Introduction

A common problem in machine learning (ML) is to evaluate the quality of a given model. A popular way to accomplish this is to train a model and then evaluate its training/testing error. There are many problems with this approach. The training/testing curves give very limited insight into the overall properties of the model; they do not take into account the (often large human and CPU/GPU) time for hyperparameter fiddling; they typically do not correlate with other properties of interest such as robustness or fairness or interpretability; and so on. A related problem, in particular in industrial-scale artificial intelligence (AI), arises when the model user is not the model developer. Then, one may not have access to the training data or the testing data. Instead, one may simply be given a model that has already been trained—a pretrained model—and need to use it as-is, or to fine-tune and/or compress it and then use it.

Naïvely—but in our experience commonly, among ML practitioners and ML theorists—if one does not have access to training or testing data, then one can say absolutely nothing about the quality of a ML model. This may be true in worst-case theory, but models are used in practice, and there is a need for a practical theory to guide that practice. Moreover, if ML is to become an industrial process, then that process will become compartmentalized in order to scale: some groups will gather data, other groups will develop models, and other groups will use those models. Users of models cannot be expected to know the precise details of how models were built, the specifics of data that were used to train the model, what was the loss function or hyperparameter values, how precisely the model was regularized, etc.

Moreover, for many large scale, practical applications, there is no obvious way to define an ideal test metric. For example, models that generate fake text or conversational chatbots may use a proxy, like perplexity, as a test metric. In the end, however, they really require human evaluation. Alternatively, models that cluster user profiles, which are widely used in areas such as marketing and advertising, are unsupervised and have no obvious labels for comparison and/or evaluation. In these and other areas, ML objectives can be poor proxies for downstream goals.

Most importantly, in industry, one faces unique practical problems such as determining whether one has enough data for a given model. Indeed, high quality, labeled data can be very expensive to acquire, and this cost can make or break a project. Methods that are developed and evaluated on any well-defined publicly available corpus of data, no matter how large or diverse or interesting, are clearly not going to be well-suited to address problems such as this. It is of great practical interest to have metrics to evaluate the quality of a trained model—in the absence of training/testing data and without any detailed knowledge of the training/testing process. There is a need for a practical theory for pretrained models which can predict how, when, and why such models can be expected to perform well or poorly.

In the absence of training and testing data, obvious quantities to examine are the weight matrices of pretrained models, e.g., properties such as norms of weight matrices and/or parameters of Power Law (PL) fits of the eigenvalues of weight matrices. Norm-based metrics have been used in traditional statistical learning theory to bound capacity and construct regularizers; and PL fits are based on statistical mechanics approaches to deep neural networks (DNNs). While we use traditional norm-based and PL-based metrics, our goals are not the traditional goals. Unlike more common ML approaches, we do not seek a bound on the generalization (e.g., by evaluating training/test errors), we do not seek a new regularizer, and we do not aim to evaluate a single model (e.g., as with hyperparameter optimization). Instead, we want to examine different models across common architecture series, and we want to compare models between different architectures themselves. In both cases, one can ask whether it is possible to predict trends in the quality of pretrained DNN models without access to training or testing data.

To answer this question, we provide a detailed empirical analysis, evaluating quality metrics for pretrained DNN models, and we do so at scale. Our approach may be viewed as a statistical meta-analysis of previously published work, where we consider a large suite of hundreds of publicly available models, mostly from computer vision (CV) and natural language processing (NLP). By now, there are many such state-of-the-art models that are publicly available, e.g., hundreds of pretrained models in CV (≥500) and NLP (≈100). (When we began this work in 2018, there were fewer than tens of such models; then in 2020, there are hundreds of such models; and we expect that in a year or two there will be an order of magnitude or more of such models.) For all these models, we have no access to training data or testing data, and we have no specific knowledge of the training/testing protocols. Here is a summary of our main results. First, norm-based metrics do a reasonably good job at predicting quality trends in well-trained CV/NLP models. Second, norm-based metrics may give spurious results when applied to poorly trained models (e.g., models trained without enough data, etc.). For example, they may exhibit what we call Scale Collapse for these models. Third, PL-based metrics can do much better at predicting quality trends in pretrained CV/NLP models. In particular, a weighted PL exponent (weighted by the log of the spectral norm of the corresponding layer) is quantitatively better at discriminating among a series of well-trained versus very-well-trained models within a given architecture series; and the (unweighted) average PL exponent is qualitatively better at discriminating well-trained versus poorly-trained models. Fourth, PL-based metrics can also be used to characterize fine-scale model properties, including what we call layer-wise Correlation Flow, in well-trained and poorly-trained models; and they can be used to evaluate model enhancements (e.g., distillation, fine-tuning, etc.). Our work provides a theoretically principled empirical evaluation—by far the largest, most detailed, and most comprehensive to date—and the theory we apply was developed previously1,2,3. Performing such a meta-analysis of previously published work is common in certain areas, but it is quite rare in ML, where the emphasis is on develo** better training protocols.

Results

After describing our overall approach, we study in detail three well-known CV architecture series (the VGG, ResNet, and DenseNet series of models). Then, we look in detail at several variations of a popular NLP architecture series (the OpenAI GPT and GPT2 series of models), and we present results from a broader analysis of hundreds of pretrained DNN models.

Overall approach

Consider the objective/optimization function (parameterized by Wls and bls) for a DNN with L layers, and weight matrices Wl and bias vectors bl, as the minimization of a general loss function \({\mathcal{L}}\) over the training data instances and labels, \(\{{{\bf{x}}}_{i},{y}_{i}\}\in {\mathcal{D}}\). For a typical supervised classification problem, the goal of training is to construct (or learn) Wl and bl that capture correlations in the data, in the sense of solving

$$\mathop{{\rm{argmin}}}\limits_{{{\bf{W}}}_{l},{{\bf{b}}}_{L}}\ \mathop{\sum }\limits_{i=1}^{N}{\mathcal{L}}({E}_{DNN}({{\bf{x}}}_{i}),{y}_{i}),$$
(1)

where the loss function \({\mathcal{L}}(\cdot ,\cdot )\) can take on a myriad of forms4, and where the energy (or optimization) landscape function

$${E}_{DNN}=f({{\bf{x}}}_{i};{{\bf{W}}}_{1},\ldots ,{{\bf{W}}}_{L},{{\bf{b}}}_{1},\ldots ,{{\bf{b}}}_{L})$$
(2)

depends parametrically on the weights and biases. For a trained model, the form of the function EDNN does not explicitly depend on the data (but it does explicitly depend on the weights and biases). The function EDNN maps data instance vectors (xi values) to predictions (yi labels), and thus the output of this function does depend on the data. Therefore, one can analyze the form of EDNN in the absence of any training or test data.

Test accuracies have been reported online for publicly available pretrained pyTorch models5. These models have been trained and evaluated on labeled data \(\{{{\bf{x}}}_{i},{y}_{i}\}\in {\mathcal{D}}\), using standard techniques. We do not have access to this data, and we have not trained any of the models ourselves. Our methodological approach is thus similar to a statistical meta-analysis, common in biomedical research, but uncommon in ML. Computations were performed with the publicly available WeightWatcher tool (version 0.2.7)6. To be fully reproducible, we only examine publicly available, pretrained models, and we provide all Jupyter and Google Colab notebooks used in an accompanying github repository7. See Supplementary Note 1 for details.

Our approach involves analyzing individual DNN weight matrices, for (depending on the architecture) fully connected and/or convolutional layers. Each DNN layer contains one or more layer 2D Nl × Ml weight matrices, Wl, or pre-activation maps, Wi,l, e.g., extracted from 2D Convolutional layers, where N > M. (We may drop the i and/or i, l subscripts below.) The best performing quality metrics depend on the norms and/or spectral properties of each weight matrix, W, and/or, equivalently, it’s empirical correlation matrix, X = WTW. To evaluate the quality of state-of-the-art DNNs, we consider the following metrics:

$$\,{\text{Frobenius}}\; {\text{Norm}}\,:\parallel {\bf{W}}{\parallel }_{F}^{2}=\parallel {\bf{X}}{\parallel }_{F}=\mathop{\sum }\limits_{i = 1}^{M}{\lambda }_{i}$$
(3)
$$\,{\text{Spectral}}\; {\text{Norm}}\,:\parallel {\bf{W}}{\parallel }_{\infty }^{2}=\parallel {\bf{X}}{\parallel }_{\infty }={\lambda }_{max}$$
(4)
$$\,{\text{Weighted}}\; {\text{Alpha}}\,:\hat{\alpha }=\alpha\, {\mathrm{log}}\,{\lambda }_{max}$$
(5)
$$\alpha {\mbox{-}}{\rm{Norm}}({\rm{or}}\,\, \alpha {\mbox{-}}{\rm{Shatten}}\; {\rm{Norm}}):\parallel {\bf{W}}{\parallel }_{2\alpha }^{2\alpha }=\parallel {\bf{X}}{\parallel }_{\alpha }^{\alpha }=\mathop{\sum }\limits_{i = 1}^{M}{\lambda }_{i}^{\alpha }.$$
(6)

To perform diagnostics on potentially problematic DNNs, we will decompose \(\hat{\alpha }\) into its two components, α and λmax. Here, λi is the ith eigenvalue of the X, λmax is the maximum eigenvalue, and α is the fitted PL exponent. These eigenvalues are squares of the singular values σi of W, \({\lambda }_{i}={\sigma }_{i}^{2}\). All four metrics can be computed easily from DNN weight matrices. The first two metrics are well-known in ML. The last two metrics deserve special mention, as they depend on an empirical parameter α that is the PL exponent that arises in the recently developed Heavy Tailed Self Regularization (HT-SR) Theory1,2,3.

In the HT-SR Theory, one analyzes the eigenvalue spectrum, i.e., the Empirical Spectral Density (ESD), of the associated correlation matrices1,2,3. From this, one characterizes the amount and form of correlation, and therefore implicit self-regularizartion, present in the DNN’s weight matrices. For each layer weight matrix W, of size N × M, construct the associated M × M (uncentered) correlation matrix X. Drop** the L and l, i indices, one has

$${\bf{X}}=\frac{1}{N}{{\bf{W}}}^{T}{\bf{W}}.$$

If we compute the eigenvalue spectrum of X, i.e., λi such that Xvi = λivi, then the ESD of eigenvalues, ρ(λ), is just a histogram of the eigenvalues, formally written as \(\rho (\lambda )=\mathop{\sum }\nolimits_{i = 1}^{M}\delta (\lambda -{\lambda }_{i}).\) Using HT-SR Theory, one characterizes the correlations in a weight matrix by examining its ESD, ρ(λ). It can be well-fit to a truncated PL distribution, given as

$$\rho (\lambda ) \sim {\lambda }^{-\alpha },$$
(7)

which is (at least) valid within a bounded range of eigenvalues λ ∈ [λmin, λmax].

The original work on HT-SR Theory considered a small number of NNs, including AlexNet and InceptionV3. It showed that for nearly every W, the (bulk and tail) of the ESDs can be fit to a truncated PL, and that PL exponents α nearly all lie within the range α ∈ (1.5, 5)1,2,3. As for the mechanism responsible for these properties, statistical physics offers several possibilities8,9, e.g., self-organized criticality10,11 or multiplicative noise in the stochastic optimization algorithms used to train these models12,13. Alternatively, related techniques have been used to analyze correlations and information propogation in actual spiking neurons14,15. Our meta-analysis does not require knowledge of mechanisms; and it is not even clear that one mechanism is responsible for every case. Crucially, HT-SR Theory predicts that smaller values of α should correspond to models with better correlation over multiple size scales and thus to better models. The notion of “size scale” is well-defined in physical systems, to which this style of analysis is usually applied, but it is less well-defined in CV and NLP applications. Informally, it would correspond to pixel groups that are at a greater distance in some metric, or between sentence parts that are at a greater distance in text. Relatedly, previous work observed that smaller exponents α correspond to more implicit self-regularization and better generalization, and that we expect a linear correlation between \(\hat{\alpha }\) and model quality1,2,3.

For norm-based metrics, we use the average of the log norm, to the appropriate power. Informally, this amounts to assuming that the layer weight matrices are statistically independent, in which case we can estimate the model complexity \({\mathcal{C}}\), or test accuracy, with a standard Product Norm (which resembles a data dependent VC complexity),

$${\mathcal{C}} \sim \parallel {{\bf{W}}}_{1}\parallel \times \parallel {{\bf{W}}}_{2}\parallel \times \cdots \times \parallel {{\bf{W}}}_{L}\parallel ,$$
(8)

where ∥ ⋅ ∥ is a matrix norm. The log complexity,

$${\mathrm{log}}\,{\mathcal{C}} \sim {\mathrm{log}}\,\parallel {{\bf{W}}}_{1}\parallel +{\mathrm{log}}\,\parallel {{\bf{W}}}_{2}\parallel +\cdots +{\mathrm{log}}\,\parallel {{\bf{W}}}_{L}\parallel =\mathop{\sum }\limits_{l}{\mathrm{log}}\,\parallel {{\bf{W}}}_{l}\parallel ,$$
(9)

takes the form of an average Log Norm. For the Frobenius Norm metric and Spectral Norm metric, we can use Eq. (9) directly (since, when taking \({\mathrm{log}}\,\parallel {{\bf{W}}}_{l}{\parallel }_{F}^{2}\), the 2 comes down and out of the sum, and thus ignoring it only changes the metric by a constant factor).

The Weighted Alpha metric is an average of αl over all layers l ∈ {1, …, l}, weighted by the size, or scale, or each matrix,

$$\hat{\alpha }=\frac{1}{L}\mathop{\sum}\limits_{l}{\alpha }_{l}{\mathrm{log}}\,{\lambda }_{max,l}\approx \langle {\mathrm{log}}\,\parallel {\bf{X}}{\parallel }_{\alpha }^{\alpha }\rangle ,$$
(10)

where L is the total number of layer weight matrices. The Weighted Alpha metric was introduced previously3, where it was shown to correlate well with trends in reported test accuracies of pretrained DNNs, albeit on a much smaller and more limited set of models than we consider here.

Based on this, in this paper, we introduce and evaluate the α-Shatten Norm metric,

$$\mathop{\sum}\limits_{l}{\mathrm{log}}\,\parallel {{\bf{X}}}_{l}{\parallel }_{{\alpha }_{l}}^{{\alpha }_{l}}=\mathop{\sum}\limits_{l}{\alpha }_{l}{\mathrm{log}}\,\parallel {{\bf{X}}}_{l}{\parallel }_{{\alpha }_{l}}.$$
(11)

For the α-Shatten Norm metric, αl varies from layer to layer, and so in Eq. (11) it cannot be taken out of the sum. For small α, the Weighted Alpha metric approximates the Log α-Shatten norm, as can be shown with a statistical mechanics and random matrix theory derivation; and the Weighted Alpha and α-Shatten norm metrics often behave like an improved, weighted average Log Spectral Norm.

Finally, although it does less well for predicting trends in state-of-the-art model series, e.g., as depth changes, the average value of α, i.e.,

$$\bar{\alpha }=\frac{1}{L}\mathop{\sum}\limits_{l}{\alpha }_{l}=\langle \alpha \rangle ,$$
(12)

can be used to perform model diagnostics, to identify problems that cannot be detected by examining training/test accuracies, and to discriminate poorly trained models from well-trained models.

One determines α for a given layer by fitting the ESD of that layer’s weight matrix to a truncated PL, using the commonly accepted Maximum Likelihood method16,17. This method works very well for exponents between α ∈ (2, 4); and it is adequate, although imprecise, for smaller and especially larger α18. Operationally, α is determined by using the WeightWatcher tool6 to fit the histogram of eigenvalues, ρ(λ), to a truncated PL,

$$\rho (\lambda ) \sim {\lambda }^{\alpha },\ \ \lambda \in [{\lambda }_{min},{\lambda }_{max}],$$
(13)

where λmax is the largest eigenvalue of X = WTW, and where λmin is selected automatically to yield the best (in the sense of minimizing the K-S distance) PL fit. Each of these quantities is defined for a given layer W matrix. See Fig. 1 for an illustration.

Fig. 1: Schematic of analyzing DNN layer weight matrices W.
figure 1

Given an individual layer weight matrix W, from either a fully connected layer or a convolutional layer, perform a Singular Value Decomposition (SVD) to obtain W = UΣVT, and examine the histogram of eigenvalues of WTW. Norm-based metrics and PL-based metrics (that depend on fitting the histogram of eigenvalues to a truncated PL) can be used to compare models. For example, one can analyze one layer of a pre-trained model, compare multiple layers of a pre-trained model, make comparisons across model architectures, monitor neural network properties during training, etc.

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To avoid confusion, let us clarify the relationship between α and \(\hat{\alpha }\). We fit the ESD of the correlation matrix X to a truncated PL, parameterized by 2 values: the PL exponent α, and the maximum eigenvalue λmax. The PL exponent α measures the amount of correlation in a DNN layer weight matrix W. It is valid for λ ≤ λmax, and it is scale-invariant, i.e., it does not depend on the normalization of W or X. The λmax is a measure of the size, or scale, of W. Multiplying each α by the corresponding \({\mathrm{log}}\,{\lambda }_{max}\) weighs “bigger” layers more, and averaging this product leads to a balanced, Weighted Alpha metric \(\hat{\alpha }\) for the entire DNN. We will see that for well-trained CV and NLP models, \(\hat{\alpha }\) performs quite well and as expected, but for CV and NLP models that are potentially problematic or less well-trained, metrics that depend on the scale of the problem can perform anomalously. In these cases, separating \(\hat{\alpha }\) into its two components, α and λmax, and examining the distributions of each, can be helpful.

Comparison of CV models

Each of the VGG, ResNet, and DenseNet series of models consists of several pretrained DNN models, with a given base architecture, trained on the full ImageNet19 dataset, and each is distributed with the current open source pyTorch framework (version 1.4)20. In addition, we examine a larger set of ResNet models, which we call the ResNet-1K series, trained on the ImageNet-1K dataset19 and provided on the OSMR Sandbox5. For these models, we first perform coarse model analysis, comparing and contrasting the four model series, and predicting trends in model quality. We then perform fine layer analysis, as a function of depth. This layer analysis goes beyond predicting trends in model quality, instead illustrating that PL-based metrics can provide novel insights among the VGG, ResNet/ResNet-1K, and DenseNet architectures.

We examine the performance of the four quality metrics—Log Frobenius norm (\(\langle {\mathrm{log}}\,\parallel {\bf{W}}{\parallel }_{F}^{2}\rangle \)), Log Spectral norm (\(\langle {\mathrm{log}}\,\parallel {\bf{W}}{\parallel }_{\infty }^{2}\rangle \)), Weighted Alpha (\(\hat{\alpha }\)), and Log α-Norm (\(\langle {\mathrm{log}}\,\parallel {\bf{X}}{\parallel }_{\alpha }^{\alpha }\rangle \))—applied to each of the VGG, ResNet, ResNet-1K, and DenseNet series. Figure 2 plots the four quality metrics versus reported test accuracies20, as well as a basic linear regression line, for the VGG series. (These test accuracies have been previously reported and made publicly available by others. We take them as given. We do not attempt to reproduce/verify them, since we do not permit ourselves access to training/test data.) Here, smaller norms and smaller values of \(\hat{\alpha }\) imply better generalization (i.e., greater accuracy, lower error). Quantitatively, Log Spectral norm is the best; but, visually, all four metrics correlate quite well with reported Top1 accuracies. The DenseNet series has similar behavior. (These and many other such plots can be seen on our publicly available repo.)

Fig. 2: Comparison of average Log Norm and Weighted Alpha quality metrics for CV models.
figure 2

Comparison of average Log Norms (in (a), (b), and (d)) and Weighted Alpha (in (c)) quality metrics versus reported test accuracy for pretrained VGG models: VGG11, VGG13, VGG16, and VGG19, with and without Batch Normalization (BN), trained on ImageNet, available in pyTorch (v1.4). Metrics fit by linear regression, RMSE, R2, and the Kendal-tau rank correlation metric reported.

Full size image

To examine visually how the four quality metrics depend on data set size on a larger, more complex model series, we next look at results on ResNet versus ResNet-1K. Figure 3 compares the Log α-Norm metric for the full ResNet model, trained on the full ImageNet dataset, against the ResNet-1K model, trained on a much smaller ImageNet-1K data set. Here, the Log α-Norm is much better than the Log Frobenius/Spectral norm metrics (although, as Table 1 shows, it is slightly worse than the Weighted Alpha metric). The ResNet series has strong correlation (RMSE of 0.66, R2 of 0.9, and Kendall-τ of −1.0), whereas the ResNet-1K series also shows good but weaker correlation (much larger RMSE of 1.9, R2 of 0.88, and Kendall-τ of −0.88).

Fig. 3: Comparison of average α-Norm quality metric for CV models.
figure 3

Comparison of average α-Norm quality metric versus reported Top1 test accuracy for the ResNet (in (a)) and ResNet-1K (in (b)) pretrained (pyTorch) models. Metrics fit by linear regression, RMSE, R2, and the Kendal-tau rank correlation metric reported.

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Table 1 Quality metrics (for RMSE, smaller is better; for R2, larger is better; for Kendall-τ rank correlation, larger magnitude is better; best is bold) for reported Top1 test error for pretrained models in each architecture series.
Full size table

See Table 1 for a summary of results for Top1 accuracies for all four metrics for the VGG, ResNet, ResNet-1K, and DenseNet series. Similar results are obtained for the Top5 accuracies. The Log Frobenius norm performs well but not extremely well; the Log Spectral norm performs very well on smaller, simpler models like the VGG and DenseNet architectures; and, when moving to the larger, more complex ResNet series, the PL-based metrics, Weighted Alpha and the Log α-Norm, perform the best. Overall, though, these model series are all very well-trodden; and our results indicate that norm-based metrics and PL-based metrics can both distinguish among a series of well-trained versus very-well-trained models, with PL-based metrics performing somewhat (i.e., quantitatively) better on the larger, more complex ResNet series.

In particular, the PL-based Weighted Alpha and Log α-Norm metrics tend to perform better when there is a wider variation in the hyperparameters, going beyond just increasing the depth. In addition, sometimes the purely norm-based metrics such as the Log Spectral norm can be uncorrelated or even anti-correlated with the test accuracy, while the PL-metrrics are positively correlated. See Supplementary Note 2 for additional details.

Going beyond coarse averages to examining quality metrics for each layer weight matrix as a function of depth (or layer id), our metrics can be used to perform model diagnostics and to identify fine-scale properties in a pretrained model. Doing so involves separating \(\hat{\alpha }\) into its two components, α and λmax, and examining the distributions of each. We provide examples of this.

Figure 4 plots the PL exponent α, as a function of depth, for each layer (first layer corresponds to data, last layer to labels) for the least accurate (shallowest) and most accurate (deepest) model in each of the VGG (no BN), ResNet, and DenseNet series. (Many more such plots are available at our repo.)

Fig. 4: PL exponent (α) versus layer id for VGG, ResNet, and DenseNet.
figure 4

PL exponent (α) versus layer id, for the least and the most accurate models in VGG (a), ResNet (b), and DenseNet (c) series. (VGG is without BN; and note that the Y axes on each plot are different.) Subfigure (d) displays the ResNet models (b), zoomed in to α ∈ [1, 5], and with the layer ids overlaid on the X-axis, from smallest to largest, to allow a more detailed analysis of the most strongly correlated layers. Notice that ResNet152 exhibits different and much more stable behavior of α across layers. This contrasts with how both VGG models gradually worsen in deeper layers and how the DenseNet models are much more erratic. In the text, this is interpreted in terms of Correlation Flow.

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In the VGG models, Fig. 4a shows that the PL exponent α systematically increases as we move down the network, from data to labels, in the Conv2D layers, starting with α ≲ 2.0 and reaching all the way to α ~ 5.0; and then, in the last three, large, fully connected (FC) layers, α stabilizes back down to α ∈ [2, 2.5]. This is seen for all the VGG models (again, only the shallowest and deepest are shown), indicating that the main effect of increasing depth is to increase the range over which α increases, thus leading to larger α values in later Conv2D layers of the VGG models. This is quite different than the behavior of either the ResNet-1K models or the DenseNet models.

For the ResNet-1K models, Fig. 4b shows that α also increases in the last few layers (more dramatically than for VGG, observe the differing scales on the Y axes). However, as the ResNet-1K models get deeper, there is a wide range over which α values tend to remain small. This is seen for other models in the ResNet-1K series, but it is most pronounced for the larger ResNet-1K (152) model, where α remains relatively stable at α ~ 2.0, from the earliest layers all the way until we reach close to the final layers.

For the DenseNet models, Fig. 4c shows that α tends to increase as the layer id increases, in particular for layers toward the end. While this is similar to the VGG models, with the DenseNet models, α values increase almost immediately after the first few layers, and the variance is much larger (in particular for the earlier and middle layers, where it can range all the way to α ~ 8.0) and much less systematic throughout the network.

Overall, Fig. 4 demonstrates that the distribution of α values among layers is architecture dependent, and that it can vary in a systematic way within an architecture series. This is to be expected, since some architectures enable better extraction of signal from the data. This also suggests that, while performing very well at predicting trends within an architecture series, PL-based metrics (as well as norm-based metrics) should be used with caution when comparing models with very different architectures.

Figure 4 can be understood in terms of what we will call Correlation Flow. Recall that the average Log α-Norm metric and the Weighted Alpha metric are based on HT-SR Theory1,2,3, which is in turn based on the statistical mechanics of heavy tailed and strongly correlated systems8,21,22,23. There, one expects that the weight matrices of well-trained DNNs will exhibit correlations over many size scales, as is well-known in other strongly correlated systems8,21. This would imply that their ESDs can be well-fit by a truncated PL, with exponents α ∈ [2, 4]. Much larger values (α ≫ 6) may reflect poorer PL fits, whereas smaller values (α ~ 2), are associated with models that generalize better.

Informally, one would expect a DNN model to perform well when it facilitates the propagation of information/features across layers. In the absence of training/test data, one might hypothesize that this flow of information leaves empirical signatures on weight matrices, and that we can quantify this by measuring the PL properties of weight matrices. In this case, smaller α values correspond to layers in which information correlations between data across multiple scales are better captured1,8. This leads to the hypothesis that small α values that are stable across multiple layers enable better correlation flow through the network. This is similar to recent work on the information bottleneck24,24,

Methods

To be fully reproducible, we only examine publicly available, pretrained models. All of our computations were performed with the WeightWatcher tool (version 0.2.7)6, and we provide all Jupyter and Google Colab notebooks used in an accompanying github repository7, which includes more details and more results.

Additional details on layer weight matrices

Recall that we can express the objective/optimization function for a typical DNN with L layers and with N × M weight matrices Wl and bias vectors bl as Eq. (2). We expect that most well-trained, production-quality models will employ one or more forms of regularization, such as Batch Normalization (BN), Dropout, etc., and many will also contain additional structure such as Skip Connections, etc. Here, we will ignore these details, and will focus only on the pretrained layer weight matrices Wl. Typically, this model would be trained on some labeled data \(\{{d}_{i},{y}_{i}\}\in {\mathcal{D}}\), using Backprop, by minimizing the loss \({\mathcal{L}}\). For simplicity, we do not indicate the structural details of the layers (e.g., Dense or not, Convolutions or not, Residual/Skip Connections, etc.). Each layer is defined by one or more layer 2D weight matrices Wl, and/or the 2D feature maps Wl,i extracted from 2D Convolutional (Conv2D) layers. A typical modern DNN may have anywhere between 5 and 5000 2D layer matrices.

For each Linear Layer, we get a single (N × M) (real-valued) 2D weight matrix, denoted Wl, for layer l. This includes Dense or Fully Connected (FC) layers, as well as 1D Convolutional (Conv1D) layers, Attention matrices, etc. We ignore the bias terms bl in this analysis. Let the aspect ratio be \(Q=\frac{N}{M}\), with Q ≥ 1. For the Conv2D layers, we have a 4-index Tensor, of the form (N × M × c × d), consisting of c × d 2D feature maps of shape (N × M). We extract nl = c × d 2D weight matrices Wl,i, one for each feature map i = [1, …, nl] for layer l.

SVD of convolutional 2D layers

There is some ambiguity in performing spectral analysis on Conv2D layers. Each layer is a 4-index tensor of dimension (w, h, in, out), with an (w × h) filter (or kernel) and (in, out) channels. When w = h = k, it gives (k × k) tensor slices, or pre-Activation Maps, Wi,L of dimension (in × out) each. We identify 3 different approaches for running SVD on a Conv2D layer:

  1. 1.

    run SVD on each pre-Activation Map Wi,L, yielding (k × k) sets of M singular values;

  2. 2.

    stack the maps into a single matrix of, say, dimension ((k × k × out) × in), and run SVD to get in singular values;

  3. 3.

    compute the 2D Fourier Transform (FFT) for each of the (in, out) pairs, and run SVD on the Fourier coefficients42, leading to ~ (k × in × out) non-zero singular values.

Each method has tradeoffs. Method (3) is mathematically sound, but computationally expensive. Method (2) is ambiguous. For our analysis, because we need thousands of runs, we select method (1), which is the fastest (and is easiest to reproduce).

Normalization of empirical matrices

Normalization is an important, if underappreciated, practical issue. Importantly, the normalization of weight matrices does not affect the PL fits because α is scale-invariant. Norm-based metrics, however, do depend strongly on the scale of the weight matrix—that is the point. To apply RMT, we usually define X with a 1/N normalization, assuming variance of σ2 = 1.0. Pretrained DNNs are typically initialized with random weight matrices W0, with \({\sigma }^{2} \sim 1/\sqrt{N}\), or some variant, e.g., the Glorot/Xavier normalization43, or a \(\sqrt{2/N{k}^{2}}\) normalization for Convolutional 2D Layers. With this implicit scale, we do not “renormalize” the empirical weight matrices, i.e., we use them as-is. The only exception is that we do rescale the Conv2D pre-activation maps Wi,L by \(k/\sqrt{2}\) so that they are on the same scale as the Linear/Fully Connected (FC) layers.

Special consideration for NLP models

NLP models, and other models with large initial embeddings, require special care because the embedding layers frequently lack the implicit \(1/\sqrt{N}\) normalization present in other layers. For example, in GPT, for most layers, the maximum eigenvalue \({\lambda }_{max} \sim {\mathcal{O}}(10-100)\), but in the first embedding layer, the maximum eigenvalue is of order N (the number of words in the embedding), or \({\lambda }_{max} \sim {\mathcal{O}}(1{0}^{5})\). For GPT and GPT2, we treat all layers as-is (although one may want to normalize the first 2 layers X by 1/N, or to treat them as outliers).