Accelerated Gossip in Networks of Given Dimension using Jacobi Polynomial Iterations

Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. We develop a method solving the gossip problem that depends only on the spectral dimension of the network, that is, in the communication network set-up, the dimension of the space in which the agents live. This contrasts with previous work that required the spectral gap of the network as a parameter, or suffered from slow mixing. Our method shows an important improvement over existing algorithms in the non-asymptotic regime, i.e., when the values are far from being fully mixed in the network. Our approach stems from a polynomial-based point of view on gossip algorithms, as well as an approximation of the spectral measure of the graphs with a Jacobi measure. We show the power of the approach with simulations on various graphs, and with performance guarantees on graphs of known spectral dimension, such as grids and random percolation bonds. An extension of this work to distributed Laplacian solvers is discussed. As a side result, we also use the polynomial-based point of view to show the convergence of the message passing algorithm for gossip of Moallemi \& Van Roy on regular graphs. The explicit computation of the rate of the convergence shows that message passing has a slow rate of convergence on graphs with small spectral gap

Accelerated Gossip in Networks of Given Dimension using Jacobi Polynomial Iterations

Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. We develop a method solving the gossip problem that depends only on the spectral dimension of the network, that is, in the communication network set-up, the dimension of the space in which the agents live. This contrasts with previous work that required the spectral gap of the network as a parameter, or suffered from slow mixing. Our method shows an important improvement over existing algorithms in the non-asymptotic regime, i.e., when the values are far from being fully mixed in the network. Our approach stems from a polynomial-based point of view on gossip algorithms, as well as an approximation of the spectral measure of the graphs with a Jacobi measure. We show the power of the approach with simulations on various graphs, and with performance guarantees on graphs of known spectral dimension, such as grids and random percolation bonds. An extension of this work to distributed Laplacian solvers is discussed. As a side result, we also use the polynomial-based point of view to show the convergence of the message passing algorithm for gossip of Moallemi & Van Roy on regular graphs. The explicit computation of the rate of the convergence shows that message passing has a slow rate of convergence on graphs with small spectral gap

Tight Nonparametric Convergence Rates for Stochastic Gradient Descent under the Noiseless Linear Model

International audienceIn the context of statistical supervised learning, the noiseless linear model assumes that there exists a deterministic linear relation $Y = \langle \theta_*, X \rangle$ between the random output $Y$ and the random feature vector $\Phi(U)$, a potentially non-linear transformation of the inputs $U$. We analyze the convergence of single-pass, fixed step-size stochastic gradient descent on the least-square risk under this model. The convergence of the iterates to the optimum $\theta_*$ and the decay of the generalization error follow polynomial convergence rates with exponents that both depend on the regularities of the optimum $\theta_*$ and of the feature vectors $\Phi(u)$. We interpret our result in the reproducing kernel Hilbert space framework. As a special case, we analyze an online algorithm for estimating a real function on the unit interval from the noiseless observation of its value at randomly sampled points; the convergence depends on the Sobolev smoothness of the function and of a chosen kernel. Finally, we apply our analysis beyond the supervised learning setting to obtain convergence rates for the averaging process (a.k.a. gossip algorithm) on a graph depending on its spectral dimension

Specific effect of a dopamine partial agonist on counterfactual learning: evidence from Gilles de la Tourette syndrome.

The dopamine partial agonist aripiprazole is increasingly used to treat pathologies for which other antipsychotics are indicated because it displays fewer side effects, such as sedation and depression-like symptoms, than other dopamine receptor antagonists. Previously, we showed that aripiprazole may protect motivational function by preserving reinforcement-related signals used to sustain reward-maximization. However, the effect of aripiprazole on more cognitive facets of human reinforcement learning, such as learning from the forgone outcomes of alternative courses of action (i.e., counterfactual learning), is unknown. To test the influence of aripiprazole on counterfactual learning, we administered a reinforcement learning task that involves both direct learning from obtained outcomes and indirect learning from forgone outcomes to two groups of Gilles de la Tourette (GTS) patients, one consisting of patients who were completely unmedicated and the other consisting of patients who were receiving aripiprazole monotherapy, and to healthy subjects. We found that whereas learning performance improved in the presence of counterfactual feedback in both healthy controls and unmedicated GTS patients, this was not the case in aripiprazole-medicated GTS patients. Our results suggest that whereas aripiprazole preserves direct learning of action-outcome associations, it may impair more complex inferential processes, such as counterfactual learning from forgone outcomes, in GTS patients treated with this medication

A Continuized View on Nesterov Acceleration

We introduce the "continuized" Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation and take gradient steps at random times. This continuized variant benefits from the best of the continuous and the discrete frameworks: as a continuous process, one can use differential calculus to analyze convergence and obtain analytical expressions for the parameters; but a discretization of the continuized process can be computed exactly with convergence rates similar to those of Nesterov original acceleration. We show that the discretization has the same structure as Nesterov acceleration, but with random parameters

Efficacy of Transcranial Direct-Current Stimulation in Catatonia: A Review and Case Series

Catatonia is a severe neuropsychiatric syndrome, usually treated by benzodiazepines and electroconvulsive therapy. However, therapeutic alternatives are limited, which is particularly critical in situations of treatment resistance or when electroconvulsive therapy is not available. Transcranial direct-current stimulation (tDCS) is a promising non-invasive neuromodulatory technique that has shown efficacy in other psychiatric conditions. We present the largest case series of tDCS use in catatonia, consisting of eight patients in whom tDCS targeting the left dorsolateral prefrontal cortex and temporoparietal junction was employed. We used a General Linear Mixed Model to isolate the effect of tDCS from other confounding factors such as time (spontaneous evolution) or co-prescriptions. The results indicate that tDCS, in addition to symptomatic pharmacotherapies such as lorazepam, seems to effectively reduce catatonic symptoms. These results corroborate a synthesis of five previous case reports of catatonia treated by tDCS in the literature. However, the specific efficacy of tDCS in catatonia remains to be demonstrated in a randomized controlled trial. The development of therapeutic alternatives in catatonia is of paramount importance

Guillain-Barré Syndrome, Greater Paris Area

We studied 263 cases of Guillain-Barré syndrome from 1996 to 2001, 40% of which were associated with a known causative agent, mainly Campylobacter jejuni (22%) or cytomegalovirus (15%). The cases with no known agent (60%) peaked in winter, and half were preceded by respiratory infection, influenzalike syndrome, or gastrointestinal illness