show how to approximate the metric by the more familiar Product Norm capacity measure, as the average of the log Frobenius norm of the layer weight matrices. Our approach requires no changes to the underlying DNN or its loss function, it does not require us to train a model (although it could be used to monitor training), and it does not even require access to the ImageNet data.

Charles H. Martin, Michael W. Mahoney

Random Matrix Theory (RMT) is applied to analyze the weight matrices of Deep Neural Networks (DNNs), including both production quality, pre-trained models such as AlexNet and Inception, and smaller models trained from scratch, such as LeNet5 and a miniature-AlexNet. Empirical and theoretical results clearly indicate that the empirical spectral density (ESD) of DNN layer matrices displays signatures of traditionally-regularized statistical models, even in the absence of exogenously specifying traditional forms of regularization, such as Dropout or Weight Norm constraints. Building on recent results in RMT, most notably its extension to Universality classes of Heavy-Tailed matrices, we develop a theory to identify \emph{5+1 Phases of Training}, corresponding to increasing amounts of \emph{Implicit Self-Regularization}. For smaller and/or older DNNs, this Implicit Self-Regularization is like traditional Tikhonov regularization, in that there is a `size scale' separating signal from noise. For state-of-the-art DNNs, however, we identify a novel form of \emph{Heavy-Tailed Self-Regularization}, similar to the self-organization seen in the statistical physics of disordered systems. This implicit Self-Regularization can depend strongly on the many knobs of the training process. By exploiting the generalization gap phenomena, we demonstrate that we can cause a small model to exhibit all 5+1 phases of training simply by changing the batch size.

Charles H. Martin, Michael W. Mahoney

Random Matrix Theory (RMT) is applied to analyze weight matrices of Deep Neural Networks (DNNs), including both production quality, pre-trained models such as AlexNet and Inception, and smaller models trained from scratch, such as LeNet5 and a miniature-AlexNet. Empirical and theoretical results clearly indicate that the DNN training process itself implicitly implements a form of Self-Regularization. The empirical spectral density (ESD) of DNN layer matrices displays signatures of traditionally-regularized statistical models, even in the absence of exogenously specifying traditional forms of explicit regularization. Building on relatively recent results in RMT, most notably its extension to Universality classes of Heavy-Tailed matrices, we develop a theory to identify 5+1 Phases of Training, corresponding to increasing amounts of Implicit Self-Regularization. These phases can be observed during the training process as well as in the final learned DNNs. For smaller and/or older DNNs, this Implicit Self-Regularization is like traditional Tikhonov regularization, in that there is a "size scale" separating signal from noise. For state-of-the-art DNNs, however, we identify a novel form of Heavy-Tailed Self-Regularization, similar to the self-organization seen in the statistical physics of disordered systems. This results from correlations arising at all size scales, which arises implicitly due to the training process itself. This implicit Self-Regularization can depend strongly on the many knobs of the training process. By exploiting the generalization gap phenomena, we demonstrate that we can cause a small model to exhibit all 5+1 phases of training simply by changing the batch size. This demonstrates that---all else being equal---DNN optimization with larger batch sizes leads to less-well implicitly-regularized models, and it provides an explanation for the generalization gap phenomena.

Charles H. Martin, Michael W. Mahoney

We describe an approach to understand the peculiar and counterintuitive generalization properties of deep neural networks. The approach involves going beyond worst-case theoretical capacity control frameworks that have been popular in machine learning in recent years to revisit old ideas in the statistical mechanics of neural networks. Within this approach, we present a prototypical Very Simple Deep Learning (VSDL) model, whose behavior is controlled by two control parameters, one describing an effective amount of data, or load, on the network (that decreases when noise is added to the input), and one with an effective temperature interpretation (that increases when algorithms are early stopped). Using this model, we describe how a very simple application of ideas from the statistical mechanics theory of generalization provides a strong qualitative description of recently-observed empirical results regarding the inability of deep neural networks not to overfit training data, discontinuous learning and sharp transitions in the generalization properties of learning algorithms, etc.