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arXiv2026

The information geometry of product-reference discrete diffusion: Interaction growth complexity and optimal scheduling

We study a class of product-reference diffusion algorithms for sampling from a discrete distribution. We show that their sampling performance can be characterized using a path-based measure of data geometry that we call the interaction growth complexity (IGC). We show that a bivariate IGC kernel gives an exact representation of both the KL discretization error and a simple one-step upper bound. The simpler univariate IGC density can be used to study the effect of stepsize choices on the iteration complexity required to obtain $ε$-accurate samples in KL divergence. Samplers that traverse the path with equi-spaced steps in log-squared-reliability-odds have performance that depends on the aggregate IGC mass, whereas refined choices of stepsizes have a lower complexity depending on a square-root functional. In the fine-grid limit, both of these characterizations become sharp. We also allow general product reference distributions and show that the reference law can substantially reshape the IGC profile and the resulting sampling complexity; in particular, references far from both the uniform and the data marginals can yield dimension-dependent improvements. Finally, the aggregate IGC mass admits bounds in terms of total correlation and dual total correlation, thereby connecting the pathwise geometry to classical measures of multivariate dependence.

stat.ML↗

arXiv2026

The Role of Pseudo-labels in Self-training Linear Classifiers on High-dimensional Gaussian Mixture Data

Self-training (ST) is a simple yet effective semi-supervised learning method. However, why and how ST improves generalization performance by using potentially erroneous pseudo-labels is still not well understood. To deepen the understanding of ST, we derive and analyze a sharp characterization of the behavior of iterative ST when training a linear classifier by minimizing the ridge-regularized convex loss on binary Gaussian mixtures, in the asymptotic limit where input dimension and data size diverge proportionally. The results show that ST improves generalization in different ways depending on the number of iterations. When the number of iterations is small, ST improves generalization performance by fitting the model to relatively reliable pseudo-labels and updating the model parameters by a large amount at each iteration. This suggests that ST works intuitively. On the other hand, with many iterations, ST can gradually improve the direction of the classification plane by updating the model parameters incrementally, using soft labels and small regularization. It is argued that this is because the small update of ST can extract information from the data in an almost noiseless way. However, in the presence of label imbalance, the generalization performance of ST underperforms supervised learning with true labels. To overcome this, two heuristics are proposed to enable ST to achieve nearly compatible performance with supervised learning even with significant label imbalance.

stat.ML↗

arXiv2026

From Zipf's Law to Neural Scaling through Heaps' Law and Hilberg's Hypothesis

We inspect the deductive connection between the neural scaling law and Zipf's law -- two statements discussed in machine learning and quantitative linguistics. The neural scaling law describes how the cross entropy rate of a foundation model -- such as a large language model -- changes with respect to the amount of training tokens, parameters, and compute. By contrast, Zipf's law posits that the distribution of tokens exhibits a power law tail. Whereas similar claims have been made in more specific settings, we show that the neural scaling law is a consequence of Zipf's law under certain broad assumptions that we reveal systematically. The derivation steps are as follows: We derive Heaps' law on the vocabulary growth from Zipf's law, Hilberg's hypothesis on the entropy scaling from Heaps' law, and the neural scaling from Hilberg's hypothesis. We illustrate these inference steps by a toy example of the Santa Fe process that satisfies all four statistical laws.

cs.IT↗

arXiv2026

Generalization Error Curves for Analytic Spectral Algorithms under Power-law Decay

The generalization error curve of certain kernel regression method aims at determining the exact order of generalization error with various source condition, noise level and choice of the regularization parameter rather than the minimax rate. In this work, under mild assumptions, we rigorously provide a full characterization of the generalization error curves of the kernel gradient descent method (and a large class of analytic spectral algorithms) in kernel regression. Consequently, we could sharpen the near inconsistency of kernel interpolation and clarify the saturation effects of kernel regression algorithms with higher qualification, etc. Thanks to the neural tangent kernel theory, these results greatly improve our understanding of the generalization behavior of training the wide neural networks. A novel technical contribution, the analytic functional argument, might be of independent interest.

cs.LG↗