arXiv subjects

Luca Saglietti

Publications and source records attributed to Luca Saglietti.

26 records · Page 2Linked to original sources

arXiv2019

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.

cond-mat.dis-nn↗

arXiv2018

Gaussian Process Prior Variational Autoencoders

Variational autoencoders (VAE) are a powerful and widely-used class of models to learn complex data distributions in an unsupervised fashion. One important limitation of VAEs is the prior assumption that latent sample representations are independent and identically distributed. However, for many important datasets, such as time-series of images, this assumption is too strong: accounting for covariances between samples, such as those in time, can yield to a more appropriate model specification and improve performance in downstream tasks. In this work, we introduce a new model, the Gaussian Process (GP) Prior Variational Autoencoder (GPPVAE), to specifically address this issue. The GPPVAE aims to combine the power of VAEs with the ability to model correlations afforded by GP priors. To achieve efficient inference in this new class of models, we leverage structure in the covariance matrix, and introduce a new stochastic backpropagation strategy that allows for computing stochastic gradients in a distributed and low-memory fashion. We show that our method outperforms conditional VAEs (CVAEs) and an adaptation of standard VAEs in two image data applications.

cs.LG↗

arXiv2018

From statistical inference to a differential learning rule for stochastic neural networks

Stochastic neural networks are a prototypical computational device able to build a probabilistic representation of an ensemble of external stimuli. Building on the relationship between inference and learning, we derive a synaptic plasticity rule that relies only on delayed activity correlations, and that shows a number of remarkable features. Our "delayed-correlations matching" (DCM) rule satisfies some basic requirements for biological feasibility: finite and noisy afferent signals, Dale's principle and asymmetry of synaptic connections, locality of the weight update computations. Nevertheless, the DCM rule is capable of storing a large, extensive number of patterns as attractors in a stochastic recurrent neural network, under general scenarios without requiring any modification: it can deal with correlated patterns, a broad range of architectures (with or without hidden neuronal states), one-shot learning with the palimpsest property, all the while avoiding the proliferation of spurious attractors. When hidden units are present, our learning rule can be employed to construct Boltzmann machine-like generative models, exploiting the addition of hidden neurons in feature extraction and classification tasks.

cond-mat.dis-nn↗

arXiv2018

On the role of synaptic stochasticity in training low-precision neural networks

Stochasticity and limited precision of synaptic weights in neural network models are key aspects of both biological and hardware modeling of learning processes. Here we show that a neural network model with stochastic binary weights naturally gives prominence to exponentially rare dense regions of solutions with a number of desirable properties such as robustness and good generalization performance, while typical solutions are isolated and hard to find. Binary solutions of the standard perceptron problem are obtained from a simple gradient descent procedure on a set of real values parametrizing a probability distribution over the binary synapses. Both analytical and numerical results are presented. An algorithmic extension aimed at training discrete deep neural networks is also investigated.

cond-mat.dis-nn↗

arXiv2016

Unreasonable Effectiveness of Learning Neural Networks: From Accessible States and Robust Ensembles to Basic Algorithmic Schemes

In artificial neural networks, learning from data is a computationally demanding task in which a large number of connection weights are iteratively tuned through stochastic-gradient-based heuristic processes over a cost-function. It is not well understood how learning occurs in these systems, in particular how they avoid getting trapped in configurations with poor computational performance. Here we study the difficult case of networks with discrete weights, where the optimization landscape is very rough even for simple architectures, and provide theoretical and numerical evidence of the existence of rare - but extremely dense and accessible - regions of configurations in the network weight space. We define a novel measure, which we call the "robust ensemble" (RE), which suppresses trapping by isolated configurations and amplifies the role of these dense regions. We analytically compute the RE in some exactly solvable models, and also provide a general algorithmic scheme which is straightforward to implement: define a cost-function given by a sum of a finite number of replicas of the original cost-function, with a constraint centering the replicas around a driving assignment. To illustrate this, we derive several powerful new algorithms, ranging from Markov Chains to message passing to gradient descent processes, where the algorithms target the robust dense states, resulting in substantial improvements in performance. The weak dependence on the number of precision bits of the weights leads us to conjecture that very similar reasoning applies to more conventional neural networks. Analogous algorithmic schemes can also be applied to other optimization problems.

stat.ML↗

arXiv2016

Learning may need only a few bits of synaptic precision

Learning in neural networks poses peculiar challenges when using discretized rather then continuous synaptic states. The choice of discrete synapses is motivated by biological reasoning and experiments, and possibly by hardware implementation considerations as well. In this paper we extend a previous large deviations analysis which unveiled the existence of peculiar dense regions in the space of synaptic states which accounts for the possibility of learning efficiently in networks with binary synapses. We extend the analysis to synapses with multiple states and generally more plausible biological features. The results clearly indicate that the overall qualitative picture is unchanged with respect to the binary case, and very robust to variation of the details of the model. We also provide quantitative results which suggest that the advantages of increasing the synaptic precision (i.e.~the number of internal synaptic states) rapidly vanish after the first few bits, and therefore that, for practical applications, only few bits may be needed for near-optimal performance, consistently with recent biological findings. Finally, we demonstrate how the theoretical analysis can be exploited to design efficient algorithmic search strategies.

cond-mat.dis-nn↗

arXiv2016

Local entropy as a measure for sampling solutions in Constraint Satisfaction Problems

We introduce a novel Entropy-driven Monte Carlo (EdMC) strategy to efficiently sample solutions of random Constraint Satisfaction Problems (CSPs). First, we extend a recent result that, using a large-deviation analysis, shows that the geometry of the space of solutions of the Binary Perceptron Learning Problem (a prototypical CSP), contains regions of very high-density of solutions. Despite being sub-dominant, these regions can be found by optimizing a local entropy measure. Building on these results, we construct a fast solver that relies exclusively on a local entropy estimate, and can be applied to general CSPs. We describe its performance not only for the Perceptron Learning Problem but also for the random $K$-Satisfiabilty Problem (another prototypical CSP with a radically different structure), and show numerically that a simple zero-temperature Metropolis search in the smooth local entropy landscape can reach sub-dominant clusters of optimal solutions in a small number of steps, while standard Simulated Annealing either requires extremely long cooling procedures or just fails. We also discuss how the EdMC can heuristically be made even more efficient for the cases we studied.

cond-mat.dis-nn↗

arXiv2015

Subdominant Dense Clusters Allow for Simple Learning and High Computational Performance in Neural Networks with Discrete Synapses

We show that discrete synaptic weights can be efficiently used for learning in large scale neural systems, and lead to unanticipated computational performance. We focus on the representative case of learning random patterns with binary synapses in single layer networks. The standard statistical analysis shows that this problem is exponentially dominated by isolated solutions that are extremely hard to find algorithmically. Here, we introduce a novel method that allows us to find analytical evidence for the existence of subdominant and extremely dense regions of solutions. Numerical experiments confirm these findings. We also show that the dense regions are surprisingly accessible by simple learning protocols, and that these synaptic configurations are robust to perturbations and generalize better than typical solutions. These outcomes extend to synapses with multiple states and to deeper neural architectures. The large deviation measure also suggests how to design novel algorithmic schemes for optimization based on local entropy maximization.

cond-mat.dis-nn↗