Hanneke, Moran, and Waknine \cite{HannekeMoranWaknine2024} asked how the agnostic PAC learning curve of the direct sum $C^r$ depends on the single-instance learning curve $\epsagn(n\mid C)$ and on $r$. We show that the single-instance learning rate does not determine the direct-sum rate. Let $\F$ be the class of the two constant binary functions and let $\G$ consist of the zero function and the identity function. Both classes have agnostic learning curve of order $n^{-1/2}$.
Prior classical-ML learning-curve work fits power laws to tree, linear, and kernel models on tabular data, but at small scale: typically one curve, one team, a handful of cells. We present a distributed classroom-scale replication: 127 graduate students each ran a fixed protocol on 3 assigned datasets, drawn from 18 tabular classification and regression datasets and 6 model families (Boosting, Random Forest, SVM, Linear/Logistic, Ridge, Lasso), yielding 11,536 training runs and 1,648 fitted power-law curves of the form error(N) = a N^(-b) + c. Three findings. (1) Power laws fit: R^2 > 0.8 on 77.7% of cells, with tree ensembles dominating at full data (Boosting 50% of datasets, RandomForest 33%; linear models underperform on classification). (2) Approximate shared exponents within a model family: for 5 of 6 families, a single family-level exponent predicts each family's cross-dataset curves nearly as well as per-dataset exponents (R^2 gap < 0.011), though AIC favors the unconstrained fit and curve collapse is partial (32-58% of points within +/-0.5 dex). We frame this as approximate predictive compressibility, not dataset-independent universality; Lasso fails outright (negative control) and Ridge is fragile under leave-one-dataset-out. (3) Replicator-implementation variance: with random_state=42 fixed, independent re-implementations of the same protocol still differ by mean CV(b) = 0.144 on the fitted exponent -- not seed variance, but the spread induced by unconstrained parts of the protocol (preprocessing, encoding, missing-value handling). We release the aggregated curves, per-cell fits, and a practical data-requirement table for N* to reach target error 0.15.
Deep learning models dependency on large-scale inertial datasets presents a significant bottleneck in inertial sensor-based classification tasks, such as human activity recognition and smartphone location recognition. In these domains, data collection requires massive recording campaigns that are complex, time-consuming, and difficult to scale. Currently, data-driven guidelines for determining the minimum sample size required to reach a desired accuracy level do not exist. To address this gap, this study presents a systematic empirical evaluation of learning curve convergence rates in inertial classification. We introduce a unified framework that analyzes classification performance under both binary and multi-class scenarios, and derive an empirical formula to estimate performance relative to dataset size. Testing across six diverse, real-world datasets totaling 102.7 hours of inertial measurements demonstrates that accuracy follows a consistent logarithmic growth pattern, regardless of task complexity. Leveraging this finding, we propose a quantitative stability point metric, defined as the sample size required for the learning curve to stabilize within a predefined mean absolute percentage deviation of its asymptotic maximum. Our analysis reveals that models often reach practical stability with substantially fewer samples than traditional heuristics suggest. Ultimately, we offer a generalizable framework to extrapolate total data requirements from small-scale pilot studies, optimizing the tradeoff between recording effort and model reliability. These findings shift the prevailing paradigm from maximizing data volume toward optimizing data efficiency, offering concrete, data-backed guidelines for planning recording campaigns in inertial sensing applications.
Hard-label classification is usually trained with smooth surrogate losses, most prominently softmax cross-entropy. We isolate an asymptotic mechanism by which this mismatch between smooth surrogate and discrete labels produces power-law learning curves in an online teacher-student model. After subtracting the mean logit, the thermodynamic-limit dynamics close in centered variables: a growing centered student-teacher alignment $D$ and the residual student variance $Δ$. At late times, examples away from teacher decision boundaries are already classified confidently and contribute exponentially little. Only boundary layers of width $O(D^{-1})$ remain active, while the noise of fixed-learning-rate online gradient descent maintains a nonzero $Δ$. As a function of the training time $α$ the late-time solution yields a $α^{-1/3}$ power law not only for the test loss but also for the generalization error $ε_g$, i.e., one minus test accuracy. This is much slower than the $α^{-1}$ Bayes-optimal reference for the same model. We further show that learning-rate schedules can improve the generalization error towards a $ε_g \sim α^{-1/2}$ power law. Simulations support the predicted order parameter dynamics and learning curves. Controlled experiments with correlated Gaussian inputs and whitened pretrained features show that data structure can dominate transients. Therefore, our result is an asymptotic, complementary mechanism rather than an alternative to spectral explanations of neural scaling laws.