Appendix E — What the Toys Cannot Show
Every result in this book is computed on a small synthetic testbed whose truth is known exactly (The Testbeds). That is what makes the results checkable, and it is also their limit: a toy shows a mechanism and its direction, not its size on real data. Each chapter ends by saying what its toy cannot show. This appendix collects those sections, generated from the chapters, so the limits can be read in one place.
Four limits recur:
- Scale. The testbeds have a few dimensions and thousands of examples; the reported large-scale results are stated from their sources, not reproduced.
- Real data. Real missingness, noise, shift and duplication mix mechanisms that a testbed isolates one at a time, and which mechanism holds is often not testable from the data.
- Real models. Most demonstrations use linear models, a memorizer or a bigram model; how much a flexible model is affected is not measured here.
- Estimated, not known, quantities. The testbeds give exact margins, posteriors, density ratios and floors; practice has to estimate them, and the estimates carry their own error.
The Data-Generating Process
T1 knows \(s(x, y)\) exactly. Real selection rules are rarely written down, usually depend on variables that are not in the dataset, and mix both cases. The demonstration also uses a correctly specified linear model. With a misspecified model, selection on \(x\) is no longer harmless, and how much it hurts depends on the model.
Labels
T1’s noise is exactly class-conditional and its classes overlap by a known amount. Real label noise mixes systematic confusions, annotator disagreement and ambiguity that no relabeling could resolve. The figure’s separability axis is unobservable in practice, and so is the noise rate.
Data Types and Their Structure
The tones differ in one obvious feature and the series has one source of dependence. Real audio needs perceptual frequency scales and longer context. Real time series mix trends, seasonality and changes in regime. Both demonstrations show the direction and the mechanism of an effect, not its size on real data.
Cleaning: Missing Values, Duplicates, Outliers, Leakage
The testbed’s missingness depends on one column through a known curve, and its duplicates are exact. Real missingness mixes mechanisms, and which one holds cannot be tested from the observed data. The demonstrations show what each treatment does under each mechanism, not which mechanism a real dataset has.
Affine Scaling
T1’s features are Gaussian or exactly log-normal, and the outliers are a clean 1% at one location. Real features mix units, have ties and gaps, and contain outliers at many scales. The demonstrations show which models are sensitive, not by how much on a given dataset.
Changing the Shape
T1’s target is exactly log-normal, so the log is exactly right and the shortfall factor is known. Real targets are only roughly log-normal, and the right transform, if one exists, is unknown. The demonstrations show what each transform aims at and when its correction holds, not which transform to choose for a given dataset.
Encoding and Features
The noise category is pure noise, and the curve is a single smooth function of one feature. Real categories carry some signal and some noise in unknown proportions, and real text needs far more than counts. The demonstrations show what each encoding can represent and how target encoding leaks, not which encoding is best for a given dataset.
Sampling and Weighting
T1’s posterior is exactly logistic, its density ratio is known in closed form, and its minority clusters are Gaussian. Real class-conditional distributions change along with the priors, density ratios must be estimated, and the geometry of a minority class is unknown. The demonstrations show what each lever targets and what it costs, not how well a given estimate of a prior or a ratio will work on real data.
Augmentation
The side label is exactly symmetric, so whether an augmentation is right can be decided by construction. Real symmetries are approximate, partial and strength-dependent, and nothing measures them except held-out data. The demonstration uses a linear model on tiny images. Networks that learn their own features may gain more, or less, from the same augmentation.
Distribution Shift
T1’s shifts are clean: one factor changes, by a known amount, at a known time. Real shifts mix kinds, drift gradually, and arrive with labels that are late, sparse or biased. The demonstrations show what each test can see in principle, not what it will catch in a given monitoring setup.
Learning Curves
The task is linear, its noise Gaussian and its model correctly specified, so the curve is smooth and its floor known. Real learning curves can be ill-behaved, “showing worse learning performance with more training data”, for example when the model is misspecified, and they change shape as hyperparameters are retuned (Viering and Loog 2021). Classical generalization bounds give worst-case rates; they are not covered here.
Scaling Laws
A bigram model on a 50-token Markov source is not a transformer on natural text. Its exponents, regimes and crossover points are properties of the toy, not predictions about real models. What the toy does show is general: any curve with a floor bends away from a pure power law, and a fit on one regime misreads the next.
Why Power Laws: The Long Tail
The learner memorizes, the labels are deterministic, and cases are exact identities. Real models share structure between cases, so one example teaches about many, and “covered” is a matter of degree. The demonstration shows how a long tail can produce power-law returns, and what selection can and cannot buy when it does. It does not show that a given real scaling law arises this way.
Choosing Data
Each testbed isolates one condition. Real selection methods face several at once, use estimated scores instead of true margins and posteriors, and run at scales where comparing with random selection is itself expensive. The large-scale results above are reported by their authors, on their data, as of their publication; they are not reproduced here.
A Data Decision Guide
Here the truth is known, so every check is exact. In practice the checks use held-out data, which carries its own noise and its own shift, and the deployment question is often vague. Real tabular data has more fields, missing for less tidy reasons; real audio has noise, reverberation and gains that differ by frequency. The procedure transfers; the numbers do not.