arXiv subjects

O. Duranthon

Publications and source records attributed to O. Duranthon.

8 recordsLinked to original sources

arXiv2026

High-Dimensional Theory of LoRA Fine-Tuning in a Solvable Attention Model

We develop a high-dimensional statistical theory of low-rank adaptation (LoRA) in attention models, capturing the interplay between pre-training and fine-tuning. We introduce a solvable framework in which a single-head attention layer is first pre-trained on a data-abundant task and subsequently adapted via a rank-one LoRA update on limited data. In the high-dimensional limit, both stages admit a sharp asymptotic characterization in terms of a finite set of order parameters, yielding explicit predictions for test errors and representation alignment. Our analysis shows that the impact of pre-training on LoRA is summarized by an effective noise term, from which we derive prescriptions for the optimal pre-training procedure. We also demonstrate a regime with a mismatch between the value of the test error and representation quality, and propose an application of our theory to active fine-tuning.

cs.LG↗

arXiv2026

Specialization of softmax attention heads: insights from the high-dimensional single-location model

Multi-head attention enables transformer models to represent multiple attention patterns simultaneously. Empirically, head specialization emerges in distinct stages during training, while many heads remain redundant and learn similar representations. We propose a theoretical model capturing this phenomenon, based on the multi-index and single-location regression frameworks. In the first part, we analyze the training dynamics of multi-head softmax attention under SGD, revealing an initial unspecialized phase followed by a multi-stage specialization phase in which different heads sequentially align with latent signal directions. In the second part, we study the impact of attention activation functions on performance. We introduce the Bayes-softmax attention, which achieves optimal prediction performance in this setting.

cs.LG↗

arXiv2025

Statistical Advantage of Softmax Attention: Insights from Single-Location Regression

Large language models rely on attention mechanisms with a softmax activation. Yet the dominance of softmax over alternatives (e.g., component-wise or linear) remains poorly understood, and many theoretical works have focused on the easier-to-analyze linearized attention. In this work, we address this gap through a principled study of the single-location regression task, where the output depends on a linear transformation of a single input token at a random location. Building on ideas from statistical physics, we develop an analysis of attention-based predictors in the high-dimensional limit, where generalization performance is captured by a small set of order parameters. At the population level, we show that softmax achieves the Bayes risk, whereas linear attention fundamentally falls short. We then examine other activation functions to identify which properties are necessary for optimal performance. Finally, we analyze the finite-sample regime: we provide an asymptotic characterization of the test error and show that, while softmax is no longer Bayes-optimal, it consistently outperforms linear attention. We discuss the connection with optimization by gradient-based algorithms.

cs.LG↗

arXiv2025

Statistical physics analysis of graph neural networks: Approaching optimality in the contextual stochastic block model

Graph neural networks (GNNs) are designed to process data associated with graphs. They are finding an increasing range of applications; however, as with other modern machine learning techniques, their theoretical understanding is limited. GNNs can encounter difficulties in gathering information from nodes that are far apart by iterated aggregation steps. This situation is partly caused by so-called oversmoothing; and overcoming it is one of the practically motivated challenges. We consider the situation where information is aggregated by multiple steps of convolution, leading to graph convolutional networks (GCNs). We analyze the generalization performance of a basic GCN, trained for node classification on data generated by the contextual stochastic block model. We predict its asymptotic performance by deriving the free energy of the problem, using the replica method, in the high-dimensional limit. Calling depth the number of convolutional steps, we show the importance of going to large depth to approach the Bayes-optimality. We detail how the architecture of the GCN has to scale with the depth to avoid oversmoothing. The resulting large depth limit can be close to the Bayes-optimality and leads to a continuous GCN. Technically, we tackle this continuous limit via an approach that resembles dynamical mean-field theory (DMFT) with constraints at the initial and final times. An expansion around large regularization allows us to solve the corresponding equations for the performance of the deep GCN. This promising tool may contribute to the analysis of further deep neural networks.

cond-mat.dis-nn↗

arXiv2024

Asymptotic generalization error of a single-layer graph convolutional network

While graph convolutional networks show great practical promises, the theoretical understanding of their generalization properties as a function of the number of samples is still in its infancy compared to the more broadly studied case of supervised fully connected neural networks. In this article, we predict the performances of a single-layer graph convolutional network (GCN) trained on data produced by attributed stochastic block models (SBMs) in the high-dimensional limit. Previously, only ridge regression on contextual-SBM (CSBM) has been considered in Shi et al. 2022; we generalize the analysis to arbitrary convex loss and regularization for the CSBM and add the analysis for another data model, the neural-prior SBM. We also study the high signal-to-noise ratio limit, detail the convergence rates of the GCN and show that, while consistent, it does not reach the Bayes-optimal rate for any of the considered cases.

cs.LG↗

arXiv2023

Optimal Inference in Contextual Stochastic Block Models

The contextual stochastic block model (cSBM) was proposed for unsupervised community detection on attributed graphs where both the graph and the high-dimensional node information correlate with node labels. In the context of machine learning on graphs, the cSBM has been widely used as a synthetic dataset for evaluating the performance of graph-neural networks (GNNs) for semi-supervised node classification. We consider a probabilistic Bayes-optimal formulation of the inference problem and we derive a belief-propagation-based algorithm for the semi-supervised cSBM; we conjecture it is optimal in the considered setting and we provide its implementation. We show that there can be a considerable gap between the accuracy reached by this algorithm and the performance of the GNN architectures proposed in the literature. This suggests that the cSBM, along with the comparison to the performance of the optimal algorithm, readily accessible via our implementation, can be instrumental in the development of more performant GNN architectures.

cs.SI↗

arXiv2023

Neural-prior stochastic block model

The stochastic block model (SBM) is widely studied as a benchmark for graph clustering aka community detection. In practice, graph data often come with node attributes that bear additional information about the communities. Previous works modeled such data by considering that the node attributes are generated from the node community memberships. In this work, motivated by a recent surge of works in signal processing using deep neural networks as priors, we propose to model the communities as being determined by the node attributes rather than the opposite. We define the corresponding model; we call it the neural-prior SBM. We propose an algorithm, stemming from statistical physics, based on a combination of belief propagation and approximate message passing. We analyze the performance of the algorithm as well as the Bayes-optimal performance. We identify detectability and exact recovery phase transitions, as well as an algorithmically hard region. The proposed model and algorithm can be used as a benchmark for both theory and algorithms. To illustrate this, we compare the optimal performances to the performance of simple graph neural networks.

cond-mat.dis-nn↗

arXiv2021

Coarse-grained quantum cellular automata

One can think of some physical evolutions as being the emergent-effective result of a microscopic discrete model. Inspired by classical coarse-graining procedures, we provide a simple procedure to coarse-grain color-blind quantum cellular automata that follow Goldilocks rules. The procedure consists in (i) space-time grouping the quantum cellular automaton (QCA) in cells of size $N$; (ii) projecting the states of a cell onto its borders, connecting them with the fine dynamics; (iii) describing the overall dynamics by the border states, that we call signals; and (iv) constructing the coarse-grained dynamics for different sizes $N$ of the cells. A byproduct of this simple toy-model is a general discrete analog of the Stokes law. Moreover we prove that in the spacetime limit, the automaton converges to a Dirac free Hamiltonian. The QCA we introduce here can be implemented by present-day quantum platforms, such as Rydberg arrays, trapped ions, and superconducting qbits. We hope our study can pave the way to a richer understanding of those systems with limited resolution.

quant-ph↗