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It has been shown that models such as deep neural networks, as well as certain so-called interpolating kernels and decision trees, can generalize well in this regime.

Recent work on pruning large language models (LLMs) (Frantar and Alistarh, 2023a; Jaiswal et al., 2023; Sun et al., 2023) has shown the ability to reduce the number of parameters significantly, without compromising performance, resulting in notable savings in memory footprint, computing time, and energy consumption. Unlike pre-LLM pruning methods (Kurtic et al., 2022; Sanh et al., 2020), existing LLM pruning approaches typically allocate the "sparsity budget" (i.e., the number of pruned parameters or pruning ratios) uniformly across layers, making it difficult to increase sparsity to very high levels. Relatively little effort has been put into developing theoretically-principled ways to compute layerwise pruning ratios. For example, the Outlier Weighed Layerwise sparsity (OWL) method (Yin et al., 2023) uses a nonuniform layerwise sparsity based on the distribution of outlier activations. However, OWL relies on heuristics related to the presence of outliers (Dettmers et al., 2022; Kovaleva et al., 2021; Puccetti et al., 2022). This can lead to suboptimal performance in the absence of outliers, and this can make it difficult to achieve very aggressive levels of sparsity. For example, Yin et al. (2023) shows that pruning LLMs to 80% sparsity often significantly degrades the prediction performance of LLMs. First two authors contributed equally.

To understand better the causes of good generalization performance in state-of-the-art neural network (NN) models, 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 identify what amounts to a Simpson's paradox: where "scale" metrics (from traditional statistical learning theory) perform well overall but perform poorly on subpartitions of the data of a given depth, when regularization hyperparameters are varied; and where "shape" metrics (from Heavy-Tailed Self Regularization theory) perform well on subpartitions of the data, when hyperparameters are varied for models of a given depth, but perform poorly overall when models with varying depths are aggregated. Our results highlight the subtly of comparing models when both architectures and hyperparameters are varied, as well as the complementary role of implicit scale versus implicit shape parameters in understanding NN model quality. Our results also suggest caution when one tries to extract causal insight with a single metric applied to aggregate data, and they highlight the need to go beyond one-size-fits-all metrics based on upper bounds from generalization theory to describe the performance of state-of-the-art NN models. Based on these findings, we present two novel shape metrics, one data-independent, and the other data-dependent, which can predict trends in the test accuracy of a series of NNs, of a fixed architecture/depth, when varying solver hyperparameters.

In many applications, one works with deep neural network (DNN) models trained by someone else. For such pretrained models, one typically does not have access to training/test data. Moreover, one does not know many details about the model, such as the specifics of the training data, the loss function, the hyperparameter values, etc. Given one or many pretrained models, can one say anything about the expected performance or quality of the models? Here, we present and evaluate empirical quality metrics for pretrained DNN models at scale. Using the open-source WeightWatcher tool, we analyze hundreds of publicly-available pretrained models, including older and current state-of-the-art models in CV and NLP. We examine norm-based capacity control metrics as well as newer Power Law (PL) based metrics (including fitted PL exponents and a Weighted Alpha metric), from the recently-developed Theory of Heavy-Tailed Self Regularization. Norm-based metrics correlate well with reported test accuracies for well-trained models across nearly all CV architecture series. On the other hand, norm-based metrics can not distinguish "good-versus-bad" models---which, arguably is the point of needing quality metrics. Indeed, they may give spurious results. PL-based metrics do much better---quantitatively better at discriminating series of "good-better-best" models, and qualitatively better at discriminating "good-versus-bad" models. PL-based metrics can also be used to characterize fine-scale properties of models, and we introduce the layer-wise Correlation Flow as new quality assessment. We show how poorly-trained (and/or poorly fine-tuned) models may exhibit both Scale Collapse and unusually large PL exponents, in particular for recent NLP models. Our techniques can be used to identify when a pretrained DNN has problems that can not be detected simply by examining training/test accuracies.

Double descent refers to the phase transition that is exhibited by the generalization error of unregularized learning models when varying the ratio between the number of parameters and the number of training samples. The recent success of highly over-parameterized machine learning models such as deep neural networks has motivated a theoretical analysis of the double descent phenomenon in classical models such as linear regression which can also generalize well in the over-parameterized regime. We build on recent advances in Randomized Numerical Linear Algebra (RandNLA) to provide the first exact non-asymptotic expressions for double descent of the minimum norm linear estimator. Our approach involves constructing what we call a surrogate random design to replace the standard i.i.d. design of the training sample. This surrogate design admits exact expressions for the mean squared error of the estimator while preserving the key properties of the standard design. We also establish an exact implicit regularization result for over-parameterized training samples. In particular, we show that, for the surrogate design, the implicit bias of the unregularized minimum norm estimator precisely corresponds to solving a ridge-regularized least squares problem on the population distribution.

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 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 these 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.

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.