Introduction
With the success and heightened adoption of neural networks for real world tasks, some questions remain poorly answered. For a given task and model architecture, how much data would one require to reach a prescribed performance level? How big a model would be needed? Addressing such questions is made especially difficult by the mounting evidence that large, deep neural networks trained on large-scale data outperform their smaller counterparts, rendering the training of high performance models prohibitively costly. Indeed, in the absence of practical answers to the above questions, surrogate approaches have proven useful. One such common approach is model scaling, where one designs and compares small-scale models, and applies the obtained architectural principles at a larger scale BIBREF0, BIBREF1, BIBREF2. Despite these heuristics being widely used to various degrees of success, the relation between the performance of a model in the small- and large-scale settings is not well understood. Hence, exploring the limitations or improving the efficiency of such methods remains subject to trial and error.
In this work we circle back to the fundamental question: what is the (functional) relation between generalization error and model and dataset sizes? Critically, we capitalize on the concept of model scaling in its strictest form: we consider the case where there is some given scaling policy that completely defines how to scale up a model from small to large scales. We include in this context all model parameters, such that traversing from one scale (in which all parameters are known) to another requires no additional resources for specifying the model (e.g., architecture search/design).
We empirically explore the behavior of the generalization error over a wide range of datasets and models in vision and language tasks. While the error landscape seems fairly complex at first glance, we observe the emergence of several key characteristics shared across benchmarks and domains. Chief among these characteristics is the emergence of regions where power-law behavior approximates the error well both with respect to data size, when holding model size fixed, and vice versa.
Motivated by these observations, we establish criteria which a function approximating the error landscape should meet. We propose an intuitive candidate for such a function and evaluate its quality, both in explaining the observed error landscapes and in extrapolating from small scale (seen) to large scale (unseen) errors. Critically, our functional approximation of the error depends on both model and data sizes. We find that this function leads to a high quality fit and extrapolation. For instance, the mean and standard deviation of the relative errors are under 2% when fitting across all scales investigated and under 5% when extrapolating from a slimmed-down model (1/16 of the parameters) on a fraction of the training data (1/8 of the examples) on the ImageNet BIBREF3 and WikiText-103 BIBREF4 datasets, with similar results for other datasets.
To the best of our knowledge, this is the first work that provides simultaneously:
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A joint functional form of the generalization error landscape—as dependent on both data and model size—with few, interpretable degrees of freedom (section SECREF5).
Direct and complete specification (via the scaling policy) of the model configuration attaining said generalization error across model and dataset sizes.
Highly accurate approximation of error measurements across model and data scales via the functional form, evaluated on different models, datasets, and tasks (section SECREF6 ).
Highly accurate error prediction from small to large model and data (section SECREF7).
We conclude with a discussion of some implications of our findings as a practical and principled tool for understanding network design at small scale and for efficient computation and trade-off design in general. We hope this work also provides a useful empirical leg to stand on and an invitation to search for a theory of generalization error which accounts for our findings.
Related work
Related work ::: Model scaling:
A number of studies have explored the effect of model scaling on performance. For instance, image classification networks can be scaled by depth BIBREF5 or width BIBREF6, BIBREF7. More recently, BIBREF8 demonstrated how scaling width, depth, and input resolution has combined positive effects larger than scaling each factor in isolation. However, this relationship has yet to be quantified in a predictive form – by how much will error change with model scaling? In this work, we focus on finding a constructive functional form for determining the model given a specified performance.
Related work ::: Data scaling:
It has long been recognized that more data improves performance, and various studies report such trends in both computer vision BIBREF9, BIBREF10 and language processing tasks BIBREF11, BIBREF12. A number of prior studies observed power-law relations between the generalization error and training data size BIBREF13, BIBREF14, BIBREF15. Most relevant to our work, BIBREF16 explored the effect of data size on the generalization error in vision, language, and speech tasks, and observed a strikingly consistent power-law behavior in a large set of experiments. However, while these studies point to the empirical existence of a power law in terms of data, they do not offer tools for predicting the performance given a specified model. Nor do they offer low-cost methods to specify the model configuration which would attain the power law with data dependency. Indeed, BIBREF16 had to search over models and their configurations at large scale to exhibit their findings, incurring prohibitive computational costs.
In contrast, we demonstrate a constructive recipe, where we directly predict the test performance at large scale and specify the full model configuration which attains it (with no need for large-scale search), given performance at small scale.
Related work ::: Predicting model performance:
Since training models at full data/model scale may be computationally prohibitive, a line of work tries to predict the performance of a given model on a given dataset, without training the model, for example by using a bank of previously trained models, dataset, and their associated performances BIBREF17. Others have proposed to estimate performance on small data BIBREF18 or model sizes BIBREF2, BIBREF19 in the context of neural architecture search (NAS). In this case, the small-scale evaluation is used to compare models at small cost, to expedite the search process; see BIBREF20 for a recent survey. Our work complements previous approaches by demonstrating a functional form that can predict large-scale performance from small-scale measurements. Moreover, our method may be integrated in NAS, addressing some of its current limitations (as discussed in section SECREF8).
Related work ::: Theoretical error bounds:
Much attention has been given to theoretical explanations of the generalization capabilities of deep neural networks BIBREF21, BIBREF22, BIBREF23, BIBREF24, BIBREF25. While fully engaging with this literature is beyond our scope, we note that recent studies have derived bounds involving power-law dependencies in both model BIBREF26 and data size BIBREF27. We leave it as an open question for future work to find theoretical explanations for the empirical behavior and the functional form we investigate in this work.
Experimental Setup
Experimental Setup ::: Notation:
Let $_n = \lbrace _i,y_i \rbrace _{i=1}^{n}$ denote a labeled (training) dataset with $n$ samples or datapoints. Let $f_m$ denote a neural network whose size is the number of parameters $m$, such that $\hat{y} = f_m()$ is the predicted label. Let $\epsilon \left(n,m \right)$ be the generalization error as a function of $n$ and $m$, measured by a performance metric (e.g., top-1 accuracy or cross-entropy loss) on a held-out test set. We refer to this error function as the error landscape.
Experimental Setup ::: Scaling Policies
Experimental Setup ::: Scaling Policies ::: Dataset scaling:
We wish to scale datasets while preserving the original distribution. For image classification, we uniformly subsample all classes by a constant ratio, thus preserving the relative sample size per class. We limit the maximal sub-sampling to avoid eradicating any class. For language modeling, where the number of classes (vocabulary items) has a very long tail distribution, we randomly sample sentences such that the total number of sampled words will be a certain fraction of the original dataset. Table TABREF9 reports the data scales we use. In all tasks the held-out test set remains untouched for evaluating the error.
Experimental Setup ::: Scaling Policies ::: Model scaling:
We are critically interested in a method where moving across scales is defined by some scaling function, such that no additional significant computation would be incurred. We thus consider the case where the model architecture is given and the model size determines how to scale it. For instance, one may scale width (number of channels in convolutional networks, hidden state size in recurrent networks), depth (number of layers), do compound scaling BIBREF8, or more generally define a function tying the model degrees of freedom and size. We focus primarily on width scaling in our experiments; the model scales are reported in Table TABREF9. We also perform selected depth scaling to demonstrate flexibility with respect to the scaling method.
Experimental Setup ::: Scaling Policies ::: Hyper-parameters:
For similar reasons we wish to avoid hyper-paramater search at large scales, and thus avoid the temptation to tune hyper-parameters accordingly (learning rate, regularization, etc.). Therefore, we hold all hyper-parameters fixed. This enables us to construct a functional form that fits the error landscape and can be used to predict the error across scales while completely defining the model attaining it. We consider pros and cons of this approach in the discussion (section SECREF8).
Experimental Setup ::: Tasks, Models, and Datasets
We experiment with both vision and language tasks. We use 6 benchmark datasets for image classification and 3 for language modeling. For image classification, we train ResNet BIBREF5 and WRN models BIBREF6. For language modeling, we train AWD-LSTM BIBREF28 and Transformer-XL models BIBREF29. Summary statistics are shown in Table TABREF9, along with the range of explored scales. Appendix SECREF9 gives additional information.
Observations on the Error Landscape
figsub:observe3dwiki103figsub:observe3dcifar10depth=44 respectively show an example test error landscape for width scaling of Transformer-XL on WikiText-103 and WRN-44-16 on CIFAR10. Various additional such landscapes are found in appendix SECREF11, showing largely consistent patterns. Examining the error landscapes yields the following observations:
Model Scaling
For a given dataset size, scaling up the model results in an initial decrease in test error, which then saturates to a level determined by the dataset size. This behavior has been noted by BIBREF8 across varied model scaling methods, although they have not engaged with the dependency on dataset size.
The rate of error decrease with model size appears well approximated by a power-law.
These two observations together can be summarized as the following relation:
where $b, \beta , c_m$ may depend on the data size $n$, s.t. as $m$ grows, $\epsilon \rightarrow c_m$. Example fits to this form (allowing $b, \beta , c_m$ to be fit per $n$) are seen in figsub:observe2dwiki103 (right) and figsub:observe2dcifar10 (right).
Data scaling
For a given model size, scaling up the dataset results in an initial increase in performance, which then saturates to a level determined by the model size.
The rate of error decrease with dataset size appears well approximated by a power-law. BIBREF16 also noted a similar relationship, but did not functionally tie the saturation level to the dataset size.
These two observations together can be summarized as the following relation:
where $a, \alpha , c_n$ may depend on the model size $m$, s.t. as $n$ grows, $\epsilon \rightarrow c_n$. Example fits to this form (allowing $a, \alpha , c_n$ to be fit per $m$) are seen in figsub:observe2dwiki103 (left) and figsub:observe2dcifar10 (left).
Joint properties The behavior of the error when scaling model size while holding data size fixed, and vice versa, extends to the entire error landscape in a well-behaved manner, such that the manifold $\epsilon (m,n)$ is smooth everywhere as a function of both model and data scales.
Functional Approximation of the Generalization Error
Functional Approximation of the Generalization Error ::: Criteria
Motivated by the above observations, we now consider a functional approximation for the error landscape. In particular, let us consider function families meeting the following criteria which augment and restrict our observations:
As either model or dataset size goes to zero, the expected performance is equivalent to a random-guess error level $\epsilon _0$.
For a given dataset size, scaling up the model will result in an initial increase in performance, which will then saturate, taking the form in (DISPLAY_FORM26).
For a given model size, scaling up the dataset will result in an initial increase in performance, which will then saturate, taking the form in (DISPLAY_FORM30).
There exists an irreducible error $\epsilon _\infty $, intrinsic to the dataset.
The function must be smooth everywhere and monotonic non-increasing in terms of model and data size (observation UNKREF31).
While there are many possible function families meeting the above criteria, below we propose a simple function family for our evaluation. We do not claim that this is in fact the true underlying dependency, but rather that it serves as a good approximation of the error landscape—consistent with these criteria.
Functional Approximation of the Generalization Error ::: Proposed Function Family
As a first insightful step, consider the implications of satisfying UNKREF35 and UNKREF36 simultaneously. By examining the limiting behavior as $m$ or $n$ grow, we have:
Thus, a consistent form satisfying UNKREF35 and UNKREF36 simultaneously is:
where $c_\infty $ is a constant not dependent on either $m$ or $n$. Let us now examine the simplified case where $a,b,\alpha ,\beta $ are constant:
where $\alpha \ge 0$ and $\beta \ge 0$ control the global rate at which error decreases with data and model size, respectively, $a>0$ and $b>0$ are a form of unit conversion between data and model sizes and error, and $c_\infty >0$ is the asymptotic lower value attainable. This function is a special case of (DISPLAY_FORM40) and meets criteria UNKREF35 and UNKREF36 by construction. Importantly UNKREF37 and UNKREF38 are also met.
However, by giving up the dependence of $a,b,\alpha ,\beta $ on $m,n$, this function does not meet criterion UNKREF33. We thus need to model the transition from the initial random-guess level to the power-law region. We propose to parameterize the transition using the following envelope (complex) function:
where $i = \sqrt{-1}$. Here the simple pole at $ \eta $ controls the transition point from the initial random-guess level $\epsilon _0$ as $(m,n)$ increase. As $(m,n)$ grow, $\tilde{\epsilon }\rightarrow c_\infty $ and the final irreducible error $\epsilon _\infty \triangleq \epsilon _0c_\infty \eta ^{-1}$ is approached. The random-guess error, $\epsilon _0$, is a known parameter determined by dataset statistics (e.g, $(N_{classes}-1) / N_{classes}$ for a balanced datasaet). Note that due to our choice of rational envelope, we can divide by a constant the form in (DISPLAY_FORM41). Without loss of generality, let us choose $a=1$.
Note that while the forms in equations DISPLAY_FORM40 and DISPLAY_FORM41 are well motivated, the approach taken for modeling the transition is solely a convenience one. In fact, the transition(s) as function of $m$ and $n$ may be captured in the functional forms of $a,b,\alpha ,\beta $ or another envelope mechanism. We leave a more refined investigation of the nature of the transitions to future work.
error landscape estimation
We wish to empirically estimate the quality of the proposed functional parameterization as a fit to the true error landscape. Let $\hat{\epsilon }(n,m ; )$ be the parametric function family ((DISPLAY_FORM42)) approximating the error landscape $\epsilon \left(n,m \right)$, where $= \lbrace \alpha ,\beta ,b,c_\infty ,\eta \rbrace $. Define the divergence $\delta (n,m;)$ as the relative difference between the estimated error $\hat{\epsilon }(m,n;)$ and the true error $\epsilon (m,n)$:
We fit a least squares regression model to find the best parameters minimizing the divergence. In this section, we fit the function given all model/data configurations $m , n$ (see Table TABREF9) and evaluate the fit quality. (In the next section, we perform extrapolation experiments, from seen to unseen points.) We perform the fit separately for each dataset and evaluate its quality by the mean $\mu $ and standard deviation $\sigma $ of the divergence $\delta $ over all points $(m,n)$. See Appendix SECREF68 for experimental details.
As fig:fit shows, estimated test accuracy is highly correlated with actual test accuracy for various datasets, with worst-case values $\mu <1\%$ and $\sigma <5\%$ . Note that the number of free parameters is small ($||\le 6$) compared to the number of points (42–49 model-data configurations), demonstrating the appropriateness of the proposed function for modeling the complex error landscape.
error landscape estimation ::: A Probe into Depth Scaling
Here we verify that our results extend to another canonical scaling policy, namely depth scaling. fig:cifar10-depth shows the error landscape with depth scaling on CIFAR10, exhibiting the same characteristics as width scaling. fig:fit-cifar10-widthfig:fit-cifar10-depth show error landscape estimation results for both cases of width and depth scaling, exhibiting small and comparable fit errors (error intervals $<1.2\%$). Since the difference in approximation quality is effectively indistinguishable when scaling depth or width orthogonally, we expect compound scaling to adhere to the same functional form. Indeed, we verified this on the publicly available (model scaling only) results for EfficientNet BIBREF8.
Extrapolation
In this section, we evaluate the ability of our functional approximation to extrapolate beyond seen model/data configurations. The primary question we ask is: can we predict the error of a large model/data configuration from the errors of smaller-scale model/data configurations? To do this, we fit the least squares regression on a subset of the configurations and predict the error on larger, unseen configurations. More formally, let $(m_i, n_j)$ denote a given model/data configuration. We first estimate parameters $_{ij}$ by fitting the function in (DISPLAY_FORM42) on all points of at most that size ($m \le m_i, n \le n_j$). Then we predict the error $\epsilon (m,n)$ in all points corresponding to larger configurations ($m > m_i, n > n_j$) using estimated $_{ij}$. Finally, we measure the divergence $\delta (m,n)$ between the estimated error and the actual error at all larger configurations. This process is illustrated in fig:extrapolation-array.
fig:extrapolation-single-vision shows the results of one such extrapolation experiment, on ImageNet. In this case, we have fit the functional form on all configurations of model size $m \le m_i = M/16 $ and data size $n \le n_j = N/8$, and predicted the error on all larger configurations. As the figure shows, the extrapolation is highly accurate, with a mean divergence of $\mu =4.5\%$ (std: $\sigma =4.7\%$). fig:extrapolation-single-language reports a similar experiment on WikiText-103. Here, again, we see very good extrapolation, with a mean divergence of $\mu =0.5\%$ (std: $\sigma =1.7\%$). Note that each extrapolation is run 10 times with different random initializations of $_{ij}$ in the least squares with negligible effect on the prediction.
In practice, we may be interested in extrapolation quality with different subsets of configurations. Appendix SECREF12 provides detailed extrapolation results on multiple subsets of configurations, for both vision and language datasets. Generally, the extrapolation performs well once not ill-posed, which may be caused by lack of signal in the region of the initial “random-guess” level, or in degenerate cases like having fewer measurements than the number of free parameters in $$.
Discussion and Conclusion
In this work, through insights gained by the joint examination of the dependencies of generalization error on both model and data size, we arrive at criteria for functions consistent with the form of the generalization error under a given scaling policy. We consider one such function and find it to be in very good agreement with the actual behavior of the error landscape. Indeed, the agreement is strong enough that extrapolation from small to large scale becomes feasible: the function predicts the behavior of the generalization error in practice for the practical case of scaling models and data. We discuss several example implications of knowing such a functional form.
Discussion and Conclusion ::: Small-scale network development:
At the core of small fidelity searches is the notion of performance rank comparison between models. However, small scale and large scale ranks are not assured to be consistent. If indeed a functional form such as empirically found in this work holds very generally, then in contrast, one can safely assess scaling rank between models at small scale, with the assurance that it remains consistent. This suggests that one would be well served by searching over scaling policies; a pertinent example of such a success is BIBREF8. The functional form also explains the limitation of small-scale search: once reaching the random-guess error level, where the sensitivity to scaling vanishes, the informativeness of ranking diminishes. Finally, the functional form allows direct usage of differntiable methods for NAS.
Discussion and Conclusion ::: Efficient computation and principled trade-off choices:
Knowing the error landscape function facilitates reasoning about the choice of $(m,n)$ attaining a specified error level. A case in point is optimizing for efficient computation which has both practical and environmental importance BIBREF30. Since the number of FLOPs during training is $\propto m\cdot n$ and a constant error contour is very well approximated by $c = \frac{1}{n^\alpha }+\frac{b}{m^\beta }$, dataset and models may be scaled down with optimal resource efficiency with no effect on performance by solving for: $n,m = _{m,n} m\cdot n ~ \text{s.t.} ~ c=1/n^\alpha + b/m^\beta $.
Discussion and Conclusion ::: Limitations:
We have made a few simplifying assumptions in our choice of approximating function, in particular in how to model the transition from the initial random-guess error level and the union of the random-guess level of the two scenarios (small model with large data and large model with small data). We leave a more detailed examination of the behavior of the transitions from random-guess error levels and refinements of the functional form to future work.
Critically, the restrictive nature of our scaling framework (all parameters and hyperparameters described by a policy) is both a blessing and a challenge. The blessing comes in fulfilling the goal of finding simultaneously both the form of the generalization error and the full specification of the model and hyperparameters that attain it across scales. The challenge is that we have demonstrated in this work only the case of constant hyper-parameters. We conjecture that the relation between model configuration and hyperparameter choice BIBREF31 may entail the potential to formulate hyperparameter-scaling policies similar in nature to the model-scaling polices, and that these too fall under the scope of the form we find in this work. This too will be the subject of future work.
We hope that this work will bring the actual functional form of the generalization error in this practical case of scaling to the fore, both in practice and as an empirical leg to stand on in the quest for its theoretical origins.
Discussion and Conclusion ::: Acknowledgments
We thank Alexander Rakhlin, Alexander Madry, Kai Xiao, Lu Mi, Viaks Garg, Dan Alistrah, and Tommi Jaakkola for helpful discussions. J.R. was partly supported by the Eli and Dorothy Berman Fellowship as well as grants NSF IIS-1447786, NSF CCF-1563880 and China-Singapore Suzhou Industrial Park. A.R. was partially supported by the Air Force Office of Scientific Research USA (FA9550-18-1-0054) though a grant to John K. Tsotsos. Y.B. was supported by the Harvard Mind, Brain, and Behavior Initiative.
Datasets and Models ::: Image Classification ::: Datasets
We evaluated our predictions on several popular image classification datasets: ImageNet BIBREF3: a large-scale recognition benchmark consisting of natural images of 1000 object categories with 1.28M training images spread roughly uniformly over the categories. It has 50K validation and 100K testing images. It has been the most popular large-scale benchmark for image classification methods for the better part of the last decade. CIFAR10/100 BIBREF37: 60K natural RGB images of 10 classes (100 for CIFAR100) with a train/test split of 50K/10K. For each of the following datasets, we use the version collated, resized, and split into train/validation/test sets by BIBREF41. DTD BIBREF34: a texture database of 47 categories and 5640 images. Aircraft BIBREF38: 10K images of 100 different aircraft classes. UCF101 BIBREF42: originally a video action recognition dataset, converted using the method of BIBREF32 into a single image per video. It contains 13,320 images of 101 action classes.
Datasets and Models ::: Image Classification ::: Models
We use different variants of the popular ResNet architecture BIBREF5. For ImageNet we use Wide-ResNet-50 BIBREF44 and build on the code from the PyTorch framework BIBREF40 to vary the model width. For all other datasets we use ResNet-18 BIBREF5 of varying widths, modified from the implementation of BIBREF36.
A ResNet is a convolutional network consisting of a single convolutional layer followed by several blocks, where within each block there are residual connections. The last block is followed by some pooling (for example, average pooling) and by a fully-connected layer, in the case of classification tasks. Typically there are four blocks. The differentiating factor between different residual networks is the width (number of filters) of the layers within each block and the number of convolutional layers within each block. Within each block there is a set of convolutions with residual connections.
Scaling the models' width is performed by multiplying the number of channels in each convolutional layer by a constant factor and rounding to the nearest integer. Depth-scaling, in the CIFAR10 case, is performed by appending extra layers within each block.
Datasets and Models ::: Image Classification ::: Training
Training is done via SGD with a momentum of 0.9, weight decay of 1e-4 and initial learning rate of 0.1. For ImageNet we train for 90 epochs, decreasing the learning rate by a multiplicative factor of 0.1 after and 30 and after 60 epochs. We use a batch size of 16. For all other vision datasets we use a batch-size of 128. We begin training with a learning rate of 0.1, run for 200 epochs, and reduce by a multiplicative factor of 0.1 after 80, 120, and 160 epochs.
Datasets and Models ::: Language Modeling ::: Datasets
We evaluate on several datasets commonly used for (word-level) language modeling: Penn Treebank BIBREF39, WikiText-2 BIBREF33, and WikiText-103 BIBREF4. The PTB is a relatively small language modeling dataset of news texts, with a vocabulary of 10K unique words and about 900K/70K/80K training/validation/test words. WikiText-2 is drawn from Wikipedia articles and it is both larger and richer, with a vocabulary of 33K words and 2M/210K/240K training/validation/test words. WikiText-103 is also based on Wikipedia, but larger still, with a vocabulary of 270K words and 100M training words (and the same validation and test sets as WikiText-2).
Datasets and Models ::: Language Modeling ::: Models
We experiment with two standard models for language modeling: Transformer-XL BIBREF29 and AWD-LSTM BIBREF28. Transformer-XL is a recent language modeling architecture that is based on transformer self-attention BIBREF43, but modified to better learn dependencies beyond a fixed length by adding a segment-level recurrence mechanism. It has achieved state-of-the-art results on multiple benchmarks. We use the official PyTorch implementation with their base configuration: 16 layers, embedding size of 410, inner dimension of 2100 in the fully-connected layers, and 10 attention heads. See the implementation for other details. For scaling experiments, we decimate the inner dimension. We use Transformer-XL for WikiText-103.
AWD-LSTM is a long short-term memory BIBREF35 language model with adaptive weight averaging. We use the official implementation with the recommended configuration: 3 layers, embedding size of 400, and hidden state size of 1150. We use AWD-LSTM for PTB and WikiText-2 and follow the recommended settings for these two datasets. For scaling experiments, we decimate the hidden state size.
Error Estimation Experiment ::: Experimental Details
In the experiment described in section SECREF6, we fit a least squares regression model to find the best parameters minimizing the divergence $\delta (m,n)$ - evaluated at configurations $m,n$ as in Table TABREF9:
We quantify the quality of the fit by the mean $\mu $ and standard deviation $\sigma $ of the fitted divergence over all points $(m,n)$. A perfect fit corresponds to $\mu ,\sigma =0$. In order to assess the uncertainty in $\mu ,\sigma $, we randomly drop each point with 50% to perform the fit on and randomly initialize $\sim U[0,1]$. We repeat this process 100 times, and perform this separately for each dataset. We report error confidence intervals of $\pm \sqrt{Var(\mu )}$, $\pm \sqrt{Var(\sigma )}$.
Error Estimation Experiment ::: Found Theta Values
. 2
Additional Error Landscape Measurements and Estimations
In this appendix, we provide error landscape measurements and estimations for all datasets, corresponding to the experiment in section SECREF6. The results are shown in 3D graphs similar to fig:landscape-3d. In each such graph, the z-axis is the logarithm of the generalization error as a function of two independent variables: the model size $m$ and the data size $n$.
The 3D graph is deliberately portrayed in log-log-log scale, as we cover a very large range of data scales and model scales and a correspondingly wide range of errors. This view is a useful one when one wishes to evaluate both large dynamic ranges (simultaneously both very large and very small values) and is especially vivid in portraying power-law like dependencies; a power-law naturally forms a straight line in a log-log view.
In each figure, subfigure (a) shows the measured error landscape is in log-log-log scale, where each point (blue dot) is the error resulting from training with a model/data configuration $m,n$. Subfigure (b) shows the best-fit estimated error landscape. The surface is a linear interpolation between the points, which is then projected on the model-error $(m,\epsilon )$, data-error $(n,\epsilon )$, and model-data $(m,n)$ planes. The contour plots on each one of these planes are the projections of the error landscape surface, and are useful in considering the behavior of the surface when holding one dimension constant.
We call to attention several interesting observations on the datasets explored:
As quantified rigorously in section SECREF6, the fits perform well across error ranges. In these surfaces, one also gets qualitative sense of the fit adequacy across the wide ranges of the dataset and model scales directly. While perhaps slightly difficult to asses the surface directly, a helpful view is to consider the similarity between the projections of the actual and projected surfaces.
With increasing model size, indeed typically the error does remain saturated. However, in one of our tested datasets (fig:appBucf101) there was a renewed slight increase. We verify that this is indeed over-fitting, in the sense that there is no corresponding increase in the training error. We note that the functional form we find can actually be used to veer clear of the $m,n$ regions where such over-fitting may occur.
The simplifying approach taken by considering the random guess levels (and associated transitions) for small models or small data as identical, seems to work fairly well with some deviation apparent by examining fig:appBwiki103. Indeed the simplification can hold well for balanced datasets, but need not for imbalanced ones such as in the task of language modeling. Thus, a relaxation of this simplification is expected to be important conceptually and practically.
Additional Extrapolation Results
Here we provide detailed extrapolation results, for all datasets. All figures are structured in a similar way. Each subplot shows estimated (y-axis) vs. actual error (x-axis) (0 to 1 scale on both axes). Each subplot is located at the coordinate of the maximal data and model given for the task of performing the fit to the functional form in (DISPLAY_FORM42). This is the point at the top-right corner of the green dots in the illustration in fig:extrapolation-array. The target is to find the error-landscape values for unseen, larger scales of both model and data (red points in the same illustration). Going from left to right in each figure indicates observed measurements of the error from models of an increasing fraction w.r.t the full size. Going from bottom-to top indicates observed measurements of the error from dataset sizes of an increasingly large fraction of the full dataset.
In each subplot, every point shows the estimated vs. actual error on a model-data configuration. Points that were given for fitting the function are colored in green, while unseen points that were not used are in red. The red points show the estimation error vs. actual error when extrapolating to all larger models and data sizes. In each subplot, the mean and standard deviation over all divergences $\delta $ at target points are given in text.
Each experiment fit of the parameters was repeated 100 times, with different random initializations of $$. The shaded bands show one standard deviation across these runs.
The quality of the extrapolation is critically dependent on the signal provided in the (green) fitted points. Two limiting factors are evident by examining the figures below, which both play a role in the well-posedness of the solution:
The proximity to the initial random guess level. Only upon transitioning from the initial error plateau, does meaningful signal about the scaling rates become available. Indeed, for scales prior still in the region or close to the initial error level, one sees poor extrapolation results; see figures FIGREF106, FIGREF107, and FIGREF109, and the vivid origin of this phenomena by examining figures FIGREF86, FIGREF83, and FIGREF89.
A second source of ill-posedness is tied to the number of configurations used for the estimation of $$. Clearly, when this is small, one cannot expect the extrapolation to be stable. In fact, at least two measurements in each scaling dimension (model/data) are needed, and no less than the number of parameters in $$ in total. Indeed, for all the plots in this appendix, the smallest scale of $m,n$ is omitted form the graph such that the lowermost row and leftmost column span exactly two model and data scales correspondingly. Of course, there is nothing tying directly the number of points and scale of configurations measured, and one can decouple these two factors by taking closer spaced samples at small scale.
When both the above factors are not limiting the measurement, one readily sees that for divergences of no more than a few percent, it is sufficient to measure model/data configurations which are far-ranged from the configurations which one wishes to extrapolate to .

Question: What is proof that proposed functional form approximates well generalization error in practice?
Answer: estimated test accuracy is highly correlated with actual test accuracy for various datasets, appropriateness of the proposed function for modeling the complex error landscape

Question: How is proposed functional form constructed for some model?
Answer: Unanswerable