Generalized Approximate Survey Propagation for High-Dimensional Estimation

In Generalized Linear Estimation (GLE) problems, we seek to estimate a signal that is observed through a linear transform followed by a component-wise, possibly nonlinear and noisy, channel. In the Bayesian optimal setting, Generalized Approximate Message Passing (GAMP) is known to achieve optimal performance for GLE. However, its performance can significantly degrade whenever there is a mismatch between the assumed and the true generative model, a situation frequently encountered in practice. In this paper, we propose a new algorithm, named Generalized Approximate Survey Propagation (GASP), for solving GLE in the presence of prior or model mis-specifications. As a prototypical example, we consider the phase retrieval problem, where we show that GASP outperforms the corresponding GAMP, reducing the reconstruction threshold and, for certain choices of its parameters, approaching Bayesian optimal performance. Furthermore, we present a set of State Evolution equations that exactly characterize the dynamics of GASP in the high-dimensional limit

Out of equilibrium Statistical Physics of learning

In the study of hard optimization problems, it is often unfeasible to achieve a full analytic control on the dynamics of the algorithmic processes that find solutions efficiently. In many cases, a static approach is able to provide considerable insight into the dynamical properties of these algorithms: in fact, the geometrical structures found in the energetic landscape can strongly affect the stationary states and the optimal configurations reached by the solvers. In this context, a classical Statistical Mechanics approach, relying on the assumption of the asymptotic realization of a Boltzmann Gibbs equilibrium, can yield misleading predictions when the studied algorithms comprise some stochastic components that effectively drive these processes out of equilibrium. Thus, it becomes necessary to develop some intuition on the relevant features of the studied phenomena and to build an ad hoc Large Deviation analysis, providing a more targeted and richer description of the geometrical properties of the landscape. The present thesis focuses on the study of learning processes in Artificial Neural Networks, with the aim of introducing an out of equilibrium statistical physics framework, based on the introduction of a local entropy potential, for supporting and inspiring algorithmic improvements in the field of Deep Learning, and for developing models of neural computation that can carry both biological and engineering interest

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