As machine learned models increase in complexity and expressive power, features of simpler models, such as intrinsic coefficient transparency and control over the shape of the modeled function are lost. On the one edge of the spectrum we have simple linear models that possess coefficient transparency, but have a limited expressive power. On the other edge we have neural networks, that have expressive power that improves with scaling, but are mostly opaque. In this work we develop the \emph{spectral neuron} concept: a scalar model given by $f(x)=λ_k (A_0 + A_1 x + ... + A_n x_n)$, with learned real symmetric matrices $A_0, ..., A_n$. The input enters the model through an affine matrix function, but the prediction is obtained by reading one of its eigenvalues. Thus, the model is nonlinear, but the source of nonlinearity is still mathematically explicit. This gives us a useful middle ground: the model can become more expressive as the matrix dimension grows, while retaining coefficient transparency through the learned matrices. For example, extremal eigenvalues yield convex or concave functions, semidefinite constraints on the coefficient matrices impose monotonicity, and the associated eigenspaces characterize local feature influence. We study coefficient transparency, feature-influence bounds, and shape-control properties of this model family, and then test whether it can be learned and scaled in practice. We develop a systematic study of this model family, bringing together spectral results from several mathematical literatures to characterize its expressivity, coefficient transparency, feature influence, and shape-control properties. Code available at https://github.com/alexshtf/spectral_neuron_paper.
Transfer learning leverages knowledge from related source domains to improve learning in a target domain. Recent theoretical advances cover a broad range of regression settings within (generalized) linear models. Despite their diversity, these methods share two common constraints: they assume a known link function or linear structure and require direct access to raw source data. To move beyond these constraints, we propose a source-data-free transfer learning framework based on the single-index model (SIM). Instead of requiring raw source data, our method transfers only summary statistics derived from a generalized Stein's lemma in a one-time communication. This design preserves privacy and avoids side effects caused by dissimilarities of unknown nonlinear link functions across domains. To capture flexible, unknown nonlinearity, we employ a multilayer perceptron guided by the pre-estimated index from the transferred statistics, which significantly mitigates overfitting. Extensive experiments on synthetic data and a real-world application demonstrate consistent improvements over existing (generalized) linear model-based approaches. The proposed framework thus offers a practical, privacy-preserving, and nonlinear-adaptive solution for transfer learning.
Alexander Hsu, Zhaiming Shen, Wenjing Liao +1cs.LG math.NA
Pre-trained transformers are able to learn from examples provided as part of the prompt without any weight updates, a remarkable ability known as in-context learning (ICL). Despite its demonstrated efficacy across various domains, the theoretical understanding of ICL is still developing. Whereas most existing theory has focused on linear models, we study ICL in the nonlinear regression setting. Through the interaction mechanism in attention, we explicitly construct transformer networks to realize nonlinear features, such as polynomial or spline bases, which span a wide class of functions. Based on this construction, we establish a framework to analyze end-to-end in-context nonlinear regression with the constructed features. Our theory provides finite-sample generalization error bounds in terms of context length and training set size. We numerically validate the theory on synthetic regression tasks.