arXiv subjects

Stefano Sarao Mannelli

Publications and source records attributed to Stefano Sarao Mannelli.

25 records · Page 2Linked to original sources

arXiv2020

Optimization and Generalization of Shallow Neural Networks with Quadratic Activation Functions

We study the dynamics of optimization and the generalization properties of one-hidden layer neural networks with quadratic activation function in the over-parametrized regime where the layer width $m$ is larger than the input dimension $d$. We consider a teacher-student scenario where the teacher has the same structure as the student with a hidden layer of smaller width $m^*\le m$. We describe how the empirical loss landscape is affected by the number $n$ of data samples and the width $m^*$ of the teacher network. In particular we determine how the probability that there be no spurious minima on the empirical loss depends on $n$, $d$, and $m^*$, thereby establishing conditions under which the neural network can in principle recover the teacher. We also show that under the same conditions gradient descent dynamics on the empirical loss converges and leads to small generalization error, i.e. it enables recovery in practice. Finally we characterize the time-convergence rate of gradient descent in the limit of a large number of samples. These results are confirmed by numerical experiments.

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arXiv2020

Post-Workshop Report on Science meets Engineering in Deep Learning, NeurIPS 2019, Vancouver

Science meets Engineering in Deep Learning took place in Vancouver as part of the Workshop section of NeurIPS 2019. As organizers of the workshop, we created the following report in an attempt to isolate emerging topics and recurring themes that have been presented throughout the event. Deep learning can still be a complex mix of art and engineering despite its tremendous success in recent years. The workshop aimed at gathering people across the board to address seemingly contrasting challenges in the problems they are working on. As part of the call for the workshop, particular attention has been given to the interdependence of architecture, data, and optimization that gives rise to an enormous landscape of design and performance intricacies that are not well-understood. This year, our goal was to emphasize the following directions in our community: (i) identify obstacles in the way to better models and algorithms; (ii) identify the general trends from which we would like to build scientific and potentially theoretical understanding; and (iii) the rigorous design of scientific experiments and experimental protocols whose purpose is to resolve and pinpoint the origin of mysteries while ensuring reproducibility and robustness of conclusions. In the event, these topics emerged and were broadly discussed, matching our expectations and paving the way for new studies in these directions. While we acknowledge that the text is naturally biased as it comes through our lens, here we present an attempt to do a fair job of highlighting the outcome of the workshop.

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arXiv2020

Complex Dynamics in Simple Neural Networks: Understanding Gradient Flow in Phase Retrieval

Despite the widespread use of gradient-based algorithms for optimizing high-dimensional non-convex functions, understanding their ability of finding good minima instead of being trapped in spurious ones remains to a large extent an open problem. Here we focus on gradient flow dynamics for phase retrieval from random measurements. When the ratio of the number of measurements over the input dimension is small the dynamics remains trapped in spurious minima with large basins of attraction. We find analytically that above a critical ratio those critical points become unstable developing a negative direction toward the signal. By numerical experiments we show that in this regime the gradient flow algorithm is not trapped; it drifts away from the spurious critical points along the unstable direction and succeeds in finding the global minimum. Using tools from statistical physics we characterize this phenomenon, which is related to a BBP-type transition in the Hessian of the spurious minima.

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arXiv2020

Passed & Spurious: Descent Algorithms and Local Minima in Spiked Matrix-Tensor Models

In this work we analyse quantitatively the interplay between the loss landscape and performance of descent algorithms in a prototypical inference problem, the spiked matrix-tensor model. We study a loss function that is the negative log-likelihood of the model. We analyse the number of local minima at a fixed distance from the signal/spike with the Kac-Rice formula, and locate trivialization of the landscape at large signal-to-noise ratios. We evaluate in a closed form the performance of a gradient flow algorithm using integro-differential PDEs as developed in physics of disordered systems for the Langevin dynamics. We analyze the performance of an approximate message passing algorithm estimating the maximum likelihood configuration via its state evolution. We conclude by comparing the above results: while we observe a drastic slow down of the gradient flow dynamics even in the region where the landscape is trivial, both the analyzed algorithms are shown to perform well even in the part of the region of parameters where spurious local minima are present.

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arXiv2020

Who is Afraid of Big Bad Minima? Analysis of Gradient-Flow in a Spiked Matrix-Tensor Model

Gradient-based algorithms are effective for many machine learning tasks, but despite ample recent effort and some progress, it often remains unclear why they work in practice in optimising high-dimensional non-convex functions and why they find good minima instead of being trapped in spurious ones. Here we present a quantitative theory explaining this behaviour in a spiked matrix-tensor model. Our framework is based on the Kac-Rice analysis of stationary points and a closed-form analysis of gradient-flow originating from statistical physics. We show that there is a well defined region of parameters where the gradient-flow algorithm finds a good global minimum despite the presence of exponentially many spurious local minima. We show that this is achieved by surfing on saddles that have strong negative direction towards the global minima, a phenomenon that is connected to a BBP-type threshold in the Hessian describing the critical points of the landscapes.

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arXiv2020

Marvels and Pitfalls of the Langevin Algorithm in Noisy High-dimensional Inference

Gradient-descent-based algorithms and their stochastic versions have widespread applications in machine learning and statistical inference. In this work we perform an analytic study of the performances of one of them, the Langevin algorithm, in the context of noisy high-dimensional inference. We employ the Langevin algorithm to sample the posterior probability measure for the spiked matrix-tensor model. The typical behaviour of this algorithm is described by a system of integro-differential equations that we call the Langevin state evolution, whose solution is compared with the one of the state evolution of approximate message passing (AMP). Our results show that, remarkably, the algorithmic threshold of the Langevin algorithm is sub-optimal with respect to the one given by AMP. We conjecture this phenomenon to be due to the residual glassiness present in that region of parameters. Finally we show how a landscape-annealing protocol, that uses the Langevin algorithm but violate the Bayes-optimality condition, can approach the performance of AMP.

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arXiv2020

Thresholds of descending algorithms in inference problems

We review recent works on analyzing the dynamics of gradient-based algorithms in a prototypical statistical inference problem. Using methods and insights from the physics of glassy systems, these works showed how to understand quantitatively and qualitatively the performance of gradient-based algorithms. Here we review the key results and their interpretation in non-technical terms accessible to a wide audience of physicists in the context of related works.

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