Diagnosing Generalization Failures from Representational Geometry Markers

Successful deep neural networks discover salient features of data. We show when and why they fail to learn out-of-distribution (OOD)-relevant representations from an in-distribution (ID) training window. This requires decoupling feature learning from data-generating-process (DGP) identifiability. From a single training window, OOD extrapolation is non-identifiable: infinitely many DGPs are $\varepsilon$-observationally equivalent on the training data but diverge arbitrarily outside it, and no in-distribution criterion alone reliably breaks the tie. A structural commitment, the feature map, label map, and model class $(\varphi, ψ, \mathcal{M})$, dictates the assumed DGP and governs OOD generalization while leaving ID performance essentially unchanged. When architecture, pretraining, augmentation, input formats, or domain knowledge implicitly inject the missing commitment, the model succeeds. When it cannot infer OOD-relevant structure from ID evidence, it fails. Changing only the representation can make the same architecture, at the same in-distribution loss, differ by ${\sim}520\times$ out of distribution. When the commitment is correct and identifiable, OOD error vanishes. For example, Fourier coordinates turn periodic extrapolation into interpolation on $\mathbb{S}^1$. The same mechanism predicts outcomes in three natural-science settings (mass-action chemistry; Kepler's-third-law exoplanet prediction, $n=2{,}362$; and cross-species coding-DNA detection) and in a 264-run positional-encoding study across Transformer, Mamba, and S4D. Finally, a controlled study shows: correct features are necessary but not sufficient. The model class must express the target, and the transformed training data must cover the relevant representation space.

Reliable generalization metrics are fundamental to the evaluation of machine learning models. Especially in high-stakes applications where labeled target data are scarce, evaluation of models' generalization performance under distribution shift is a pressing need. We focus on two practical scenarios: (1) Before deployment, how to select the best model for unlabeled target data? (2) After deployment, how to monitor model performance under distribution shift? The central need in both cases is a reliable and label-free proxy metric. Yet existing proxy metrics, such as model confidence or accuracy-on-the-line, are often unreliable as they only assess model output while ignoring the internal mechanisms that produce them. We address this limitation by introducing a new perspective: using the inner workings of a model, i.e., circuits, as a predictive metric of generalization performance. Leveraging circuit discovery, we extract the causal interactions between internal representations as a circuit, from which we derive two metrics tailored to the two practical scenarios. (1) Before deployment, we introduce Dependency Depth Bias, which measures different models' generalization capability on target data. (2) After deployment, we propose Circuit Shift Score, which predicts a model's generalization under different distribution shifts. Across various tasks, both metrics demonstrate significantly improved correlation with generalization performance, outperforming existing proxies by an average of 13.4% and 34.1%, respectively. Our code is available at https://github.com/deep-real/GenCircuit.

Grokking describes a delayed generalization phenomenon in which a neural network achieves perfect training accuracy long before validation accuracy improves, followed by an abrupt transition to strong generalization. Existing detection signals are indirect: weight norm reflects parameter-space regularization and consistently lags the transition, while GrokFast's slow gradient EMA, used without gradient amplification, is unstable across seeds with standard deviation exceeding mean lead time. We propose the Inter/Intra-class Distance Ratio (ILDR), a geometric metric computed on second-to-last layer representations as the ratio of inter-class centroid separation to intra-class scatter. ILDR provides an early detection signal: it rises and crosses a threshold at 2.5 times its baseline before the grokking transition appears in validation accuracy, indicating early geometric reorganization in representation space. Grounded in Fisher's linear discriminant criterion, ILDR requires no eigendecomposition and runs in O(|C|^2 + N). It is evaluated exclusively on held-out data, making it robust to memorization effects. Across modular arithmetic and permutation group composition (S5), ILDR leads the grokking transition by 9 to 73 percent of the training budget, with lead time increasing with task algebraic complexity. Over eight random seeds, ILDR leads by 950 +/- 250 steps with a coefficient of variation of 26 percent, and post-grokking variance drops by 1696 times, consistent with a sharp phase transition in representation space. Using ILDR as an early stopping trigger reduces training by 18.6 percent on average. Optimizer interventions triggered at the ILDR threshold demonstrate bidirectional control over the transition, suggesting ILDR tracks representational conditions underlying generalization rather than a downstream correlate.