Shaping the learning landscape in neural networks around wide flat minima

Learning in Deep Neural Networks (DNN) takes place by minimizing a non-convex high-dimensional loss function, typically by a stochastic gradient descent (SGD) strategy. The learning process is observed to be able to find good minimizers without getting stuck in local critical points, and that such minimizers are often satisfactory at avoiding overfitting. How these two features can be kept under control in nonlinear devices composed of millions of tunable connections is a profound and far reaching open question. In this paper we study basic non-convex one- and two-layer neural network models which learn random patterns, and derive a number of basic geometrical and algorithmic features which suggest some answers. We first show that the error loss function presents few extremely wide flat minima (WFM) which coexist with narrower minima and critical points. We then show that the minimizers of the cross-entropy loss function overlap with the WFM of the error loss. We also show examples of learning devices for which WFM do not exist. From the algorithmic perspective we derive entropy driven greedy and message passing algorithms which focus their search on wide flat regions of minimizers. In the case of SGD and cross-entropy loss, we show that a slow reduction of the norm of the weights along the learning process also leads to WFM. We corroborate the results by a numerical study of the correlations between the volumes of the minimizers, their Hessian and their generalization performance on real data.Comment: 37 pages (16 main text), 10 figures (7 main text

Computational core and fixed-point organisation in Boolean networks

In this paper, we analyse large random Boolean networks in terms of a constraint satisfaction problem. We first develop an algorithmic scheme which allows to prune simple logical cascades and under-determined variables, returning thereby the computational core of the network. Second we apply the cavity method to analyse number and organisation of fixed points. We find in particular a phase transition between an easy and a complex regulatory phase, the latter one being characterised by the existence of an exponential number of macroscopically separated fixed-point clusters. The different techniques developed are reinterpreted as algorithms for the analysis of single Boolean networks, and they are applied to analysis and in silico experiments on the gene-regulatory networks of baker's yeast (saccaromices cerevisiae) and the segment-polarity genes of the fruit-fly drosophila melanogaster.Comment: 29 pages, 18 figures, version accepted for publication in JSTA

Deploy-As-You-Go Wireless Relay Placement: An Optimal Sequential Decision Approach using the Multi-Relay Channel Model

We use information theoretic achievable rate formulas for the multi-relay channel to study the problem of as-you-go deployment of relay nodes. The achievable rate formulas are for full-duplex radios at the relays and for decode-and-forward relaying. Deployment is done along the straight line joining a source node and a sink node at an unknown distance from the source. The problem is for a deployment agent to walk from the source to the sink, deploying relays as he walks, given that the distance to the sink is exponentially distributed with known mean. As a precursor, we apply the multi-relay channel achievable rate formula to obtain the optimal power allocation to relays placed along a line, at fixed locations. This permits us to obtain the optimal placement of a given number of nodes when the distance between the source and sink is given. Numerical work suggests that, at low attenuation, the relays are mostly clustered near the source in order to be able to cooperate, whereas at high attenuation they are uniformly placed and work as repeaters. We also prove that the effect of path-loss can be entirely mitigated if a large enough number of relays are placed uniformly between the source and the sink. The structure of the optimal power allocation for a given placement of the nodes, then motivates us to formulate the problem of as-you-go placement of relays along a line of exponentially distributed length, and with the exponential path-loss model, so as to minimize a cost function that is additive over hops. The hop cost trades off a capacity limiting term, motivated from the optimal power allocation solution, against the cost of adding a relay node. We formulate the problem as a total cost Markov decision process, establish results for the value function, and provide insights into the placement policy and the performance of the deployed network via numerical exploration.Comment: 21 pages. arXiv admin note: substantial text overlap with arXiv:1204.432

Sequential Decision Algorithms for Measurement-Based Impromptu Deployment of a Wireless Relay Network along a Line

We are motivated by the need, in some applications, for impromptu or as-you-go deployment of wireless sensor networks. A person walks along a line, starting from a sink node (e.g., a base-station), and proceeds towards a source node (e.g., a sensor) which is at an a priori unknown location. At equally spaced locations, he makes link quality measurements to the previous relay, and deploys relays at some of these locations, with the aim to connect the source to the sink by a multihop wireless path. In this paper, we consider two approaches for impromptu deployment: (i) the deployment agent can only move forward (which we call a pure as-you-go approach), and (ii) the deployment agent can make measurements over several consecutive steps before selecting a placement location among them (which we call an explore-forward approach). We consider a light traffic regime, and formulate the problem as a Markov decision process, where the trade-off is among the power used by the nodes, the outage probabilities in the links, and the number of relays placed per unit distance. We obtain the structures of the optimal policies for the pure as-you-go approach as well as for the explore-forward approach. We also consider natural heuristic algorithms, for comparison. Numerical examples show that the explore-forward approach significantly outperforms the pure as-you-go approach. Next, we propose two learning algorithms for the explore-forward approach, based on Stochastic Approximation, which asymptotically converge to the set of optimal policies, without using any knowledge of the radio propagation model. We demonstrate numerically that the learning algorithms can converge (as deployment progresses) to the set of optimal policies reasonably fast and, hence, can be practical, model-free algorithms for deployment over large regions.Comment: 29 pages. arXiv admin note: text overlap with arXiv:1308.068

Controlling overestimation of error covariance in ensemble Kalman filters with sparse observations: A variance limiting Kalman filter

We consider the problem of an ensemble Kalman filter when only partial observations are available. In particular we consider the situation where the observational space consists of variables which are directly observable with known observational error, and of variables of which only their climatic variance and mean are given. To limit the variance of the latter poorly resolved variables we derive a variance limiting Kalman filter (VLKF) in a variational setting. We analyze the variance limiting Kalman filter for a simple linear toy model and determine its range of optimal performance. We explore the variance limiting Kalman filter in an ensemble transform setting for the Lorenz-96 system, and show that incorporating the information of the variance of some un-observable variables can improve the skill and also increase the stability of the data assimilation procedure.Comment: 32 pages, 11 figure