arXiv subjects

Luca Saglietti

Publications and source records attributed to Luca Saglietti.

At least 19 recordsLinked to original sources

arXiv2026

Replication and Information Extraction in a Minimal Agent-Environment Model

How can information be extracted from data without explicit guidance or rewards? We investigate this question in a minimal setting where a classifying agent is exposed to a stream of structured data produced by a generative environment, and evolves by seeking consistency of its own labels in time. We find that imposing a simplicity bias on the classification rule can drive the dynamics toward label-coherent steady states, which we coin functional replicators. Remarkably, these persistent labeling rules align with the latent structure of the data. Using analytical tools from statistical mechanics, we characterize this spontaneous learning phase transition. Extending the analysis to a population of agents that pool labels from one another, we show that interaction reshapes the learning phase boundary, in some regimes enabling spontaneous learning that no isolated agent can achieve, while suppressing it in others. Our minimal framework thus opens a route to decentralized learning through label exchange alone, requiring no access to the internal weights of other agents.

cond-mat.dis-nn↗

arXiv2026

The Interplay of Data Structure and Imbalance in the Learning Dynamics of Diffusion Models

Real-world datasets are inherently heterogeneous, yet how per-class structural differences and sampling imbalance shape the training dynamics of diffusion models-and potentially exacerbate disparities-remains poorly understood. While models typically transition from an initial phase of generalization to memorizing the training set, existing theory assumes homogeneous data, leaving open how class imbalance and heterogeneity reshape these dynamics. In this work, we develop a high-dimensional analytical framework to study class-dependent learning in score-based diffusion models. Analyzing a random-features model trained on Gaussian mixtures, we derive the feature-covariance spectrum to characterize per-class generalization and memorization times. We reveal the explicit hierarchy governing these dynamics: class variance is the primary determinant of learning order-consistently favoring higher-variance classes-while centroid geometry plays a secondary role. Sampling imbalance acts as a modulator that can reverse this ordering and, under strong imbalance, forces minority classes to acquire distinct, delayed speciation times during backward diffusion. Together, these results suggest that diffusion models can memorize some classes while others remain insufficiently learned. We validate our theoretical predictions empirically using U-Net models trained on Fashion MNIST.

stat.ML↗

arXiv2026

Escape dynamics and implicit bias of one-pass SGD in overparameterized quadratic networks

We analyze the one-pass stochastic gradient descent dynamics of a two-layer neural network with quadratic activations in a teacher--student framework. In the high-dimensional regime, where the input dimension $N$ and the number of samples $M$ diverge at fixed ratio $α= M/N$, and for finite hidden widths $(p,p^*)$ of the student and teacher, respectively, we study the low-dimensional ordinary differential equations that govern the evolution of the student--teacher and student--student overlap matrices. We show that overparameterization ($p>p^*$) only modestly accelerates escape from a plateau of poor generalization by modifying the prefactor of the exponential decay of the loss. We then examine how unconstrained weight norms introduce a continuous rotational symmetry that results in a nontrivial manifold of zero-loss solutions for $p>1$. From this manifold the dynamics consistently selects the closest solution to the random initialization, as enforced by a conserved quantity in the ODEs governing the evolution of the overlaps. Finally, a Hessian analysis of the population-loss landscape confirms that the plateau and the solution manifold correspond to saddles with at least one negative eigenvalue and to marginal minima in the population-loss geometry, respectively.

cond-mat.dis-nn↗

arXiv2026

Biased Generalization in Diffusion Models

Generalization in generative modeling is defined as the ability to learn an underlying distribution from a finite dataset and produce novel samples, with evaluation largely driven by held-out performance and perceived sample quality. In practice, training is often stopped at the minimum of the test loss, taken as an operational indicator of generalization. We challenge this viewpoint by identifying a phase of biased generalization during training, in which the model continues to decrease the test loss while favoring samples with anomalously high proximity to training data. By training the same network on two disjoint datasets and comparing the mutual distances of generated samples and their similarity to training data, we introduce a quantitative measure of bias and demonstrate its presence on real images. We then study the mechanism of bias, using a controlled hierarchical data model where access to exact scores and ground-truth statistics allows us to precisely characterize its onset. We attribute this phenomenon to the sequential nature of feature learning in deep networks, where coarse structure is learned early in a data-independent manner, while finer features are resolved later in a way that increasingly depends on individual training samples. Our results show that early stopping at the test loss minimum, while optimal under standard generalization criteria, may be insufficient for privacy-critical applications.

cs.LG↗

arXiv2025

On the Bias of Next-Token Predictors Toward Systematically Inefficient Reasoning: A Shortest-Path Case Study

Recent advances in natural language processing highlight two key factors for improving reasoning in large language models (LLMs): (i) allocating more test-time compute tends to help on harder problems but often introduces redundancy in the reasoning trace, and (ii) compute is most effective when reasoning is systematic and incremental, forming structured chains of thought (CoTs) akin to human problem-solving. To study these factors in isolation, we introduce a controlled setting based on shortest-path tasks in layered graphs. We train decoder-only transformers on question-trace-answer triples using a custom tokenizer, comparing models trained on optimal bottom-up dynamic programming traces with those trained on longer, valid traces involving backtracking. Surprisingly, with the same training-token budget, models trained on inefficient traces generalize better to unseen graphs. This benefit is not due to length alone-injecting arbitrary redundancy into reasoning traces fails to help and can even hurt performance. Instead, we find that generalization correlates with the model's confidence in next-token prediction, suggesting that long, coherent, and locally incremental traces make the training signal easier to optimize.

cs.CL↗

arXiv2025

How transformers learn structured data: insights from hierarchical filtering

Understanding the learning process and the embedded computation in transformers is becoming a central goal for the development of interpretable AI. In the present study, we introduce a hierarchical filtering procedure for data models of sequences on trees, allowing us to hand-tune the range of positional correlations in the data. Leveraging this controlled setting, we provide evidence that vanilla encoder-only transformers can approximate the exact inference algorithm when trained on root classification and masked language modeling tasks, and study how this computation is discovered and implemented. We find that correlations at larger distances, corresponding to increasing layers of the hierarchy, are sequentially included by the network during training. By comparing attention maps from models trained with varying degrees of filtering and by probing the different encoder levels, we find clear evidence of a reconstruction of correlations on successive length scales corresponding to the various levels of the hierarchy, which we relate to a plausible implementation of the exact inference algorithm within the same architecture.

cs.LG↗

arXiv2025

Isolating the hard core of phaseless inference: the Phase selection formulation

Real-valued Phase retrieval is a non-convex continuous inference problem, where a high-dimensional signal is to be reconstructed from a dataset of signless linear measurements. Focusing on the noiseless case, we aim to disentangle the two distinct sub-tasks entailed in the Phase retrieval problem: the hard combinatorial problem of retrieving the missing signs of the measurements, and the nested convex problem of regressing the input-output observations to recover the hidden signal. To this end, we introduce and analytically characterize a two-level formulation of the problem, called ``Phase selection''. Within the Replica Theory framework, we perform a large deviation analysis to characterize the minimum mean squared error achievable with different guesses for the hidden signs. Moreover, we study the free-energy landscape of the problem when both levels are optimized simultaneously, as a function of the dataset size. At low temperatures, in proximity to the Bayes-optimal threshold -- previously derived in the context of Phase retrieval -- we detect the coexistence of two free-energy branches, one connected to the random initialization condition and a second to the signal. We derive the phase diagram for a first-order transition after which the two branches merge. Interestingly, introducing an $L_2$ regularization in the regression sub-task can anticipate the transition to lower dataset sizes, at the cost of a bias in the signal reconstructions which can be removed by annealing the regularization intensity. Finally, we study the inference performance of three meta-heuristics in the context of Phase selection: Simulated Annealing, Approximate Message Passing, and Langevin Dynamics on the continuous relaxation of the sign variables. With simultaneous annealing of the temperature and the $L_2$ regularization, they are shown to approach the Bayes-optimal sample efficiency.

cond-mat.dis-nn↗

arXiv2024

Bias-inducing geometries: an exactly solvable data model with fairness implications

Machine learning (ML) may be oblivious to human bias but it is not immune to its perpetuation. Marginalisation and iniquitous group representation are often traceable in the very data used for training, and may be reflected or even enhanced by the learning models. In the present work, we aim at clarifying the role played by data geometry in the emergence of ML bias. We introduce an exactly solvable high-dimensional model of data imbalance, where parametric control over the many bias-inducing factors allows for an extensive exploration of the bias inheritance mechanism. Through the tools of statistical physics, we analytically characterise the typical properties of learning models trained in this synthetic framework and obtain exact predictions for the observables that are commonly employed for fairness assessment. Despite the simplicity of the data model, we retrace and unpack typical unfairness behaviour observed on real-world datasets. We also obtain a detailed analytical characterisation of a class of bias mitigation strategies. We first consider a basic loss-reweighing scheme, which allows for an implicit minimisation of different unfairness metrics, and quantify the incompatibilities between some existing fairness criteria. Then, we consider a novel mitigation strategy based on a matched inference approach, consisting in the introduction of coupled learning models. Our theoretical analysis of this approach shows that the coupled strategy can strike superior fairness-accuracy trade-offs.

cs.LG↗

arXiv2024

Tilting the Odds at the Lottery: the Interplay of Overparameterisation and Curricula in Neural Networks

A wide range of empirical and theoretical works have shown that overparameterisation can amplify the performance of neural networks. According to the lottery ticket hypothesis, overparameterised networks have an increased chance of containing a sub-network that is well-initialised to solve the task at hand. A more parsimonious approach, inspired by animal learning, consists in guiding the learner towards solving the task by curating the order of the examples, i.e. providing a curriculum. However, this learning strategy seems to be hardly beneficial in deep learning applications. In this work, we undertake an analytical study that connects curriculum learning and overparameterisation. In particular, we investigate their interplay in the online learning setting for a 2-layer network in the XOR-like Gaussian Mixture problem. Our results show that a high degree of overparameterisation -- while simplifying the problem -- can limit the benefit from curricula, providing a theoretical account of the ineffectiveness of curricula in deep learning.

stat.ML↗

arXiv2024

The phase diagram of compressed sensing with $\ell_0$-norm regularization

Noiseless compressive sensing is a two-steps setting that allows for undersampling a sparse signal and then reconstructing it without loss of information. The LASSO algorithm, based on $\lone$ regularization, provides an efficient and robust to address this problem, but it fails in the regime of very high compression rate. Here we present two algorithms based on $\lzero$-norm regularization instead that outperform the LASSO in terms of compression rate in the Gaussian design setting for measurement matrix. These algorithms are based on the Approximate Survey Propagation, an algorithmic family within the Approximate Message Passing class. In the large system limit, they can be rigorously tracked through State Evolution equations and it is possible to exactly predict the range compression rates for which perfect signal reconstruction is possible. We also provide a statistical physics analysis of the $\lzero$-norm noiseless compressive sensing model. We show the existence of both a replica symmetric state and a 1-step replica symmmetry broken (1RSB) state for sufficiently low $\lzero$-norm regularization. The recovery limits of our algorithms are linked to the behavior of the 1RSB solution.

cs.IT↗

arXiv2024

The twin peaks of learning neural networks

Recent works demonstrated the existence of a double-descent phenomenon for the generalization error of neural networks, where highly overparameterized models escape overfitting and achieve good test performance, at odds with the standard bias-variance trade-off described by statistical learning theory. In the present work, we explore a link between this phenomenon and the increase of complexity and sensitivity of the function represented by neural networks. In particular, we study the Boolean mean dimension (BMD), a metric developed in the context of Boolean function analysis. Focusing on a simple teacher-student setting for the random feature model, we derive a theoretical analysis based on the replica method that yields an interpretable expression for the BMD, in the high dimensional regime where the number of data points, the number of features, and the input size grow to infinity. We find that, as the degree of overparameterization of the network is increased, the BMD reaches an evident peak at the interpolation threshold, in correspondence with the generalization error peak, and then slowly approaches a low asymptotic value. The same phenomenology is then traced in numerical experiments with different model classes and training setups. Moreover, we find empirically that adversarially initialized models tend to show higher BMD values, and that models that are more robust to adversarial attacks exhibit a lower BMD.

cs.LG↗

arXiv2023

The star-shaped space of solutions of the spherical negative perceptron

Empirical studies on the landscape of neural networks have shown that low-energy configurations are often found in complex connected structures, where zero-energy paths between pairs of distant solutions can be constructed. Here we consider the spherical negative perceptron, a prototypical non-convex neural network model framed as a continuous constraint satisfaction problem. We introduce a general analytical method for computing energy barriers in the simplex with vertex configurations sampled from the equilibrium. We find that in the over-parameterized regime the solution manifold displays simple connectivity properties. There exists a large geodesically convex component that is attractive for a wide range of optimization dynamics. Inside this region we identify a subset of atypical high-margin solutions that are geodesically connected with most other solutions, giving rise to a star-shaped geometry. We analytically characterize the organization of the connected space of solutions and show numerical evidence of a transition, at larger constraint densities, where the aforementioned simple geodesic connectivity breaks down.

cond-mat.dis-nn↗

arXiv2023

Optimal transfer protocol by incremental layer defrosting

Transfer learning is a powerful tool enabling model training with limited amounts of data. This technique is particularly useful in real-world problems where data availability is often a serious limitation. The simplest transfer learning protocol is based on ``freezing" the feature-extractor layers of a network pre-trained on a data-rich source task, and then adapting only the last layers to a data-poor target task. This workflow is based on the assumption that the feature maps of the pre-trained model are qualitatively similar to the ones that would have been learned with enough data on the target task. In this work, we show that this protocol is often sub-optimal, and the largest performance gain may be achieved when smaller portions of the pre-trained network are kept frozen. In particular, we make use of a controlled framework to identify the optimal transfer depth, which turns out to depend non-trivially on the amount of available training data and on the degree of source-target task correlation. We then characterize transfer optimality by analyzing the internal representations of two networks trained from scratch on the source and the target task through multiple established similarity measures.

cs.LG↗

arXiv2022

An Analytical Theory of Curriculum Learning in Teacher-Student Networks

In humans and animals, curriculum learning -- presenting data in a curated order - is critical to rapid learning and effective pedagogy. Yet in machine learning, curricula are not widely used and empirically often yield only moderate benefits. This stark difference in the importance of curriculum raises a fundamental theoretical question: when and why does curriculum learning help? In this work, we analyse a prototypical neural network model of curriculum learning in the high-dimensional limit, employing statistical physics methods. Curricula could in principle change both the learning speed and asymptotic performance of a model. To study the former, we provide an exact description of the online learning setting, confirming the long-standing experimental observation that curricula can modestly speed up learning. To study the latter, we derive performance in a batch learning setting, in which a network trains to convergence in successive phases of learning on dataset slices of varying difficulty. With standard training losses, curriculum does not provide generalisation benefit, in line with empirical observations. However, we show that by connecting different learning phases through simple Gaussian priors, curriculum can yield a large improvement in test performance. Taken together, our reduced analytical descriptions help reconcile apparently conflicting empirical results and trace regimes where curriculum learning yields the largest gains. More broadly, our results suggest that fully exploiting a curriculum may require explicit changes to the loss function at curriculum boundaries.

cs.LG↗

arXiv2022

Probing transfer learning with a model of synthetic correlated datasets

Transfer learning can significantly improve the sample efficiency of neural networks, by exploiting the relatedness between a data-scarce target task and a data-abundant source task. Despite years of successful applications, transfer learning practice often relies on ad-hoc solutions, while theoretical understanding of these procedures is still limited. In the present work, we re-think a solvable model of synthetic data as a framework for modeling correlation between data-sets. This setup allows for an analytic characterization of the generalization performance obtained when transferring the learned feature map from the source to the target task. Focusing on the problem of training two-layer networks in a binary classification setting, we show that our model can capture a range of salient features of transfer learning with real data. Moreover, by exploiting parametric control over the correlation between the two data-sets, we systematically investigate under which conditions the transfer of features is beneficial for generalization.

cs.LG↗

arXiv2021

Large Deviations of Semi-supervised Learning in the Stochastic Block Model

In community detection on graphs, the semi-supervised learning problem entails inferring the ground-truth membership of each node in a graph, given the connectivity structure and a limited number of revealed node labels. Different subsets of revealed labels can in principle lead to higher or lower information gains and induce different reconstruction accuracies. In the framework of the dense stochastic block model, we employ statistical physics methods to derive a large deviation analysis for this problem, in the high-dimensional limit. This analysis allows the characterization of the fluctuations around the typical behaviour, capturing the effect of correlated label choices and yielding an estimate of their informativeness and their rareness among subsets of the same size. We find theoretical evidence of a non-monotonic relationship between reconstruction accuracy and the free energy associated to the posterior measure of the inference problem. We further discuss possible implications for active learning applications in community detection.

cond-mat.dis-nn↗

arXiv2020

Solvable Model for Inheriting the Regularization through Knowledge Distillation

In recent years the empirical success of transfer learning with neural networks has stimulated an increasing interest in obtaining a theoretical understanding of its core properties. Knowledge distillation where a smaller neural network is trained using the outputs of a larger neural network is a particularly interesting case of transfer learning. In the present work, we introduce a statistical physics framework that allows an analytic characterization of the properties of knowledge distillation (KD) in shallow neural networks. Focusing the analysis on a solvable model that exhibits a non-trivial generalization gap, we investigate the effectiveness of KD. We are able to show that, through KD, the regularization properties of the larger teacher model can be inherited by the smaller student and that the yielded generalization performance is closely linked to and limited by the optimality of the teacher. Finally, we analyze the double descent phenomenology that can arise in the considered KD setting.

cs.LG↗

arXiv2019

Large deviations for the perceptron model and consequences for active learning

Active learning is a branch of machine learning that deals with problems where unlabeled data is abundant yet obtaining labels is expensive. The learning algorithm has the possibility of querying a limited number of samples to obtain the corresponding labels, subsequently used for supervised learning. In this work, we consider the task of choosing the subset of samples to be labeled from a fixed finite pool of samples. We assume the pool of samples to be a random matrix and the ground truth labels to be generated by a single-layer teacher random neural network. We employ replica methods to analyze the large deviations for the accuracy achieved after supervised learning on a subset of the original pool. These large deviations then provide optimal achievable performance boundaries for any active learning algorithm. We show that the optimal learning performance can be efficiently approached by simple message-passing active learning algorithms. We also provide a comparison with the performance of some other popular active learning strategies.

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