Vicente Conde Mendes, Lorenzo Bardone, Cédric Koller et al.
Many real-world datasets contain hidden structure that cannot be detected by simple linear correlations between input features. For example, latent factors may influence the data in a coordinated way, even though their effect is invisible to covariance-based methods such as PCA. In practice, nonlinear neural networks often succeed in extracting such hidden structure in unsupervised and self-supervised learning. However, constructing a minimal high-dimensional model where this advantage can be rigorously analyzed has remained an open theoretical challenge. We introduce a tractable high-dimensional spiked model with two latent factors: one visible to covariance, and one statistically dependent yet uncorrelated, appearing only in higher-order moments. PCA and linear autoencoders fail to recover the latter, while a minimal nonlinear autoencoder provably extracts both. We analyze both the population risk, and empirical risk minimization. Our model also provides a tractable example where self-supervised test loss is poorly aligned with representation quality: nonlinear autoencoders recover latent structure that linear methods miss, even though their reconstruction loss is higher.
Julius Girardin, Emanuele Troiani, Yizhou Xu et al.
Understanding how performance scales jointly with model size and data is a central problem in modern machine learning. Existing theoretical works on scaling laws typically describe generalization as a function of data or compute, often in fixed-feature or infinite-width regimes and for online SGD. Here, we instead study how generalization scales with the number of trainable parameters and the number of samples in a feature-learning model. We analyze $\ell_2$-regularized empirical test error minimization in a quadratic two-layer network in a finite-sample setting with structured data. This setting allows for an explicit characterization of the generalization error as a function of the number of samples, model width, and regularization. Our results reveal a phase diagram with distinct scaling regimes as the number of parameters varies. In particular, the generalization error follows data-dependent power laws controlled by the spectral structure of the target. We further characterize the transitions between regimes, including the onset of interpolation, and their impact on generalization.
Fabrizio Boncoraglio, Emanuele Troiani, Vittorio Erba et al.
We introduce the attention-indexed model (AIM), a theoretical framework for analyzing learning in deep attention layers. Inspired by multi-index models, AIM captures how token-level outputs emerge from layered bilinear interactions over high-dimensional embeddings. Unlike prior tractable attention models, AIM allows full-width key and query matrices, aligning more closely with practical transformers. Using tools from statistical mechanics and random matrix theory, we derive closed-form predictions for Bayes-optimal generalization error and identify sharp phase transitions as a function of sample complexity, model width, and sequence length. We propose a matching approximate message passing algorithm and show that gradient descent can reach optimal performance. AIM offers a solvable playground for understanding learning in self-attention layers, that are key components of modern architectures.
Vittorio Erba, Emanuele Troiani, Luca Biggio et al.
Current progress in artificial intelligence is centered around so-called large language models that consist of neural networks processing long sequences of high-dimensional vectors called tokens. Statistical physics provides powerful tools to study the functioning of learning with neural networks and has played a recognized role in the development of modern machine learning. The statistical physics approach relies on simplified and analytically tractable models of data. However, simple tractable models for long sequences of high-dimensional tokens are largely underexplored. Inspired by the crucial role models such as the single-layer teacher-student perceptron (aka generalized linear regression) played in the theory of fully connected neural networks, in this paper, we introduce and study the bilinear sequence regression (BSR) as one of the most basic models for sequences of tokens. We note that modern architectures naturally subsume the BSR model due to the skip connections. Building on recent methodological progress, we compute the Bayes-optimal generalization error for the model in the limit of long sequences of high-dimensional tokens, and provide a message-passing algorithm that matches this performance. We quantify the improvement that optimal learning brings with respect to vectorizing the sequence of tokens and learning via simple linear regression. We also unveil surprising properties of the gradient descent algorithms in the BSR model.
Leonardo Defilippis, Yizhou Xu, Julius Girardin et al.
Neural scaling laws underlie many of the recent advances in deep learning, yet their theoretical understanding remains largely confined to linear models. In this work, we present a systematic analysis of scaling laws for quadratic and diagonal neural networks in the feature learning regime. Leveraging connections with matrix compressed sensing and LASSO, we derive a detailed phase diagram for the scaling exponents of the excess risk as a function of sample complexity and weight decay. This analysis uncovers crossovers between distinct scaling regimes and plateau behaviors, mirroring phenomena widely reported in the empirical neural scaling literature. Furthermore, we establish a precise link between these regimes and the spectral properties of the trained network weights, which we characterize in detail. As a consequence, we provide a theoretical validation of recent empirical observations connecting the emergence of power-law tails in the weight spectrum with network generalization performance, yielding an interpretation from first principles.
Matteo Vilucchio, Emanuele Troiani, Vittorio Erba et al.
We study robust linear regression in high-dimension, when both the dimension $d$ and the number of data points $n$ diverge with a fixed ratio $α=n/d$, and study a data model that includes outliers. We provide exact asymptotics for the performances of the empirical risk minimisation (ERM) using $\ell_2$-regularised $\ell_2$, $\ell_1$, and Huber losses, which are the standard approach to such problems. We focus on two metrics for the performance: the generalisation error to similar datasets with outliers, and the estimation error of the original, unpolluted function. Our results are compared with the information theoretic Bayes-optimal estimation bound. For the generalization error, we find that optimally-regularised ERM is asymptotically consistent in the large sample complexity limit if one perform a simple calibration, and compute the rates of convergence. For the estimation error however, we show that due to a norm calibration mismatch, the consistency of the estimator requires an oracle estimate of the optimal norm, or the presence of a cross-validation set not corrupted by the outliers. We examine in detail how performance depends on the loss function and on the degree of outlier corruption in the training set and identify a region of parameters where the optimal performance of the Huber loss is identical to that of the $\ell_2$ loss, offering insights into the use cases of different loss functions.