This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.

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 measure, 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 that 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 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.machine learning | neural networks | statistical physics | optimization T here is increasing evidence that artificial neural networks perform exceptionally well in complex recognition tasks (1). Despite huge numbers of parameters and strong nonlinearities, learning often occurs without getting trapped in local minima with poor prediction performance (2). The remarkable output of these models has created unprecedented opportunities for machine learning in a host of applications. However, these practical successes have been guided by intuition and experiments, whereas obtaining a complete theoretical understanding of why these techniques work seems currently out of reach, due to the inherent complexity of the problem. In other words, in practical applications, large and complex architectures are trained on big and rich datasets using an array of heuristic improvements over basic stochastic gradient descent (SGD). These heuristic enhancements over a stochastic process have the general purpose of improving the convergence and robustness properties (and therefore the generalization properties) of the networks, with respect to what would be achieved with a pure gradient descent (GD) on a cost function.There are many parallels between the studies of algorithmic stochastic processes and out-of-equilibrium processes in complex systems. Examples include jamming processes in physics, local search alg...

In the course of evolution, proteins show a remarkable conservation of their three-dimensional structure and their biological function, leading to strong evolutionary constraints on the sequence variability between homologous proteins. Our method aims at extracting such constraints from rapidly accumulating sequence data, and thereby at inferring protein structure and function from sequence information alone. Recently, global statistical inference methods (e.g. direct-coupling analysis, sparse inverse covariance estimation) have achieved a breakthrough towards this aim, and their predictions have been successfully implemented into tertiary and quaternary protein structure prediction methods. However, due to the discrete nature of the underlying variable (amino-acids), exact inference requires exponential time in the protein length, and efficient approximations are needed for practical applicability. Here we propose a very efficient multivariate Gaussian modeling approach as a variant of direct-coupling analysis: the discrete amino-acid variables are replaced by continuous Gaussian random variables. The resulting statistical inference problem is efficiently and exactly solvable. We show that the quality of inference is comparable or superior to the one achieved by mean-field approximations to inference with discrete variables, as done by direct-coupling analysis. This is true for (i) the prediction of residue-residue contacts in proteins, and (ii) the identification of protein-protein interaction partner in bacterial signal transduction. An implementation of our multivariate Gaussian approach is available at the website http://areeweb.polito.it/ricerca/cmp/code.